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abstract: 'The formulation of noncommutative quantum mechanics as a quantum system represented in the space of Hilbert-Schmidt operators is used to systematically derive, using the standard time slicing procedure, the path integral action for a particle moving in the noncommutative plane and in the presence of a magnetic field and an arbitrary potential. Using this action, the equation of motion and the ground state energy for the partcle are obtained explicitly. The Aharonov-Bohm phase is derived using a variety of methods and several dualities between this system and other commutative and noncommutative systems are demonstrated. Finally, the equivalence of the path integral formulation with the noncommutative Schrödinger equation is also established.'
author:
- 'Sunandan Gangopadhyay$^{a,b,c}$[^1], Frederik G Scholtz $^{a,d}$[^2]'
title: 'Path integral action of a particle in a magnetic field in the noncommutative plane and the Aharonov-Bohm effect'
---
The idea of noncommutative spacetime [@snyder] and its possible physical consequences in quantum mechanics [@duval]-[@sgthesis], field theories [@witten]-[@sghazra] as well as their phenomenological implications in the standard model of particle physics [@chams]-[@lebed] has been an active area of research for quite some time. Futhermore, the framework of noncommutative geometry [@connes] provides a useful mathematical setting for the analysis of matrix models in string theory [@connes1]. In a recent paper [@fgs], attention has been paid to the formal and interpretational aspects of noncommutative quantum mechanics. It has been discussed that noncommutative quantum mechanics can be viewed as a quantum system represented in the space of Hilbert-Schmidt operators acting on noncommutative configuration space. Based on this formalism, the path integral formulation and derivation of the action for a particle moving in a noncommutative plane was done in [@sgfgs] using coherent states. The results obtained were indeed found to be in agreement with those found by other means [@muthukumar], [@fgs]. It is worth mentioning that the noncommutative version of the path integral was also derived in [@spal] but it did not reflect the breaking of the time reversal symmetry that one would expect. The same trick of using coherent states to define the propagation kernel was employed in [@tan], however, the final expressions for the propagator and the resulting physics were quite different from [@spal]. In [@acat], a phase-space path integral formulation of noncommutative quantum mechanics was carried out and its equivalence to the operatorial formulation was shown. However, the explicit form of the action derived in [@sgfgs] was not obtained here.
In this paper, we proceed to derive the path integral representation and the action for a particle in a magnetic field moving in a noncommutative plane. From this action, we obtain the equation of motion for the particle and compute the ground state energy for the particle in a magnetic field in the presence of a harmonic oscillator potential. We then move on to investigate the Aharonov-Bohm effect (studied earlier by comparing the solutions of the Schrödinger equation in the noncommutative plane in the presence and absence of the magnetic field [@harms]) in the path integral approach. Similar studies using the path integral formulation of quantum mechanics has also been made earlier in [@tur]. However, the results obtained are upto linear order in the noncommutative parameter. In this context, it is observed that when the noncommutative parameter and magnetic field are related in a specific way, the action for a particle in this magnetic field and noncommutative plane can be mapped to a particle of zero mass moving in a magnetic field in the commutative plane. This relation between the noncommutative parameter and magnetic field has also been found earlier in the context of the Landau level problem in the noncommutative plane [@nair], quantum mechanics on the noncommutative torus [@poly] and in the study of exotic Galilean symmetry in the noncommutative plane [@duval1]. Further, the relation between the noncommutative parameter and magnetic field leads to a second class constrained system which upon quantization yields a noncommutative algebra. This is in fact an fairly old and well known result [@jackiw; @jack], obtained here for the first time on the level of the action within a path integral setting. The Aharonov-Bohm phase-difference can then be very easily computed by following the standard arguements. Obtaining this phase-difference in general (without any specific relation between the magnetic field and noncommutative parameter) is rather subtle and needs to be done with great care. We also substantiate our results (obtained by the path integral approach) by computing this phase-difference through the transportation of the particle in a closed loop around a magnetic field confined in a solenoid. We also find that the action obtained can be mapped to the action for a particle in a harmonic oscillator potential (whose frequency is determined by the magnetic field) moving in the noncommutative plane and also to a commutative Landau problem. Finally, we discuss the equivalence of the path integral formulation with the noncommutative Schrödinger equation.
To begin our discussion, we present a brief review of the formalism of noncommutative quantum mechanics developed recently in [@fgs] before constructing the path integral representation of a particle in a magnetic field on the noncommutative plane. It was suggested in these papers that one can give precise meaning to the concepts of the classical configuration space and the Hilbert space of a noncommutative quantum system. The first step is to define classical configuration space. In two dimensions, the coordinates of noncommutative configuration space satisfy the commutation relation $$[\hat{x}, \hat{y}] = i\theta
\label{1}$$ for a constant $\theta$ that we can take, without loss of generality, to be positive. The above commutation relation is invariant with respect to $SL(2, R)$ transformation in $(\hat{x}, \hat{y})$-plane, in particular to rotations in this plane. The annihilation and creation operators are defined by $\hat b = \frac{1}{\sqrt{2\theta}} (\hat{x}+i\hat{y})$, $\hat{b}^\dagger =\frac{1}{\sqrt{2\theta}} (\hat{x}-i\hat{y})$ and satisfy the Fock algebra $[\hat{b}, \hat{b}^\dagger ] = 1$. The noncommutative configuration space is then isomorphic to the boson Fock space $$\mathcal{H}_c = \textrm{span}\{ |n\rangle=
\frac{1}{\sqrt{n!}}(\hat{b}^\dagger)^n |0\rangle\}_{n=0}^{n=\infty}
\label{3}$$ where the span is taken over the field of complex numbers.
The next step is to introduce the Hilbert space of the noncommutative quantum system, which is taken to be: $$\mathcal{H}_q = \left\{ \psi(\hat{x},\hat{y}):
\psi(\hat{x},\hat{y})\in \mathcal{B}
\left(\mathcal{H}_c\right),\;
{\rm tr_c}(\psi^\dagger(\hat{x},\hat{y})
\psi(\hat{x},\hat{y})) < \infty \right\}.
\label{4}$$ Here ${\rm tr_c}$ denotes the trace over noncommutative configuration space and $\mathcal{B}\left(\mathcal{H}_c\right)$ the set of bounded operators on $\mathcal{H}_c$. This space has a natural inner product and norm $$\left(\phi(\hat{x}, \hat{y}), \psi(\hat{x},\hat{y})\right) =
{\rm tr_c}(\phi(\hat{x}, \hat{x})^\dagger\psi(\hat{x}, \hat{y}))
\label{inner}$$ and forms a Hilbert space [@hol]. To distinguish states in the noncommutative configuration space from those in the quantum Hilbert space, states in the noncommutative configuration space are denoted by $|\cdot\rangle$ and states in the quantum Hilbert space by $\psi(\hat{x},\hat{y})\equiv |\psi)$. Assuming commutative momenta, a unitary representation of the noncommutative Heisenberg algebra in terms of operators $\hat{X}$, $\hat{Y}$, $\hat{P}_x$ and $\hat{P}_y$ acting on the states of the quantum Hilbert space (\[4\]) is easily found to be $$\begin{aligned}
\hat{X}\psi(\hat{x},\hat{y}) &=& \hat{x}\psi(\hat{x},\hat{y})\quad,\quad
\hat{Y}\psi(\hat{x},\hat{y}) = \hat{y}\psi(\hat{x},\hat{y})\nonumber\\
\hat{P}_x\psi(\hat{x},\hat{y}) &=& \frac{\hbar}{\theta}[\hat{y},\psi(\hat{x},\hat{y})]\quad,\quad
\hat{P}_y\psi(\hat{x},\hat{y}) = -\frac{\hbar}{\theta}[\hat{x},\psi(\hat{x},\hat{y})]~.
\label{action}\end{aligned}$$ The minimal uncertainty states on noncommutative configuration space, which is isomorphic to boson Fock space, are well known to be the normalized coherent states [@klaud] $$\label{cs}
|z\rangle = e^{-z\bar{z}/2}e^{z b^{\dagger}} |0\rangle$$ where, $z=\frac{1}{\sqrt{2\theta}}\left(x+iy\right)$ is a dimensionless complex number. These states provide an overcomplete basis on the noncommutative configuration space. Corresponding to these states we can construct a state (operator) in quantum Hilbert space as follows $$|z, \bar{z} )=\frac{1}{\sqrt{\theta}}|z\rangle\langle z|.
\label{csqh}$$ These states have the property $$\hat{B}|z, \bar{z})=z|z, \bar{z})~;~\hat{B}=\frac{1}{\sqrt{2\theta}}(\hat{X}+i\hat{Y}).
\label{p1}$$ Writing the trace in terms of coherent states (\[cs\]) and using $|\langle z|w\rangle|^2=e^{-|z-w|^2}$ it is easy to see that $$(z, \bar{z}|w, \bar{w})=\frac{1}{\theta}tr_{c}
(|z\rangle\langle z|w\rangle\langle w|)=
\frac{1}{\theta}|\langle z|w\rangle|^2=\frac{1}{\theta}e^{-|z-w|^2}
\label{p2}$$ which shows that $|z, \bar{z})$ is indeed a Hilbert-Schmidt operator. The ‘position’ representation of a state $|\psi)=\psi(\hat{x},\hat{y})$ can now be constructed as $$(z, \bar{z}|\psi)=\frac{1}{\sqrt\theta}tr_{c}
(|z\rangle\langle z| \psi(\hat{x},\hat{y}))=
\frac{1}{\sqrt\theta}\langle z|\psi(\hat{x},\hat{y})|z\rangle.
\label{posrep}$$ We now introduce the momentum eigenstates normalised such that $(p'|p)=\delta(p-p')$ $$\begin{aligned}
|p)&=&\sqrt{\frac{\theta}{2\pi\hbar^{2}}}e^{i\sqrt{\frac{\theta}{2\hbar^2}}
(\bar{p}b+pb^\dagger)}~;~\hat{P}_i |p)=p_i |p)\\
p_x&=&{\mbox Re}\,p~,~p_y={\mbox Im}\,p\nonumber
\label{eg}\end{aligned}$$ satisfying the completeness relation $$\begin{aligned}
\int d^{2}p~|p)(p|=1_{Q}~.
\label{eg5}\end{aligned}$$ We now observe that the wave-function of a “free particle" on the noncommutative plane is given by [@sgfgs] $$\begin{aligned}
(z, \bar{z}|p)=\frac{1}{\sqrt{2\pi\hbar^{2}}}
e^{-\frac{\theta}{4\hbar^{2}}\bar{p}p}
e^{i\sqrt{\frac{\theta}{2\hbar^{2}}}(p\bar{z}+\bar{p}z)}~.
\label{eg3}\end{aligned}$$ The completeness relations for the position eigenstates $|z,\bar{z})$ (which is an important ingredient in the construction of the path integral representation) reads $$\begin{aligned}
\int \frac{dzd\bar{z}}{\pi}~|z, \bar{z})\star(z, \bar{z}|=1_{Q}
\label{eg6}\end{aligned}$$ where the star product between two functions $f(z, \bar{z})$ and $g(z, \bar{z})$ is defined as $$\begin{aligned}
f(z, \bar{z})\star g(z, \bar{z})=f(z, \bar{z})
e^{\stackrel{\leftarrow}{\partial_{\bar{z}}}
\stackrel{\rightarrow}{\partial_z}} g(z, \bar{z})~.
\label{eg7}\end{aligned}$$ This can be proved by using eq.(\[eg3\]) and computing $$\begin{aligned}
\int \frac{dzd\bar{z}}{\pi}
(p'|z, \bar{z})\star(z, \bar{z}|p)=
e^{-\frac{\theta}{4\hbar^{2}}(\bar{p}p+\bar{p}'p')}
e^{\frac{\theta}{2\hbar^{2}}\bar{p}p'}\delta(p-p')=(p'|p)~.
\label{eg8}\end{aligned}$$ Thus, the position representation of the noncommutative system maps quite naturally to the Voros plane. With the above formalism and the completeness relations for the momentum and the position eigenstates (\[eg5\], \[eg6\]) in place, we now proceed to write down the path integral for the propagation kernel on the two dimensional noncommutative plane. This reads (upto constant factors) $$\begin{aligned}
(z_f, t_f|z_0, t_0)&=&\lim_{n\rightarrow\infty}\int
\prod_{j=1}^{n}(dz_{j}d\bar{z}_{j})~(z_f, t_f|z_n, t_n)\star_n
(z_n, t_n|....|z_1, t_1)\star_1(z_1, t_1|z_0, t_0)~.
\label{pint1}\end{aligned}$$
The Hamiltonian (acting on the quantum Hilbert space) for a particle in a magnetic field in the presence of a potential on the noncommutative plane reads $$\begin{aligned}
\hat{H}=\frac{(\hat{\vec{P}} -e\hat{\vec{A}})^2}{2m}+:V(\hat{B}^{\dagger},\hat{B}):
\label{hamil}\end{aligned}$$ where $V(\hat{X},\hat{Y})$ is the normal ordered potential expressed in terms of the annihilation and creation operators ($\hat{B}$, $\hat{B}^{\dagger}$). In the symmetric gauge [^3] $$\begin{aligned}
\hat{\vec{A}}=\left(-\frac{B}{2}\hat{Y}, \frac{B}{2}\hat{X}\right)
\label{gauge}\end{aligned}$$ (note that $B$ refers here to the magnetic field and not the annihilation operator $\hat{B}$. This slight abuse of notation will not create confusion in what follows) the above Hamiltonian takes the form $$\begin{aligned}
\hat{H}=\frac{\hat{\vec{P}}^2}{2m}+\frac{e^2 B^2}{8m}(\hat{X}^2 +\hat{Y}^2)-\frac{eB}{2m}(\hat{X}\hat{P}_y -\hat{Y}\hat{P}_x)+:V(\hat{B}^{\dagger},\hat{B}):~.
\label{hamil1}\end{aligned}$$ With this Hamiltonian, we now compute the propagator over a small segment in the above path integral (\[pint1\]). With the help of eq(s) (\[eg5\]) and (\[eg3\]), we have $$\begin{aligned}
(z_{j+1}, t_{j+1}|z_j, t_j)&=&(z_{j+1}|e^{-\frac{i}{\hbar}\hat{H}\tau}|z_j)\nonumber\\
&=&(z_{j+1}|1-\frac{i}{\hbar}\hat{H}\tau +O(\tau^2)|z_j)\nonumber\\
&=&\int_{-\infty}^{+\infty}d^{2}p_j~e^{-\frac{\theta}{2\hbar^{2}}\bar{p}_j p_{j}}
e^{i\sqrt{\frac{\theta}{2\hbar^{2}}}\left[p_{j}(\bar{z}_{j+1}-\bar{z}_{j})+\bar{p}_{j}(z_{j+1}-z_{j})\right]}\nonumber\\
&&\times e^{-\frac{i}{\hbar}\tau[\frac{\bar{p}_j p_{j}}{2m}+\frac{e^2 B^2 \theta}{8m}(2\bar{z}_{j+1}z_{j}+1)
+\frac{ieB}{2m}\sqrt{\frac{\theta}{2}}(p_j \bar{z}_{j+1}-\bar{p}_j z_j)-\frac{eB\hbar}{2m}+V(\bar{z}_{j+1}, z_{j})]}+O(\tau^2)~.
\label{pint2}\end{aligned}$$ Substituting the above expression in eq.(\[pint1\]) and computing the star products explicitly, we obtain (apart from a constant factor) $$\begin{aligned}
(z_f, t_f|z_0, t_0)=&&\lim_{n\rightarrow\infty}\int \prod_{j=1}^{n} (dz_{j}d\bar{z}_{j})
\prod_{j=0}^{n}d^{2}p_{j}\nonumber\\
&&\exp\left(\sum_{j=0}^{n}\left[\frac{i}{\hbar}\sqrt{\frac{\theta}{2}}\left[p_{j}\left\{\left(1-\frac{ieB\tau}{2m}\right)\bar{z}_{j+1}-\bar{z}_{j}\right\}+\bar{p}_{j}\left\{z_{j+1}-\left(1-\frac{ieB\tau}{2m}\right)z_{j}\right\}\right] +\alpha p_{j}\bar{p}_{j}
-\frac{i}{\hbar}\tau V(\bar{z}_{j+1},z_{j})\right]\right.\nonumber\\
&&\left.~~~~~~~~~~~~~~~~+\frac{\theta}{2\hbar^{2}}\sum_{j=0}^{n-1}p_{j+1}\bar{p}_{j}\right)
\label{pint3}\end{aligned}$$ where $\alpha=-\left(\frac{i\tau}{2m\hbar}+\frac{\theta}{2\hbar^{2}}\right)$. Making the identification $p_{n+1}=p_{0}$, the integrand of the above integral can be cast in the following form :\
$\exp\left(-\vec{\partial}_{z_{f}}\vec{\partial}_{\bar{z}_{0}}\right)$ $$\times\exp\left(\sum_{j=0}^{n}\left[\frac{i}{\hbar}\sqrt{\frac{\theta}{2}}\left[p_{j}\left\{\left(1-\frac{ieB\tau}{2m}\right)\bar{z}_{j+1}-\bar{z}_{j}\right\}+\bar{p}_{j}\left\{z_{j+1}-\left(1-\frac{ieB\tau}{2m}\right)z_{j}\right\}\right]
+\alpha p_{j}\bar{p}_{j}-\frac{i}{\hbar}\tau V(\bar{z}_{j+1},z_{j})+\frac{\theta}{2\hbar^{2}}p_{j+1}\bar{p}_{j}\right]
\right).$$ The purpose of the boundary operator in the above expression is to cancel an additional coupling which has been introduced between $p_0$ and $p_n$. The introduction of this coupling makes it easy to perform the momentum integral since it is of the Gaussian form $\exp(\sum_{i,j}p_{i}A_{i,j}\bar{p}_j)$, where $A$ is a $D\times D$ ($D=n+1=T/\tau$, $T=t_{f}-t_{0}$) dimensional matrix given by $$\begin{aligned}
A_{lr}=\alpha\delta_{l,r}+\frac{\theta}{2\hbar^{2}}\delta_{l+1,r}~.
\label{matrix}\end{aligned}$$ A simple inspection shows that the eigenvalues and the normalised eigenvectors of the matrix $A$ are given by $$\begin{aligned}
\lambda_{k}&=&\alpha+\frac{\theta}{2\hbar^2}e^{2\pi ik/D}\quad;\quad k\in[0,n]\nonumber\\
u_{k}&=&\frac{1}{\sqrt{D}}(1\quad e^{2\pi ik/D}\quad e^{4\pi ik/D}....)^{T}~.
\label{evalues}\end{aligned}$$ Since the real part of the eigenvalues of $A$ are nonpositive, one can carry out the momentum integral, to obtain $$\begin{aligned}
(z_f, t_f|z_0, t_0)&=&\lim_{n\rightarrow\infty}N\int\prod_{j=1}^{n}(dz_{j}d\bar{z}_{j})
\exp\left(-\vec{\partial}_{z_{f}}\vec{\partial}_{\bar{z}_{0}}\right)\nonumber\\
&&~~~~~~~~~~~~~~~~~~~~~\times\exp\left(\frac{\theta}{2\hbar^{2}}\sum_{l=0}^{n}\sum_{r=0}^{n}
\left\{\left(1-\frac{ieB\tau}{2m}\right)\bar{z}_{l+1}-\bar{z}_{l}\right\}A^{-1}_{lr}\left\{z_{r+1}-\left(1-\frac{ieB\tau}{2m}\right)z_{r}\right\}\right)\nonumber\\
&&~~~~~~~~~~~~~~~~~~~~~\times\exp\left(-\frac{i}{\hbar}\tau \sum_{j=0}^{n}V(\bar{z}_{j+1},z_{j})\right).
\label{pintegral1}\end{aligned}$$ The inverse of the matrix $A$ is easily obtained as $A^{-1}_{lr}=\sum_{k=0}^{n}\lambda_{k}^{-1}e^{2\pi i(l-r)k/D}$ leading to [^4] $$\begin{aligned}
(z_f, t_f|z_0, t_0)=&&\lim_{n\rightarrow\infty}N\int\prod_{j=1}^{n}(dz_{j}d\bar{z}_{j})
\exp\left(-\vec{\partial}_{z_{f}}\vec{\partial}_{\bar{z}_{0}}\right)
\exp\left(\frac{\theta\tau}{2\hbar^{2}T}\sum_{l,r,k=0}^{n}
\tau\left[\dot{\bar{z}}(l\tau)-\frac{ieB}{2m}\bar{z}(l\tau)\right]\left[\alpha+\frac{\theta}{2\hbar^{2}}e^{-\tau\partial_{(r\tau)}}\right]^{-1}\right.\nonumber\\
&&\left.~~~~~~~~~~~~~~~~~~~~~~~~~~~~\times[e^{2\pi i(l-r)k\tau/T}]\times\tau\left[\dot{z}(r\tau)+\frac{ieB}{2m}z(r\tau)\right]\right)\times\exp\left(-\frac{i}{\hbar}\tau \sum_{j=0}^{n}V(\bar{z}_{j},z_{j})+O(\tau^{2})\right)
\nonumber\\
=&&\lim_{n\rightarrow\infty}N\int\prod_{j=1}^{n}(dz_{j}d\bar{z}_{j})\exp\left(-\vec{\partial}_{z_{f}}\vec{\partial}_{\bar{z}_{0}}\right)\nonumber\\
&&\times\exp\left(\frac{\theta}{2\hbar^{2}T}\sum_{l,r,k=0}^{n}
\tau\left[\dot{\bar{z}}(l\tau)-\frac{ieB}{2m}\bar{z}(l\tau)\right]\left[-\frac{i}{2m\hbar}-\frac{\theta}{2\hbar^{2}}\partial_{(r\tau)}+O(\tau)\right]^{-1}\right.\nonumber\\
&&\left.~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \times[e^{2\pi i(l-r)k\tau/T}]\tau\left[\dot{z}(r\tau)+\frac{ieB}{2m}z(r\tau)\right]\right)\times\exp\left(-\frac{i}{\hbar}\tau \sum_{j=0}^{n}V(\bar{z}_{j},z_{j})+O(\tau^{2})\right)
\label{pintegral2}\end{aligned}$$ where in the first line, we have used the fact that $z_{l}=z(l\tau)$ and $z_{l+1}-z_{l}=\tau\dot{z}(l\tau)+O(\tau^{2})$. Taking the limit $\tau\rightarrow 0$ and performing the sum over $k$, we finally arrive at the path integral representation of the propagator $$\begin{aligned}
(z_f, t_f|z_0, t_0)&=&N\exp\left(-\vec{\partial}_{z_{f}}\vec{\partial}_{\bar{z}_{0}}\right)\int_{z(t_0)=z_0}^{z(t_f)=z_f }\mathcal{D}z\mathcal{D}\bar{z}
\exp({\frac{i}{\hbar}S})
\label{pintegral3}\end{aligned}$$ where $S$ is the action given by $$\begin{aligned}
S=\int_{t_{0}}^{t_{f}}dt \left[\frac{\theta}{2}\left\{\dot{\bar{z}}(t)-\frac{ieB}{2m}\bar{z}(t)\right\}\left(\frac{1}{2m}+\frac{i\theta}{2\hbar}
\partial_{t}\right)^{-1}
\left\{\dot{z}(t)+\frac{ieB}{2m}z(t)\right\}-\frac{e^2 B^2 \theta}{4m}\bar{z}(t)z(t)- V(\bar{z}(t),z(t))\right]~.
\label{action_ncqm}\end{aligned}$$
We now compute the ground state energy for the particle in a magnetic field and in the presence of a harmonic oscillator potential $V=\frac{1}{2}m\omega^2(\hat{X}^{2}+\hat{Y}^{2})$ from the above path integral representation of the transition amplitude. Using the normal ordered form of this potential in terms of the creation and annihilation operators, that is $:V:=m\omega^2 \theta \hat{B}^{\dagger}\hat{B}$, the action (\[action\_ncqm\]) reads $$\begin{aligned}
S=\int_{t_{0}}^{t_{f}}dt~\theta\left[\frac{1}{2}\left\{\dot{\bar{z}}(t)-\frac{ieB}{2m}\bar{z}(t)\right\}\left(\frac{1}{2m}+\frac{i\theta}{2\hbar}
\partial_{t}\right)^{-1}
\left\{\dot{z}(t)+\frac{ieB}{2m}z(t)\right\}-\left(\frac{e^2 B^2 }{4m}+m\omega^2\right)\bar{z}(t)z(t)\right]~.
\label{action_nchar}\end{aligned}$$ The equation of motion following from the above action is of the following form $$\begin{aligned}
\ddot{z}(t)+i\left\{\frac{eB}{m}\left(1+\frac{eB\theta}{4\hbar}\right)+\frac{m\omega^2 \theta}{\hbar}\right\}\dot{z}(t)+\omega^2 z(t)=0~.
\label{harsol}\end{aligned}$$ It is easy to see that the above equation reduces to the equation of motion for the particle in the harmonic oscillator in the $B\rightarrow0$ limit [@sgfgs]. Making an ansatz of the solution of the above equation in the form $z(t)\sim e^{-i\gamma t}$ leads to the following ground state energy eigenvalues for the particle $$\begin{aligned}
\gamma=\frac{1}{2}\left\{\frac{m\omega^{2}\theta}{\hbar}+\frac{eB}{m}\left(1+\frac{eB\theta}{4\hbar}\right)\pm\sqrt{\left(\frac{m\omega^{2}\theta}{\hbar}+\frac{eB}{m}\left(1+\frac{eB\theta}{4\hbar}\right)\right)^2+4\omega^{2}}\right\}~.
\label{energyeig}\end{aligned}$$ In the $\omega\rightarrow0$ limit, the above expression yields the two frequencies for the particle in a magnetic field on the noncommutative plane to be $$\begin{aligned}
\gamma=\frac{eB}{m}\left(1+\frac{eB\theta}{4\hbar}\right), 0~.
\label{enereigval}\end{aligned}$$ This ground state energy, computed from the path integral formalism, matches with those obtained by the canonical approach as we show in an appendix. In the $B\rightarrow0$ limit, the above expression yields the two frequencies for a particle in a harmonic oscillator potential on the noncommutative plane [@sgfgs].
With the above results in place, we now move on to study the Aharonov-Bohm effect. To proceed, we first observe that the action (\[action\_ncqm\]) can be recast in the following form $$\begin{aligned}
S=\int_{t_{0}}^{t_{f}}dt~\left[\theta m\left(1+\frac{eB\theta}{2\hbar}\right)^2
\dot{\bar z}(t)\left(1+\frac{i\theta m}{\hbar}\partial_{t}\right)^{-1}\dot{z}(t)+ieB\theta\left(1+\frac{eB\theta}{4\hbar}\right)\dot{\bar{z}}(t)z(t)-V(\bar{z}(t),z(t))\right]~.
\label{action_mag}\end{aligned}$$ Setting $V=0$, we find that the above action can be mapped to a particle of zero mass moving in the commutative plane and in a magnetic field given by $$\begin{aligned}
B=-\frac{2\hbar}{e\theta}.
\label{choice}\end{aligned}$$ The above choice for the magnetic field has also been observed earlier in the literature in different contexts [@nair],[@poly],[@duval1],[@schapos], e.g. in [@nair],[@poly] they were found to be critical values where the density of states for a charged particle in a magnetic field with a harmonic oscillator potential becomes infinite.
Indeed, with this choice of the magnetic field: $$\begin{aligned}
S&=&\frac{ieB\theta}{2}\int_{t_{0}}^{t_{f}}dt~\dot{\bar{z}}(t)z(t)\nonumber\\
&=&-\frac{eB}{4}\int_{t_{0}}^{t_{f}}dt~[\dot{x}(t)y(t)-\dot{y}(t)x(t)]\nonumber\\
&=&\frac{e}{2}\int_{\vec{x}_{0}}^{\vec{x}_{f}}\vec{A}.d\vec{x}
\label{action_map}\end{aligned}$$ where the second line is true upto boundary terms. It is evident from the first line that this is a constrained system with the following second class constraints $$\begin{aligned}
\Omega_{1}&=&p_x +\frac{eB}{4}y\approx0\nonumber\\
\Omega_{2}&=&p_y -\frac{eB}{4}x\approx0~.
\label{constr}\end{aligned}$$ Introducing the Dirac bracket and replacing $\{. ,.\}_{DB}\rightarrow\frac{1}{i\hbar}[. ,.]$ yield the following noncommutative algebra $$\begin{aligned}
[x_i, x_j]=-i\frac{2\hbar}{eB}\epsilon_{ij}=i\theta\epsilon_{ij}~;~[x_i , p_j]=\frac{i\hbar}{2}\delta_{ij}~;
~[p_i , p_j]=-i\hbar\frac{eB}{8}\epsilon_{ij}=\frac{i\hbar^2}{4\theta}\epsilon_{ij}~;~(i, j=1, 2)
\label{nc_alg}\end{aligned}$$ where we have used eq.(\[choice\]). It is to be noted that this noncommutativity was observed earlier in [@jackiw; @jack] by noting that in the limit $m\rightarrow0$, the $y$-coordinate is effectively constrained to the momentum canonical conjugate to the $x$-coordinate. However, in the path integral approach, the mass zero limit arises naturally.
With the usual Aharonov-Bohm experimental set up, one can now easily read off the Aharonov-Bohm phase-difference $\phi$ from the action (\[action\_map\]) following the discussion in [@sakurai] to be $$\begin{aligned}
\phi=\frac{eBA}{2\hbar}=\frac{e\Phi}{2\hbar}.
\label{phasediff}\end{aligned}$$ Here $A$ is the area enclosed by the loop around which the particle is transported and the magnetic field is non-vanishing and, correspondingly, $\Phi$ is the total magnetic flux enclosed by this loop.
In general it is, however, not this easy to obtain the Aharonov-Bohm phase-difference from the action (\[action\_ncqm\]) (for V=0) (without making the choice (\[choice\]) for the magnetic field) due to the presence of the non-local time derivative operator. To proceed in the general case, consider the experimental setup shown in figure 1.
{width="10cm" height="10cm"}
The transition amplitude for a particle originating at S to arrive at point B is the sum over all paths connecting these two points. From the setup in figure 1, these can naturally be divided into two classes, those paths that pass through slit A and those that pass through slit O and the total transition amplitude therefore consists of the sum of these two contributions. Since the action is quadratic, the transition amplitude can be computed exactly in a saddle point (or classical) approximation in which case the phase of the transition amplitude is simply given by the action evaluated on the classical path divided by $\hbar$. Hence, we can compute these two contributions by evaluating them for two classical paths passing through the two slits. The phase difference is then, of course, obtained by computing the difference between these two phases. The result for the action for a classical path starting from the source S ($z_{-T}=\frac{1}{\sqrt{2\theta}}(x_s, -y_s)$ at time $t=-T$), passing through a slit A ($z_2 =\frac{1}{\sqrt{2\theta}}(x_2, 0)$ at time $t=0$) and reaching a point B ($z_T =\frac{1}{\sqrt{2\theta}}(x_f, y_f)$ at time $t=T$) located on the screen reads $$\begin{aligned}
S_1 &=& \frac{i\gamma\theta m}{2(1-\cos\gamma T)}\left\{(z_{-T}\bar{z}_2 -c.c) +(\bar{z}_{-T} z_2 e^{i\gamma T}-c.c)
+|z_2|^2 (1-e^{i\gamma T})+|z_{-T}|^2(e^{-i\gamma T}-1)\right.\nonumber\\
&&\left.~~~~~~~~~~~~~~~~~~(\bar{z}_{T}z_2 -c.c) +(z_{T} \bar{z}_2 e^{i\gamma T}-c.c)
+|z_{2}|^2(e^{-i\gamma T}-1) +|z_T|^2 (1-e^{i\gamma T})\right\}
\label{class_action1}\end{aligned}$$ where $\gamma=\frac{eB}{m}\left(1+\frac{eB\theta}{4\hbar}\right)$. The result for the action evaluated on a classical path starting from the same point S (at $t=-T$), passing through a slit O ($z_0 =\frac{1}{\sqrt{2\theta}}(0, 0)$ at time $t=0$) and ending at the same point B (at $t=T$) on the screen can be obtained just by replacing $z_2$ by $z_0$ in the above expression. Taking the difference of these two classical actions and noting that the boundary operator $e^{-\stackrel{\rightarrow}{\partial}_{z_{T}}\stackrel{\rightarrow}{\partial}_{\bar{z}_{-T}}}$ has no effect on this difference, we get $$\begin{aligned}
S_1 -S_2&=& \frac{m\gamma}{2}\cot\frac{\gamma T}{2}[x_2 (x_2 -x_s -x_f)]+\frac{eB}{\hbar}\left(1+\frac{eB\theta}{4\hbar}\right)\times A.
\label{class_action2}\end{aligned}$$ Here $A$ is, as before, the area enclosed by the closed loop and in which the magnetic field is non-vanishing. Clearly the only topological term in the above expression is the second term, which must be identified with the Aharonov-Bohm-phase. We remark that in the above calculation one should actually also integrate over the intermediate times at which the particles pass through the slits (to sum over all paths), but one can easily check that this only leads to a multiplicative factor that does not affect the Aharonov-Bohm phase. The result reduces to eq.(\[phasediff\]) for the choice of the noncommutative parameter $\theta$ in eq.(\[choice\]).
An elegant way of obtaining the Aharonov-Bohm-phase is by transporting a particle in a closed loop. This can be done by the action of a chain of translation operators on the wave-function as follows $$\begin{aligned}
e^{-\frac{i}{\hbar}\hat{\pi}_{y}\Delta y}e^{-\frac{i}{\hbar}\hat{\pi}_{x}\Delta x}e^{\frac{i}{\hbar}\hat{\pi}_{y}\Delta y}e^{\frac{i}{\hbar}\hat{\pi}_{x}\Delta x}\Psi~.
\label{loop1}\end{aligned}$$ Now using the identity $\hat{S}^{-1}e^{\hat{A}}\hat{S}=e^{\hat{S}^{-1}\hat{A}\hat{S}}$ and the Baker-Campbell-Hausdorff formula [@sakurai], the above expression can be simplified to $$\begin{aligned}
e^{\frac{i}{\hbar}\Delta x \Delta y eB\left(1+\frac{eB\theta}{4\hbar}\right)}\Psi~.
\label{loop2}\end{aligned}$$ The AB-phase can immediately be read off from the above expression and agrees with that obtained from eq.(\[class\_action2\]). There exists another interesting connection between the action of (\[action\_ncqm\]) and the action of a harmonic oscillator in the noncommutative plane. Setting $V=0$ and making the following change of variables $$\begin{aligned}
z(t)=\zeta(t)e^{-\frac{ieB}{2m}t}
\label{harm1}\end{aligned}$$ eq.(\[action\_ncqm\]) can be recast in the following form $$\begin{aligned}
S=\int_{t_0}^{t_f}dt~\left[\frac{\theta}{2}e^{\frac{ieB}{2m}t}\dot{\bar\zeta}(t)\left(\frac{1}{2m}+\frac{i\theta}{2\hbar}
\partial_{t}\right)^{-1}\left(\dot{\zeta}(t)e^{-\frac{ieB}{2m}t}\right)-\frac{e^2 B^2 \theta}{4m}
\bar{\zeta}(t)\zeta(t)\right]~.
\label{harm2}\end{aligned}$$ Now making a Fourier transform of $\dot{\zeta}(t)$, the above expression simplies to $$\begin{aligned}
S=\int_{t_0}^{t_f}dt~\left[\frac{\theta}{2}\dot{\bar\zeta}(t)\left(\frac{1}{2m}+\frac{eB\theta}{4m\hbar}+\frac{i\theta}{2\hbar}
\partial_{t}\right)^{-1}\dot{\zeta}(t)-\frac{e^2 B^2 \theta}{4m}
\bar{\zeta}(t)\zeta(t)\right]~.
\label{harm3}\end{aligned}$$ The above action is that of a noncommutative harmonic oscillator with the following identifications $$\begin{aligned}
\frac{1}{2M}&=&\frac{1}{2m}+\frac{eB\theta}{4m\hbar},\nonumber\\
M\Omega^2 &=&\frac{e^2 B^2}{4m}~.
\label{harm4}\end{aligned}$$ The equation of motion following from this action reads $$\begin{aligned}
\ddot{\zeta}(t)+\frac{ie^2 B^2 \theta}{4M\left(1+\frac{eB\theta}{2\hbar}\right)\hbar}\dot{\zeta}(t)+\frac{e^2 B^2}{4M^2 \left(1+\frac{eB\theta}{2\hbar}\right)}\zeta(t)=0~.
\label{harm5}\end{aligned}$$ The ground state energy of this harmonic oscillator can be obtained by substituting the ansatz $\zeta(t)=e^{-i\Gamma t}$ in the above equation and solving for $\Gamma$ or by simply using the formula for the ground state energy of a harmonic oscillator [@sgfgs] which yields $$\begin{aligned}
\Gamma&=&\frac{1}{2\hbar}\left\{M\Omega^2 \theta \pm\Omega\sqrt{M^2 \Omega^2 \theta^2 +4\hbar^2}\right\}\nonumber\\
&=&\frac{eB}{2m}+\frac{e^2 B^2 \theta}{4m\hbar}~,~-\frac{eB}{2m}
\label{harm6}\end{aligned}$$ where we have used eq.(\[harm4\]) to obtain the final result. The ground state energy for the problem of the particle in a magnetic field in the noncommutative plane can now be obtained (as implied by eq.(\[harm1\])) by shifting the above energy by $\frac{eB}{2m}$ which gives the result obtained earlier (\[enereigval\]).
Another interesting link between the problem of a particle moving in a magnetic field in the noncommutative plane and a particle moving in a magnetic field in the commutative plane can be obtained from the following change of variables $$\begin{aligned}
u=\left(1+\frac{im\theta}{\hbar}\partial_{t}\right)^{-1} z~.
\label{change}\end{aligned}$$ Using this we find that the action (\[action\_mag\]) (for $V=0$) can be rewritten in the following form $$\begin{aligned}
S=\int_{t_{0}}^{t_{f}}dt~\theta m\left[\dot{\bar z}(t)\dot{u}(t)+\frac{ieB}{m}\left(1+\frac{eB\theta}{4\hbar}\right)\dot{\bar{z}}(t)u(t)\right]~.
\label{action_mag1}\end{aligned}$$ The above action shows that the problem of a particle in a magnetic field in the noncommutative plane can be mapped to a problem of a particle in a different magnetic field in the commutative plane. Although this is true on the level of the actions, one must realize that the map between transition amplitudes is more subtle as the boundary conditions on the path integral are also affected by this change of variables. Indeed, note that the boundary condition on $u$ depends on all higher order derivatives of $z$ at the boundary. This simply implies the expected, namely, that the commutative transition amplitude is only uniquely determined once all higher order derivatives of $z$ is specified at the boundary. This is in line with the analysis carried out in [@scholtz]. This does, however, indicate an interesting duality between a quantum Hall system and a particle in a magnetic field in the noncommutative plane. Alternatively, one can start from the problem of a particle in a magnetic field $B$ (kept fixed) and get a one parameter family of problems of a particle in a magnetic field $B^{\star}$ in the noncommutative plane where $B$ and $B^{\star}$ are related by $$\begin{aligned}
B=B^{\star} \left(1+\frac{eB^{\star} \theta}{4\hbar}\right)~.
\label{one_param}\end{aligned}$$ Solving for $B^{\star}$ gives $$\begin{aligned}
B^{\star}(\theta)=-\frac{2\hbar}{e\theta}\left\{1-\sqrt{1+\frac{eB\theta}{\hbar}}\right\}~.
\label{solution}\end{aligned}$$ Expectedly, this solution reduces to the appropriate $B$ and $\theta$ zero limits, i.e. $B^{\star}\rightarrow0$ for $B\rightarrow0$ and $B^{\star}\rightarrow B$ for $\theta\rightarrow0$.
Finally, we show the equivalence between the path integral formulation in the noncommutative plane and the noncommutative Schrödinger equation. To proceed, we use the fact that the transition amplitude is the propagator which gives the propagation of the wave-function in the following way $$\begin{aligned}
\psi(z_f , \epsilon)&=&(z_f , \epsilon|\psi)=\int \frac{d^2 z_i}{\pi}(z_f , \epsilon|z_i , 0)\star_{z_i}(z_i , 0|\psi)\nonumber\\
&=&\int \frac{d^2 z_i}{\pi}(z_f , \epsilon|z_i , 0)\star_{z_i}\psi(z_i, \bar{z}_i , 0)~.
\label{sc1}\end{aligned}$$ For infinitesimal $\epsilon$ and $\hat{H}=\frac{\hat{P}_{i}^2}{2m}+:V(B^{\dagger}, B):$, the infintesimal transition amplitude $(z_f , \epsilon|z_i , 0)$ upto $\mathcal{O}(\epsilon)$ reads [@sgfgs] $$\begin{aligned}
(z_f , \epsilon|z_i , 0)=\frac{\theta}{2\hbar^2 \alpha}\left\{1-\frac{i\epsilon}{\hbar}V(\bar{z}_{f}, z_{i})\right\}e^{-\frac{\theta}{2\hbar^2 \alpha}|z_f -z_i|^2}
\label{sc2}\end{aligned}$$ where $\alpha=\frac{i\epsilon}{2m\hbar}+\frac{\theta}{2\hbar^2}$. Substituting this expression in eq.(\[sc1\]), we get $$\begin{aligned}
\psi(z_f , \epsilon)=\frac{\theta}{2\pi \hbar^2 \alpha}\int d^2 z_i \left\{1-\frac{i\epsilon}{\hbar}V(\bar{z}_{f}, z_{i})\right\}e^{-\frac{\theta}{2\hbar^2 \alpha}|z_f -z_i|^2}\star_{z_i}\psi(z_i, \bar{z}_i , 0)~.
\label{sc3}\end{aligned}$$ Making a change of variables to $$\begin{aligned}
z_i = z_f +\eta
\label{sc4}\end{aligned}$$ the above expression can be recast in the form $$\begin{aligned}
\psi(z_f , \epsilon)=\frac{\theta}{2\pi \hbar^2 \alpha}\int d^2 \eta \left\{1-\frac{i\epsilon}{\hbar}V(\bar{z}_{f}, z_{f}+\eta)\right\}e^{-\frac{\theta}{2\hbar^2 \alpha}|\eta|^2}\star_{\eta}\psi(z_f +\eta, \bar{z}_f +\bar\eta , 0)~.
\label{sc5}\end{aligned}$$ Using the form of the star product and the fact that $f(z+\eta)=e^{\eta\stackrel{\rightarrow}{\partial_z}}f(z)$, the above equation can be simplified to $$\begin{aligned}
\psi(z_f , \epsilon)=\frac{\theta}{2\pi \hbar^2 \alpha}\int d^2 \eta~e^{-\frac{\theta}{2\hbar^2 \alpha}|\eta|^2}e^{\frac{i\epsilon}{2m\hbar\alpha}\eta\stackrel{\rightarrow}{\partial}_{z_{f}}}
e^{\bar{\eta}\stackrel{\rightarrow}{\partial}_{\bar{z}_f}}\left[1-\frac{i\epsilon}{\hbar}V(\bar{z}_f , z_f)
e^{\eta\stackrel{\leftarrow}{\partial}_{z_f}} \right]\psi(z_f , \bar{z}_f, 0)~.
\label{sc6}\end{aligned}$$ We now expand the exponential involving $\epsilon$ in the exponent in a power series (keeping terms upto $\mathcal{O}(\epsilon)$) and perform the $\eta$ integral to get $$\begin{aligned}
\psi(z_f , \epsilon)=\psi(z_f , 0) +\frac{i\epsilon\hbar}{m\theta}\frac{\partial^{2}}{\partial_{\bar{z}_f} \partial_{z_f}}\psi(z_f , 0)-\frac{i\epsilon}{\hbar}\frac{\theta}{2\hbar^2 \alpha}V(\bar{z}_f , z_f)\star_{z_f}
\psi(z_f , 0)~.
\label{sc7}\end{aligned}$$ In the limit $\epsilon\rightarrow0$, we get the time dependent Schrödinger equation in NC plane $$\begin{aligned}
i\hbar\partial_{t}\psi(z_f , t)=\left[-\frac{\hbar^2}{m\theta}\frac{\partial^{2}}{\partial_{\bar{z}_f} \partial_{z_f}}+V(\bar{z}_f , z_f)\star_{z_f}\right]\psi(z_f , t)~.
\label{sc8}\end{aligned}$$
To summarise: In this paper, we have systematically derived the path integral representation of the propagation kernel for a particle in a magnetic field in the presence of an arbitrary potential moving in the the noncommutative plane using the recently proposed formulation of noncommutative quantum mechanics. From the path integral, we have obtained the action for the particle in noncommutative quantum mechanics. This is one of the important results in our paper. The equation of motion of the particle is obtained from this action and was used to compute the ground state energy of the particle and to show that the result is in conformity with the results obtained by other methods. We then investigated the Aharonov-Bohm effect using the path integral formulation. In this context we found an interesting connection (observed earlier in different contexts in [@nair]-[@duval1]) with a particle of zero mass moving in a magnetic field related in a particular way to the noncommutative parameter. The Aharonov-Bohm phase-difference in this case is easy to read off. Although the computation is in general a bit more subtle, this computation has also been performed and found to be in agreement with other techniques of computation. We also observed interesting connections of this action with those of a noncommutative harmonic oscillator and a commutative Landau problem. Finally, we discussed the equivalence of the path integral formulation and the noncommutative Schrödinger equation.\
Appendix {#appendix .unnumbered}
========
In this appendix, we obtain the energy spectrum of the particle in a magnetic field in the noncommutative plane by the canonical method [@mezincescu]. The noncommuting coordinates can be expressed in terms of commuting coordinates and their momenta in the form $$\begin{aligned}
\hat{X}&=&X-\frac{\theta}{2\hbar}P_{y}\nonumber\\
\hat{Y}&=&Y+\frac{\theta}{2\hbar}P_{x}~;~\hat{P}_{i}=P_i.
\label{app1}\end{aligned}$$ Under this change of variables, the Hamiltonian (\[hamil1\]) can be rewritten as $$\begin{aligned}
\hat{H}=\frac{1}{2m}\left\{h_{1}^{2}\vec{X}^{2}+h_{2}^{2}\vec{P}^{2}+h_{3}(XP_{y}-YP_{x})\right\}
\label{app2}\end{aligned}$$ where $$\begin{aligned}
h_{1}^{2}&=&m^2 \omega_{c}^2 ~;~\omega_{c}=\frac{eB}{2m}\nonumber\\
h_{2}^{2}&=&\left(1+\frac{m\omega_{c}\theta}{2\hbar}\right)^2 \nonumber\\
h_{3}&=&2m\omega_{c}\left(1+\frac{m\omega_{c}\theta}{2\hbar}\right)=2h_1 h_2~.
\label{app3}\end{aligned}$$ Introducing the operators $$\begin{aligned}
a_{x}=\frac{ih_2 P_x +h_1 X}{\sqrt{h_3 \hbar}}~;~a_{y}=\frac{ih_2 P_y +h_1 Y}{\sqrt{h_3 \hbar}}
\label{app4}\end{aligned}$$ where $[a_x , a_{x}^{\dagger}]=1=[a_y , a_{y}^{\dagger}]$ leads to $$\begin{aligned}
\hat{H}=\frac{h_3 \hbar}{2m}\{(a_{x}^{\dagger}a_x +a_{y}^{\dagger}a_y +1 )+i(a_{y}^{\dagger}a_x -a_{x}^{\dagger}a_y)\}~.
\label{app5}\end{aligned}$$ Finally, defining $$\begin{aligned}
a_{\pm}=\frac{a_x \pm i a_y}{\sqrt{2}}
\label{app6}\end{aligned}$$ where $[a_+ , a_{+}^{\dagger}]=1=[a_{-} , a_{-}^{\dagger}]$ leads to the following second quantized Hamiltonian from which the energy spectrum can easily be read off: $$\begin{aligned}
:\hat{H}:&=&\frac{h_3 \hbar}{m}a_{-}^{\dagger}a_{-}=\frac{eB}{m}\left(1+\frac{eB\theta}{4\hbar}\right)\hbar a_{-}^{\dagger}a_{-}~.
\label{app7}\end{aligned}$$ For the sake of completeness, we also obtain the Heisenberg equations of motion for the noncommutative variable $\hat{Z}$ which reads $$\begin{aligned}
\dot{\hat Z}(t)&=&\frac{1}{i\hbar}[\hat Z, \hat H]\nonumber\\
&=&\frac{1}{m}\left(1+\frac{eB\theta}{2\hbar}\right)\left\{\frac{\hat{P}}{\sqrt{2\theta}}-\frac{ieB}{2}\hat Z\right\}~.
\label{he1}\end{aligned}$$ Similarly, one gets $$\begin{aligned}
\dot{\hat P}(t)&=&-\sqrt{2\theta}\frac{e^2 B^2}{4m}\hat{Z}-\frac{ieB}{2m}\hat{P}~.
\label{he2}\end{aligned}$$ Differentiating eq.(\[he1\]) and combining it with eq(s)(\[he1\], \[he2\]) leads to $$\begin{aligned}
\ddot{\hat Z}(t)+\frac{ieB}{m}\left(1+\frac{eB\theta}{4\hbar}\right)\dot{\hat{Z}}=0
\label{he3}\end{aligned}$$ which is the equation of motion for the particle. 0.5 cm [**[Acknowledgements]{}**]{} : This work was supported under a grant of the National Research Foundation of South Africa. The authors would also like to thank the referees for useful comments.
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[^1]: e-mail:sunandan.gangopadhyay@gmail.com
[^2]: e-mail: fgs@sun.ac.za
[^3]: Note that any asymmetric gauge linear in $\hat{x}$, $\hat{y}$ is related to a symmetric one by a gauge transformation represented by rotation in the $(\hat{x}, \hat{y})$-plane and hence leads to the same Hamiltonian.
[^4]: There is a sign error in the inverse of the matrix $A$ in [@sgfgs]. However, the final conclusions regarding the spectrum of the harmonic oscillator in the noncommutative plane remains unaltered.
|
---
abstract: 'We consider stationary driven systems in contact with a thermal equilibrium bath. There is a constant (Joule) heat dissipated from the steady system to the environment as long as all parameters are unchanged. As a natural generalization from equilibrium thermodynamics, the nonequilibrium heat capacity measures the excess in that dissipated heat when the temperature of the thermal bath is changed. To improve experimental accessibility we show how the heat capacity can also be obtained from the response of the instantaneous heat flux to small periodic temperature variations.'
author:
- Christian Maes
- Karel Netočný
title: Nonequilibrium calorimetry
---
Calorimetry out of equilibrium
==============================
Standard thermodynamics deals with equilibrium systems and their energy exchange with the environment as external parameters like the volume or the temperature are changed. If these changes are slow (quasistatic and along equilibria), then the entire scheme simplifies as described by the thermodynamic laws for reversible processes. There heat and entropy get proportional (Clausius heat theorem). As the ability of a specific system to exchange heat and store energy is usually given in terms of the heat capacity (the heat contribution from temperature changes) and the latent heat (when temperature is unchanged), those quantities also yield important information about the internal structure of the equilibrium system. In that way, calorimetry has provided crucial information about microscopic structures and the nature of the physical states.
Concerning possible generalizations, the first (and by now more standard) option is to go beyond quasistatic processes and towards a time-resolved thermodynamics. That can be done within the framework of linear response and it naturally leads to a frequency-dependent generalization of the heat capacity and related thermodynamic quantities. That was mostly studied for equilibrium systems, with the zero-frequency limit recovering the usual reversible thermodynamic picture. See e.g. [@ND] and also the very recent [@dio] in the context of stochastic thermodynamics. In the present paper we suggest a similar step forward in the study of thermodynamic processes connecting steady states of nonequilibrium systems.\
It is important to realize that in driven nonequilibrium systems the heat exchange runs upon a dissipation background. It means that the total heat exchanged with the environment goes virtually to infinity when making the process slower and slower because of its DC-component coming from the steady dissipation. Hence, we are really interested not in the (eventually diverging) total heat but in its *excess* part coming from changes in the temperature and/or other parameters.
On the theoretical side, a natural question arises whether such an excess heat is well defined in the quasistatic regime, in the sense of being essentially insensitive of the actual speed of the process as long as it is slow enough. This question has been answered in the affirmative; see [@eu; @jir]. It allows to consistently construct a generalization of the heat capacity to nonequilibrium steady states. We have checked via examples that such a steady heat capacity exhibits some new features when far from thermal equilibrium. For example, it can take negative values. Nevertheless, some more systematic understanding of how these properties reflect the structure of nonequilibrium steady states is still lacking. Response to temperature variations in the general context of fluctuation–dissipation relations has also been discussed in [@yol].
Towards the experimental realization, there are other problems. First, one may want to measure the excess heat directly along a relaxation process to the new steady condition after making a small sudden change of temperature (or other parameters). One then needs to extract the transient part of the dissipated heat, i.e., the one obtained after subtracting the steady “background” dissipation. As a possible variation, instead of measuring the heat directly, it can also be accessed indirectly from measuring the (excess) work done by the driving forces. Yet, main issues to be solved here include the finding of the experimentally most feasible systems on which the temperature can be manipulated on time scales comparable with those of the system itself. The present paper seeks an alternative route: to extend the frequency-dependent calorimetry to truly nonequilibrium systems and to extract the quasistatic excess from its low-frequency behavior. That is the main purpose of the present paper.\
We start in the next Section with the definition of (nonequilibrium) heat capacity. We also include in Section \[mar\] some relevant formulae how to rewrite that specifically for processes modeled as Markov dynamics, in terms of dissipated power. The main result of the paper is in Section \[di\] which describes the method of measuring heat capacity via temperature modulation and for which we believe the problem of excess (as a difference between very large quantities) may be avoided. Instead of making the difference of time-extensive heats, we consider there the heat flux as function of time. The heat capacity of nonequilibrium steady states then also appears as the static limit of a nonequilibrium frequency-dependent heat capacity.
Nonequilibrium theory
=====================
We refer to [@eu; @jir] for the initial theory and basic examples of nonequilibrium heat capacities. The basic idea builds on concepts from steady state thermodynamics as in [@oon; @kom2]. The result of the present paper is to see in Section \[di\] that the specific heat of a system under steady dissipative conditions can be measured by following the dissipated power as a function of time. We start however next with the basic formul[æ]{} which rigorously connect the nonequilibrium heat capacity with the excess heat.
Quasistatic excess heat
-----------------------
Consider a generic thermodynamic system on which external forces perform some work $W$ and which exchanges heat $Q$ with an (equilibrium) heat bath at a temperature $T$, so that $W + Q = \De U$ is the energy balance. We assume that the external forces maintain the system under fixed nonequilibrium conditions before time zero, so that they perform work $W_{[-t,0]} = w^{(T)}\,t$ at constant power $w^{(T)} > 0$, which passes through the system and then dissipates as heat $-Q_{[-t,0]} = -q^{(T)}\,t$ at rate $-q^{(T)} = w^{(T)}$. We explicitly indicate the dependence on the temperature $T$ playing the role of a control parameter. We remark that this always means the (well-defined) temperature of the equilibrium heat bath to which the system dissipates.
Both heat and work are time-extensive and nonzero because of assumed nonequilibrium conditions. In applications to fluctuating mesoscopic systems, heat and work can be physically well-defined per trajectory when the system is weakly coupled to the environment, but the heat capacity involves taking statistical averages over possible system trajectories; see Appendix \[mar\].\
Assume now that we make a measurement of the heat under slow temperature changes starting from time zero. A general quasistatic process can be decomposed in many elementary processes, each one consisting of a tiny sudden warming up (or cooling down) and then followed by a relaxation to new steady conditions. We could sum all the elementary contributions but clearly, for both theoretical and experimental purposes, it is enough to concentrate on one such an elementary process.
Before the sudden change of temperature from $T$ to $T + \de T$ at time zero, the system was in the steady state corresponding to the bath temperature $T$. After the change, it undertakes a relaxation to the new steady state at $T + \de T$. That is a transient process and the heat $Q_{[0,t]}$ is no longer purely extensive but it contains a transient part as well. The latter can be extracted by comparing with the steady heat under the new stationary conditions, which is $q^{(T+\de T)}\,t$. That transient contribution along the complete relaxation process, $$\label{qex}
\de Q^{\text{ex}} = \lim_{t \to \infty} \Bigl( Q_{[0,t]} - q^{(T+\de T)}\,t \Bigr)$$ is called an *excess heat*. Note that we really have to subtract the steady heat as corresponding to the *new* temperature $T + \de T$ since the dissipation rate can be (and typically is) temperature-dependent. Under equilibrium conditions the latter would be just zero and the excess heat coincides with the total heat exchange along the elementary process. In contrast, out of equilibrium we take the difference of large (in the limit, infinite) quantities. In practice, one surely performs no time limit but, instead, let the relaxation run till it is “essentially finished”. If $\tau$ is a characteristic time of relaxation then the excess heat $\de Q^\text{ex}$ is to be compared with the steady heat $q^{(T)} \tau$. Obviously, if $|q^{(T)}|\tau \gg |\de Q^\text{ex}|$ then one can hardly expect the excess heat to be distinguishable against the steady dissipation background.
Steady heat capacity
--------------------
The steady heat capacity quantifies the extra heat needed for the system to accommodate to a unit temperature change, $$\label{hca}
C(T) = \frac{\de Q^\text{ex}}{\de T}$$ Analogously, one can consider more general quasistatic processes including also the change of other thermodynamic parameters, which would then lead to a nonequilibrium generalization of the latent heat (capacities). All these quantities naturally supplement the incoming heat flux $q^{(T)}$ and provide a more complete characterization of the nonequilibrium steady state and its thermal sensitivity to external perturbations.\
Although heat is a primary quantity here, we can as well consider the *excess work* defined analogously as $$\label{wex}
\de W^\text{ex} = \lim_{t \to \infty} \Bigl( W_{[0,t]} - w^{(T+\de T)}\,t \Bigr)$$ where always $W_{[0,t]} + Q_{[0,t]} = U(t) - U(0)$. Since the steady power on the system is just $w^{(T+\de T)} = -q^{(T+ \de T)}$, we can relate with in the balance $\de W^\text{ex} + \de Q^\text{ex} = {\textrm{d}}U$. Hence, the steady heat capacity can also be written in the form $$\label{tex}
C(T) = \frac{\partial U}{\partial T} - \frac{\de W^\text{ex}}{\de T}$$ where the first term is a usual temperature-energy response. Under equilibrium conditions such as constant volume and/or other thermodynamic coordinates, the second term vanishes. In this case the familiar equilibrium formula is recovered, namely that the equilibrium heat capacity coincides with the temperature-energy response coefficient. In contrast, the nonequilibrium contribution cannot be reduced to such a simple “thermodynamic” form and it depends on dynamical details of the system.\
In Appendix \[mar\] we derive an explicit form of the nonequilibrium heat capacity for general Markov systems obeying the local detailed balance principle. The result reads that besides the steady-state average energy $U = \langle E(x) \rangle_T$, with $E(x)$ the energy function on mesoscopic states $x$, we need still another function $V^T(x)$, $\langle V^T(x) \rangle_T = 0$, which encapsulates the effect of nonequilibrium driving forces. In total, $$\label{c-mar}
C(T) = \frac{{\textrm{d}}\langle E(x) \rangle_{T}}{{\textrm{d}}T} -
\Bigl\langle \frac{{\textrm{d}}V^{T}(x)}{{\textrm{d}}T}\, \Bigr\rangle_T$$
An important feature of the new function $V^T(x)$ is that it depends both on the state $x$ and the bath temperature $T$. Suppose we can approximately write $V^T(x) \simeq \Phi(x) - \langle \Phi(x) \rangle_T$ with a temperature-independent “potential” $\Phi(x)$. Then $$C(T) \simeq \frac{{\textrm{d}}\langle \tilde E(x) \rangle_{T}}{{\textrm{d}}T}\,,\qquad
\tilde E(x) = E(x) + \Phi(x)$$ and we obtain an approximate formula resembling the equilibrium form for the modified energy function $\tilde E(x)$. Indeed, this is a viable simplification, e.g., in the regimes of very low or very high temperature, see [@jir] for specific examples. However, such a decomposition of the function $V^T(x)$ is not possible in general.
Temperature–heat response {#di}
=========================
In order to overcome possible experimental difficulties with measuring the excess heat above the steady dissipation background, we next discuss an alternative but theoretically equivalent scenario within the framework of time-resolved calorimetry.
The heat can generally be resolved into the time-dependent flux as $Q_{[0,t]} = \int_0^t J^Q_s\,{\textrm{d}}s$. Initially we have the steady heat current $J_0^Q = q^{(T)}$ into the system (equal to minus the steady rate of dissipation at temperature $T$). Let us now modulate the temperature, $T_s = T + h_s$, at times $s \geq 0$. Within the linear response theory the heat current at time $t>0$ is $$\label{adm}
J^Q_t = J^Q_0 + \lambda_\infty\,h_t + \int_0^t \lambda_s\,h_{t-s}\,{\textrm{d}}s$$ The function $\lambda_t$ is a temporal temperature-heat “admittance”, assumed to decay fast enough in time; $\lambda_\infty$ accounts for the immediate, non-delayed response. The latter naturally emerges in Markov systems with discrete states as a consequence of temporal coarse-graining; for a more general discussion on delayed and non-delayed contributions in the linear theories see, e.g., Section 3.1.2 in [@kubo]. For other considerations of fluctuation-response relations for thermal perturbations in overdamped diffusions, see [@yol].\
Let us again take the special case where the temperature suddenly changes at time zero from $T$ to $T+\de T$ (i.e., $h_s = \de T$ for $s>0$). In the limit $t \to \infty$ the system approaches a new steady state at bath temperature $T + \de T$ with the steady heat current $J_t^Q \rightarrow J_\infty^Q = q^{(T+\de T)}$. From – the heat capacity $C(T)$ satisfies $$\int_0^\infty ( J^Q_t - J^Q_\infty)\,{\textrm{d}}t = C(T)\,\de T$$ so that yields $$\label{c-ka-relation}
C(T) = -\int_0^\infty {\textrm{d}}t \int_t^\infty \lambda_s\,{\textrm{d}}s = -\int_0^\infty t \lambda_t\,{\textrm{d}}t$$ which expresses the heat capacity in terms of the admittances. On the other hand, the shift in steady heat currents is, again from , $$\label{sig}
J_\infty^Q - J_0^Q = B(T) \,\de T\,,\qquad
B(T) = \lambda_\infty + \int_0^\infty \lambda_t\,{\textrm{d}}t$$ This way both response coefficients $B(T)$ and $C(T)$ derive from the admittance $\lambda_s$ and they capture different aspects of the temperature-heat response.
As a more experimentally feasible protocol we consider the harmonic temperature oscillations $h_s = \epsilon \,\sin(\omega s)$ with some small amplitude $\epsilon$ and frequency $\omega$. Provided the admittance $\lambda_s$ decays asymptotically as $O(e^{-\gamma s})$ with some $\gamma > 0$, the heat current at large times obtains the form $$\label{gaga}
J^Q_t = q^{(T)} + \epsilon\,[\si_1(\om) \sin(\om t) + \si_2(\om) \cos(\om t)] + O(e^{-\ga t})$$ defining $\si_{1,2}(\om)$ as the in- and out-phase components of the temperature-sensitivity of the dissipation. Comparing with , they are related to the admittance $\lambda_t$ by the Fourier-Laplace transform $$ \si_1(\om) + i\, \si_2(\om) = \lambda_\infty + \int_0^\infty e^{-i\om\,t}\lambda_t\,{\textrm{d}}t
$$From we get $\si_1(\om = 0) = B(T)$, $\si_2(\om = 0) = 0$, and from , $$\frac{\partial\si_1}{\partial\om}\Bigr|_{\om = 0} = 0\,,\qquad
\frac{\partial\si_2}{\partial\om}\Bigr|_{\om = 0} = C(T)
$$ Hence, combining that with , the low-frequency asymptotics of the heat current response is $$J^Q_t = J_0^Q + \epsilon\,[B(T)\, \sin(\om t) + C(T)\,\om \cos(\om t) + O(\om^2)] + O(e^{-\ga t})$$ We see that the nonequilibrium heat capacity, as originally defined via the excess heat, provides the leading low-frequency (out-phase) correction to the steady (in-phase) linear temperature-heat relation. This also indicates how the steady heat capacity can be detected and measured from the response to slow periodic temperature variations.
Note that this is nothing but a frequency-dependent calorimetry restricted to low frequencies, see e.g. [@dio], the only difference being that in the usual equilibrium setup $J_0^Q= q^{(T)}$ vanishes. In contrast, around a steady nonequilibrium the latter provides the dominant (for $\omega \to 0$) contribution to the heat flux, whereas the heat capacity becomes the next correction.
Conclusions
===========
Thermal properties of nonequilibria appear essential in the program of steady state thermodynamics. Calorimetry of nonequilibrium systems may be developed to provide a useful characterization of the change in a material’s thermal properties when driven away from equilibrium conditions, [@bioc; @b2]. Nonequilibrium heat capacities can be consistently defined in terms of the notion of excess heat, or from how the steady dissipated power varies with temperature. We have seen how temperature modulation for nonequilibria gives access to that information via the time-dependence of the instantaneous heat flux.
[**Acknowledgments:**]{} KN acknowledges the support from the Grant Agency of the Czech Republic, grant no. 17-06716S.
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Heat capacity of Markov systems {#mar}
===============================
In this section we derive formula for the nonequilibrium heat capacity of Markov systems with discrete states, which are often used as models in stochastic thermodynamics. For further details see [@eu; @jir]. We consider a system with finitely many states $x$ which are uniquely associated to an energy level $E(x)$. The system is in contact with an equilibrium bath at temperature $T$ and it is also driven by external forces. It means that whenever there occurs a transition $x\rightarrow y$, the driving forces perform some work ${{\mathcal W}}(x,y) = -{{\mathcal W}}(y,x)$ on the system and some heat ${{\mathcal Q}}(x,y)$ enters the system from the bath. They are related by the energy balance $$E(y) - E(x) = {{\mathcal W}}(x,y) + {{\mathcal Q}}(x,y)$$ As we want to model a genuine nonequilibrium system, we assume that ${{\mathcal W}}(x,y)$ cannot be written as a difference of some potential which could then be included in the energy function $E(x)$.
The Markov dynamics is introduced via transition rates $k^T(x,y)$ for each admissible transition $x \rightarrow y$. For thermodynamic consistency, they have to satisfy the local detailed balance principle [@der; @mn], $$\frac{k^T(x,y)}{k^T(y,x)} = \exp \Bigl[ -\frac{{{\mathcal Q}}(x,y)}{k_B T} \Bigr]$$ In particular, the rates depend on the bath temperature as indicated in our notation.
Recall that we want to compute the (average) excess work along the relaxation process started from the steady state distribution at bath temperature $T$ and then running under the dynamics corresponding to the temperature $T + \de T$. The work done by driving forces can be obtained for any trajectory of the system by summing up contributions ${{\mathcal W}}(x_j,x_{j+1})$ from all subsequent transitions $x_j \rightarrow x_{j+1}$ along that trajectory. To find its statistical average it is useful to introduce the instantaneous power: Given the system at $x$ and attached to the equilibrium bath at temperature $T + \de T$, the average power of driving forces, i.e. the work per unit time, is $${{\mathcal P}}^{T+\de T}(x) = \sum_{y \neq x} k^{T+\de T}(x,y)\,{{\mathcal W}}(x,y)$$ Then the average work is $$W_{[0,t]} = \Bigl\langle \int_0^t {{\mathcal P}}^{T+\de T}(x_t)\,{\textrm{d}}t \Bigr\rangle_{T \rightarrow T+\de T}$$ where $\langle \cdot \rangle_{T \rightarrow T+\de T}$ stands for averaging with respect to the process started from the steady state at $T$ at $t=0$ and then running dynamics with the transition rules $k^{T+\de T}(x,y)$. Analogously, the steady state power is given by the stationary average $$w^{(T+\de T)} = \langle {{\mathcal P}}^{T+\de T}(x) \rangle_{T + \de T} =
\sum_x {{\mathcal P}}^{T+\de T}(x)\,\rho_{T + \de T}(x)$$ where $\rho_{T + \de T}(x)$ is the stationary distribution given the bath is at temperature $T + \de T$. Hence the excess work is $$\begin{split}
\de W^\text{ex} &= \Bigl\langle \int_0^\infty
\Bigl[ {{\mathcal P}}^{T+\de T}(x_t) - \bigl\langle {{\mathcal P}}^{T+\de T}(x) \bigr\rangle_{T + \de T} \Bigr]\,{\textrm{d}}t \Bigr\rangle_{T \rightarrow T+\de T}
\end{split}$$ This can be somewhat simplified by introducing the function $$V^{T+\de T}(x) = \Bigl\langle \int_0^\infty
\Bigl[ {{\mathcal P}}^{T+\de T}(x_t) - \bigl\langle {{\mathcal P}}^{T+\de T}(x) \bigr\rangle_{T + \de T} \Bigr]\,{\textrm{d}}t \, \Bigl| \bigr. \, x_0 = x \Bigr\rangle_{T+\de T}$$ where the conditional average means that the process starts from $x$ and then runs according to the dynamics at $T + \de T$. Note that it now depends only on a single temperature (in this case $T + \de T$), and by construction $\langle V^{T}(x) \rangle_{T} = 0$ for any $T$. This finally yields $$\de W^\text{ex} = \langle V^{T+\de T}(x) \rangle_T =
\Bigl\langle \frac{{\textrm{d}}V^T(x)}{{\textrm{d}}T} \Bigr\rangle_T \de T$$ Together with $U = \langle E(x) \rangle_T$ we obtain $$C(T) = \frac{{\textrm{d}}\langle E(x) \rangle_T}{{\textrm{d}}T} -
\Bigl\langle \frac{{\textrm{d}}V^T(x)}{{\textrm{d}}T} \Bigr\rangle_T$$ which is formula . A more rigorous derivation employs the quasistatic limit of any smooth time dependence of temperature, see [@eu]. The function $V^T(x) $ can be conveniently computed in terms of the backward Kolmogorov generator, see [@jir]. For diffusion processes similar expressions hold, as for example made explicit in Eq. (3.5) in [@mcl]. For dissipative mechanical systems we have the usual expression for the power ${\cal P}^T(p,q) = F(q)\cdot p$ for nonconservative force $F$ (which can depend implicitly on $T$) and states $x=(p,q)$ in phase space.
Close to equilibrium, when ${{\mathcal W}}(x,y) = \ve {{\mathcal W}}_1(x,y)$ with $\ve$ a small parameter, more explicit expressions for the heat capacity $C(T)$ can be obtained by invoking McLennan ensembles to approximate $\langle\cdot\rangle_T-$expectations, see [@mcl]. As it happens, in linear order around equilibrium (up to order $\ve$), the correction to the Gibbs ensemble is exactly given by $V^T$: $$\rho_T(x) = \frac{1}{Z} \exp \{-\beta [E(x) + V^T(x) + O(\ve^2)]\}$$ Per consequence, close to equilibrium the heat capacity is given by $$\label{thre}
C(T) = \frac{{\textrm{d}}\langle E(x) \rangle_T}{{\textrm{d}}T}\ -
\frac{\langle E(x) \rangle_T -\langle E(x) \rangle_T^{\text{eq}}}{T} +
O(\ve^2)$$ where $\langle \cdot \rangle^\text{eq}$ is the average under the equilibrium Gibbs ensemble $(\ve = 0)$.
|
---
abstract: 'We investigate the performance of entangled coherent state for quantum enhanced phase estimation. An exact analytical expression of quantum Fisher information is derived to show the role of photon losses on the ultimate phase sensitivity. We find a transition of the sensitivity from the Heisenberg scaling to the classical scaling due to quantum decoherence of the photon state. This quantum-classical transition is uniquely determined by the number of photons being lost, instead of the number of incident photons or the photon loss rate alone. Our results also reveal that a crossover of the sensitivity between the entangled coherent state and the NOON state can occur even for very small photon loss rate.'
author:
- 'Y. M. Zhang, X. W. Li'
- 'W. Yang'
- 'G. R. Jin'
title: 'Quantum Fisher information of entangled coherent state in the presence of photon losses: exact solution'
---
Introduction
============
The estimation of parameters characterizing dynamical processes is essential to science and technology. A typical parameter estimation consists of three steps. Firstly, the input state $|\psi _{\mathrm{in}}\rangle $ of the sensor is prepared. Secondly, the sensor undergoes the $\phi$-dependent dynamical process $\hat{U}(\phi )$ and evolve to the output state $|\psi\rangle$. Finally, a measurement is made on the output state and the outcome $x$ is used by suitable data processing to produce an unbiased estimator $\hat{\phi}(x)$ of the parameter $\phi $. The precision of the estimation is quantified by the standard deviation $\delta \phi =\langle (\hat{\phi}(x)-\phi
)^{2}\rangle $, which is determined by the input state $|\psi _{\mathrm{in}}\rangle $ [@Caves; @Yurke; @Holland; @Wineland1; @Wineland2; @Mitchell; @Giovannetti], the nature of the dynamical process $\hat{U}(\phi )$ [@Luis; @Rey; @Boixo08; @Choi; @Woolley; @Liu], the observable being measured [@Bollinger; @Campos; @Dowling; @Kim; @Lucke], and the specific data processing technique. The precision of the estimator $\hat{\phi}_{\mathrm{opt}}(x)$ from optimal data processing is limited by the Cramér-Rao inequality [@Helstrom; @Holevo] as $\delta \phi _{\mathrm{opt}}\geq 1/\sqrt{F(\phi )}$, where $F(\phi )$ is the classical Fisher information, determined by $|\psi
_{\mathrm{in}}\rangle $, $\hat{U}(\phi )$, and the measurement scheme. Given $|\psi _{\mathrm{in}}\rangle $ and $\hat{U}(\phi )$, maximizing $F(\phi )$ over all possible measurements gives the quantum Fisher information (QFI) $F_{Q}$ and hence the quantum Cramér-Rao bound $\delta \phi _{\min }=1/
\sqrt{F_{Q}}$ [@Braunstein94; @Peeze; @Sun; @Kacprowicz; @Braunstein; @Zhong] on the attainable precision to estimate the phase $\phi$.
In general, the best precision $\delta \phi_{\min}$ improves with increasing amount of resources $N$ employed in the measurement, e.g., the number of photons in optical phase estimation or the total duration of measurements in high-precision magnetic field or electric field sensing. For separable input states, the QFI $F_{Q}\sim N$ gives the classical limit $\delta
\phi_{\min}\sim1/\sqrt{N}$, in agreement with classical central limit theorem. To obtain an enhanced precision, it is necessary to utilize quantum resources such as coherence, entanglement, and squeezing in the input state for maximizing the QFI and hence the precision. This is a central issue in quantum metrology [@Paris; @Ensher; @XWang; @Caves13]. In the absence of noise, it has been well established that by utilizing quantum entanglement, the QFI can be enhanced up to $F_{Q}\sim N^{2}$ and hence the precision $\delta \phi_{\min}\sim1/N$, beating the the Heisenberg limit [@Wineland1; @Wineland2; @Mitchell; @Giovannetti; @Dowling2; @Gerry1; @Gerry2; @Lee]. This limit is ultimate estimation precision allowed by quantum resource with definite particle number. In the presence of noises, however, it is not clear whether the Heisenberg limit can still be achieved [@Dorner; @Escher2; @Demkowicz], and whether entanglement is still a useful resource for quantum metrology.
A paradigmatic example is the estimation of relative phase shift between the two modes propagating on different arms of the Mach-Zehnder interferometer (MZI). Precise phase estimation is important for multiple areas of scientific research [@Dowling], such as imaging, sensing, and information processing. In the absence of noise, the classical limit $\delta \phi_{\min}\sim1/\sqrt{ \bar{n}}$ ($\bar{n}$ is the average number of photons) for classical coherent state can be dramatically improved by using nonclassical states of the light. The maximally entangled NOON states $\sim|N,0\rangle_{1,2}+|0,N \rangle_{1,2}$ (also called the GHZ state in atomic spectroscopy) has been prepared in experiments for pursuing the Heisenberg-limited phase estimation [@Wineland1; @Wineland2; @Mitchell]. However, the NOON states are extremely fragile to photon losses [@Enk; @Dorner; @Lee2; @Cooper; @Escher2; @Joo; @Demkowicz; @Cooper1; @Jarzyna; @Knysh]. In a lossy interferometer, it has been shown that a transition of the precision from the Heisenberg limit to the shot-noise limit can occur with the increase of particle number $N$ [@Escher2; @Demkowicz].
Recently, a specific coherent superposition of the NOON states, the entangled coherent state (ECS) $\sim |\alpha ,0\rangle _{1,2}+|0,\alpha
\rangle _{1,2}$, was proposed as the input state for enhanced precision [@Joo]. In the absence of photon losses, the precision of the ECS can surpass that of the NOON state (i.e., the Heisenberg limit, $\delta \phi _{\min }=1/\bar{n}$). In the presence of photon losses, numerical simulation suggests that the ECS outperforms the NOON state for photon numbers $\bar{n}\lesssim
5 $. For a small photon number $\bar{n}\sim 5$, the precision is better than the classical limit by a factor $\sqrt{\bar{n}}\sim 2$. To achieve more significant enhancement for practical applications, a much larger photon numbers are required. The performance with a large amount of resources is an important benchmark for a realistic quantum enhanced estimation scheme. Therefore, a careful analysis of the QFI and the ultimate precision for the input ECS with large $\bar{n}$ is necessary.
In this paper, we present an exact analytical result of the QFI for the entangled coherent state with arbitrary $\bar{n}$, which provides counter-intuitive physics that is inaccessible from previous numerical simulations. To understand why the ECS is better than the NOON state, we first consider an arbitrary superposition of the NOON states and find the QFI $F_{Q}\geq \bar{n}^{2}$, leading to a sub-Heisenberg limited sensitivity $\delta \phi _{\min }\leq 1/\bar{n}$. Next, we investigate the role of photon losses on the QFI and hence the ultimate precision of the ECS. An exact result of the QFI is derived, which is the sum of the classical term $\propto \bar{n}$ and the Heisenberg term $\propto \bar{n}^{2}$. We show that the photon losses suppresses exponentially off-diagonal (coherence) part of the reduced density matrix $\hat{\rho}$ and hence the Heisenberg term, while leaving the classical term largely unchanged. The loss-induced quantum decoherence leads to a transition of the estimation precision from the Heisenberg scaling to the classical scaling as the number of lost photons $R\bar{n}$ increases, where $R$ is the photon loss rate and $\bar{n}$ is the mean photon number of the initial ECS. This behavior is in sharp contrast to the NOON state, for which the photon losses eliminate completely the phase information stored in the coherence part of $\hat{\rho}$. The ultimate precision of the NOON state gets even worse than the classical limit when $R\bar{n}\gg 1$. Surprisingly, we find that the precision of the NOON state may be better than that of the ECS within the crossover region at $R\bar{n}\sim 1$. This is because although the classical term of the ECS is robust against the photon losses, the Heisenberg term decays about twice as quick as that of the NOON state.
Sub-Heisenberg limited phase sensitivity with a Superposition of NOON states
============================================================================
Firstly, let us consider an *arbitrary* coherent superposition of the NOON states as the input state after the first beam splitter of a two-mode MZI, $$\left\vert \psi _{\mathrm{in}}\right\rangle =\sum_{n=0}^{\infty }c_{n}\frac{\left\vert n\right\rangle _{1}+\left\vert n\right\rangle _{2}}{\sqrt{2}},
\label{in}$$where, for brevity, we introduce the notations $|n\rangle _{1}\equiv
|n\rangle _{1}|0\rangle _{2}$ and $|n\rangle _{2}\equiv |0\rangle
_{1}|n\rangle _{2}$, representing $n$ photons in the mode $1$ (or $2$) and the other mode in vacuum. To analyze possible achievable phase sensitivity with $|\psi _{\mathrm{in}}\rangle $, we directly evaluate the QFI of the outcome state after phase accumulation $|\psi (\phi )\rangle =\hat{U}(\phi
)|\psi _{\mathrm{in}}\rangle =e^{i\phi \hat{G}}|\psi _{\mathrm{in}}\rangle $, where $\hat{G}$ is the generator of phase shift. For a lossless MZI, $|\psi \rangle $ is a pure state and the QFI is given by the well-known formula [@Braunstein94; @Peeze; @Sun; @Kacprowicz; @Braunstein; @Zhong]: $F_{Q}=4(\langle \psi ^{\prime }|\psi ^{\prime }\rangle -|\langle \psi
^{\prime }|\psi \rangle |^{2})=4(\langle \hat{G}^{2}\rangle -\langle \hat{G}\rangle ^{2})$, where $|\psi ^{\prime }\rangle \equiv \partial |\psi \rangle/\partial \phi $ and the expectation values are taken with respect to $|\psi
_{\mathrm{in}}\rangle $. Considering a linear phase-shift generator $\hat{G}=\hat{n}_{2}$ [@Dorner; @Joo], with the photon number operators $\hat{n}_{2}=\hat{a}_{2}^{\dag }\hat{a}_{2}$ and $\hat{n}_{1}=\hat{a}_{1}^{\dag}\hat{a}_{1}$, we obtain the QFI $$F_{Q}=4(\langle \hat{n}_{2}^{2}\rangle -\langle \hat{n}_{2}\rangle
^{2})=2\langle \hat{n}^{2}\rangle -\langle \hat{n}\rangle ^{2},
\label{FQ-pure}$$where we have used the relation $\langle \hat{n}_{1}^{l}\rangle =\langle
\hat{n}_{2}^{l}\rangle =\langle \hat{n}^{l}\rangle /2$ (for $l=1$, $2$, $\cdots$), which, together with $\langle \hat{n}_{1}\hat{n}_{2}\rangle =0$ are valid for Eq. (\[in\]). Since $\langle \hat{n}^{2}\rangle \geq \langle
\hat{n}\rangle ^{2}$, we have $F_{Q}\geq \bar{n}^{2}$, where $\bar{n}=\langle \hat{n}\rangle $ is the mean photon number of $|\psi _{\mathrm{in}}\rangle $. This inequality also applies to another kind of phase-shift generator $\hat{G}=(\hat{n}_{2}-\hat{n}_{1})/2$, for which $F_{Q}=\langle
\hat{n}^{2}\rangle $. They suggest that a sub-Heisenberg limited phase sensitivity $\delta \phi _{\min }<1/\bar{n}$ can be achievable with an arbitrary coherent superposition of the NOON states, as Eq. (\[in\]). The equality $\delta \phi _{\min }=1/\bar{n}$, known as the Heisenberg limit, is attained by the NOON state [@Wineland1; @Wineland2; @Mitchell; @Giovannetti; @Dowling2; @Gerry1; @Gerry2; @Lee] $(|N\rangle _{1}+|N\rangle _{2})/\sqrt{2}$ with $\bar{n}=N$.
Next, we review the recently proposed ECS state [@Ono; @Gerry]: ${\mathcal{N}}_{\alpha }(|\alpha \rangle _{1}+|\alpha \rangle _{2})$ as a special case of the superposition of NOON states, where ${\mathcal{N}}_{\alpha
}=[2(1+e^{-|\alpha |^{2}})]^{-1/2}$ is the normalization constant and $|\alpha \rangle _{1}\equiv |\alpha \rangle _{1}\vert 0\rangle
_{2} $ denotes a coherent state in the sensor mode 1 and vacuum in the sensor mode 2 and similarly for $|\alpha \rangle _{2}\equiv |\alpha \rangle
_{2}\vert 0\rangle _{1}$. The ECS can be generated by passing a coherent state $|\alpha /\sqrt{2}\rangle _{1}$ and a coherent state superposition $\sim |\alpha /\sqrt{2}\rangle _{2}+|-\alpha /\sqrt{2}\rangle
_{2}$ (experimentally available for $\alpha \approx 2$ [@Ourjoumtsev]) through a 50:50 beam splitter [@Joo]. In the absence of photon losses, using the ECS as the input state and considering the phase accumulation dynamics $\hat{U}(\phi )=e^{i\phi \hat{n}_{2}}$ [@Dorner; @Joo], we obtain $\bar{n}=\langle \hat{n}\rangle =2{\mathcal{N}}_{\alpha }^{2}|\alpha |^{2}$, $\langle \hat{n}^{2}\rangle =2{\mathcal{N}}_{\alpha }^{2}|\alpha
|^{2}(|\alpha |^{2}+1)$, and the quantum Fisher information $$F_{Q}=2\bar{n}[1+w(\bar{n}e^{-\bar{n}})]+\bar{n}^{2}, \label{FQ_ECS0}$$where we have used $\bar{n}=|\alpha |^{2}/(1+e^{-|\alpha |^{2}})$ and hence $w(\bar{n}e^{-\bar{n}})=\bar{n}e^{-|\alpha |^{2}}$. Here, $w(z)$ denotes the Lambert W function (also called the product logarithm), which gives the principal solution for $w$ in $z=we^{w}$. For mean photon number $\bar{n}\approx |\alpha |^{2}\gg 1$, we have $w(\bar{n}e^{-\bar{n}})\approx 0$ and $
F_{Q}\approx \bar{n}(\bar{n}+2)$. From Fig. \[fig1\](a), one can find that $\delta \phi _{\min }$ of the ECS (the blue solid line) is better than that of the NOON (the blue dashed line), especially for a modest photon number. A recent numerical simulation shows that this improved sensitivity of the ECS can be maintained in the presence of the photon losses for $\bar{n}\lesssim
5 $ [@Joo]. However, the performance of the ECS with larger $\bar{n}$ remains unclear.
![(Color online) The ultimate precision $\delta\phi_{\min}$ against the number of photons $\bar{n}$ or $N$ (a) and the transmission rate $T$ (b) for the NOON (dashed) and the ECS (solid) states. In (a), $T=1$ (blue lines), $0.9$ (red lines), and $0.8$ (black lines). Two arrows indicate $N_{\text{opt}}=-2/\ln T$ with $T=0.8$ and $0.9$. In (b), $\bar{n}=4$ (red lines) and $20$ (blue lines). A crossover of $\delta\phi_{\min}$ between the ECS and the NOON states occurs for $\bar{n}$ (or $N$)$=20$ and $T\in(0.85, 1)$. Shaded area in (a): Region for the sensitivity worse than the shot-noise limit $1/\sqrt{\bar{n}}$.[]{data-label="fig1"}](fig1.eps){width="0.95\columnwidth"}
Quantum Fisher information and ultimate precision of Entangled coherent state with photon losses
================================================================================================
In this section, we present an exact analytical expression of the QFI $F_{Q}$ and hence the ultimate precision $\delta \phi _{\min }=1/\sqrt{F_{Q}}$ for the ECS in the presence of photon losses. This provides detailed information for the performance of the ECS in the quantum phase estimation, especially those at relatively large photon numbers, that are inaccessible from the previous numerical simulation. Firstly, we derive exact analytical expression of the quantum Fisher information for the input ECS based upon a general formula of the QFI. This formula decomposes the total QFI into three physically intuitive contributions. Next, we present the QFI of the NOON state. Finally, by comparing with the NOON state, we discuss the key features of the ECS and provides a simple physics picture.
The photon losses can be modeled by inserting two identical beam splitters $\hat{B}_{k,k^{\prime }}=\exp [i(\theta /2)(\hat{a}_{k^{\prime }}^{\dag }\hat{a}_{k}+h.c.)]$ that couples two sensor modes $k=1$, $2$ and two environment modes $k^{\prime }=1^{\prime }$, $2^{\prime }$ that are initially in the vacuum [@Enk; @Dorner; @Lee2; @Cooper; @Escher2; @Joo; @Demkowicz; @Cooper1; @Jarzyna; @Knysh]. The action of beam splitters transforms the sensor mode $\hat{a}_{k}^{\dagger}$ into a linear combination of $\hat{a}_{k}^{\dagger}$ and $\hat{a}_{k^{\prime }}^{\dagger }$: $\hat{B}_{k,k^{\prime }}\hat{a}_{k}^{\dagger }\hat{B}_{k,k^{\prime }}^{-1}=\sqrt{T}\hat{a}_{k}^{\dagger }+i\sqrt{R}\hat{a}_{k^{\prime }}^{\dagger }$, where $T=\cos ^{2}(\theta /2)$ and $R=1-T$ are transmission and absorption (loss) rates of the photons, respectively. More specially, $T=1$ (i.e., $R=0$ ) means no photon loss and $T=0$ ($R=1$) corresponds to complete photon loss. For the input ECS state, using $\hat{U}(\phi )|\alpha \rangle _{2}=|\alpha e^{i\phi }\rangle _{2}$ and $\hat{B}_{k,k^{\prime }}|\alpha \rangle _{k}=|\sqrt{T}\alpha \rangle _{k}|i\sqrt{R}\alpha \rangle _{k^{\prime }}$, we obtain the outcome state $$\begin{aligned}
|\psi (\phi )\rangle &=&{\mathcal{N}}_{\alpha }\hat{B}_{1,1^{\prime }}\hat{B}%
_{2,2^{\prime }}\hat{U}(\phi )(|\alpha \rangle _{1}+|\alpha \rangle
_{2})|0\rangle _{1^{\prime }}|0\rangle _{2^{\prime }} \\
&=&{\mathcal{N}}_{\alpha }\left( |\sqrt{T}\alpha \rangle _{1}|E^{(1)}\rangle
+|\sqrt{T}\alpha e^{i\phi }\rangle _{2}|E^{(2)}\rangle \right) ,\end{aligned}$$where the environment states are given by $|E^{(1)}\rangle \equiv |i\sqrt{R}\alpha \rangle _{1^{\prime }}|0\rangle _{2^{\prime }}$ and $|E^{(2)}\rangle\equiv |0\rangle _{1^{\prime }}|i\sqrt{R}\alpha e^{i\phi }\rangle
_{2^{\prime }}$. Tracing over them, we obtain the reduced density matrix of the sensor modes $$\begin{aligned}
\hat{\rho} &=&{\mathcal{N}}_{\alpha }^{2}\left\{ |\sqrt{T}\alpha \rangle
_{11}\langle \sqrt{T}\alpha |+|\sqrt{T}\alpha e^{i\phi }\rangle _{22}\langle
\sqrt{T}\alpha e^{i\phi }|\right. \notag \\
&&\left. +\langle E^{(2)}|E^{(1)}\rangle \left( |\sqrt{T}\alpha \rangle
_{12}\langle \sqrt{T}\alpha e^{i\phi }|+h.c.\right) \right\} , \label{RHO}\end{aligned}$$where $\langle E^{(2)}|E^{(1)}\rangle =\langle E^{(1)}|E^{(2)}\rangle
=e^{-R|\alpha |^{2}}$. Compared with the lossless case (i.e., $T=1$), the amplitudes in the sensor modes are reduced from $|\alpha |$ to $\sqrt{T}|\alpha|$. More importantly, the photon losses suppresses the off-diagonal coherence between the two sensor states by a factor $\langle
E^{(2)}|E^{(1)}\rangle $. We will show below that this decoherence effect significantly degrades the estimation precision of the ECS.
Since $\hat{\rho}$ is a mixed state, to obtain the QFI one has to diagonalize it as $\hat{\rho}=\sum_{m}\lambda _{m}|\lambda _{m}\rangle
\langle \lambda _{m}|$, where $\{|\lambda _{m}\rangle \}$ forms an ortho-normalized and complete basis, with $\lambda _{m}$ being the weight of $|\lambda _{m}\rangle $. According to the well-known formula [Braunstein94,Peeze,Sun,Kacprowicz,Braunstein,Zhong]{}, the QFI is given by $$F_{Q}=\sum_{m,n}\frac{2}{\lambda _{m}+\lambda _{n}}\left\vert \left\langle
\lambda _{m}\right\vert \hat{\rho}^{\prime }|\lambda _{n}\rangle \right\vert
^{2}, \label{QFI-mix}$$where the prime denotes the derivation about $\phi $, such as $\hat{\rho}%
^{\prime }=\partial \hat{\rho}/\partial \phi $, $\lambda _{m}^{\prime
}=\partial \lambda _{m}/\partial \phi $, and $|\lambda _{m}^{\prime }\rangle
=\partial |\lambda _{m}\rangle /\partial \phi $. Typically, the dimension of entire Hilbert space and hence the basis $\{|\lambda _{m}\rangle\}$ is huge, but only a small subset has nonzero weights. Therefore, using the completeness and the ortho-normalization of $\{|\lambda _{m}\rangle\}$, we can express the QFI in terms of the subset $\{|\lambda _{i}\rangle\}$ with $\lambda _{i}\neq 0$ (see Appendix): $$F_{Q}=\sum_{i}\frac{(\lambda _{i}^{\prime })^{2}}{\lambda _{i}}%
+\sum_{i}\lambda _{i}F_{Q,i}-\sum_{i\neq j}\frac{8\lambda _{i}\lambda _{j}}{%
\lambda _{i}+\lambda _{j}}\left\vert \left\langle \lambda _{i}^{\prime
}\right\vert \lambda _{j}\rangle \right\vert ^{2}, \label{QFI-mix3}$$which contains three kinds of contributions. The first term is the classical Fisher information for the probability distribution $P(i|\phi )\equiv
\lambda _{i}(\phi )$. The second term is a weighted average over the quantum Fisher information $F_{Q,i}=4(\langle \lambda _{i}^{\prime }|\lambda
_{i}^{\prime }\rangle -|\langle \lambda _{i}^{\prime }|\lambda _{i}\rangle
|^{2})$ for each pure state in the subset $\{|\lambda _{i}(\phi )\rangle\}$ with $\lambda _{i}\neq 0$. The last term reduces the QFI and hence the estimation precision below the pure-state case. If the phase shift $\phi$ comes into the reduced density matrix $\hat{\rho}$ through the weights $\lambda _{i}(\phi )$ only, then the last two terms of Eq. (\[QFI-mix3\]) give vanishing contribution to the QFI. While for $\phi$-independent weights, however, the first term vanishes.
Compared with Eq. (\[QFI-mix\]) that relies on the complete basis, our formula Eq. (\[QFI-mix3\]), defined within a truncated Hilbert space, has the advantages of faster convergence and numerical stability, especially when the reduced density matrix $\hat{\rho}$ has some eigenvectors with extremely small but nonvanishing weights.
For the input ECS, we note that the reduced density matrix $\hat{\rho}$ only contains two sensor states $|\sqrt{T}\alpha \rangle _{1}$ and $|\sqrt{T}\alpha e^{i\phi }\rangle _{2}$ \[see Eq. (\[RHO\])\]. This feature enables us to expand $\hat{\rho}$ in terms of two eigenvectors with nonzero eigenvalues (see Append. A), $$\hat{\rho}=\lambda _{+}\left\vert \lambda _{+}(\phi )\right\rangle \langle
\lambda _{+}(\phi )|+\lambda _{-}\left\vert \lambda _{-}(\phi )\right\rangle
\langle \lambda _{-}(\phi )|, \label{diag}$$where the eigenvalues $\lambda _{\pm }={\mathcal{N}}_{\alpha }^{2}(1\pm
e^{-R|\alpha |^{2}})(1\pm e^{-T|\alpha |^{2}})$ are $\phi $-independent and obey $\lambda _{-}+\lambda _{+}=1$. The phase-dependent eigenvectors are given by $$|\lambda _{\pm }(\phi )\rangle =\eta _{\pm }\left[ \pm |\sqrt{T}\alpha
\rangle _{1}+|\sqrt{T}\alpha e^{i\phi }\rangle _{2}\right] ,
\label{eigenvectors}$$with the normalization factors $\eta _{\pm }=1/\sqrt{2(1\pm e^{-T|\alpha
|^{2}})}$. It is easy to prove that $\langle \lambda _{\pm }|\lambda _{\pm
}\rangle =1$ and $\langle \lambda _{+}|\lambda _{-}\rangle =\langle \lambda
_{+}|\hat{\rho}|\lambda _{-}\rangle =0$. Using Eq. (\[QFI-mix3\]), we obtain exact analytical expression of the QFI (see Append. B): $$F_{Q}=F_{Q}^{\mathrm{cl}}+F_{Q}^{\mathrm{HL}}, \label{FQ_ECS}$$where the classical term $F_{Q}^{\mathrm{cl}}=2\bar{n}T[1+Tw(\bar{n}e^{-\bar{n}})]$, and the Heisenberg term $$F_{Q}^{\mathrm{HL}}=(\bar{n}T)^{2}\left( \frac{e^{-2R|\alpha
|^{2}}-e^{-2T|\alpha |^{2}}}{1-e^{-2T|\alpha |^{2}}}\right) . \label{FQHL}$$In the absence of photon losses (i.e., $R=0$ and $T=1$), our result reduces to the lossless case, i.e., Eq. (\[FQ\_ECS0\]). Compared with it, we find that the photon losses leads to two effects on the QFI (and hence the estimation precision). Firstly, it trivially reduces the photon number from $\bar{n}$ in the input state to $\bar{n}T$ in the output state. Secondly, it exponentially suppresses the QFI from $F_{Q}^{\mathrm{HL}}\sim(\bar{n}T)^{2}$ to the classical scaling $\sim 2\bar{n}T$ (see below).
For a comparison, we also employ Eq. (\[QFI-mix3\]) to derive the QFI for the NOON state $(|N\rangle _{1}+|N\rangle _{2})/\sqrt{2}$. It is easy to write down the reduced density matrix in a diagonal form: $$\hat{\rho}=\sum_{n=0}^{N-1}\lambda _{n}(\left\vert n\right\rangle
_{11}\left\langle n\right\vert +\left\vert n\right\rangle _{22}\left\langle
n\right\vert )+T^{N}\left\vert \psi _{\mathrm{NOON}}\right\rangle
\left\langle \psi _{\mathrm{NOON}}\right\vert , \label{N00N}$$where the first part is an incoherent mixture of Fock states $|n\rangle _{1}$ and $|n\rangle _{2}$ with $\phi $-independent weights $\lambda _{n}=\tbinom{N}{n}T^{n}R^{N-n}/2$. The phase information is stored in the second part, $|\psi _{\mathrm{NOON}}\rangle =(|N\rangle _{1}+e^{iN\phi }|N\rangle _{2})/\sqrt{2}$, which, for the lossless case, gives the QFI $N^{2}$. Therefore, according to Eq. (\[QFI-mix3\]), the total QFI is equal to the QFI of $|\psi _{\mathrm{NOON}}\rangle $ times its weight $T^{N}$, namely $$F_{Q,\mathrm{NOON}}=N^{2}T^{N}, \label{FQ_NOON}$$in agreement with Ref. . With increasing photon number $N$, the ultimate precision $\delta \phi _{\min }=T^{-N/2}/N$ shows a global minimum at $N_{\mathrm{opt}}=-2/\ln T\approx 2/R$ (as $T=1-R\approx e^{-R}$ for small $R$), indicated by the arrows of Fig. \[fig1\](a).
In Fig. \[fig1\](a), we plot $\delta \phi_{\min}$ of the ECS (the NOON) state as a function of number of photons $\bar{n}$ ($N$) for the transmission rates $T=0.8$, $0.9$, and $1$ (from top to bottom). Regardless of $T$, one can find that $\delta \phi_{\min}$ of the input ECS decreases monotonically with the increase of $\bar{n}$. While for the NOON state, however, $\delta \phi_{\min}$ reaches its minimum at $N_{\mathrm{opt}}$ and then grows rapidly. In Fig. \[fig1\](b), we show $\delta \phi_{\min}$ against $T$ for $\bar{n}$ ($N$)$=4$ and $20$. It is interesting to note that a crossover of $\delta \phi_{\min}$ between the ECS and the NOON states occurs for large $\bar{n}$ and $T$ (say, $T>0.85$).
We now analyze the QFI under practical conditions: $T\sim 1$ ($R\sim 0$) and $|\alpha |^{2}\gg 1$, for which $w(\bar{n}e^{-\bar{n}})\approx 0$ and hence $|\alpha |^{2}\approx \bar{n}$. In addition, the exponential term $e^{-2T|\alpha |^{2}}$ of Eq. (\[FQHL\]) is negligible. As a result, the QFI reduces to $$F_{Q}=F_{Q}^{\mathrm{cl}}+F_{Q}^{\mathrm{HL}}\approx 2\bar{n}T+(\bar{n}%
T)^{2}e^{-2R\bar{n}}, \label{FQ_ECS_APPROX}$$where the exponential term $e^{-2R\bar{n}}=|\langle E^{(2)}|E^{(1)}\rangle
|^{2}$, quantifies the off-diagonal coherence of the sensor states. When the number of photons being lost $R\bar{n}=(1-T)\bar{n}\ll 1$, the Heisenberg term $F_{Q}^{\mathrm{HL}}\approx (\bar{n}T)^{2}e^{-2R\bar{n}}$ dominates and the ultimate precision obeys $\delta \phi _{\min }^{\mathrm{HL}}\approx e^{R\bar{n}}/(\bar{n}T)$. With the increase of $R\bar{n}$, the classical term $F_{Q}^{\mathrm{cl}}\approx 2\bar{n}T$ becomes important. As $R\bar{n}\gg 1$, a complete decoherence of the two sensor states occurs due to $|\langle
E^{(2)}|E^{(1)}\rangle |^{2}\rightarrow 0$, leading to the completely mixed state $$\hat{\rho}\approx \frac{1}{2}(|\sqrt{T}\alpha \rangle _{11}\langle \sqrt{T}%
\alpha |+|\sqrt{T}\alpha e^{i\phi }\rangle _{22}\langle \sqrt{T}\alpha
e^{i\phi }|), \label{ECS_RHO}$$where the first term $|\sqrt{T}\alpha \rangle _{11}\langle \sqrt{T}\alpha |$ carries no phase information and hence $F_{Q,1}=0$, and the $\phi$-dependent second term $|\sqrt{T}\alpha e^{i\phi }\rangle _{22}\langle \sqrt{T}\alpha e^{i\phi }|$ produces the pure-state QFI $F_{Q,2}\approx 4\bar{n}T$. Therefore, according to Eq. (\[QFI-mix3\]), the total QFI of $\hat{\rho}$ reads $F_{Q}\approx \sum_{i}\lambda _{i}F_{Q,i}\approx 2\bar{n}T$, which in turn gives the classical scaling of the sensitivity $\delta \phi _{\min
}\approx \delta \phi _{\min }^{\mathrm{cl}}\approx 1/\sqrt{2\bar{n}T}$. In Fig. \[fig2\], we present the log-log plot of $\delta \phi _{\min }$ for the loss rate $R=0.1$ and $0.01$. As shown by the red solid line, the simple formula of Eq. (\[FQ\_ECS\_APPROX\]) agrees quite well with the exact result (the solid circles). They both show a turning point at $\bar{n}\sim 1/R$. Indeed, the crossover of the quantum-classical transition takes place when the Heisenberg term $F_{Q}^{\mathrm{HL}}$ is comparable with the classical term $F_{Q}^{\mathrm{cl}}$, i.e., $R\bar{n}\sim 1$.
![(Color online) log-log plot of $\delta\phi_{\min}$ for $T=0.9$ (a) and $T=0.99$ (b). Black dotted line: the classical limit $1/\sqrt{2T\bar{n}}$; Blue dashed line: $\delta\phi_{\min}$ of the NOON state; Red solid line (circles): approximated (exact) $\delta\phi_{\min}$ of the ECS; Red dot-dashed line: $\delta\phi_{\min}$ of the ECS in the absence of photon losses (i.e, $T=1$), given by Eq. (\[FQ\_ECS0\]). The two vertical lines at $\bar{n}=6.4$ and $23.5$ in (a) and $\bar{n}=14.8$ and $561$ in (b) show the crossover of $\delta\phi_{\min}$ between the ECS and the NOON states.[]{data-label="fig2"}](fig2.eps){width="\columnwidth"}
For the ECS with large photon losses, i.e., $R\bar{n}\gg 1$, the ultimate precision $\delta \phi _{\min }$ obeys the classical scaling $1/\sqrt{2\bar{n}T}$, which is conformed by Fig. \[fig2\]. The precision of the NOON state is optimal at $\bar{n}=-2/\ln T\approx 2/R$ and then rapidly degrades below the classical limit \[see the dashed lines, also Fig. \[fig1\](a)\]. This is in sharp contrast to the ECS state. Qualitatively, different behaviors of the two states arises from the different influences of photon losses:
1. For the ECS $\sim |\alpha \rangle _{1}+|\alpha \rangle _{2}$, the off-diagonal coherence between the two sensor states $|\sqrt{T}\alpha
\rangle _{1}$ and $|\sqrt{T}\alpha e^{i\phi }\rangle _{2}$ is exponentially suppressed by the photon losses, but the diagonal components of $\hat{\rho}$ still carries the phase information \[see Eq. (\[ECS\_RHO\])\], which contributes the QFI $F_{Q}\approx 2\bar{n}T$.
2. For the NOON state $\sim |N\rangle _{1}+|N\rangle _{2}$, the phase information is stored only in the coherence part of $\hat{\rho}$ \[see Eq. (\[N00N\])\], which decays with the photon losses as $T^{N}\approx e^{-RN}$ for small $R$. When the lost photon number $RN\gg 1$, the information about the phase shift $\phi $ is completely eliminated.
From Fig. \[fig1\], we have observed the crossover of $\delta \phi _{\min}$ between the ECS and the NOON states, which can be understood by simply comparing the QFIs for the two states. Without the photon losses, the ultimate precision of the ECS always surpass those of the NOON states because $F_{Q}=F_{Q}^{\mathrm{cl}}+F_{Q}^{\mathrm{HL}}>F_{Q,\mathrm{NOON}}$ (as $F_{Q}^{\mathrm{HL}}=F_{Q,\mathrm{NOON}}=\bar{n}^{2}$). In the presence of moderate photon losses, the Heisenberg term $F_{Q}^{\mathrm{HL}}\approx (\bar{n}T)^{2}e^{-2R\bar{n}}$ decays more quickly than that of the NOON state $F_{Q,\mathrm{NOON}}\approx \bar{n}^{2}e^{-R\bar{n}}$. This makes it possible for the NOON state to outperform the ECS when the quantum contribution $F_{Q}^{\mathrm{HL}}$ dominates the classical contribution $F_{Q}^{\mathrm{cl}}$. From Fig. \[fig2\], one can find that the NOON states with $\bar{n}\in (6.4$, $23.5)$ for $R=0.1$ and $\bar{n}\in(14.8$, $561)$ for $R=0.01$ are preferable, within the vertical lines of Fig. \[fig2\].
In general, the crossover condition can be obtained by equating Eq. (\[FQ\_ECS\_APPROX\]) and Eq. (\[FQ\_NOON\]). This gives a transcendental equation: $\bar{n}T^{\bar{n}-1}\approx 2+\bar{n}Te^{-2R\bar{n}}$, as illuminated by the red solid curve in Fig. \[fig3\]. It shows that the NOON states outperforms the ECS inside the crossover region, while the ECS prevails outside. The upper and the lower boundaries of the region are well fitted by $\bar{n}_{u}\approx 3.2T^{6}/R^{1.15}$ (the black dashed line) and $\bar{n}_{l}\approx 1.4T^{-3}/R^{1/2}$ (the blue dash-dotted line), respectively. The upper boundary corresponds to $F_{Q,\mathrm{NOON}}\approx
F_{Q}^{\mathrm{cl}}$. As shown in Fig. \[fig3\], we find that the crossover of $\delta \phi _{\min }$ between the ECS and the NOON states takes place for $T\in (0.854,1)$. For such a relatively low loss rate ($0<R<0.15$), the precision of the NOON state could surpass that of the ECS over a wider range of $\bar{n}$ until the classical term $F_{Q}^{\mathrm{cl}}$ begins to dominate. However, the NOON states with $\bar{n}>\bar{n}_{u}$ ceases to be optimal and its precision gets even worse than the classical limit [@Dorner]. From Fig. \[fig3\], we also note that no crossover occurs for $T\lesssim 0.854$ and the ultimate precision of the ECS is always better than that of the NOON state \[see also the black lines of Fig. \[fig1\](a)\].
![(Color online) The crossover region in which $\delta\phi_{\min}$ of the NOON state is preferable. Red solid line: $\bar{n}T^{
\bar{n}-1}=2+\bar{n}Te^{-2R\bar{n}}$ for $R=1-T$ and $T\in(0.85,1)$; Black dashed and blue dot-dashed lines: $\bar{n}_{u}\simeq 3.2T^{6}/R^{1.15}$ and $\bar{ n}_{l}\simeq1.4T^{-3}/R^{0.5}$, fitting very well with the boundary of the crossover region (Open circles). The critical point of the crossover: ($T$, $\bar{n}$ )=($0.854$, $8.58$), as indicated by the arrow.[]{data-label="fig3"}](fig3.eps){width="\columnwidth"}
Conclusion
==========
By considering a superposition of NOON states as input" of a lossless optical interferometer, we have shown that the quantum Fisher information $F_{Q}\geq \bar{n}^{2}$ and therefore the ultimate precision of phase sensitivity can be better than the Heisenberg limit.
As a special case of the superposed state, an entangled coherent state $\propto |\alpha ,0\rangle _{1,2}+|0,\alpha \rangle _{1,2}$ has been investigated. Exact result of the quantum Fisher information is obtained to investigate the role of photon losses on the lower bound of phase sensitivity $\delta \phi _{\min }$. Without the photon losses, i.e., the absorption rate $R=0$ and the transmission rate $T=1$, we confirm that the input ECS always outperform the NOON state [@Joo]. In the presence of photon losses, the transition of $\delta \phi _{\min }$ from the Heisenberg scaling to the classical limit occurs due to the loss-induced quantum decoherence between the sensor states. The quantum-classical transition depends upon the number of photons being lost $R\bar{n}$, but rather the total photon number $\bar{n}$ or the lose rate $R$ alone. For a given transmission rate $T\in (0.85, 1)$, we also find that there exists a crossover of $\delta \phi _{\min}$ between the ECS and the NOON states. The NOON state is preferable in the crossover region, i.e., $\bar{n}T^{\bar{n}-1}\gtrsim 2+\bar{n}Te^{-2R\bar{n}}$. For $R\bar{n}\gg 1$, however, the precision of the NOON state degrades below the classical limit; While for the ECS state, $\delta \phi _{\min }$ obeys the classical limit $1/\sqrt{2T\bar{n}}$, better than that of the NOON state.
We thank Professor D. L. Zhou and Professor J. P. Dowling for helpful discussions. This work is supported by Natural Science Foundation of China (NSFC, Contract Nos. 11174028 and 11274036), the Fundamental Research Funds for the Central Universities (Contract No. 2011JBZ013), and the Program for New Century Excellent Talents in University (Contract No. NCET-11-0564). X.W.L is partially supported by National Innovation Experiment Program for University Students.
Eigenvalues and Eigenvectors of the reduced density matrix
==========================================================
We present a general method to diagonalize the reduced density matrix likes Eq. (\[RHO\]). The eigenvector of $\hat{\rho}$ can be spanned as $|\lambda(\phi)\rangle=\sum_{j}c_{j}|\Phi_{j}\rangle$, where the states $|\Phi_{j}\rangle$ are not necessary orthogonal. Using the eigenvalue equation: $\hat{\rho}|\lambda(\phi)\rangle=\lambda|\lambda(\phi)\rangle$, or equivalently $\sum_{j}\langle \Phi_{i}|\hat{\rho}|\Phi_{j}\rangle
c_{j}=\lambda \sum_{j}\langle \Phi_{i}|\Phi_{j}\rangle c_{j} $, we can determine the eigenvalue $\lambda$ and the amplitudes $c_{j}$. It is convenient to write down the eigenvalue equation in a matrix form: $\boldsymbol{\rho}\mathbf{c}=\lambda \mathbf{Ac}$, where the elements of $\boldsymbol{\rho}$ and $\mathbf{A}$ are $\rho_{ij}=\langle \Phi_{i}|\hat{\rho }|\Phi_{j}\rangle$ and $A_{ij}=\langle \Phi_{i}|\Phi_{j}\rangle$, and $\mathbf{c}=(c_{1},c_{2},\cdots)^{T}$. Multiplying the inverse matrix $\mathbf{A}^{-1}$ on the left, we can rewrite the eigenvalue equation as $$\boldsymbol{\tilde{\rho}}\mathbf{c}\equiv \mathbf{A}^{-1}\boldsymbol{\rho}
\mathbf{c}=\lambda \mathbf{c}, \label{eigen eq}$$ where $\boldsymbol{\tilde{\rho}}=\mathbf{A}^{-1}\boldsymbol{\rho}$.
Using the above formula, we now diagonalize the reduced density operator of Eq. (\[RHO\]). Firstly, we expand the eigenvectors as $|\lambda (\phi
)\rangle =c_{1}|\Phi_{1}\rangle +c_{2}|\Phi_{2}\rangle $, where $|\Phi
_{1}\rangle =|\sqrt{T}\alpha \rangle _{1}=|\sqrt{T}\alpha \rangle
_{1}|0\rangle_{2}$ and $|\Phi_{2}\rangle =|\sqrt{T}\alpha e^{i\phi
}\rangle_{2}=|0\rangle_{1}|\sqrt{T}\alpha e^{i\phi}\rangle_{2}$. It is easy to obtain the matrix $$\boldsymbol{\tilde{\rho}}={\mathcal{N}}_{\alpha }^{2}\left(
\begin{array}{cc}
1+e^{-|\alpha |^{2}} & e^{-T|\alpha |^{2}}+e^{-R|\alpha |^{2}} \\
e^{-T|\alpha |^{2}}+e^{-R|\alpha |^{2}} & 1+e^{-|\alpha |^{2}}%
\end{array}
\right) ,$$ where $T$ ($R=1-T$) are the transmission (absorption) rate of the photons and ${\mathcal{N}}_{\alpha}^2=1/[2(1+e^{-|\alpha |^{2}})]$. Next, from the equation $|\lambda \mathbf{I}-\boldsymbol{\tilde{\rho}}|=0$, we obtain the eigenvalues $$\lambda _{\pm }={\mathcal{N}}_{\alpha}^{2}\left[ \left( 1+e^{-|\alpha
|^{2}}\right) \pm \left( e^{-T|\alpha |^{2}}+e^{-R|\alpha |^{2}}\right)\right] , \label{lambda}$$ which obeys $\lambda_{-}+\lambda_{+}=1$. Substituting $\lambda_{\pm}$ into Eq. (\[eigen eq\]), or $(\lambda \mathbf{I}-\boldsymbol{\tilde{\rho}})
\mathbf{c}=0$, we further obtain the amplitudes $c_{1}=\pm c_{2}$, i.e., the eigenvectors $|\lambda_{\pm}(\phi)\rangle \varpropto (\pm |\sqrt{T} \alpha
\rangle_{1}+|\sqrt{T}\alpha e^{i\phi}\rangle_{2}) $, as Eq. (\[eigenvectors\]).
Derivations of the quantum Fisher information
=============================================
Firstly, we derive the general expression of the QFI \[i.e., Eq. ([QFI-mix3]{})\]. For a mixed state $\hat{\rho}=\sum_{m}\lambda _{m}|\lambda
_{m}\rangle \langle \lambda _{m}|$, the QFI is given by the well-known formula of Eq. (\[QFI-mix\]), where the eigenvectors of the reduced density matrix $\{|\lambda _{m}\rangle\}$ span an otho-normalized and complete basis. In general, the dimension of the entire Hilbert space is huge. However, there exists a much smaller subset $\{|\lambda _{i}\rangle\}$ with nonzero weights $\lambda _{i}$. It is convenient to express the QFI in terms of this subset only. For this purpose, we divide the complete basis $\{|\lambda _{m}\rangle \}$ into two subsets: $\{\vert \lambda _{i}\rangle \}$ and $\{\vert\lambda _{\bar{\imath}}\rangle\}$, with $\lambda _{i}\neq 0$ and $\lambda _{\bar{\imath}}=0$, respectively. Using the completeness relation $\sum_{\bar{\imath}}\vert
\lambda _{\bar{\imath}}\rangle \langle \lambda _{\bar{\imath}}\vert
=1-\sum_{i}\vert \lambda _{i}\rangle\langle\lambda_{i}\vert$, Eq. (\[QFI-mix\]) can be rewritten as$$\begin{aligned}
F_{Q}& =&\sum_{i}\frac{2\langle \lambda _{i}\vert (\hat{\rho}^{\prime
})^{2}|\lambda _{i}\rangle }{\lambda _{i}}+\sum_{j}\frac{2\langle \lambda
_{j}\vert (\hat{\rho}^{\prime })^{2}|\lambda _{j}\rangle }{\lambda _{j}}
\notag \\
&& +\sum_{i,j}2\left( \frac{1}{\lambda _{i}+\lambda _{j}}-\frac{1}{\lambda
_{i}}-\frac{1}{\lambda _{j}}\right) \left\vert \left\langle \lambda
_{i}\right\vert \hat{\rho}^{\prime }|\lambda _{j}\rangle \right\vert ^{2},
\label{QFI4}\end{aligned}$$where only the subset $\{|\lambda _{i}\rangle \}$ with $\lambda _{i}\neq 0$ is involved. Since $\{|\lambda _{i}\rangle \}$ are ortho-normalized, i.e., $\langle \lambda _{i}|\lambda _{j}\rangle =\delta _{i,j}$, we have $\langle\lambda _{i}|\lambda _{j}^{\prime }\rangle +\langle \lambda _{i}^{\prime}|\lambda _{j}\rangle =0$, and hence $$\begin{aligned}
\langle \lambda _{i}|(\hat{\rho}^{\prime })^{2}|\lambda _{i}\rangle &=&(
\lambda _{i}^{\prime }) ^{2}+\lambda _{i}^{2}\langle \lambda _{i}^{\prime
}|\lambda _{i}^{\prime }\rangle \\
&&+\sum_{l}( \lambda _{l}^{2}-2\lambda _{i}\lambda _{l}) |\langle \lambda
_{i}^{\prime }|\lambda _{l}\rangle |^{2}, \\
\vert \langle \lambda _{i}\vert \hat{\rho}^{\prime }\vert \lambda
_{j}\rangle \vert ^{2} &=&(\lambda _{i}^{\prime })^{2}\delta _{i,j}+(\lambda
_{i}-\lambda _{j})^{2}|\langle \lambda _{i}^{\prime }|\lambda _{j}\rangle
|^{2}.\end{aligned}$$Substituting them into Eq. (\[QFI4\]), and using $|\langle \lambda
_{j}^{\prime }|\lambda _{i}\rangle |^{2}=|\langle \lambda _{i}^{\prime
}|\lambda _{j}\rangle |^{2}$, we obtain the general formula of the QFI as main text of Eq. (\[QFI-mix3\]).
Now we apply the general formula to calculate the QFI of the ECS state. Since the eigenvalues of the reduced density matrix $\lambda_{\pm}$ are phase-independent, the first term of Eq. (\[QFI-mix3\]) vanishes. From Eq. (\[eigenvectors\]), it is easy to obtain the derivation of the eigenvectors $$|\lambda _{\pm }^{\prime }\rangle =\eta _{\pm }\frac{\partial }{\partial
\phi }|\sqrt{T}\alpha e^{i\phi }\rangle _{2}=\eta _{\pm }\sum_{n=0}^{\infty
}ind_{n}(\alpha \sqrt{T}e^{i\phi })\left\vert n\right\rangle _{2},
\label{derivation}$$where the normalization factors $\eta _{\pm }=1/\sqrt{2(1\pm e^{-T|\alpha
|^{2}})}$ and the probability amplitudes of coherent state $d_{n}(\alpha
)\equiv \langle n|\alpha \rangle =\alpha ^{n}e^{-\frac{1}{2}|\alpha |^{2}}/\sqrt{n!}$, which satisfiy $\sum_{n}|d_{n}(\alpha )|^{2}=1$ and $$\sum_{n=0}^{+\infty }n\left\vert d_{n}(\alpha )\right\vert ^{2}=|\alpha |^{2}
\text{, \ }\sum_{n=0}^{+\infty }n^{2}\left\vert d_{n}(\alpha )\right\vert
^{2}=|\alpha |^{2}(1+|\alpha |^{2}).$$Therefore, combining Eq. (\[eigenvectors\]) and Eq. (\[derivation\]), we obtain $$\langle \lambda _{\pm }|\lambda _{\pm }^{\prime }\rangle =\eta _{\pm
}^{2}\sum_{n=0}^{\infty }in\left\vert d_{n}(\alpha e^{i\phi }\sqrt{T}
)\right\vert ^{2}=i\eta _{\pm }^{2}|\alpha |^{2}T, \label{innerP1}$$and similarly, $\langle \lambda _{\mp }|\lambda _{\pm }^{\prime }\rangle
=i\eta _{+}\eta _{-}|\alpha |^{2}T$, as well as $\langle \lambda _{\pm
}^{\prime }|\lambda _{\pm }^{\prime }\rangle =\eta _{\pm }^{2}|\alpha
|^{2}T(1+|\alpha |^{2}T)$. These results enable us to calculate the remaining terms of Eq. (\[QFI-mix3\]): $$\begin{aligned}
\sum_{i=\pm }\lambda _{i}F_{Q,i} &=&4\lambda _{+}\eta _{+}^{2}|\alpha
|^{2}T(1+|\alpha |^{2}T-\eta _{+}^{2}|\alpha |^{2}T) \notag \\
&&+4\lambda _{-}\eta _{-}^{2}|\alpha |^{2}T(1+|\alpha |^{2}T-\eta
_{-}^{2}|\alpha |^{2}T), \label{t2}\end{aligned}$$and $$\sum_{i=\pm ,j=\mp }\frac{8\lambda _{i}\lambda _{j}}{\lambda _{i}+\lambda
_{j}}\left\vert \left\langle \lambda _{i}^{\prime }\right\vert \lambda
_{j}\rangle \right\vert ^{2}=16\lambda _{+}\lambda _{-}\eta _{+}^{2}\eta
_{-}^{2}|\alpha |^{4}T^{2}, \label{t3}$$due to $\lambda _{+}+\lambda _{-}=1$. Finally, we get the exact result of the QFI for the input ECS: $$F_{Q}=4{\mathcal{N}}_{\alpha }^{2}|\alpha |^{2}T\left[ 1+|\alpha |^{2}T-{\
\mathcal{N}}_{\alpha }^{2}|\alpha |^{2}T\left( 1+\frac{1-e^{-2R|\alpha |^{2}}}{1-e^{-2T|\alpha |^{2}}}\right) \right] ,$$where we have used the relations: $\lambda _{+}\eta _{+}^{2}+\lambda
_{-}\eta _{-}^{2}={\mathcal{N}}_{\alpha }^{2}$ and $4\lambda _{+}\lambda
_{-}\eta _{+}^{2}\eta _{-}^{2}={\mathcal{N}}_{\alpha }^{4}[1-e^{-2R|\alpha
|^{2}}]$, as well as $$\lambda _{+}\eta _{+}^{4}+\lambda _{-}\eta _{-}^{4}=\frac{{\mathcal{N}}%
_{\alpha }^{2}}{2}\frac{1-e^{-|\alpha |^{2}}}{1-e^{-2T|\alpha |^{2}}}.$$Using $\bar{n}=2{\mathcal{N}}_{\alpha }^{2}|\alpha |^{2}$ and hence $|\alpha|^{2}=\bar{n}+w(\bar{n}e^{-\bar{n}})$, the QFI can be further simplified as Eq. (\[FQ\_ECS\]).
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|
---
abstract: 'The gapless fermionic excitations in superfluid $^3$He-A have a “relativistic” spectrum close to the gap nodes. They are the counterpart of chiral particles (left-handed and right-handed) in high energy physics above the electroweak transition. We discuss the effective gravity and effective gauge fields induced by these massless fermions in the low-energy corner. The interaction of the chiral fermions with the gauge field in $^3$He-A is discussed in detail. It gives rise to the effect of axial anomaly: Conversion of charge from the coherent motion of the condensate (vacuum) to the quasiparticles (matter). The charge of the quasiparticles is thus not conserved. In other words, matter can be created without creating antimatter. This effect is instrumental for vortex dynamics, in which the vortex is the mediator of conversion of linear momentum from the condensate to the normal component via spectral flow in the vortex core. The same effect leads to the instability of the counterflow in $^3$He-A, in which the flow of the normal component (incoherent degrees of freedom) is transformed to the order parameter texture (coherent degrees of freedom). We discuss the analogues of these phenomena in high energy physics. The conversion of the momentum from the vortex to the heat bath is equivalent to the nonconservation of baryon number in the presence of textures and cosmic strings. The counterflow instability is equivalent to the generation of the hypermagnetic field via the axial anomaly. We discuss also an analogue of axions and different sources of the mass of the “hyperphoton” in $^3$He-A.'
address: |
Low Temperature Laboratory, Helsinki University of Technology, Box 2200, FIN-02015 HUT, Finland\
and\
Landau Institute for Theoretical Physics, Moscow, Russia
author:
- 'G.E. Volovik'
title: ' Axial anomaly in $^3$He-A: Simulation of Baryogenesis and Generation of primordial magnetic fields in Manchester and Helsinki'
---
\[@twocolumnfalse
Keywords: superfluid $^3$He, chiral anomaly, gap nodes, effective field theory, effective gravity
\]
Introduction
============
Effective electrodynamics and gravity in $^3$He-A.
--------------------------------------------------
Many aspects of high energy physics can be modelled in condensed matter [@Wilczek]. Superfluid $^3$He-A provides a rich source for such a modelling. The most pronounced property of this superfluid is that in addition to the numerous bosonic fields (collective modes of the order parameter) it contains gapless fermionic quasiparticles. Close to the gap nodes, the points in momentum space where the energy is zero (Fig. \[ChiralFermions\]), the energy spectrum of quasiparticles is linear in momentum ${\bf p}$. This simple circumstance has far-reaching consequences: The low energy fermions and some of the bosons obey “relativistic” equations, while their interaction with the superfluid vacuum mimics that of elementary particles with gauge fields. This illustrates the principle [@Nielsen] that the effective physics in a low energy corner becomes more symmetric than in the general case. In $^3$He-A we have two low energy corners, at ${\bf p}\approx \pm p_F{\hat{\bf l}}$, where $p_F$ is the Fermi momentum and ${\hat{\bf l}}$ the unit vector specifying the direction of the nodes. This picture does not depend on details of the underlying microscopic interactions of atoms, whose only role is to produce values of “fundamental constants”, such as the “speed of light” and the “Planck energy”.
Close to the gap node the square of the quasiparticle energy $E$ is generally a quadratic form of the deviation of the momentum ${\bf p}$ from the position of the nodes $ \pm p_F{\hat{\bf l}}$: $$E_\pm^2({\bf p})=g^{ik}(p_i \mp p_F{\hat l}_i) (p_k \mp p_F{\hat l}_k)~.
\label{E^2form}$$ Let us introduce an effective vector potential of the “electromagnetic field” $${\bf A}= p_F{\hat{\bf l}}~,
\label{vector potential}$$ and the “electric charge” $e$, with $e=+ 1$ for the quasiparticles in the vicinity of the node at $p_F{\hat{\bf l}}$ and $e=-1$ for the quasiparticles in the vicinity of the opposite node, at $-p_F{\hat{\bf l}}$. Then one obtains a spectrum of relativistic fermions moving on the gravitational and electromagnetic background, determined by the metric tensor $g^{ik}$ and the vector potential ${\bf A}$: $$E^2({\bf p})=g^{ik}(p_i -eA_i) (p_k-eA_k)~.
\label{E^2relativistic}$$
The symmetric matrix $g^{ik}$, which gives the contravariant components of the metric tensor, is generally determined by the directions of the principal axes forming the orthonormal basis (${\hat{\bf e}}_1,{\hat{\bf
e}}_2,{\hat{\bf e}}_3$) and by the “speeds of light” along these directions: $$g^{ik}=c_1^2 \hat e_1^i \hat e_1^k+c_2^2 \hat e_2^i \hat e_2^k+c_3^2 \hat e_3^i
\hat e_3^k~.
\label{3speeds}$$
It is important that the effective gauge field ${\bf A}$ and the effective metric $g^{ik}$ depend on space and time, since the order parameter in general and the ${\hat{\bf l}}$-vector in particular are not fixed in $^3$He-A and can form different types of textures. The quasiparticles view the order parameter textures as a curved Lorentzian space-time and [*simultaneously*]{} as gauge fields. These fields are dynamical: The Effective Lagrangian for “electromagnetic” and “gravitational” fields can be obtained by integrating over the fermionic field. The same principle was used by Sakharov and Zeldovich to obtain an Effective Gravity [@Sakharov] and Effective Electrodynamics [@Zeldovich] from vacuum fluctuations. In some special cases (and in this review we consider just such a case) the main contribution to the effective action comes from the vacuum fermions whose momenta ${\bf p}$ are concentrated near the gap nodes, i.e. from the “relativistic” fermions. In these (and only in these) cases one obtains an effective Lagrangian which gives Maxwell equations for the ${\bf A}$-field. Since the “photons” are thus constructed from the fermionic degrees of freedom, the metric $g^{ik}$, which governs the propagation of “photons”, is the same as the metric governing the dynamics of the underlying fermionic quasiparticles. Following the title of the Laughlin talk at this Symposium, this provides an example of a “Gauge Theory from Nothing” [@Laughlin].
From Eq.(\[E\^2relativistic\]) it follows that $g^{00}=-1$ and $g^{0i}=0$, but this is not the general case: Typically all the components of the dynamical metric tensor $g_{\mu\nu}$ depend on the position in space-time. In some cases the effective metric is not trivial giving rise to conical singularities [@VolovikGravity1997], event horizons and ergoregions [@JacobsonVolovik; @RotatingCore]. This also allows to simulate quantum gravity. Note that the primary quantities in this effective (quantum) gravity are the [*contravariant*]{} components $g^{\mu\nu}$. They appear in the low-energy corner of the fermionic spectrum and represent the low-energy properties of the quantum vacuum. The geometry of the effective space-time, in which the free quasiparticles follow a geodesic, is determined by the inverse metric $g_{\mu\nu}$ and thus is a secondary object. In a similar manner the effective Lorentzian space-time comes from the spectrum of the sound waves propagating on the background of a moving inhomogeneous liquid [@UnruhSonic; @Jacobson1991; @Visser1997]. The difference to the case of superfluid liquid $^3$He-A is that ordinary liquids are essentially dissipative classical systems and thus cannot serve as a model of the quantum vacuum.
The above mechanism of the generation of the gauge field [**A**]{} and gravity $g^{\mu\nu}$ is valid for a general system with point gap nodes. In the particular case of $^3$He-A the initial “nonrelativistic” fermionic spectrum has the form $$E^2({\bf p})= v_F^2(p-p_F)^2 +{\Delta_0^2\over
p_F^2}({\hat{\bf l}} \times {\bf p})^2,
\label{AphaseSpectrum}$$ where $\Delta_0$ is the gap amplitude; $v_F=p_F/m^*$ is the Fermi velocity and $m^*$ the effective mass of the quasiparticle in the normal Fermi-liquid state, which is typically about 3–6 times the bare mass $m_3$ of the $^3$He atom.
In the low energy corner one obtains the “relativistic” spectrum of Eq. (\[E\^2relativistic\]) with the following values of the “fundamental constants” [@exotic; @VolovikVachaspati] $$c_1=c_2={\Delta_0\over
p_F}\equiv c_{\perp} ~,~c_3=v_F\equiv c_{\parallel}~.
\label{SpeedsInAphase}$$ The space characterizing the motion of quasiparticles in $^3$He-A, i.e. the space in which the quasiparticles move along the geodesic curves (in the absence of other forces) has an uniaxial anisotropy, with the anisotropy axis along ${\hat{\bf l}}$: $${\hat{\bf e}}_3={\hat{\bf l}}~.
\label{e3}$$ The speed of a “light” along the ${\hat{\bf l}}$-vector, $c_{\parallel}$, is about 3 orders of magnitude larger than that in the transverse direction: $ c_{\parallel}\gg c_{\perp}$. Another important fundamental constant, $\Delta_0$, plays the role of the Planck energy cut-off, as will be illustrated later on.
Chiral fermions in $^3$He-A
---------------------------
The chiral properties of the fermionic spectrum is revealed after the square root of Eq. (\[E\^2relativistic\]) is taken. This procedure is not unambiguous: One has to use the underlying BCS theory of Cooper pairing, which leads to the superfluid A-phase state in $^3$He. In BCS theory one obtains the Bogoliubov-Nambu Hamiltonian for fermions, which in the low-energy corner transforms into the Weyl Hamiltonian for massles chiral particles. It is represented by the proper square root of Eq. (\[E\^2relativistic\]): $${\cal H}=-e\sum_a c_a \tau^a \hat e^i_a(p_i- eA_i) ~,
\label{WeylHamiltonian}$$ where $\tau^a$ are the Pauli matrices acting in the Bogolibov-Nambu particle-hole space.
The more close inspection of the BCS theory for $^3$He-A reveals that the order parameter contains 18 degrees of freedom and thus 18 propagating collective modes. Six of these collectives modes, which represent propagating oscillations of position of nodes and of the slopes of the energy spectrum at the nodes, are shown in Fig. \[CollectiveModes\] together with their analogs in relativistic theories.
The important property of this Hamiltonian is that the sign of the “electric” charge $e$ simultaneously determines the chirality of the fermions. This is clearly seen with a simple isotropic example having $c_1=c_2=c_3=c$: $${\cal H}=-e c\vec\tau\cdot({\bf p}- e{\bf A}) ~.
\label{WeylIsotropic}$$ A particle with positive (negative) $e$ is left-handed (right-handed): Its Bogoliubov spin $\vec \tau$ is antiparallel (parallel) to the momentum ${\bf
p}$, if ${\cal H}$ is positive definite. Thus the field ${\bf A}$ corresponds to the axial field in relativistic theories. The symmetry between left and right is broken in $^3$He-A.
Rather few systems have (3+1)-dimensional chiral fermions as excitations. The superfluid $^3$He-A and the Standard Model of the electroweak interactions are among these exotic systems. This is why $^3$He-A is the best condensed matter system for the simulation of effects related to the chiral nature of the fermions, especially of the chiral anomaly. There are other condensed matter systems with chiral fermions, but these fermions occupy a space-time of 2+1 or 1+1 dimensions. Examples are the (2+1)-dimensional fermions in high-temperature superconductors [@SimonLee]; (1+1)-dimensional chiral edge states in the quantum Hall effect [@WenEdgeStates; @StoneEdgeStates], and in superconductors with broken time-reversal symmetry [@LaughlinT-symmetry; @VolovikT-symmetry]. Finally fermionic excitations in the core of quantized vortices bear this property, too [@RotatingCore]. The gap nodes in (3+1)-dimensional theories can appear also in different types of “color superfluidity” – quark condensates in dense baryonic matter [@ColorSuperconductivity1; @ColorSuperconductivity2] (The quark condensate phase analogous to the superfluid $^3$He-B, where color and flavour are locked together instead of spin and orbital momenta, while the gap is isotropic and thus has no nodes, was also discussed [@ColorSuperconductivity3]).
The spectrum of fermionic excitations of the electroweak vacuum in the present Universe contains one branch of chiral particles: The left-handed neutrino branch (Fig. \[Metal-Insulator\]). The right-handed neutrino is not present (or interacts with other matter different from the left-handed one). This is a remarkable manifestation of the violation of the left-right symmetry in the electroweak vacuum. Another symmetry, which is broken in the present Universe, is the $SU(2)$ symmetry of weak interactions. In the symmetric state of the early Universe, the left leptons (neutrino and left electron) formed a $SU(2)$ doublet, while the right electron is in a $SU(2)$ singlet. During the cooldown of the Universe the phase transition occured, at which the $SU(2)\times U(1)$ symmetry was broken to the electromagnetic $U(1)$ symmetry. As a consequence, the left and right electrons were hybridized forming the present electronic spectrum with the gap $\Delta=m_ec^2$. The electric properties of the vacuum thus exhibited the metal-insulator phase transition: The “metallic” state of the vacuum with the Fermi point in the elecronic spectrum was transformed to the insulating state with the gap. Recent numerical calculations suggest that this transformation occurs either by the first order phase transition or by continuous cross-over without any real symmetry breaking [@Kajantie].
The similarity between the chiral fermions in electroweak theory and in $^3$He-A has also a topological origin. The gap nodes – zeroes in the particle (quasiparticle) spectrum – are characterized by a topological invariant in 4-momentum space belonging to the third homotopy group $\pi_3$ [@exotic]: $$N_{\rm top} = {1\over{24\pi^2}}e_{\mu\nu\lambda\gamma}~
{\bf tr}\int_{\sigma}~ dS^{\gamma}
~ {\cal G}\partial_{p^\mu} {\cal G}^{-1}
{\cal G}\partial_{p^\nu} {\cal G}^{-1} {\cal G}\partial_{p^\lambda} {\cal
G}^{-1}~.
\label{TopInvariant}$$ Here $${\cal G}(p_\mu)= {1\over ip_0 +{\cal H}}
\label{GreenFunction}$$ is the Green’s function and $\sigma$ is the 3-dimensional surface around the point node in the 4-momentum space. For the relativistic chiral particle the node is at $p_0=0$, ${\bf p}=0$, while in $^3$He-A the nodes are at $p_0=0$, ${\bf
p}= \pm p_F \hat{\bf l}$. In all cases the topological invariant is nonzero: $N_{\rm top} =\pm 1$, and the sign of $N_{\rm top}$ depends on chirality.
The topological stability – the conservation of the topological invariant in Eq. (\[TopInvariant\]) – is important for the fermionic system. It implies that under a deformation of the system (under a continuous change of the system parameters) the gap nodes in momentum space can arise or disappear only in pairs (node-“antinode” pairs). This topological stability, which does not depend on the details and the symmetry of the system, provides topological conservation of chirality: The algebraic number of the chiral fermions, i.e. the number of the right fermionic species minus the number of the left fermionic species, is conserved: $\Delta N = N_{FR} - N_{FL}=\sum N_{\rm top}$. In $^3$He-A one has $ N_{FR}
= N_{FL}=1$ and thus $\Delta N =\sum N_{\rm top}=0$.
In the relativistic theories the electroweak transition $SU(2)\times U(1)
\rightarrow U(1)$ satisfies this topological rule: If the right neutrinos are absent, the algebraic number of chiral fermions per each generation is $\Delta N = -1$ in both phases: in the symmetric phase $SU(2)\times U(1)$ one has $\Delta N =7-8= -1$ and in the broken symmetry phase $U(1)$ one has $\Delta N =0-1= -1$. In this case the conservation of the topological invariant provides the zero mass for neutrino. In the unification theories, the $SU(5)$ symmetry breaking pattern with $N_{FL}=10+5$ left fermions in one generation does not satisfy this topological rule. The rule holds only if one doubles the number of fermions and considers right antiparticles as independent particles: in this case $\Delta N =15-15= 0$. In the $SU(4)\times SU_L(2)\times SU_R(2)$ theory with $ N_{FR} = N_{FL}=8$ the topological rule is satisfied without the doubling of fermions: one has $\Delta N =0$ throughout all the route of the symmetry breaking to $SU(3)\times U(1)$.
It is important that if the vacuum is characterized by nonzero topological charge, $\sum N_{\rm top}\neq 0$, the system has massless fermions. This means that the problem of the neutrino mass is directly related to the momentum space topology of the vacuum.
AXIAL ANOMALY
=============
Adler-Bell-Jackiw equation
--------------------------
Chiral fermions interacting with gauge fields exhibit the effect of chiral anomaly, the nonconservation of matter charge due to the interaction of matter with the quantum vacuum. The origin for the axial anomaly can be seen from the behavior of the chiral particle in a constant magnetic field, ${\bf A}=(1/2){\bf B}\times {\bf r}$. The Hamiltonians for the right particle with the electric charge $e_R$ and for the left particle with the electric charge $e_L$ are $${\cal H}= c\vec\tau\cdot({\bf p}- e_R{\bf A}) ~, {\cal H}= -c\vec\tau\cdot({\bf
p}- e_L{\bf A}) ~.
\label{WeylForLeftRight}$$ Fig. \[ChiralAnomaly\] shows the energy spectrum in a magnetic field ${\bf B}$ along $z$; the thick lines show the occupied negative-energy states. Motion of the particles in the plane perpendicular to ${\bf B}$ is quantized into the Landau levels shown. The free motion is thus effectively reduced to one-dimensional motion along ${\bf B}$ with momentum $p_z$. Because of the chirality of the particles the lowest ($n=0$) Landau level is asymmetric. It crosses zero only in one direction: $E=cp_z$ for the right particle and $E=-cp_z$ for the left one. If we now apply an electric field ${\bf E}$ along $z$, particles are pushed from negative to positive energy levels according to the equation of motion $\dot p_z =e_{R} E_z$ ($\dot p_z
=e_{L} E_z$) and the whole Dirac sea moves up (down) creating particles and electric charge from the vacuum. This motion of particles along the “anomalous” branch of the spectrum is called [*spectral flow*]{}. The rate of particle production is proportional to the density of states at the Landau level, which is $\propto \vert e_R{\bf B}\vert$ ($\vert e_L{\bf B}\vert)$, so that the rate of production of particle number $n=n_R+n_L$ and of charge $Q=n_Re_R+n_Le_L)$ from the vacuum is $$\dot{n} ={1\over {4\pi^2}} (e_R^2-e_L^2){\bf E} \cdot {\bf
B} ~,~\dot{Q}={1\over {4\pi^2}} (e_R^3-e_L^3){\bf E} \cdot
{\bf B} ~.
\label{ChargeParticlProduction}$$
This is an anomaly equation for the production of particles from vacuum of the type found by Adler[@Adler1969] and by Bell and Jackiw[@BellJackiw1969] in the context of neutral pion decay. We see that for particle or charge creation from “nothing” it is necessary to have an asymmetric branch of the dispersion relation $E(p)$ which crosses the axis from negative to positive energy. Additionally, the symmetry between the left and right particles has to be violated: $e_R\neq e_L$ for the charge creation and $e_R^2\neq e_L^2$ for the particle creation.
Anomalous nucleation of baryonic charge
---------------------------------------
In the standard electroweak model there is an additional accidental global symmetry $U(1)_B$ whose classically conserved charge is the baryon number $Q_B$. Each of the quarks is assigned $Q_B=1/3$ while the leptons (neutrino and electron) have $Q_B=0$. This baryonic number is not conserved due to the axial anomaly. There are two gauge fields whose “electric” and “magnetic” fields become a source for baryoproduction: The hypercharge field $U(1)$ and the weak field $SU(2)$. The corresponding hypercharges $Y$ and weak charges $W$ of the left $u$ and $d$ quarks are $$Y_{dL}=Y_{uL}=1/6~,~W_{dL}=-W_{uL}= 1/2 ~,
\label{ChargesOfLeftQuarks}$$ whereas for the right $u$ and $d$ quarks one has $$Y_{uR}= 2/3 ~,~Y_{dR}= -1/3~,~ W_{dR}=W_{uR}= 0 ~ .
\label{ChargesOfRightQuarks}$$
Let us first consider the effect of the hypercharge field. Since the number of different species of quarks carrying the baryonic charge is $3N_F$ (3 colours $\times$ $N_F$ generations of fermions) and the baryonic charge of the quark is $Q_B=1/3$, the production rate of baryonic charge in the presence of hyperelectric and hypermagnetic fields is $${ N_F\over 4\pi^2}(Y_{dR}^2 +Y_{uR}^2 - Y_{dL}^2 - Y_{uL}^2)~{\bf B}_Y\cdot
{\bf
E}_{Y}.
\label{BarProdByHypercharge}$$ Since the hypercharges of left and right quarks are different (see Eqs. (\[ChargesOfLeftQuarks\],\[ChargesOfRightQuarks\])), one obtains a nonzero production of baryons by the hypercharge field $${N_F\over 8\pi^2}{\bf B}_Y\cdot {\bf E}_{Y}.
\label{BarProdByHypercharge2}$$
The weak electric and magnetic fields also contribute to the production of the baryonic charge: $${N_F\over 4\pi^2} ( W_{dR}^2 +
W_{uR}^2 - W_{dL}^2 - W_{uL}^2)~{\bf B}^a_W\cdot {\bf E}_{aW} ~,
\label{BarProdByWeak}$$ which gives $$- {N_F\over 8\pi^2}{\bf B}^a_W\cdot {\bf E}_{aW}.
\label{BarProdByHypercharge3}$$
Thus the total rate of baryon production in the Standard model takes the form $$\dot Q_B= {{N_F} \over {8 \pi^2}} \left (
- {\bf B}^a_W\cdot {\bf E}_{aW} +
{\bf B}_Y\cdot {\bf E}_{Y} \right).
\label{TotalBaryoProduction}$$ The first term comes from nonabelian $SU(2)$ fields, it shows that the nucleation of baryons occurs when the topological charge of the vacuum changes, say, by sphaleron or due to de-linking of linked loops of the cosmic strings. The second, nontopological, term describes the exchange of the baryonic charge between the hypermagnetic field and the fermionic degrees of freedom.
Anomalous nucleation of linear momentum in $^3$He-A
---------------------------------------------------
The anomaly equation which describes the nucleation of fermionic charges in the presence of magnetic and electric fields describes both the production of the baryons in the electroweak vacuum (baryogenesis) and the production of the linear momentum in the superfluid $^3$He-A (momentogenesis). In $^3$He-A the effective $U(1)$ gauge field is generated by the moving ${\hat{\bf l}}$-texture. According to Eq. (\[vector potential\]), the time and space dependent ${\hat{\bf l}}$ vector, associated with the motion of the so-called continuous vortex (see below), produces a force on the excitations equivalent to that of an “electric” (or “hyperelectric”) field ${\bf E}=p_F \partial_t {\hat {\bf l}}$ and a “magnetic” (or “hypermagnetic” field) ${\bf B}=p_F{\bf \nabla}\times {\hat {\bf l}}$ acting on particles of unit charge. Equation (\[ChargeParticlProduction\]) can then be applied to calculate the rate at which left-handed and right-handed quasiparticles are created by spectral flow. What we are interested in is the production of the particle momentum due to spectral flow: $$\dot{\bf P}= {1\over {4\pi^2}}({\bf P}_R-{\bf P}_L) ~(
{\bf E} \cdot {\bf B} \, \, ) ~~.
\label{MomentoProduction1}$$ Since the right and left particle have opposite momenta ${\bf P}_R=p_F{\hat {\bf l}}=-{\bf P}_L$, excitation momentum is created at a rate $$\dot{\bf P}= { p_F^3\over {2\pi^2}} \hat {\bf l} ~(
\partial_t \hat {\bf l} \cdot (\vec\nabla\times \hat {\bf l} )\, \, ) ~~.
\label{MomentoProduction2}$$
However, the total linear momentum of the liquid has to be conserved. Therefore Eq. (\[MomentoProduction2\]) implies that in the presence of a time-dependent texture momentum is transferred from the superfluid ground state (analogue of vacuum) to the heat bath of excitations forming the normal component (analogue of matter).
SPECTRAL FLOW FORCE on VORTEX
=============================
Continuous vortex and baryogenesis in textures
----------------------------------------------
The anomalous production of linear momentum leads to an additional force acting on the continuous vortex in $^3$He-A (Fig. \[ContinuousMomentogenesis\]).
The continuous vortex, first discussed by Chechetkin [@Chechetkin] and Anderson and Toulouse[@AT] (ATC vortex), has in its simplest realization the following distribution of the ${\hat{\bf l}}$-field ($\hat{\bf z}$, ${\hat{\bf r}}$ and ${\hat{\bf \phi}}$ are unit vectors of the cylindrical coordinate system) $${\hat{\bf l}}(r,\phi)={\hat{\bf z}} \cos\eta(r) + {\hat{\bf
r}} \sin\eta(r)~,
\label{lTextureContVortex}$$ where $\eta(r)$ changes from $\eta(0)=\pi$ to $\eta(\infty)=0$ in the so called soft core of the vortex. The superfluid velocity ${\bf v}_s$ in superfliud $^3$He-A is determined by the twist of the triad ${\hat{\bf e}}_1,{\hat{\bf e}}_2,{\hat{\bf e}}_3$ and corresponds to torsion in the tetrad formalism of gravity (the space-time dependent rotation of vectors ${\bf m}=c_\perp {\hat{\bf e}}_1$ and ${\bf n}=c_\perp {\hat{\bf e}}_2$ about axis $\hat l$ in Fig. \[CollectiveModes\]): $${\bf v}_s ={\hbar\over 2 m_3} \hat e_1^i\vec \nabla \hat
e_2^i ~.
\label{v_s}$$ In comparison to a more familiar singular vortex, the continuous vortex has a regular superfluid velocity field $${\bf v}_s(r,\phi)= -{\hbar\over 2 m_3 r}[1+\cos\eta(r)]{\hat {\bf \phi}}~~,
\label{v_sContVortex}$$ with no singularity on the vortex axis.
The stationary vortex generates a “magnetic” field. If the vortex moves with a constant velocity ${\bf v}_L$ it also generates an “electric” field, since ${\hat{\bf l}}$ depends on ${\bf r}-{\bf v}_Lt$: $${\bf B}=p_F\vec \nabla \times {\hat {\bf l}}~~,~~{\bf
E}=\partial_t {\bf A}=-p_F({\bf v}_L\cdot
\vec\nabla){\hat {\bf l}}
\label{EfieldContVortex}$$
The net production of the quasiparticle momenta by the spectral flow in the moving vortex means, if the vortex moves with respect to the system of quasiparticles (the normal component of liquid or matter, whose flow is characterised by the normal velocity ${\bf v}_n$), that there is a force acting between the normal component and the vortex. Integration of the anomalous momentum transfer in Eq.(\[MomentoProduction2\]) over the cross-section of the soft core of the moving ATC vortex gives the following force acting on the vortex (per unit length) from the system of quasiparticles [@Volovik1992]: $${\bf F}_{sf}=\int d^2 r{ p_F^3\over {2\pi^2}} \hat {\bf l} ~(
\partial_t \hat {\bf l} \cdot (\vec\nabla\times \hat {\bf l} ))=-2\pi \hbar
C_0{\hat {\bf z}} \times ({\bf v}_L-{\bf v}_n) ,
\label{SpFlowForce}$$ where $$C_0= p_F^3/3\pi^2~.
\label{C0}$$ Note that this spectral-flow force is transverse to the relative motion of the vortex and thus is nondissipative (reversible). In this derivation it was assumed that the quasiparticles and their momenta, created by the spectral flow from the vacuum, are finally absorbed by the normal component. The time delay in the process of absorption and also the viscosity of the normal component lead to a dissipative (friction) force between the vortex and the normal component: ${\bf F}_{fr}=-\gamma ({\bf v}_L-{\bf v}_n)$. There is no momentum exchange between the vortex and the normal component if they move with the same velocity.
Another important property of the spectral-flow force (\[SpFlowForce\]) is that it does not depend on the details of the vortex structure: The result for ${\bf F}_{sf}$ is robust against any deformation of the ${\hat{\bf l}}$-texture which does not change the asymptote, i.e. the topology of the vortex. In this respect this force resembles another force, which acts on the vortex moves with respect to the superfluid vacuum. This is the well-known Magnus force: $${\bf F}_{M}= 2\pi \hbar
n_3 {\hat {\bf z}} \times ({\bf v}_L-{\bf v}_s(\infty)) ~.
\label{MagnusForce}$$ Here $n_3$ is the particle density (here the number density of $^3$He atoms) and ${\bf v}_s(\infty)$ is the uniform velocity of the superfluid vacuum far from the vortex.
The balance between all the forces acting on the vortex, ${\bf F}_{sf}$, ${\bf
F}_{fr}$, ${\bf F}_{M}$ and some other forces, including any external force and the so-called Iordanskiǐ force coming from the gravitational analog of the Aharonov-Bohm effect [@AB], determines the velocity of the vortex and causes it to be a linear combination of ${\bf
v}_s(\infty)$ and ${\bf v}_n$. Due to this balance the fermionic charge (the linear momentum), which is transferred from the fermionic heat bath to the vortex texture, is further transferred from the vortex texture to the superfluid motion. Thus the vortex texture serves as intermediate object for the momentum exchange between the fermionic matter and the superfluid vacuum. In this respect the texture corresponds to the sphaleron or to the cosmic string in relativistic theories.
The result (\[SpFlowForce\]) for the spectral-flow force, derived for the ATC vortex from the axial anomaly equation (\[MomentoProduction2\]), was confirmed in a microscopic theory, which took into accout the discreteness of the quasiparticle spectrum in the soft core [@Kopnin1993]. This was also confirmed in experiments on vortex dynamics in $^3$He-A [@BevanNature; @BevanJLTP].
In such experiments a uniform array of vortices is produced by rotating the whole cryostat. In equilibrium the vortices and the normal component (heat bath) of the fluid rotate together with the cryostat. An electrostatically driven vibrating diaphragm produces an oscillating superflow, which via the Magnus force generates the vortex motion, while the normal component remains clamped due to its high viscosity. This creates a motion of vortices with respect both to the heat bath and the superfluid vacuum. The vortex velocity ${\bf v}_L$ is determined by the overall balance of forces acting on the vortices, which in the absence of the external forces can be expressed in terms of the two parameters, so-called mutual friction parameters: [@BevanNature] $$\hat{\bf z}\times ({\bf v}_L-{\bf v}_s(\infty))+
d_\perp \hat{\bf z}\times({\bf
v}_n-{\bf v}_L)+d_\parallel({\bf v}_n-{\bf v}_L) =0~~.
\label{ForceBalance}$$ Measurement of the damping of the diaphragm resonance and of the coupling between different eigenmodes of vibrations enables both parameters, $d_\perp$ and $d_\parallel$, to be deduced.
From the above theory of spectral flow in the $^3$He-A vortex texture it follows that the parameter, which characterizes the transverse forces acting on the vortex, is given by $$d_\perp\approx {C_0-n_3+n_s(T)\over n_s(T)}~,
\label{d_perpAphase}$$ where $n_s(T)$ is the density of the superfluid component. In this equation the parameter $C_0$ from Eq.(\[C0\]) arises through the axial anomaly, the particle density $n_3$ stems from the Magnus force and the superfluid density $n_s(T)$ from the combined effect of Magnus and Iordanskiǐ forces. The effect of the chiral anomaly is crucial for the parameter $d_\perp$ since $C_0$ is comparable with $n_3$, since $C_0=p_F^3/3\pi^2$ is the particle density of liquid $^3$He in the normal state. The difference between $C_0$ and $n_3$ is thus determined by the tiny effect of superfluidity on the particle density and is extremely small: $n_3 -C_0\sim n_3 (\Delta_0/v_Fp_F)^2=n_3(c_\perp/c_\parallel)^2 \ll
n_3$. Because of the axial anomaly one must have $d_\perp\approx 1$ for all practical temperatures, even including the region close to $T_c$, where the superfluid component $n_s(T) \sim n_3 (1-T^2/T_c^2)$ is small. $^3$He-A experiments, made in the whole temperature range where $^3$He-A is stable, gave precisely this value within experimental uncertainty, $|1-d_\perp|<0.005$[@BevanNature].
This provides an experimental verification of the Adler-Bell-Jackiw axial anomaly equation (\[ChargeParticlProduction\]), applied to $^3$He-A, and thus supports the idea that baryonic charge (and also leptonic charge) can be generated by electroweak fields.
Singular vortex and baryogenesis by cosmic strings
--------------------------------------------------
There are many different scenarios of the electroweak baryogenesis [@Dolgov; @Turok]. In some of them the baryonic charge is created in the cores of topological objects, in particular in the core of cosmic strings. While a weak and hypercharge magnetic flux is always present in the core of electroweak strings, a weak and hypercharge electric field can be present along the string if the string is moving across a background electromagnetic field [@ewitten] or in certain other processes such as the de-linking of two linked loops [@tvgf; @jgtv]. Parallel electric and magnetic fields in the string change the baryonic charge and can lead to cosmological baryogenesis [@barriola] and to the presence of antimatter in cosmic rays [@gstv].
Again the axial anomaly is instrumental for the baryoproduction in the core of cosmic strings. But now the effect cannot be described by the anomaly equation (\[ChargeParticlProduction\]). This equation was derived using the energy spectrum of the free massless fermions in the presence of the homogeneous electric and magnetic fields. But in cosmic strings these fields are no more homogeneous. Moreover the massless fermions exist only in the vortex core as bound states in the potential well produced by the order parameter (Higgs) field. Thus the consideration of baryoproduction should be essentially different: the spectral flow phenomenon has to be studied using an exact spectrum of the massless bound states, namely fermion zero modes on strings.
A similar situation takes place in condensed matter, where the counterpart of the cosmic string is the conventional quantized vortex with a singular core (Fig. \[VortexString\]). The vortices with singular cores are: (i) Abrikosov vortices in superconductors; (ii) vortices in superfluid $^3$He-B; and (iii) such vortices in $^3$He-A which, as distinct from the continuous vortices, belong to the nontrivial elements of the $\pi_1$ homotopy group. It appears that the momentogenesis due to the axial anomaly also takes place here, but as distinct from the case of the continous ATC vortex in $^3$He-A, it cannot be described by the continuous anomaly equation of the type of Eq. (\[MomentoProduction1\]). For its description one should consider the spectral properties of the fermion zero modes localized in the singular vortex core. The main difference between fermion zero modes in relativistic strings and in the conventional condensed matter vortices is the following. In strings the anomalous branch $E(p_z)$ which crosses zero and gives rise to the spectral flow from the negative vaccum energy levels to the positive matter energy levels of is given in terms of a continuous variable – the linear momentum $p_z$ along the string. In contrast, in the case of condensed-matter vortices (Fig. \[SingularMomentogenesis\]) the branch $E(L_z)$ is “crossing” zero as a function of the [*discrete*]{} angular momentum $\hbar L_z$ ($L_z$ can be integral or half-odd integral). The level flow along the discrete energy levels is suppressed and is determined by the interlevel distance $\hbar\omega_0$ and the level width $\hbar/\tau$ resulting from the scattering of core excitations by free excitations in the heat bath outside the core (or by impurities in superconductors).
This suppression of spectral flow results in a renormalization of the spectral-flow parameter, which is roughly [@KopninVolovikParts; @StoneSpectralFlow] $$\tilde C_0 \sim {C_0\over 1+\omega_0^2\tau^2}~,
\label{C_=Renormalized}$$ and the Eq. (\[d\_perpAphase\]) becomes: $$d_\perp\approx {\tilde C_0-n_3+n_s(T)\over n_s(T)}
\label{d_perpBphase}$$ If $\omega_0\tau \ll 1$, the levels overlap and spectral flow is allowed. In the opposite limit $\omega_0\tau \gg 1$ it is completely suppressed. The parameter $\omega_0\tau $ depends on temperature and this allows us to check Eq. (\[d\_perpBphase\]) experimentally. This has been done in an experiment in Manchester on the dynamics of singular vortices in $^3$He-B [@BevanNature; @BevanJLTP]. An equation of the type of (\[d\_perpBphase\]) has been verified in a broad temperature range, which included both extreme limits, $\omega_0\tau \ll 1$ and $\omega_0\tau \gg 1$.
MAGNETIC FIELD from FERMIONIC CHARGE
====================================
A recent scenario of the generation of primordial magnetic fields by Joyce and Shaposhnikov [@JoyceShaposhnikov; @GiovanniniShaposhnikov] is based on an effect, which is the inverse to that discussed in the previous section. The axial anomaly gives rise to a transformation of an excess of chiral particles into a hypermagnetic field. In $^3$He-A language this process describes the collapse of excitation momenta (fermionic charges) towards the formation of textures. These textures are the counterpart of the hypermagnetic field in the Joyce-Shaposhnikov scenario [@NaturePrimordial] (Figs. \[Counterflow\],\[PrimordialField\]). Such a collapse of quasiparticle momentum was recently observed in the rotating cryostat of the Helsinki Low Temperature Laboratory [@Experiment; @NaturePrimordial].
Effective Lagrangian at low $T$
-------------------------------
To study the instability of the superflow and relate it to the problem of magnetogenesis, let us start with the relevant Effective Lagrangian for superfluid dynamics at low $T$ and find the correspondence to the effective Lagrangian for the system of chiral fermions interacting with the magnetic or hypermagnetic field via the axial anomaly conversion process. For the hydrodynamic action in $^3$He-A we shall use the known results collected in the book [@VollhardtWolfle].
Consider a superfluid moving with respect to the walls of container. The normal component of the liquid is clamped by the vessel walls due to its high viscosity, so that the normal velocity ${\bf v}_n=0$ in the reference frame moving with the vessel. If the superfluid velocity ${\bf v}_s$ of the condensate in Eq. (\[v\_s\]) is nonzero in this reference frame, one has a nonzero counterflow of the superfluid and normal components with relative velocity ${\bf w}={\bf v}_s - {\bf v}_n$. This relative velocity provides a nonzero fermionic charge of matter, as will be seen below, and the flow instability leads to the transformation of this charge to the analogue of the hypermagnetic field. Let us choose the axis $z$ along the velocity ${\bf w}$ of the counterflow. In equilibrium the unit orbital vector ${\hat{\bf l}}$ is oriented along the counterflow: ${\hat{\bf l}}_0= {\hat{\bf z}}$. The stability problem is investigated using the quadratic form of the deviations of the superfluid velocity and the ${\hat{\bf l}}$-vector from their equilibrium values: $${\bf {\hat l}} = {\bf {\hat l}}_0 + \delta {\bf {\hat l}}({\bf r},t)
-{1\over 2}{\bf {\hat l}}_0 (\delta {\bf {\hat l}}({\bf r},t))^2~,~ {\bf v}_s =
{\bf w}_0 + \delta {\bf v}_s({\bf r},t)\ .
\label{oscillating-l}$$ The instability of the counterflow towards generation of the inhomogenity $\delta {\bf {\hat l}}({\bf r},t)$, corresponds to the generation of the magnetic field ${\bf B}=p_F{\bf \nabla}\times \delta {\hat {\bf
l}}$ from the chiral fermions.
There are 3 terms in the energy of the liquid, which are relevant for our consideration of stability of superflow at low $T$: $$\begin{aligned}
\nonumber
F={1\over 2}m_3n_s^{ij}v_{si}v_{sj} + C_0 ({\bf v}_s\cdot {\bf {\hat
l}})({\bf {\hat l}}\cdot (\nabla\times{\bf {\hat l}}))\\
+K_b({\bf {\hat
l}}\times (\nabla\times{\bf {\hat l}}))^2
\label{F}\end{aligned}$$ (1) The first term in Eq. (\[F\]) is the kinetic energy of superflow with $n_s^{ij}$ being the anisotropic tensor of superfluid density. At low $T$ one has $$n_s^{ij}\approx n_3 \delta^{ij}- n_{n\parallel}{\hat
l}^{i}{\hat l}^{j}~,~ n_{n\parallel}
\approx { m^*\over 3m_3} p_F^3
{T^2\over \Delta_0^2} ~~.
\label{LowTnormalDensity}$$
\(2) The second term in Eq. (\[F\]) is the anomalous interaction of the superflow with the ${\hat{\bf l}}$-texture, coming from the axial anomaly [@exotic]. The anomaly parameter $C_0$ at $T=0$ is the same as in Eq. (\[C0\]).
\(3) Finally the third term is the relevant part of the energy of ${\hat{\bf l}}$-texture. There are two other terms in the textural energy [@VollhardtWolfle], containing $({\bf {\hat l}}\cdot
(\nabla\times{\bf {\hat l}}))^2$ and $(\nabla\cdot{\bf {\hat l}})^2$, but they are not important for the stability problem: The instability starts when the $z$-dependent disturbances begin to grow. Therefore we are interested only in $z$-dependent ${\hat{\bf l}}$-textures, which in a quadratic approximation contribute the term $({\bf {\hat l}}\times
(\nabla\times{\bf {\hat l}}))^2$. The rigidity $K_b$ at low $T$ is logarithmically divergent $$K_b=
{{p_F^2v_F}\over {24\pi^2\hbar}}~{\rm ln}~ \left ( {\Delta_0^2\over T^2 }~
\right )~~,
\label{Kb}$$ which we shall later relate to the zero charge effect in relativistic theories [@exotic].
There is also a topological connection between ${\bf {\hat l}}$ and ${\bf
v}_s $, since ${\bf v}_s$ in Eq. (\[v\_s\]) represents torsion of the dreibein ${\hat{\bf e}}_1,{\hat{\bf e}}_2,{\hat{\bf e}}_3$ field. This leads to a nonlinear connection, the so-called Mermin-Ho relation [@VollhardtWolfle], which in our geometry gives $$\delta {\bf v}_s = {\hbar\over 2m_3} {\bf {\hat z}}\partial_z \Phi +
{\hbar\over 4m_3} \delta {\bf {\hat l}}\times
\partial_z
\delta{\bf {\hat l}}
\label{MerminHo}$$ The three variables, the potential $\Phi$ of the flow velocity and the two components $\delta {\bf {\hat l}}\perp {\bf {\hat l}}_0$ of the unit vector ${\bf {\hat l}}$, are just another presentation of 3 rotational degrees of freedom of the dreibein ${\hat{\bf e}}_1,{\hat{\bf e}}_2,{\hat{\bf e}}_3$ (the rotation of vectors ${\bf m}=c_\perp {\hat{\bf e}}_1$, ${\bf n}=c_\perp {\hat{\bf e}}_2$ and $\hat l$ in Fig. \[CollectiveModes\]). Whereas $\delta {\bf {\hat l}}$ is responsible for the effective vector potential of the (hyper) magnetic field, the variable $\Phi$ – the angle of rotation of vectors ${\bf m}=c_\perp {\hat{\bf e}}_1$ and ${\bf n}=c_\perp {\hat{\bf e}}_2$ about axis $\hat l$ in Fig. \[CollectiveModes\] – represents an [*axion*]{} field as we shall see later.
Let us expand the energy in terms of small perturbations $\delta\hat{\bf l}$. Adding terms with time derivatives we obtain the following Lagrangian for $\Phi$ and $\delta {\bf {\hat l}}$: $$L=F_0+ L_{\delta {\bf {\hat l}}}+ L_{\Phi} ~.
\label{L}$$ Here $F_0$ is the initial homogeneous flow energy $$F_0={1\over 2}m_3n_3{\bf w}_0^2 - { m^*\over 6m_3} p_F^3
{T^2\over \Delta_0^2}({\bf w}_0 \cdot {\bf {\hat l}}_0)^2 \,,
\label{F0}$$ and $ L_{\delta {\bf {\hat l}}}$ is the textural Lagrangian of order $(\delta\hat{\bf l})^2$: $$\begin{aligned}
L_{\delta {\bf {\hat l}}}={{p_F^2}\over {24\pi^2\hbar v_F}}~{\rm ln} \left (
{\Delta_0^2\over T^2 }~
\right )~ \left[v_F^2(\partial_z \delta{\bf {\hat l}})^2 - (\partial_t
\delta{\bf {\hat l}})^2\right]
\label{Ll1}\\
+ ~{p_F^3 \over 2\pi^2}~({\hat{\bf
l}}_0\cdot{\bf w}_0) (\delta{\hat{\bf l}}\cdot {\bf\nabla}\times
\delta{\hat{\bf
l}})
\label{Ll2}\\
+~{ m^*\over 6} p_F^3
{T^2\over \Delta_0^2}({\bf w}_0 \cdot {\bf {\hat l}}_0)^2~(\delta{\hat{\bf
l}})^2
\label{Ll3}\end{aligned}$$ The first term, Eq.(\[Ll1\]), describes the propagation of textural waves (the so-called orbital waves which play the part of the hyperphoton, see below). The Eq.(\[Ll3\]) gives the mass of the hyperphoton. The term in Eq.(\[Ll2\]) is the Chern-Simons term in action (see below) which is the consequence of the axial anomaly and thus contains the same factor ${p_F^3
\over 2\pi^2}=(3/2)C_0$ as in Eq. (\[MomentoProduction2\]). To obtain this factor from the hydrodynamic action for $^3$He-A one should collect all the relevant terms: (i) The factor $C_0$ comes from Eq. (\[F\]). (ii) The factor $n_3/2$ – from Eq.(\[MerminHo\]). And (iii) the factor $-(n_3-C_0)/2$ – from the intrinsic angular momentum. Altogether they give $C_0 +n_3/2 - (n_3-C_0)/2=(3/2)C_0=p_F^3
/ 2\pi^2$ in Eq. (\[Ll2\]). We do not discuss the problem of intrinsic angular momentum, though it is clearly related to the axial anomaly and spectral flow [@OrbitalMomentum]. Here it is important that the contribution of the intrinsic angular momentum to the hydrodynamic action is [@exotic] $${1 \over 2}~(n_3-C_0)~\left({\hat{\bf
l}}_0\cdot (\delta{\hat{\bf l}}\times (\partial_t +
{\bf w}\cdot {\bf\nabla})
\delta{\hat{\bf l}})\right)
~,
\label{IntrinsicMomentum}$$ and this gives the required factor $-(n_3-C_0)/2$.
$ L_{\Phi}$ is the variation of the Lagrangian for superflow. At low $T$ one has $$\begin{aligned}
L_{\Phi}={\hbar^2\over 8 m_3}n_3 \left[(\partial_z \Phi)^2 -
{1\over s^2}(\partial_t
\Phi)^2 \right]
\label{Lphi1}\\
+ ~{3\hbar\over 4m_3} C_0 ~ \partial_z \Phi~ (\delta{\hat{\bf l}}\cdot
{\bf\nabla}\times
\delta{\hat{\bf l}})
\label{Lphi2}\end{aligned}$$ The first two terms of this Lagrangian, contained in Eq. (\[Lphi1\]), describe the propagation of sound waves (phonons), and $s$ is the speed of sound. We shall later relate the sound waves to axions, because of their coupling with the density of topological charge in Eq. (\[Lphi2\]).
Let us now establish all these correspondences step by step.
Fermionic charge and Chern-Simons energy
----------------------------------------
In the presence of counterflow, ${\bf w} ={\bf v}_s-{\bf
v}_n$, of the motion of the superfluid component of $^3$He-A with respect to the normal fraction, the energy of quasiparticles is Doppler shifted by an amount ${\bf p}\cdot{\bf w}$, which is $\approx \pm p_F({\hat{\bf l}}_0\cdot{\bf
w}_0)$ near the nodes. The counterflow therefore produces an effective chemical potential for the relativistic fermions in the vicinity of both nodes (Fig. \[Counterflow\]): $$\mu_R= - p_F({\hat{\bf l}}_0\cdot{\bf w}_0) ~,~\mu_L=-\mu_R~.
\label{ChemicalPotentials}$$
According to our analogy the relevant fermionic charge of our system, which is anomalously conserved and which corresponds to the number of right fermions, is the momentum of quasiparticles along ${\hat{\bf l}}$ divided by $p_F$. Since the momentum density of quasiparticles is ${\bf P}= - n_{n\parallel} {\bf w}$, the density of the fermionic charge is $${P\over p_F}= - {n_{n\parallel}\over p_F} {\hat{\bf l}}_0\cdot{\bf w}_0 ~.
\label{FermionCharge1}$$ Using Eq. (\[LowTnormalDensity\]) for $n_{n\parallel}$, Eq. (\[SpeedsInAphase\]) and Eq.(\[3speeds\]) for the metric tensor, and Eq. (\[ChemicalPotentials\]) for the chemical potential, one obtains a very simple covariant expression for the density of the fermionic charge $$n_R\equiv{P\over
p_F}={1\over 3} T^2 \mu_R \sqrt{-g}~.
\label{FermionCharge2}$$ Here $g$ is the determinant of the metric tensor $g_{\mu\nu}$ $$\sqrt{-g}={1\over c_\parallel
c_\perp^2}={m^*p_F\over \Delta_0^2}~~.
\label{DetG}$$
Eq. (\[FermionCharge2\]) represents the number density of chiral right-handed massless electrons induced by the chemical potential $\mu_R$ at temperature $T$. This is the starting point of the Joyce-Shaposhnikov scenario of magnetogenesis. It is assumed there that at an early stage of the universe, possibly at the Grand Unification epoch ($10^{-35}$ s after the big bang), an excess of chiral right-handed electrons, $e_R$, is somehow produced due to parity violation.
The equilibrium relativistic energy of the system of right electrons also appears to be completely equivalent to the kinetic energy of the quasiparticles in the counterflow in Eq. (\[F0\]) $$\epsilon_R = {1\over 6}
T^2 \mu_R^2\sqrt{-g} \equiv {1\over 2} m_3 n_{n\parallel}({\bf w}_0 \cdot {\bf
{\hat l}}_0)^2 ~,
\label{FermionEnergy}$$ The difference in the sign between Eqs. (\[F0\]) and (\[FermionEnergy\]) is the usual difference between the thermodynamic potentials at fixed chemical potential and at fixed particle number (fixed velocity and fixed momentum correspondingly).
Due to the “inverse” axial anomaly the leptonic charge (excess of right electrons) can be transferred to the “inhomogeneity” of the vacuum. This inhomogeneity, which absorbs the fermionic charge, arises as a hypermagnetic field configuration. Thus the charge absorbed by the hypermagnetic field, ${\bf\nabla}\times {\bf A}$, can be expressed in terms of its helicity, $$n_R \{{\bf A}\}={1\over 2\pi^2} {\bf A}\cdot ({\bf\nabla}\times {\bf A})~~.
\label{anomaly}$$ The right-hand side is the so called Chern-Simons (or topological) charge of the magnetic field.
When this charge is transformed from the fermions to the hypermagnetic field, the energy stored in the fermionic system decreases. This leads to a energy gain which is equal to the Chern-Simons charge multiplied by the chemical potential: $$F_{CS}= n_R \{{\bf A}\} \mu_R=
{1\over 2\pi^2} \mu_R {\bf A}\cdot ({\bf\nabla}\times {\bf A}) ~~.
\label{csenergy1}$$ The translation to the language of $^3$He-A, according to the dictionary in Fig. (\[PrimordialField\]), gives the following energy change, if the texture is formed from the counterflow, $$F_{CS}= {p_F^3\over 2\pi^2}({\hat{\bf
l}}_0\cdot{\bf w}_0) (\delta{\hat{\bf l}}\cdot {\bf\nabla}\times
\delta{\hat{\bf
l}}) ~~.
\label{csenergy2}$$ This exactly coincides with Eq. (\[Ll2\]).
The Chern-Simons term in Eqs. (\[csenergy1\],\[csenergy2\]) can have arbitrary sign. It is positive if the conterflow is increased and negative if the counterflow is reduced. Thus one can have an energy gain from the transformation of the counterflow (fermionic charge) to the texture (hypermagnetic field). This energy gain is however to be compared with the positive energy terms in Eq. (\[Ll1\]) and Eq. (\[Ll3\]). Let us consider these two terms in more detail.
Maxwell Lagrangian for hypermagnetic and hyperelectric fields.
--------------------------------------------------------------
The Lagrangian for the $\delta{\hat{\bf l}}$-texture in Eq. (\[Ll1\]) is completely equivalent to the conventional Maxwell Lagrangian for the (hyper-) magnetic and electric fields. For example the textural energy, written in covariant form, corresponds to the magnetic energy: $$\begin{aligned}
F_{\rm magn}= {\rm ln}~ \left ( {\Delta_0^2\over T^2 }~ \right )
{{p_F^2v_F}\over {24\pi^2\hbar}}~
(\partial_z \delta \hat {\bf l})^2
\label{magenergy1}\\
~~\equiv
{{\sqrt{-g}}\over {2\gamma^2} }g^{ij}g^{kl}F_{ik}F_{jl}~~.
\label{magenergy2}\end{aligned}$$ Here, $F_{ik}= \nabla_i A_k -\nabla_k A_i $, and $\gamma^2$ is a running coupling constant, which is logarithmically divergent because of vacuum polarization in a complete analogy with the fine structure constant $e^2/4\pi
\hbar c$: $$\gamma^{-2} ={1\over 12\pi^2} {\rm ln}~ \left ({\Delta_0^2\over T^2 }~ \right
) ~.
\label{RunningCoupling}$$ Eq. (\[magenergy2\]) transforms to Eq. (\[magenergy1\]) if one takes into account that in our geometry the “hypermagnetic” field ${\bf B}\perp {\hat{\bf l}}_0$.
The gap amplitude $\Delta_0$, constituting the ultraviolet cut-off in the logarithmically divergent magnetic energy, plays the part of the Planck energy scale. Note that $\Delta_0$ has a parallel with the Planck energy in some other situations, too. For example the analogue of the cosmological constant, which arises in the effective gravity of $^3$He-A, has the value $\Delta_0^4/12\pi^2$ [@Volovik1986].
Mass of hyperphoton
-------------------
The “hyperphoton” in $^3$He-A has a mass. There are several sources of this mass.
\(i) The value of the mass of the “hyperphoton” is seen from Eq. (\[Ll3\]), if it is written in covariant form: $$F_{mass} (T,\mu_R)={1\over 6} \sqrt{-g} g^{ik} A_i A_k {T^2 \mu_R^2\over
\Delta_0^2} ~~.
\label{MassEnergy1}$$ Thus the mass is $$M_{ph}^2={\gamma^2\over 3}~ {T^2 \mu_R^2 \over \Delta_0^2}
~~,
\label{Mass1}$$ In $^3$He-A this mass is physical, though it contains the “Planck” energy cut-off $\Delta_0$: The “hyperphoton mass” is the gap in the spectrum of orbital waves, propagating oscillations of $\delta \hat{\bf l}$, which correspond just to the hyperphoton. This mass appears due to the presence of counterflow, which provides the restoring force for oscillations of $\delta \hat{\bf l}$.
For the relativistic counterpart of $^3$He-A, the Eq. (\[Mass1\]) suggests that the mass of the hyperphoton could arise if both the temperature $T$ and the chemical potential $\mu_R$ are finite. Of course, in the case of exact local $U(1)$ symmetry, the mass of the hyperphoton should be zero. But in an effective theory, the local $U(1)$ symmetry appears only in the low-energy corner and thus is approximate. It can be violated (not spontaneously) at higher energy leading to a nonzero hyperphoton mass which depends on the cut-off parameter. And in fact the mass in Eq. (\[Mass1\]) disappears in the limit of an infinite cut-off parameter or is small, if the cut-off is of Planck scale. The $^3$He-A thus provides an illustration of how the terms of order $(T/E_{\rm Planck})^2$ appear in the effective quantum field theory [@Jegerlehner].
\(ii) In the collisionless regime $\omega\tau \gg 1$, a nonzero mass term is present even in the absence of the counterflow, ${\bf w}=0$. It corresponds to the high-frequency photon mass in the relativistic plasma, calculated by Weldon [@Weldon]: $$M^2_{ph}(\omega\tau \gg 1)={N_F\over 18}~ \gamma^2 T^2.
\label{Mass2rel}$$ Here $N_F$ is the number of fermionic species and $\gamma$ again is the running coupling constant. This can be easily translated to $^3$He-A language, since mass is a covariant quantity. Substituting the running coupling from Eq. (\[RunningCoupling\]) and taking into account that the number of the fermionic species in $^3$He-A is $N_F=N_{FR} +N_{FL}= 2$, one obtains the gap in the spectrum of the high-frequency orbital waves (called also the normal flapping mode) $$M^2_{\rm orb~waves}(\omega\tau \gg 1)={4\pi^2\over 3} {T^2 \over \ln
(\Delta_0^2/T^2)}~.
\label{Mass2Aphase}$$ This coincides with Eq. (11.76b) of Ref.[@VollhardtWolfle] for the normal flapping mode. Note that in $^3$He-A this gap in the spectrum, corresponding to the relativistic plasma oscillations, was obtained by Wölfle already in 1975 [@Wolfle].
The corresponding mass term in the Lagrangian for the gauge bosons is $$F_{mass} (T,\omega\tau \gg 1)= {N_F\over 36} T^2 \sqrt{-g}
g^{ik}A_iA_k~~.
\label{MassEnergy2}$$ which is valid both for the proper relativistic theory with chiral fermions and for $^3$He-A, where $N_F=2$.
\(iii) There is also the topological mass of the “photon” in $^3$He-A, which comes from the axial anomaly and intrinsic angular momentum [@exotic; @Volovik1975; @LeggettTakagi]. It is rather small.
\(iv) The tiny mass coming from the spin-orbital interaction in $^3$He-A [@LeggettTakagi] is described by the energy term [@VollhardtWolfle] $$-g_D(\hat {\bf l}\cdot \hat {\bf d})^2~,
\label{SpinOrbit}$$ where $\hat {\bf d}$ is the unit vector of the spontaneous anisotropy in spin space. This term has no counterpart in relativistic theories but is important in NMR experiments on $^3$He-A (see below).
Here we discussed how the “photon” mass in $^3$He-A is influenced by variuos external and internal factors: counterflow (chemical potential), temperature, anomaly, spin-orbital interaction, Planck cut-off parameter. These factors also influence the speed of “light” in $^3$He-A and this occurs essentially in the same manner as in relativistic theories (see [@DittrichGies] for references on the modification of the speed of light by electromagnetic fields, temperature, gravitational background, and other external environments). The only difference is that in $^3$He-A the environment modifies the Planck cut-off parameter as well [@GravitationalConstant], which gives an extra dependence of the “photon” mass and the speed of “light” on the environment.
Instability towards magnetogenesis
----------------------------------
For us the most important property of the axial anomaly term in Eqs. (\[csenergy1\],\[csenergy2\]) is that it is linear in the derivatives of $\delta {\bf {\hat l}}$. Its sign thus can be negative, while its magnitude can exceed the positive quadratic term in eq. (\[magenergy1\]). This leads to the helical instability towards formation of the inhomogeneous $\delta {\bf {\hat
l}}$-field. During this instability the kinetic energy of the quasiparticles in the counterflow (analogue of the energy stored in the fermionic degrees of freedom) is converted into the energy of the inhomogeneity $ {\nabla}_z
\delta{\hat{\bf l}}$, which is the analogue of the magnetic energy of the hypercharge field.
This instability can be found by investigation of the eigenvalues of the quadratic form describing the energy in terms of ${\bf
A}=p_F\delta{\hat{\bf l}}$ in Eq.(\[Ll1\]-\[Ll3\]). Using the covariant form of this equation one obtains the following $2\times
2$ matrix for two components of the vector potential, $A_x=A_{x0}e^{iqz}$ and $A_y=A_{y0}e^{iqz}$ : $$\left( \matrix
{M_{ph}^2 + c_\parallel^2q^2
& {\gamma^{2}\over 2\pi^2}\mu_R c_\parallel q \cr {\gamma^{2}\over 2\pi^2}\mu_R
c_\parallel q & M_{ph}^2 + c_\parallel^2q^2 \cr }
\right) ~~.
\label{Matrix}$$
This matrix is applied both to the Joyce-Shaposhnikov scenario and to the instability of the $^3$He-A superflow. This is one of the rare cases when the equation of motion for the $\hat {\bf l}$-vector reduces to relativistic (Maxwell + Chern-Simons) equations. This stems from the fact that for the investigation of the stability one needs an energy which is quadratic in terms of the small deviations of the vector potential (vector $\hat {\bf l}$) from the uniform background. In our geometry: (i) The equilibrium unit vector $\hat {\bf l}_0$ is oriented in one direction (along the velocity), which means that the background metric is constant in space. (ii) Small deviations $\delta \hat {\bf
l} \equiv {\bf A}/p_F$ of the vector $\hat {\bf l}$ from equilibrium are perpendicular to the flow, while the relevant coordinate dependence (i.e. that which leads to instability) is the $z$-dependence along the flow. Thus there are no derivatives in $x$ and $y$ in the relevant Lagrangian, while ${\bf A}$ contains only the transverse components. (iii) The Lagrangian is quadratic in the gauge field ${\bf A}\equiv p_F \delta \hat {\bf l}$, while the metric enters only as a constant (though anisotropic) background. All these facts conspire to produce a complete analogy with the relativistic theory. Such a geometry, in which the analogy is exact, is really unique, and it might be called a miracle that it indeed does occur in a real experimental situation.
The quadratic form in Eq. (\[Matrix\]) becomes negative if $${\mu_R\over M_{ph}}> {4\pi^2\over \gamma^2 } ~~.
\label{StabilityCondition1}$$ Inserting the photon mass from Eq. (\[Mass1\]), one finds that the uniform counterflow becomes unstable towards the nucleation of the texture if $${T\over \Delta_0} \ln^{1/2} \left({\Delta_0^2\over T^2}\right)< {3\over
2\pi } ~~.
\label{StabilityCondition2}$$ If this condition is fulfilled, the instability occurs for any value of the counterflow (any value of the chemical potential $\mu_R$ of right electrons).
In relativistic theories, where $\Delta_0$ is the Planck energy, this condition is always fulfilled. Thus the excess of the fermionic charge is always unstable towards nucleation of the hypermagnetic field. In the scenario of the magnetogenesis developed by Joyce and Shaposhnikov [@JoyceShaposhnikov; @GiovanniniShaposhnikov], this instability is responsible for the genesis of the hypermagnetic field well above the electroweak transition. The role of the subsequent electroweak transition is to transform this hypermagnetic field to the conventional (electromagnetic $U(1)$) magnetic field due to the electroweak symmetry breaking.
In $^3$He-A the Eq.(\[StabilityCondition2\]) shows that the instability always occurs if the temperature is low enough compared to $\Delta_0$ ( $\Delta_0 \sim 2 T_c$. What happens at $T \sim T_c$ is not clear from Eq.(\[StabilityCondition2\]), since our analysis works only in the limit $T\ll
\Delta_0$. So, the rigorous theory is required, which holds at any $T$. The helical instability in $^3$He-A has been intensively discussed theoretically (see, e.g., [@ThesisVollhardt]). According to a rigorous theory, which takes into account the Fermi-liquid parameters, the counterflow is unstable at any $T$ if the spin-orbital coupling in Eq. (\[SpinOrbit\]) is neglected, i.e. $g_D=0$, but is stable at $T$ above about $0.8 T_c$ if the spin-orbital coupling is taken into account and the stiffness of the spin vector $\hat
{\bf d}$ suppresses the instability (see Sec. 7.10.1 in the book [@VollhardtWolfle]). The result of the helical instabilty can be either the formation of the helical texture with small opening angle or the complete collapse of the counterflow. In the first case only some part of the counterflow momentum transforms to the momentum of the helix. In the second case the collapse of the counterflow leads to the formation of continuous ATC vortices and thus the whole counterflow momentum is transformed to the momentum carried by the vortex texture. Experimentally the second scenario, with formation of vortices, is realized [@Experiment].
“Magnetogenesis” in $^3$He-A
----------------------------
In various experiments [@Experiment; @BigBangNature1; @NaturePrimordial] the flow instability has been measured using NMR techniques, which means that one needs an external (real) magnetic field ${\bf H}$. Such a field adds an additional mass to the “hypercharge gauge field” ${\bf A}$ due to the spin-orbital interaction in Eq. (\[SpinOrbit\]). Even at low $T$ the instability then occurs only above some critical value of the counterflow velocity $w_0$ (or correspondingly chemical potential of right electrons $\mu_R$). The critical value $\mu_{R}^{cr}$ depends on $T$ and $H$ and approaches the value of order $p_F\sqrt{g_D/\rho_s}$ in the limit of large $H$.
When this helical instability develops in $^3$He-A, the final result is the formation of the ${\hat{\bf l}}$-texture which corresponds to the free energy minimum in the rotating vessel. This is the periodic ${\hat{\bf
l}}$-texture, whose elementary cell represents the Anderson-Toulouse-Chechetkin (ATC) continuous vortex in Fig. \[ContinuousMomentogenesis\]. The presence of ATC vortices and their number is extracted from the NMR absorption spectrum, which contains the satellite peaks coming from different types of vortices [@PhaseDiagram]. The position of the satellite peak indicates the type of vortex, while the intensity is proportional to the number of vortices of this type. The satellite peak for the ATC vortices is shown in Fig. \[experiment\].
In the experiment carried out in Helsinki the initial state did not contain vortices. Then the vessel was put into rotation with some angular velocity $\Omega$. If the velocity is small enough, one has only counterflow and no vortex texture. This means that there is a nonzero “chemical potential of right electrons”, $\mu_R=p_F\Omega r$, where $r$ is the distance from the axis of the rotating vessel, while the “hypermagnetic” field is absent. Accelerating the vessel further one finally reaches the critical value $\mu_{R}^{cr}$ at the wall of container, $r=R$, where the counterflow is maximal. At this moment the instability occurs, which is observed by the Helsinki group as a jump in the height of the vortex peak (see Fig. \[experiment2\]). The peak height jumps from zero to the magnitude corresponding to a vortex array with nearly the equilibrium number of vortex lines. This means that counterflow has been essentially removed. The counterflow (which carried the fermionic charge of matter) has thus been converted to a vortex ${\hat{\bf l}}$-texture (hypermagnetic field).
The magnitude of $\mu_R^{cr}$ found from experiments [@Experiment] is in good quantitative agreement with the theoretical estimation of the mass of the “hyperphoton” determined by the spin-orbit interaction in Eq. (\[SpinOrbit\]): $\mu_{R}^{cr} \sim
p_F\sqrt{g_D/\rho_s}$. Thus the Helsinki experiments model the nucleation of the hypermagnetic field for different masses of the “hyperphoton”. The flow instability in the limit when the contribution to the “hyperphoton” mass from the real external magnetic field $H$ is zero has also been investigated: First the field $H$ was turned off and after the instability had occurred the field was switched on again and the created “hypermagnetic field” was measured. In this case it was observed that $\mu_R^{cr}$ was significantly reduced.
AXION in $^3$He-A
=================
We discussed how the quasiparticles and the ${\hat{\bf l}}$-texture can exchange the fermionic charge – the linear momentum – due to the axial anomaly. There is yet another phenomenon: The ${\hat{\bf l}}$-texture and the moving superfluid vacuum can also exchange momentum. Thus the ${\hat{\bf l}}$-texture serves as an intermediate object which allows to transfer the fermionic charge from the condensate (vacuum) motion to the quasiparticles (matter), as was discussed in Sec.IIIA. In this sense the ${\hat{\bf l}}$-texture plays the same role as quantized vortices in superfluids and superconductors. This again shows the common properties of continuous ${\hat{\bf l}}$-textures (with continuous vorticity) and quantized singular vortices, which are related to the gap nodes: In the ${\hat{\bf l}}$-textures the gap nodes are lying in momentum space, while in the most symmetric quantized vortices of conventional superconductors and also in the most symmetric cosmic strings the nodes are in real space – in the cores of vortices, where the symmetry is restored and fermions are massless. The transformation between the real-space zeroes and the momentum-space zeros [@Zeroes1] actually occurs when the singular core of the vortex experiences an additional symmetry breaking, as was observed for the $^3$He-B vortices. The relation of both types of zeroes to the axial anomaly was discussed in [@Zeroes2].
Consider now the exchange between the superfluid vacuum and the texture. The momentum density of the superfluid vacuum along the equilibrium ${\hat{\bf l}}_0$-vector is $m_3n_3v_{sz}$. The momentum exchange follows from the anomalous nonconservation of the momentum Eq. (\[MomentoProduction2\]): $$m_3n_3(\partial_t v_{sz} + \partial_z \mu_3)= {p_F\over 2\pi^2}
\left(\partial_t{\bf A}\cdot {\bf\nabla}\times{\bf A}\right)~.
\label{CondensateMomentumNonconservation}$$ where $\mu_3$ is the real chemical potential of $^3$He atoms, which also determines the speed of sound in $^3$He-A: $s^2=n_3 d\mu_3/dn_3$; ${\bf A}=p_F
\delta {\hat{\bf l}}$. In what follows, we consider only the $z$- and $t$-dependence of all variables.
Let’s introduce a variable $\theta$ which is dual to the potential $\Phi$ of the superflow: $$\partial_t\theta= -{p_F\over 2m_3}\partial_z \Phi=- p_F
v_{sz}~,
\label{AxionField1}$$ $$\partial_z\theta= -{p_F\over 2m_3s^2}\partial_t \Phi= {p_F\over
s^2}\delta\mu_3 ~,
\label{AxionField2}$$ We can now write down the Lagrangian whose variation gives rise to the anomalous nonconservation of the condensate momentum in Eq. (\[CondensateMomentumNonconservation\]): $${1\over 2\pi^2} \theta \left(\partial_t\vec A\cdot \vec\nabla\times\vec
A\right) + {n_3m_3\over 2 p_F^2} \left( s^2 (\partial_z\theta)^2-
(\partial_t\theta)^2\right)~.
\label{AxionAction}$$ This is nothing but the action for the axion field $\theta$, which interacts with the CP violating combination $F^{\mu\nu}\tilde F_{\mu\nu} \propto {\bf
E}\cdot {\bf B}$ [@Axions]. The Joice-Shaposhnikov scenario of the exponential growth of magnetic field can be also realized if instead of the excess of the right electrons one has the time dependent axionic field [@CarollField]. The role of the chemical potential is now played by $\partial_t \theta$. In our case of superflow this again corresponds to the superfluid velocity according to Eq.(\[AxionField1\]).
In $^3$He-A the axion corresponds to sound waves – propagating oscillations of two conjugated variables, the phase $\Phi$, related to rotations of the fundamental triad, and the particle density $n_3$. The anomalous first term in Eq. (\[AxionAction\]) can be also obtained from Eq. (\[Lphi2\]). The speed of sound is $s^2=(1/3)v_F^2(1+F_0)(1+F_1/3)$, where $F_0$ and $F_1$ are Fermi-liquid parameters, and is the same in superfluid $^3$He-A ($T<T_c$) and in normal liquid $^3$He ($T>T_c$). In distinction from the orbital waves (“electromagnetic waves”), the speed of sound $s$ is isotropic and does not coincide with any of the two speeds of light, $c_\perp$ or $c_\parallel$, though one can expect that the axion propagating along $z$ must have the parallel speed of “light” $c_\parallel=v_F$. What is the reason? It is a property of the superfluid $^3$He-A:
Both modes, “photon” (orbital wave) and “axion” (sound wave) are collective bosonic excitations of the fermionic system and are obtained by integration over the fermions. In the case of “photons” the relevant region of the integration over the fermions is concentrated close to the gap nodes due to the logarithmic divergency. Near the nodes the fermions are relativistic and are described by the Lorentzian metric $g^{\mu\nu}$. It follows that the effective “photons” are described by the same metric and therefore the speed of light is the same as the speed of the massless fermion propagating in the same direction. On the other hand, the relevant region of the integration, which is responsible for the spectrum of the “axion” mode, is far from the gap nodes. Consequently the axion spectrum does not even depend on the existence of the gap nodes and induces its own effective metric.
DISCUSSION
==========
In principle one can introduce a model system with favourable parameters, such that for all collective modes the integration over the fermions is concentrated mostly in the region where the fermions are Lorentzian. In this case the low energy dynamics of photons, axions, gravitons, etc., will be determined by the same Lorentzian metric as that of the fermions. In the low-energy corner, one then obtains the effective relativistic quantum field theory and effective quantum gravity with the same speed of light for all bosons and fermions.
It is quite possible that in this ideal case the cosmological constant vanishes. This follows from the fact that Eq. (\[E\^2relativistic\]) for the spectrum of massless quasiparticles can be multiplied by an arbitrary scaling factor $a^2$, which does not change the energy spectrum, but changes the contravariant metric tensor: $g^{\mu\nu} \rightarrow a^2
g^{\mu\nu}$. Since physics cannot depend on such formal conformal transformation, the effective low-energy Lagrangian for gravity cannot depend on $a^2$ and thus the cosmological term $\int d^3xdt ~\Lambda\sqrt{-g}$ is prohibited (a discussion of the role of the scale invariance for vanishing cosmological constant is found in Ref.[@AdlerCosmConstant]).
This situation is somewhat similar to that which occurs in the normal Fermi-liquid where the role of the parameter $a^{-1}$ is played by the quasiparticle spectral weight $Z$ – the residue of the Green’s function at the quasiparticle pole. The low-energy properties of this system, described by the Landau phenomenological Fermi-liquid theory, do not depend on $Z$. The Landau Fermi-liquid differs from our system only in the topology of the spectrum of the low-lying fermionic excitations: The Fermi-surface instead of the Fermi-points – the gap nodes, is present there.
Note that the Fermi-surface and the point node are the only topologically stable features of the fermionic spectrum. They are described by $\pi_1$ and $\pi_3$ topological invariants respectively and thus are robust to any modification of the system. These two classes exhaust the topologically stable gapless Fermi systems. In Landau theory, which deals with the Fermi-surface class of Fermi liquids, the low-energy bosonic collective modes are related to the dynamical deformations of the Fermi surface. In the point-node class of Fermi liquids, the corresponding collective motion comes from the dynamics of the nodes. This dynamics gives rise to effective gravity and effective electromagnetic fields.
The fundamental constants in these effective theories are determined by the position of the node, $p_F$, and by the slopes of the energy $E$ of the quasiparticle as a function of its momentum ${\bf p}$ at the node. There are three such parameters in $^3$He-A: the Fermi velocity $v_F$, the Fermi momentum $p_F$ and the gap amplitude $\Delta_0$. They give the parallel speed of light $c_\parallel=v_F$; the transverse speed of light $c_\perp=\Delta_0/ p_F$; the Planck energy $\Delta_0$; the running coupling constant in Eq. (\[RunningCoupling\]); the masses of the hyperphoton in Eqs. (\[Mass1\],\[Mass2rel\]); gravitational constant $G\sim
\Delta_0^{-2}$ [@GravitationalConstant] and cosmological constant $\sim \Delta_0^4$ [@Volovik1986]; etc. Since all 3 initial parameters are in principle temperature dependent, the fundamental constants are not constants in the effective theories. For example the speed of light depends on temperature and also on the photon energy: $\delta c/c \sim
(E/E_{\rm Planck})^2$. The larger (linear) effect, $\delta c/c \sim
E/E_{\rm Planck} $, was discussed in [@Amelino].
We discussed only 3 experiments in superfluid $^3$He-A related to the properties of the electroweak vacuum. In all of them the chiral anomaly is an important mechanism. It regulates the nucleation of the fermionic charge from the vacuum, as observed in Manchester [@BevanNature], and the inverse process of the nucleation of the effective magnetic field from the fermion current, as observed in Helsinki [@Experiment; @NaturePrimordial].
There are many other connections between superfluid $^3$He and different branches of physics which should be explored. For example, we can simulate phenomena related to the effective gravity, such as the cosmological constant, quantum properties of the event horizon, vacuum instability in strong gravity, torsion strings and even inflation. In principle a nonequilibrium vacuum state can be constructed in which the speed of light $c_\perp=\Delta_0/p_F$ decreases exponentially with time. In cosmological language this implies inflation, since the length scale in the spatial metric $g_{ik}$ is growing exponentially. This would allow for a study of the development of perturbations during inflation.
Till now we considered the properites related to one pair of nodes only. If one takes into account that in $^3$He-A there is a two-fold degeneracy related to two spin projection of the $^3$He atom, the number of fermiomic and bosonic degrees of freedom increases. It appears that with these new degrees of freedom, the system transforms to a $SU(2)$ gauge theory: The conventional spin degrees of freedom of $^3$He atoms form the $SU(2)$ isospin, while some collective modes of the order parameter (the spin-orbital waves) behave as $SU(2)$ gauge bosons [@exotic].
There are several ways of extending the model, in which higher local and global symmetry groups can naturally arise. (1) One can imagine an initial normal state of condensed matter consisting of $n=3,4,$ etc. degenerate sheets of the Fermi-surface. Then the superconducting/superfluid Cooper pairing will lead to $n$-fold degeneracy of gap nodes, which in turn gives rise to the effective local $SU(n)$ group in the low-energy corner. (2) The number of gap nodes on each Fermi-surface can be also larger than 2. For example the so called $\alpha$-state of $^3$He [@VollhardtWolfle] contains 8 gap nodes per Fermi-surface and thus 8 elementary relativistic fermions in the vicinity of the nodes. The fluctuations of positions of these nodes are equivalent to several gauge fields. In high-T$_c$ superconductivity each Fermi-sheet (actually the Fermi-circle since this kind of superconductivity effectively occurs in the two-dimensional space of the CuO plane) contains 4 gap nodes. The corresponding Weyl-like Hamiltonian for 4 fermions and the corresponding gauge fields have been discussed for this material in Ref.[@Nersesyan]. Thus in principle it appears possible to construct a model which has as many fermionic and bosonic degrees of freedom as needed in Grand Unified Theories, including the different generations of fermions. Of course, the construction of a suitable condensed matter system corresponding to such a model is not a simple undertaking.
Acknowledgements
================
I thank Uwe Fischer, Henry Hall, John Hook, Ted Jacobson, Michael Joyce, Tom Kibble, Matti Krusius, Robert Laughlin, Misha Shaposhnikov, Alexei Starobinsky, Tanmay Vachaspati and Shuqian Ying for fruitful discussions.
F. Wilczek, “The Future of Particle Physics as a Natural Science”, hep-ph/9702371.
H.B. Nielsen, “Field theories without fundamental (gauge) symmetries”, preprint NBI-HE-83-41.
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abstract: 'Let $G$ be a real reductive Lie group and $H$ a closed reductive subgroup of $G$. We investigate the deformation of “standard” compact quotients of $G/H$, *i.e.*, of quotients of $G/H$ by discrete subgroups $\Gamma$ of $G$ that are uniform lattices in a closed reductive subgroup $L$ of $G$ acting properly and cocompactly on $G/H$. For $L$ of real rank $1$, we prove that after a small deformation in $G$, such a group $\Gamma$ remains discrete in $G$ and its action on $G/H$ remains properly discontinuous and cocompact. More generally, we prove that the properness of the action of any convex cocompact subgroup of $L$ on $G/H$ is preserved under small deformations, and we extend this result to reductive homogeneous spaces $G/H$ over any local field. As an application, we obtain compact quotients of ${\mathrm{SO}}(2n,2)/{\mathrm{U}}(n,1)$ by Zariski-dense discrete subgroups of ${\mathrm{SO}}(2n,2)$ acting properly discontinuously.'
address: 'Département de Mathématiques, Bâtiment 425, Faculté des Sciences d’Orsay, Université Paris-Sud 11, 91405 Orsay Cedex, France'
author:
- Fanny Kassel
title: Deformation of proper actions on reductive homogeneous spaces
---
Introduction
============
Let $G$ be a real connected reductive linear Lie group and $H$ a closed connected reductive subgroup of $G$. We are interested in the compact quotients of $G/H$ by discrete subgroups $\Gamma$ of $G$. We ask that the action of $\Gamma$ on $G/H$ be properly discontinuous in order for the quotient $\Gamma\backslash G/H$ to be Hausdorff. This imposes strong restrictions on $\Gamma$ when $H$ is noncompact. For instance, if $\mathrm{rank}_{{\mathbb{R}}}(G)=\mathrm{rank}_{{\mathbb{R}}}(H)$, then all discrete subgroups of $G$ acting properly discontinuously on $G/H$ are finite: this is the Calabi-Markus phenomenon [@kob89]. Usually the action of $\Gamma$ on $G/H$ is also required to be free, so that $\Gamma\backslash G/H$ be a manifold, but this condition is not very restrictive: if $\Gamma$ acts properly discontinuously and cocompactly on $G/H$, then it is finitely generated, hence virtually torsion-free by Selberg’s lemma [@sel].
In this paper we investigate the deformation of compact quotients $\Gamma\backslash G/H$ in the important case when $\Gamma$ is “standard”, *i.e.*, when $\Gamma$ is a uniform lattice in some closed reductive subgroup $L$ of $G$ acting properly and cocompactly on $G/H$. Most of our results hold for reductive homogeneous spaces over any local field, but in this introduction we first consider the real case.
Deformation of compact quotients in the real case
-------------------------------------------------
Let $G$ be a real reductive linear Lie group and $H$ a closed reductive subgroup of $G$. In all known examples, if $G/H$ admits a compact quotient, then there is a closed reductive subgroup $L$ of $G$ that acts properly and cocompactly on $G/H$. For instance, $L={\mathrm{U}}(n,1)$ acts properly and transitively on the $(2n+1)$-dimensional anti-de Sitter space $G/H={\mathrm{SO}}(2n,2)/{\mathrm{SO}}(2n,1)$ (see Section \[quotients compacts Zariski-denses\]). Any torsion-free uniform lattice $\Gamma$ of such a group $L$ acts properly discontinuously, freely, and cocompactly on $G/H$; we will say that the corresponding compact quotient $\Gamma\backslash G/H$ is *standard*. Note that $L$ always admits torsion-free uniform lattices by [@bor63]. Kobayashi and Yoshino conjectured that any reductive homogeneous space $G/H$ admitting compact quotients admits standard ones ([@ky], Conj. 3.3.10); this conjecture remains open.
Of course, nonstandard compact quotients may also exist: this is the case for instance for $G/H=(G_0\times G_0)/\Delta_{G_0}$, where $G_0$ is any reductive Lie group and $\Delta_{G_0}$ is the diagonal of $G_0\times G_0$ (see [@ghy], [@gol], [@kob98], [@sal00]). But in general we know only standard examples. In order to construct nonstandard ones, it is natural, given a reductive subgroup $L$ of $G$ acting properly and cocompactly on $G/H$, to slightly deform torsion-free uniform lattices $\Gamma$ of $L$ in $G$ and to see whether they remain discrete in $G$ and their action on $G/H$ remains proper, free, and cocompact.
Since our goal is to obtain nonstandard quotients, we are not really interested in trivial deformations of $\Gamma$, *i.e.*, in deformations by conjugation, for which the quotient $\Gamma\backslash G/H$ will remain standard. The problem therefore boils down to the case when $L$ is reductive of real rank $1$. Indeed, by classical preliminary reductions, that is, considering separately the different irreducible quasifactors of $\Gamma$ and possibly passing to subgroups of finite index, we may assume that $\Gamma$ is irreducible and that $L$ has no compact factor. If $L$ is semisimple of real rank $\geq 2$, then Margulis’s superrigidity theorem implies that $\Gamma$ is locally rigid in $G$ ([@mar91], Cor. IX.5.9).
We prove the following result.
\[proprete, groupes de Lie\] Let $G$ be a real reductive linear Lie group, $H$ and $L$ two closed reductive subgroups of $G$. Assume that $\mathrm{rank}_{{\mathbb{R}}}(L)=1$ and that $L$ acts properly and cocompactly on $G/H$. For any torsion-free uniform lattice $\Gamma$ of $L$, there is a neighborhood $\mathcal{U}\subset{\mathrm{Hom}}(\Gamma,G)$ of the natural inclusion such that any $\varphi\in\mathcal{U}$ is injective and $\varphi(\Gamma)$ is discrete in $G$, acting properly discontinuously and cocompactly on $G/H$.
We denote by ${\mathrm{Hom}}(\Gamma,G)$ the set of group homomorphisms from $\Gamma$ to $G$, endowed with the compact-open topology. In the real case, the fact that $\varphi(\Gamma)$ remains discrete in $G$ for $\varphi\in{\mathrm{Hom}}(\Gamma,G)$ close to the natural inclusion is a general result of Guichard ([@gui], Th. 2).
Theorem \[proprete, groupes de Lie\] improves a result of Kobayashi ([@kob98], Th. 2.4), who considered particular homomorphisms of the form $\gamma\mapsto\gamma\psi(\gamma)$, where $\psi : \Gamma\rightarrow\nolinebreak Z_G(L)$ is a homomorphism with values in the centralizer of $L$ in $G$.
By [@ky], Cor. 3.3.7, Theorem \[proprete, groupes de Lie\] applies to the following triples $(G,H,L)$:
1. $({\mathrm{SO}}(2n,2),{\mathrm{SO}}(2n,1),{\mathrm{U}}(n,1))$ for $n\geq 1$,
2. $({\mathrm{SO}}(2n,2),{\mathrm{U}}(n,1),{\mathrm{SO}}(2n,1))$ for $n\geq 1$,
3. $({\mathrm{U}}(2n,2),{\mathrm{Sp}}(n,1),{\mathrm{U}}(2n,1))$ for $n\geq 1$,
4. $({\mathrm{SO}}(8,8),{\mathrm{SO}}(8,7),\mathrm{Spin}(8,1))$,
5. $({\mathrm{SO}}(8,{\mathbb{C}}),{\mathrm{SO}}(7,{\mathbb{C}}),\mathrm{Spin}(7,1))$,
6. $({\mathrm{SO}}^{\ast}(8),{\mathrm{U}}(1,3),\mathrm{Spin}(6,1))$,
7. $({\mathrm{SO}}^{\ast}(8),\mathrm{Spin}(6,1),{\mathrm{U}}(1,3))$,
8. $({\mathrm{SO}}^{\ast}(8),{\mathrm{SO}}^{\ast}(6)\times{\mathrm{SO}}^{\ast}(2),\mathrm{Spin}(6,1))$,
9. $({\mathrm{SO}}(4,4),\mathrm{Spin}(4,3),{\mathrm{SO}}(4,1))$,
10. $({\mathrm{SO}}(4,3),\mathrm{G}_{2(2)},{\mathrm{SO}}(4,1))$.
As mentioned above, our aim is to deform standard compact quotients of $G/H$ into nonstandard ones, which are in some sense more generic. The best that we may hope for is to obtain Zariski-dense discrete subgroups of $G$ acting properly discontinuously, freely, and cocompactly on $G/H$. Of course, even when $L$ has real rank $1$, nontrivial deformations in $G$ of uniform lattices $\Gamma$ of $L$ do not always exist. For instance, if $L$ is semisimple, noncompact, with no quasisimple factor locally isomorphic to ${\mathrm{SO}}(n,1)$ or ${\mathrm{SU}}(n,1)$, then the first cohomology group $H^1(\Gamma,{\mathfrak{g}})$ vanishes by [@rag], Th. 1. This, together with [@wei64], implies that $\Gamma$ is locally rigid in $G$. (Here ${\mathfrak{g}}$ denotes the Lie algebra of $G$.)
For $(G,H,L)=({\mathrm{SO}}(2n,2),{\mathrm{SO}}(2n,1),{\mathrm{U}}(n,1))$ with $n\geq 2$, uniform lattices $\Gamma$ of $L$ are not locally rigid in $G$, but a small deformation of $\Gamma$ will never provide a Zariski-dense subgroup of $G$. Indeed, by [@rag] and [@wei64] there is a neighborhood in ${\mathrm{Hom}}(\Gamma,G)$ of the natural inclusion whose elements are all homomorphisms of the form $\gamma\mapsto\nolinebreak\gamma\psi(\gamma)$, where $\psi : \Gamma\rightarrow{\mathrm{SO}}(2n,2)$ is a homomorphism with values in the center of ${\mathrm{U}}(n,1)$.
On the other hand, for $(G,H,L)=({\mathrm{SO}}(2n,2),{\mathrm{U}}(n,1),{\mathrm{SO}}(2n,1))$ with $n\geq 1$, there do exist small Zariski-dense deformations of certain uniform lattices of $L$ in $G$ (see Section \[quotients compacts Zariski-denses\]): such deformations can be obtained by a bending construction due to Johnson and Millson [@jm]. Theorem \[proprete, groupes de Lie\] therefore implies the following result on the compact quotients of the homogeneous space $G/H={\mathrm{SO}}(2n,2)/{\mathrm{U}}(n,1)$.
\[quotients compacts de SO(2n,2)/SU(n,1)\] For any $n\geq 1$, there is a Zariski-dense discrete subgroup of ${\mathrm{SO}}(2n,2)$ acting properly discontinuously, freely, and cocompactly on ${\mathrm{SO}}(2n,2)/{\mathrm{U}}(n,1)$.
Note that by [@ky], Prop. 3.2.7, the homogeneous space ${\mathrm{SO}}(2n,2)/{\mathrm{U}}(n,1)$ is a pseudo-Riemannian symmetric space of signature $(2n,n^2-1)$.
The existence of compact quotients of reductive homogeneous spaces by Zariski-dense discrete subgroups was known so far only for homogeneous spaces of the form $(G_0\times G_0)/\Delta_{G_0}$.
Deformation of properly discontinuous actions over a general local field
------------------------------------------------------------------------
We prove that the properness of the action is preserved under small deformations not only for real groups, but more generally for algebraic groups over any local field ${\mathbf{k}}$. By a local field we mean ${\mathbb{R}}$, ${\mathbb{C}}$, a finite extension of ${\mathbb{Q}}_p$, or the field ${\mathbb{F}}_q((t))$ of formal Laurent series over a finite field ${\mathbb{F}}_q$. Moreover we relax the assumption that $\Gamma$ is a torsion-free uniform lattice of $L$, in the following way.
\[proprete, groupes algebriques\] Let ${\mathbf{k}}$ be a local field, $G$ the set of ${\mathbf{k}}$-points of a reductive algebraic ${\mathbf{k}}$-group $\mathbf{G}$, and $H$ (resp. $L$) the set of ${\mathbf{k}}$-points of a closed reductive subgroup $\mathbf{H}$ (resp. $\mathbf{L}$) of $\mathbf{G}$. Assume that $\mathrm{rank}_{{\mathbf{k}}}(\mathbf{L})=1$ and that $L$ acts properly on $G/H$. If ${\mathbf{k}}={\mathbb{R}}$ or ${\mathbb{C}}$, let $\Gamma$ be a torsion-free convex cocompact subgroup of $L$; if ${\mathbf{k}}$ is non-Archimedean, let $\Gamma$ be any torsion-free finitely generated discrete subgroup of $L$. Then there is a neighborhood $\mathcal{U}\subset{\mathrm{Hom}}(\Gamma,G)$ of the natural inclusion such that any $\varphi\in\mathcal{U}$ is injective and $\varphi(\Gamma)$ is discrete in $G$, acting properly discontinuously on $G/H$.
Recall that for ${\mathbf{k}}={\mathbb{R}}$ or ${\mathbb{C}}$, a discrete subgroup of $L$ is called *convex cocompact* if it acts cocompactly on the convex hull of its limit set in the symmetric space of $L$. In particular, any uniform lattice of $L$ is convex cocompact.
For ${\mathbf{k}}={\mathbb{R}}$ or ${\mathbb{C}}$, Theorem \[proprete, groupes de Lie\] follows from Theorem \[proprete, groupes algebriques\] and from a cohomological argument due to Kobayashi (see Section \[Proprete and deformations\]). This argument does not transpose to the non-Archimedean case.
Note that in characteristic zero, every finitely generated subgroup of $L$ is virtually torsion-free by Selberg’s lemma ([@sel], Lem. 8), hence the “torsion-free” assumption in Theorem \[proprete, groupes algebriques\] may easily be removed in this case.
Translation in terms of a Cartan projection
-------------------------------------------
Let ${\mathbf{k}}$ be a local field and $G$ the set of ${\mathbf{k}}$-points of a connected reductive algebraic ${\mathbf{k}}$-group. Fix a Cartan projection $\mu : G\rightarrow E^+$ of $G$, where $E^+$ is a closed convex cone in a real finite-dimensional vector space $E$ (see Section \[Preliminaires\]). For any closed subgroup $H$ of $G$, the *properness criterion* of Benoist ([@ben96], Cor. 5.2) and Kobayashi ([@kob96], Th. 1.1) translates the properness of the action on $G/H$ of a subgroup $\Gamma$ of $G$ in terms of $\mu$. Using this criterion (see Subsection \[Proprete and deformations\]), Theorem \[proprete, groupes algebriques\] is a consequence of the following result, where we fix a norm $\Vert\cdot\Vert$ on $E$.
\[varphi ne change pas beaucoup mu\] Let ${\mathbf{k}}$ be a local field, $G$ the set of ${\mathbf{k}}$-points of a connected reductive algebraic ${\mathbf{k}}$-group $\mathbf{G}$, and $L$ the set of ${\mathbf{k}}$-points of a closed reductive subgroup $\mathbf{L}$ of $\mathbf{G}$ of ${\mathbf{k}}$-rank $1$. If ${\mathbf{k}}={\mathbb{R}}$ or ${\mathbb{C}}$, let $\Gamma$ be a convex cocompact subgroup of $L$; if ${\mathbf{k}}$ is non-Archimedean, let $\Gamma$ be any finitely generated discrete subgroup of $L$. For any $\varepsilon>0$, there is a neighborhood $\mathcal{U}_{\varepsilon}\subset{\mathrm{Hom}}(\Gamma,G)$ of the natural inclusion and a constant $C_{\varepsilon}\geq 0$ such that $$\big\Vert\mu(\varphi(\gamma)) - \mu(\gamma)\big\Vert \leq \varepsilon \Vert\mu(\gamma)\Vert + C_{\varepsilon}$$ for all $\varphi\in\mathcal{U}_{\varepsilon}$ and all $\gamma\in\Gamma$.
Ideas of proofs
---------------
The core of the paper is the proof of Theorem \[varphi ne change pas beaucoup mu\]. We start by recalling, in Section \[Preliminaires\], that certain linear forms $\ell$ on $E$ are connected to representations $(V,\rho)$ of $\mathbf{G}$ by relations of the form $$\ell(\mu(g)) = \log\Vert\rho(g)\Vert_{_V}$$ for all $g\in G$, where $\Vert\cdot\Vert_{_V}$ is a certain fixed norm on $V$. We are thus led to bound ratios of the form $\Vert\rho(\varphi(\gamma))\Vert_{_V}/\Vert\rho(\gamma)\Vert_{_V}$, where $\gamma\in\Gamma\smallsetminus\{ 1\} $ and where $\varphi\in{\mathrm{Hom}}(\Gamma,G)$ is close to the natural inclusion of $\Gamma$ in $G$.
In order to bound these ratios we look at the dynamics of $G$ acting on the projective space ${\mathbb{P}}(V)$, notably the dynamics of the elements $g\in G$ that are *proximal* in ${\mathbb{P}}(V)$. By definition, such elements $g\in G$ admit an attracting fixed point and a repelling projective hyperplane in ${\mathbb{P}}(V)$. In Section \[Dynamique proximale\] we consider products $z_1k_2z_2\ldots k_nz_n$ of proximal elements $z_i$ having a common attracting fixed point $x_0^+$ and a common repelling hyperplane $X_0^-$, with isometries $k_i$ such that $k_i\cdot x_0^+$ remains bounded away from $X_0^-$. We estimate the contraction power of such a product in terms of the contraction powers of the $z_i$.
In Section \[Partie produit transverse\] we see how such dynamical considerations apply to the elements $\gamma\in\Gamma$ and their images $\varphi(\gamma)$ under a small deformation $\varphi\in\nolinebreak{\mathrm{Hom}}(\Gamma,G)$. We use Guichard’s idea [@gui] of writing every element $\gamma\in\Gamma$ as a product $\gamma_0\ldots\gamma_n$ of elements of a fixed finite subset $F$ of $\Gamma$, where the norms $\Vert\mu(\gamma_i)\Vert$ and $\Vert\mu(\gamma_i\gamma_{i+1})-\mu(\gamma_i)-\mu(\gamma_{i+1})\Vert$ are controlled for all $i$.
In Section \[Demonstration des theoremes\] we combine the results of Sections \[Dynamique proximale\] and \[Partie produit transverse\] by carefully choosing the finite subset $F$ of $\Gamma$ in order to get a sharp control of the ratios $\Vert\rho(\varphi(\gamma))\Vert_{_V}/\Vert\rho(\gamma)\Vert_{_V}$, or equivalently of $\ell(\mu(\varphi(\gamma))-\mu(\gamma))$ for $\gamma\in\Gamma\smallsetminus\{ 1\} $. From this we deduce Theorem \[varphi ne change pas beaucoup mu\].
At the end of Section \[Demonstration des theoremes\] we explain how Theorems \[proprete, groupes de Lie\] and \[proprete, groupes algebriques\] follow from Theorem \[varphi ne change pas beaucoup mu\]. Finally, in Section \[quotients compacts Zariski-denses\] we establish Corollary \[quotients compacts de SO(2n,2)/SU(n,1)\] by relating Theorem \[proprete, groupes de Lie\] to Johnson and Millson’s bending construction.
Acknowledgements {#acknowledgements .unnumbered}
----------------
I warmly thank Yves Benoist and Olivier Guichard for fruitful discussion.
Cartan projections, maximal parabolic subgroups, and representations {#Preliminaires}
====================================================================
Throughout the paper, ${\mathbf{k}}$ denotes a local field, *i.e.*, ${\mathbb{R}}$, ${\mathbb{C}}$, a finite extension of ${\mathbb{Q}}_p$, or the field ${\mathbb{F}}_q((t))$ of formal Laurent series over a finite field ${\mathbb{F}}_q$. If ${\mathbf{k}}={\mathbb{R}}$ or ${\mathbb{C}}$, we denote by $|\cdot|$ the usual absolute value on ${\mathbf{k}}$. If ${\mathbf{k}}$ is non-Archimedean, we denote by $\mathcal{O}$ the ring of integers of ${\mathbf{k}}$, by $q$ the cardinal of its residue field, by $\pi$ a uniformizer, by $\omega$ the (additive) valuation on ${\mathbf{k}}$ such that $\omega(\pi)=1$, and by $|\cdot| = q^{-\omega(\cdot)}$ the corresponding (multiplicative) absolute value. If $\mathbf{G}$ is an algebraic group, we denote by $G$ the set of its ${\mathbf{k}}$-points and by ${\mathfrak{g}}$ its Lie algebra.
In this section, we recall a few well-known facts on connected reductive algebraic ${\mathbf{k}}$-groups and their Cartan projections.
Weyl chambers {#chambre de Weyl}
-------------
Fix a connected reductive algebraic ${\mathbf{k}}$-group $\mathbf{G}$. The derived group $\mathbf{D(G)}$ is semisimple, the identity component $\mathbf{Z(G)}^{\circ}$ of the center of $\mathbf{G}$ is a torus, which is trivial if $\mathbf{G}$ is semisimple, and $\mathbf{G}$ is the almost product of $\mathbf{D(G)}$ and $\mathbf{Z(G)}^{\circ}$. Recall that the ${\mathbf{k}}$-split ${\mathbf{k}}$-tori of $\mathbf{G}$ are all conjugate over ${\mathbf{k}}$. Fix such a torus $\mathbf{A}$ and let $\mathbf{N}$ (resp. $\mathbf{Z}$) denote its normalizer (resp. centralizer) in $\mathbf{G}$. The group $X(\mathbf{A})$ of ${\mathbf{k}}$-characters of $\mathbf{A}$ and the group $Y(\mathbf{A})$ of ${\mathbf{k}}$-cocharacters are both free ${\mathbb{Z}}$-modules of rank $\mathrm{rank}_{{\mathbf{k}}}(\mathbf{G})$ and there is a perfect pairing $$\langle\cdot\,,\cdot\rangle : X(\mathbf{A})\times Y(\mathbf{A})\longrightarrow{\mathbb{Z}}.$$ Note that $\mathbf{A}$ is the almost product of $(\mathbf{A}\cap\mathbf{D(G)})^{\circ}$ and $(\mathbf{A}\cap\mathbf{Z(G)})^{\circ}$, hence $X(\mathbf{A})\otimes_{{\mathbb{Z}}}{\mathbb{R}}$ is the direct sum of $X((\mathbf{A}\cap\mathbf{D(G)})^{\circ})\otimes_{{\mathbb{Z}}}{\mathbb{R}}$ and $X((\mathbf{A}\cap\nolinebreak\mathbf{Z(G)})^{\circ})\otimes_{{\mathbb{Z}}}\nolinebreak{\mathbb{R}}$.
The set $\Phi=\Phi(\mathbf{A},\mathbf{G})$ of restricted roots of $\mathbf{A}$ in $\mathbf{G}$, *i.e.*, the set of nontrivial weights of $\mathbf{A}$ in the adjoint representation of $\mathbf{G}$, is a root system of $X((\mathbf{A}\cap\mathbf{D(G)})^{\circ})\otimes_{{\mathbb{Z}}}{\mathbb{R}}$. For $\alpha\in\Phi$, let $\check{\alpha}$ be the corresponding coroot: by definition, $\langle\alpha,\check{\alpha}\rangle=2$ and $s_{\alpha}(\Phi)=\Phi$, where $s_{\alpha}$ is the reflection of $X(\mathbf{A})\otimes_{{\mathbb{Z}}}{\mathbb{R}}$ mapping $x$ to $x - \langle x,\check{\alpha}\rangle\,\alpha$. The group $W=N/Z$ is finite and identifies with the Weyl group of $\Phi$, generated by the reflections $s_{\alpha}$.
Similarly, $E=Y(\mathbf{A})\otimes_{{\mathbb{Z}}}{\mathbb{R}}$ is the direct sum of $E_D=Y((\mathbf{A}\cap\mathbf{D(G)})^{\circ})\otimes_{{\mathbb{Z}}}{\mathbb{R}}$ and $E_Z=Y((\mathbf{A}\cap\nolinebreak\mathbf{Z(G)})^{\circ})\otimes_{{\mathbb{Z}}}{\mathbb{R}}$. The group $W=N/Z$ acts trivially on $E_Z$ and identifies with the Weyl group of the root system $\check{\Phi}=\{ \check{\alpha},\ \alpha\in\Phi\} $ of $E_D$. We refer to [@bot] for proofs and more detail.
If ${\mathbf{k}}$ is non-Archimedean, set $A^{\circ}=A$; if ${\mathbf{k}}={\mathbb{R}}$ or ${\mathbb{C}}$, set $$A^{\circ} = \big\{ a\in A\,,\quad \chi(a)\in\, ]0,+\infty[ \quad\forall\chi\in X(\mathbf{A})\big\} .$$ Choose a basis $\Delta$ of $\Phi$ and let $$\begin{array}{lcclcl}
& A^+ & = & \big\{ a\in A^{\circ}\!, & \ |\alpha(a)|\geq 1 & \forall\alpha\in\Delta\big\} \\
\mathrm{\big(resp.}\quad & E^+ & = & \big\{ x\in E, & \langle\alpha,x\rangle\geq 0 & \forall\alpha\in\Delta\big\} \mathrm{\big)}
\end{array}$$ denote the corresponding closed positive Weyl chamber in $A^{\circ}$ (resp. in $E$). The set $E^+$ is a closed convex cone in the real vector space $E$. If ${\mathbf{k}}={\mathbb{R}}$ or ${\mathbb{C}}$, then $E$ identifies with ${\mathfrak{a}}$ and $E^+$ with $\log A^+\subset{\mathfrak{a}}$, and we endow $E$ with the Euclidean norm $\Vert\cdot\Vert$ induced by the Killing form of ${\mathfrak{g}}$. If ${\mathbf{k}}$ is non-Archimedean, we endow $E$ with any $W$-invariant Euclidean norm $\Vert\cdot\Vert$.
Cartan decompositions and Cartan projections {#Projection de Cartan}
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If ${\mathbf{k}}={\mathbb{R}}$ or ${\mathbb{C}}$, there is a maximal compact subgroup $K$ of $G$ such that the Cartan decomposition $G=KA^+K$ holds: for $g\in G$, there are elements $k_g,\ell_g\in K$ and a unique $a_g\in A^+$ such that $g = k_g a_g \ell_g$ ([@hel], Chap. 9, Th. 1.1). Setting $\mu(g)=\log a_g$ defines a map $\mu : G\rightarrow E^+\simeq\log A^+$, which is continuous, proper, and surjective. It is called the *Cartan projection* with respect to the Cartan decomposition $G=KA^+K$.
If ${\mathbf{k}}$ is non-Archimedean, let $\operatorname{Res}: X(\mathbf{Z})\rightarrow X(\mathbf{A})$ denote the restriction homomorphism, where $X(\mathbf{Z})$ is the group of ${\mathbf{k}}$-characters of $\mathbf{Z}$. There is a unique group homomorphism $\nu : Z\rightarrow E$ such that $$\langle\operatorname{Res}(\chi),\nu(z)\rangle = -\,\omega(\chi(z))$$ for all $\chi\in X(\mathbf{Z})$ and $z\in Z$. Let $Z^+\subset Z$ denote the inverse image of $E^+$ under $\nu$. The *Cartan decomposition* $G = KZ^+K$ holds: for $g\in G$, there are elements $k_g,\ell_g\in K$ and $z_g\in Z^+$ such that $g = k_g z_g \ell_g$, and $\nu(z_g)$ is uniquely defined. Setting $\mu(g)=\nu(z_g)$ defines a map $\mu : G\rightarrow E^+$, which is continuous and proper, and whose image $\mu(G)$ is the intersection of $E^+$ with a lattice of $E$. It is called the *Cartan projection* with respect to the Cartan decomposition $G=KZ^+K$. For proofs and more detail we refer to the original articles [@bt1] and [@bt2], but the reader may also find [@rou] a useful reference.
A geometric interpretation
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If ${\mathbf{k}}={\mathbb{R}}$ or ${\mathbb{C}}$, let $X=G/K$ denote the Riemannian symmetric space of $G$ and $d$ its distance; set $x_0=K\in X$. Since $G$ acts on $X$ by isometries, we have $$\label{prelim mu distance, reel}
\Vert\mu(g)\Vert = d(x_0,g\cdot x_0)$$ for all $g\in G$. If ${\mathbf{k}}$ is non-Archimedean, let $X$ denote the *Bruhat-Tits building* of $G$: it is a metric space on which $G$ acts properly by isometries with a compact fundamental domain (see [@bt1] or [@rou]). When $\mathrm{rank}_{{\mathbf{k}}}(\mathbf{G})=1$, it is a bipartite simplicial tree (see [@ser77], § II.1, for the case of $G={\mathrm{SL}}_2({\mathbf{k}})$). The group $K$ is the stabilizer of some point $x_0\in X$, and we have $$\label{prelim mu distance, ultrametrique}
\Vert\mu(g)\Vert = d(x_0,g\cdot x_0)$$ for all $g\in G$, where $d$ denotes the distance on $X$. It follows from (\[prelim mu distance, reel\]) and (\[prelim mu distance, ultrametrique\]) that in both cases (Archimedean or not), $$\label{inegalite triangulaire pour mu}
\Vert\mu(gg')\Vert \,\leq\, \Vert\mu(g)\Vert + \Vert\mu(g')\Vert$$ for all $g,g'\in G$. In fact, the following stronger inequalities hold (see for instance [@kas08], Lem. 2.3): for all $g,g'\in G$, $$\label{inegalite fine pour mu}
\left \{
\begin{array}{c @{\ \leq\ } c}
\Vert\mu(gg')-\mu(g')\Vert & \Vert\mu(g)\Vert,\\
\Vert\mu(gg')-\mu(g)\Vert & \Vert\mu(g')\Vert.
\end{array}
\right.$$
Maximal parabolic subgroups {#Sous-groupes paraboliques maximaux}
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For $\alpha\in\Phi$, let $\mathbf{U}_{\alpha}$ denote the corresponding unipotent subgroup of $\mathbf{G}$, with Lie algebra ${\mathfrak{u}}_{\alpha}={\mathfrak{g}}_{\alpha}\oplus{\mathfrak{g}}_{2\alpha}$, where $${\mathfrak{g}}_{i\alpha} = \big\{ X\in{\mathfrak{g}},\quad \operatorname{Ad}(a)(X)=\alpha(a)^iX\quad \forall a\in A\big\}$$ for $i=1,2$. For any subset $\theta$ of $\Delta$, let $\mathbf{P}_{\theta}$ denote the corresponding *standard* parabolic subgroup of $\mathbf{G}$, with Lie algebra $${\mathfrak{p}}_{\theta} = {\mathfrak{z}}\oplus \Big(\bigoplus_{\beta\in\Phi^+} {\mathfrak{u}}_{\beta}\Big) \oplus \Big(\bigoplus_{\beta\in{\mathbb{N}}(\Delta\smallsetminus\theta)} {\mathfrak{u}}_{-\beta}\Big).$$ Every parabolic ${\mathbf{k}}$-subgroup $\mathbf{P}$ of $\mathbf{G}$ is conjugate over ${\mathbf{k}}$ to a unique standard one. In particular, the maximal proper parabolic ${\mathbf{k}}$-subgroups of $\mathbf{G}$ are the conjugates of the groups $\mathbf{P}_{\alpha}=\mathbf{P}_{\{ \alpha\} }$, where $\alpha\in\Delta$.
Fix $\alpha\in\Delta$. Since $\mathbf{P}_{\alpha}$ is its own normalizer in $\mathbf{G}$, the *flag variety* $\mathbf{G}/\mathbf{P}_{\alpha}$ parametrizes the set of parabolic ${\mathbf{k}}$-subgroups that are conjugate to $\mathbf{P}_{\alpha}$. It is a projective variety, which is defined over ${\mathbf{k}}$. Let $\mathbf{N}_{\alpha}^-$ denote the unipotent subgroup of $\mathbf{G}$ generated by the groups $\mathbf{U}_{-\beta}$ for $\beta\in\alpha +\Phi^+$, with Lie algebra $${\mathfrak{n}}_{\alpha}^- = \bigoplus_{\beta\in\Phi^+} {\mathfrak{u}}_{-(\alpha+\beta)}.$$ Let $W_{\alpha}$ be the subgroup of $W$ generated by the reflections $s_{\beta}$ for $\beta\in\Delta\smallsetminus\{ \alpha\} $. The *Bruhat decomposition* $$\mathbf{G}/\mathbf{P}_{\alpha} = \coprod_{w\in W/W_{\alpha}} \mathbf{N}_{\alpha}^- w \mathbf{P}_{\alpha}$$ holds, where the projective subvariety $\mathbf{N}_{\alpha}^- w \mathbf{P}_{\alpha}$ has positive codimension whenever $wW_{\alpha}\neq W_{\alpha}$. We refer to [@bot] for proofs and more detail.
Representations of $\mathbf{G}$ {#Representations de G}
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For $\alpha\in\Delta$, let $\omega_{\alpha}\in X(\mathbf{A})$ denote the corresponding fundamental weight: by definition, $\langle\omega_{\alpha},\check{\alpha}\rangle=1$ and $\langle\omega_{\alpha},\check{\beta}\rangle=\nolinebreak 0$ for all $\beta\in\Delta\smallsetminus\{ \alpha\} $. By [@tit71], Th. 7.2, there is an irreducible ${\mathbf{k}}$-representation $(\rho_{\alpha},V_{\alpha})$ of $\mathbf{G}$ whose highest weight $\chi_{\alpha}$ is a positive multiple of $\omega_{\alpha}$ and whose highest weight space $x_{\alpha}^+$ is a line. The point $x_{\alpha}^+\in{\mathbb{P}}(V_{\alpha})$ is the unique fixed point of $P_{\alpha}$ in ${\mathbb{P}}(V_{\alpha})$. The map from $\mathbf{G}/\mathbf{P}_{\alpha}$ to ${\mathbb{P}}(V_{\alpha})$ sending $g\mathbf{P}_{\alpha}$ to $\rho_{\alpha}(g)(x_{\alpha}^+)$ is a closed immersion. We denote the set of restricted roots of $(\rho_{\alpha},V_{\alpha})$ by $\Lambda_{\alpha}$ and, for every $\lambda\in\Lambda_{\alpha}$, the weight space of $\lambda$ by $(V_{\alpha})_{\lambda}$.
If ${\mathbf{k}}={\mathbb{R}}$ (resp. if ${\mathbf{k}}={\mathbb{C}}$), then the weight spaces are orthogonal with respect to some $K$-invariant Euclidean (resp. Hermitian) norm $\Vert\cdot\Vert_{\alpha}$ on $V_{\alpha}$. The corresponding operator norm $\Vert\cdot\Vert_{\alpha}$ on ${\mathrm{End}}(V_{\alpha})$ satisfies $$\label{norme des representations and Cartan projection, cas reel}
\Vert\rho_{\alpha}(g)\Vert_{\alpha} = e^{\langle\chi_{\alpha},\mu(g)\rangle}$$ for all $g\in G$. If ${\mathbf{k}}$ is non-Archimedean, then there is a $K$-invariant ultrametric norm $\Vert\cdot\Vert_{\alpha}$ on $V_{\alpha}$ such that $$\bigg\Vert\sum_{\lambda\in\Lambda_{\alpha}} v_{\lambda}\bigg\Vert_{\alpha} = \max_{\lambda\in\Lambda_{\alpha}} \Vert v_{\lambda}\Vert_{\alpha}$$ for all $(v_{\lambda})\in \prod_{\lambda\in\Lambda_{\alpha}}(V_{\alpha})_{\lambda}$ and such that the restriction of $\rho_{\alpha}(z)$ to $(V_{\alpha})_{\lambda}$ is a homothety of ratio $q^{\langle\lambda,\nu(z)\rangle}$ for all $z\in Z$ and all $\lambda\in\Lambda_{\alpha}$ ([@qui], Th. 6.1). The corresponding operator norm $\Vert\cdot\Vert_{\alpha}$ on ${\mathrm{End}}(V_{\alpha})$ satisfies $$\label{norme des representations and Cartan projection, cas ultrametrique}
\Vert\rho_{\alpha}(g)\Vert_{\alpha} = q^{\langle\chi_{\alpha},\mu(g)\rangle}$$ for all $g\in G$.
The example of $\mathbf{SL}_n$
------------------------------
Let $\mathbf{G}=\mathbf{SL}_n$ for some integer $n\geq 2$. The group $\mathbf{A}$ of diagonal matrices with determinant $1$ is a maximal ${\mathbf{k}}$-split ${\mathbf{k}}$-torus of $\mathbf{G}$ which is its own centralizer, *i.e.*, $\mathbf{Z}=\mathbf{A}$. The corresponding root system $\Phi$ is the set of linear forms $\varepsilon_i-\varepsilon_j$, $1\leq i\neq j\leq n$, where $$\varepsilon_i\big({\mathrm{diag}}(a_1,\ldots,a_n)\big) = a_i.$$ The roots $\varepsilon_i-\varepsilon_{i+1}$, for $1\leq i\leq n-1$, form a basis $\Delta$ of $\Phi$. If ${\mathbf{k}}$ is Archimedean (resp. non-Archimedean), the corresponding positive Weyl chamber is $$\begin{aligned}
A^+ & = & \big\{ {\mathrm{diag}}(a_1,\ldots,a_n)\in A,\ \ \! a_i\in\, ]0,+\infty[\ \forall i\ \,\mathrm{and}\ a_1\geq\ldots\geq a_n\big\} \\
\mathrm{\big(resp.}\quad A^+ & = & \big\{ {\mathrm{diag}}(a_1,\ldots,a_n)\in A,\ |a_1|\geq\ldots\geq|a_n|\big\} \mathrm{\big).}\end{aligned}$$ Set $K={\mathrm{SO}}(n)$ (resp. $K={\mathrm{SU}}(n)$, resp. $K={\mathrm{SL}_n}(\mathcal{O})$) if ${\mathbf{k}}={\mathbb{R}}$ (resp. if ${\mathbf{k}}=\nolinebreak{\mathbb{C}}$, resp. if ${\mathbf{k}}$ is non-Archimedean). The Cartan decomposition $G=KA^+K$ holds. If ${\mathbf{k}}={\mathbb{R}}$ (resp. if ${\mathbf{k}}=\nolinebreak{\mathbb{C}}$) it follows from the polar decomposition in ${\mathrm{GL}_n(\mathbb{R})}$ (resp. in ${\mathrm{GL}_n(\mathbb{C})}$) and from the reduction of symmetric (resp. Hermitian) matrices. If ${\mathbf{k}}$ is non-Archimedean, it follows from the structure theorem for finitely generated modules over a principal ideal domain. The real vector space $$E = \big\{ (x_1,\ldots,x_n)\in{\mathbb{R}}^n,\ x_1+\ldots+x_n=0\big\}\ \simeq\ {\mathbb{R}}^{n-1}$$ and its closed convex cone $$E^+ = \big\{ (x_1,\ldots,x_n)\in E,\ x_1\geq\ldots\geq x_n\big\}$$ do not depend on ${\mathbf{k}}$. Let $\mu : G\rightarrow E^+$ denote the Cartan projection with respect to the Cartan decomposition $G=KA^+K$. If ${\mathbf{k}}={\mathbb{R}}$ or ${\mathbb{C}}$, then $\mu(g)=(\frac{1}{2}\log x_i)_{1\leq i\leq n}$ where $x_i$ is the $i$-th eigenvalue of $^t\!\overline{g}g$. If ${\mathbf{k}}$ is non-Archimedean and if $m$ is any integer such that $\pi^mg\in{\mathrm{M}}_n(\mathcal{O})$, then $\mu(g)=(\omega(x_{m,i})-m)_{1\leq i\leq n}$ where $x_{m,i}$ is the $i$-th invariant factor of $\pi^mg$.
Fix a simple root $\alpha=\varepsilon_{i_0}-\varepsilon_{i_0+1}\in\Delta$. The parabolic group $\mathbf{P}_{\alpha}$ is defined by the vanishing of the $(i,j)$-matrix entries for $1\leq j\leq i_0<i\leq n$. The flag variety $\mathbf{G}/\mathbf{P}_{\alpha}$ is the Grassmannian $\mathcal{G}(i_0,n)$ of $i_0$-dimensional subspaces of the affine space $\mathbb{A}^n$. The Lie algebra ${\mathfrak{n}}_{\alpha}^-$ is defined by the vanishing of the $(i,j)$-matrix entries for $1\leq i\leq i_0$ and for $i_0+1\leq i,j\leq n$. The decomposition $$\mathbf{G}/\mathbf{P}_{\alpha} = \coprod_{w\in W/W_{\alpha}} \mathbf{N}_{\alpha}^- w \mathbf{P}_{\alpha}$$ is the decomposition of the Grassmannian $\mathcal{G}(i_0,n)$ into Schubert cells. The representation $(\rho_{\alpha},V_{\alpha})$ is the natural representation of $\mathbf{SL}_n$ in the wedge product $\Lambda^{i_0}{\mathbb{A}}^n$. Its highest weight is the fundamental weight $$\omega_{\alpha}=\varepsilon_1+\ldots+\varepsilon_{i_0}$$ associated with $\alpha$. The embedding of the Grassmannian $\mathcal{G}(i_0,n)$ into the projective space ${\mathbb{P}}(V_{\alpha})={\mathbb{P}}(\Lambda^{i_0}{\mathbb{A}}^n)$ is the Plücker embedding.
Dynamics in projective spaces {#Dynamique proximale}
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In this section we look at the dynamics of certain endomorphisms of ${\mathbf{k}}$-vector spaces in the corresponding projective spaces, where ${\mathbf{k}}$ is a local field. In Subsection \[Proximalite et normes\] we start by recalling the notion of proximality. We then consider products of the form $z_1k_2z_2\ldots k_nz_n$, where the $z_i$ are proximal elements with a common attracting fixed point $x_0^+$ and a common repelling hyperplane $X_0^-$, and the $k_i$ are isometries such that $k_i\cdot x_0^+$ remains bounded away from $X_0^-$. We estimate the contraction power of such a product in terms of the contraction powers of the $z_i$. In Subsection \[Projection de Cartan and poids fondamentaux\] we consider a connected reductive algebraic ${\mathbf{k}}$-group $\mathbf{G}$ and apply the result of Subsection \[Proximalite et normes\] to the representations $(V_{\alpha},\rho_{\alpha})$ of $\mathbf{G}$ introduced in Subsection \[Representations de G\]. From (\[norme des representations and Cartan projection, cas reel\]) and (\[norme des representations and Cartan projection, cas ultrametrique\]) we get an upper bound for $|\langle\chi_{\alpha},\mu(g_1\ldots g_n)-\mu(g_1)-\ldots-\mu(g_n)\rangle|$ for elements $g_1,\ldots,g_n\in G$ satisfying certain contractivity and transversality conditions.
Proximality in projective spaces and norm estimates {#Proximalite et normes}
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Let ${\mathbf{k}}$ be a local field and $V$ be a finite-dimensional vector space over ${\mathbf{k}}$. Given a basis $(v_1,\ldots,v_n)$ of $V$, we define the norm $$\label{definition norme avec une base}
\bigg\Vert\sum_{1\leq j\leq n} t_j\,v_j\bigg\Vert_{_V} = \sup_{1\leq j\leq n} |t_j|$$ on $V$, and we keep the notation $\Vert\cdot\Vert_{_V}$ for the corresponding operator norm on ${\mathrm{End}}(V)$. We endow the projective space ${\mathbb{P}}(V)$ with the distance $$d(x_1,x_2) = \inf\big\{ \Vert v_1-v_2\Vert_{_V},\ v_i\in x_i\ \mathrm{and}\ \Vert v_i\Vert_{_V} =1\ \forall i=1,2\big\} .$$ Recall that an element $g\in{\mathrm{End}}(V)\smallsetminus\{ 0\} $ is called *proximal* if it has a unique eigenvalue of maximal absolute value and if this eigenvalue has multiplicity 1. (The eigenvalues of $g$ belong to a finite extension ${\mathbf{k}}_g$ of ${\mathbf{k}}$ and we consider the unique extension to ${\mathbf{k}}_g$ of the absolute value $|\cdot|$ on ${\mathbf{k}}$.) If $g$ is proximal, then its maximal eigenvalue belongs to ${\mathbf{k}}$; we denote by $x_g^+\in{\mathbb{P}}(V)$ the corresponding eigenline and by $X_g^-$ the image in ${\mathbb{P}}(V)$ of the unique $g$-invariant complementary subspace of $x_g^+$ in $V$. Note that $g$ acts on ${\mathbb{P}}(V)$ by contracting ${\mathbb{P}}(V)\smallsetminus X_g^-$ towards $x_g^+$. For $\varepsilon>0$, we will say that $g$ is $\varepsilon$-*proximal* if it satisfies the two following additional conditions:
1. $d(x_g^+,X_g^-)\geq 2\varepsilon$,
2. for any $x\in{\mathbb{P}}(V)$, if $d(x,X_g^-)\geq\varepsilon$, then $d(g\cdot x,x_g^+)\leq\varepsilon$.
We will need the following lemma.
\[produits d’elements proximaux decales\] Let $X_0^-$ be a projective hyperplane of ${\mathbb{P}}(V)$, let $x_0^+\in{\mathbb{P}}(V)\smallsetminus\nolinebreak X_0^-$, and let $\varepsilon>0$ such that $d(x_0^+,X_0^-)\geq 2\varepsilon$. Then there exists $r_{\varepsilon}>0$ such that for any isometries $k_2,\ldots,k_n\in{\mathrm{End}}(V)$ with $d(k_i\cdot x_0^+,X_0^-)\geq 2\varepsilon$ and for any $\varepsilon$-proximal endomorphisms $z_1,\ldots,z_n\in{\mathrm{End}}(V)$ with $x_{z_i}^+=x_0^+$ and $X_{z_i}^-=X_0^-$, inducing a homothety of ratio $\Vert z_i\Vert_{_V}$ on the line $x_0^+$, we have $$e^{-(n-1)\,r_{\varepsilon}}\cdot\prod_{i=1}^n \Vert z_i\Vert_{_V} \,\leq\, \Vert z_1k_2z_2\ldots k_n z_n\Vert_{_V} \,\leq\, \prod_{i=1}^n \Vert z_i\Vert_{_V}.$$
Since the operator norm $\Vert\cdot\Vert_{_V}$ on ${\mathrm{End}}(V)$ is submultiplicative and $k_i$ is an isometry of $V$ for all $i$, $$\Vert z_1k_2z_2\ldots k_n z_n\Vert_{_V} \,\leq\, \prod_{i=1}^n \Vert z_i\Vert_{_V}.$$ Let us prove the left-hand inequality. Let $v_0\in V\smallsetminus\{ 0\} $ satisfy $x_0^+={\mathbf{k}}v_0$ and let $V_0$ be the hyperplane of $V$ such that $X_0^-={\mathbb{P}}(V_0)$. Set $$\begin{aligned}
b_{\varepsilon} & = & \{ x\in{\mathbb{P}}(V),\ d(x,x_0^+)\leq\varepsilon\} \quad\quad\ \,\\
\mathrm{and}\quad\quad B_{\varepsilon} & = & \{ x\in{\mathbb{P}}(V),\ d(x,X_0^-)\geq\varepsilon\} .\quad\quad\ \,\end{aligned}$$ Note that the set of unitary vectors $v\in V$ with ${\mathbf{k}}v\in B_{\varepsilon}$ is compact and that the map sending $v\in V$ to $t\in{\mathbf{k}}$ such that $v\in tv_0+V_0$ is continuous, hence there exists $r_{\varepsilon}>0$ such that $$\label{definition de r_epsilon}
v\in \big[e^{-\frac{r_{\varepsilon}}{2}},e^{\frac{r_{\varepsilon}}{2}}\big]\,\Vert v\Vert_{_V}\,v_0 + V_0$$ for all $v\in V\smallsetminus\{ 0\} $ with ${\mathbf{k}}v\in B_{\varepsilon}$. Set $$h_j = z_jk_{j+1}z_{j+1}\ldots k_nz_n\in{\mathrm{End}}(V)$$ for $1\leq j\leq n$. We claim that $h_j\cdot B_{\varepsilon}\subset b_{\varepsilon}$ and $$\label{inegalite h_j}
\Vert h_j\cdot v_0\Vert_{_V} \,\geq\, e^{-(n-j)\,r_{\varepsilon}}\cdot\prod_{i=j}^n \Vert z_i\Vert_{_V}$$ for all $j$. This follows from an easy descending induction on $j$. Indeed, for all $i$ we have $k_i\cdot b_{\varepsilon}\subset B_{\varepsilon}$ since $k_i$ is an isometry of $V$ and $d(k_i\cdot x_0^+,X_0^-)\geq 2\varepsilon$, and $z_i\cdot B_{\varepsilon}\subset b_{\varepsilon}$ since $z_i$ is $\varepsilon$-proximal with $x_{z_i}^+=x_0^+$ and $X_{z_i}^-=X_0^-$. By (\[definition de r\_epsilon\]), we have $k_{j+1}h_{j+1}\cdot v_0\in t_jv_0+V_0$ for some $t_j\in{\mathbb{R}}$ with $$|t_j|\ \geq\ e^{-\frac{r_{\varepsilon}}{2}}\,\Vert k_{j+1}h_{j+1}\cdot v_0\Vert_{_V}\ =\ e^{-\frac{r_{\varepsilon}}{2}}\,\Vert h_{j+1}\cdot v_0\Vert_{_V}.$$ By the inductive assumption, $$|t_j|\ \geq\ e^{-(n-j-\frac{1}{2})\,r_{\varepsilon}}\cdot\prod_{i=j+1}^n \Vert z_i\Vert_{_V}.$$ By hypothesis, $z_j$ preserves $V_0$ and induces a homothety of ratio $\Vert z_j\Vert_{_V}$ on the line $x_0^+$, hence $h_j\cdot v_0=z_jk_{j+1}h_{j+1}\cdot v_0\in\Vert z_j\Vert_{_V}\,t_jv_0+V_0$, where $$\Vert z_j\Vert_{_V}\,|t_j|\ \geq\ e^{-(n-j-\frac{1}{2})\,r_{\varepsilon}}\cdot\prod_{i=j}^n \Vert z_i\Vert_{_V}.$$ Inequality (\[inegalite h\_j\]) follows, using (\[definition de r\_epsilon\]) again.
Cartan projection along the fundamental weights {#Projection de Cartan and poids fondamentaux}
-----------------------------------------------
Lemma \[produits d’elements proximaux decales\] implies the following result.
\[mu d’un produit transverse selon alpha\] Let ${\mathbf{k}}$ be a local field and $G$ the set of ${\mathbf{k}}$-points of a connected reductive algebraic ${\mathbf{k}}$-group. Let $G=KA^+K$ or $G=KZ^+K$ be a Cartan decomposition and $\mu : G\rightarrow E^+$ the corresponding Cartan projection. Fix $\alpha\in\Delta$ and let $\mathcal{C}_{\alpha}$ be a compact subset of $N_{\alpha}^-$. Then there exist $r_{\alpha},R_{\alpha}>0$ such that for all $g_1,\ldots,g_n\in G$ with $\langle\alpha,\mu(g_i)\rangle\geq R_{\alpha}$ and $\ell_{g_i}k_{g_{i+1}}\in\mathcal{C}_{\alpha} P_{\alpha}$, we have $$\bigg|\Big\langle\chi_{\alpha},\mu(g_1\ldots g_n) - \sum_{i=1}^n \mu(g_i)\Big\rangle\bigg| \leq n r_{\alpha}.$$
We keep notation from Section \[Preliminaires\]. In particular, for $g\in G$ we write $g = k_g z_g \ell_g$ with $k_g,\ell_g\in K$ and $z_g\in Z^+$, as in Subsection \[Projection de Cartan\]. Given a simple root $\alpha\in\Delta$, we denote by $\mathbf{N}_{\alpha}^-$ (resp. by $\mathbf{P}_{\alpha}$) the unipotent (resp. parabolic) subgroup of $\mathbf{G}$ introduced in Subsection \[Sous-groupes paraboliques maximaux\], and by $\chi_{\alpha}$ the highest weight of the representation $(V_{\alpha},\rho_{\alpha})$ introduced in Subsection \[Representations de G\].
Precisely, Proposition \[mu d’un produit transverse selon alpha\] follows from Lemma \[produits d’elements proximaux decales\], from (\[norme des representations and Cartan projection, cas reel\]) and (\[norme des representations and Cartan projection, cas ultrametrique\]), and from the following lemma.
\[remarques V\_alpha\] Let $x_{\alpha}^+\in{\mathbb{P}}(V_{\alpha})$ be the highest weight line $(V_{\alpha})_{\chi_{\alpha}}$, and let $X_{\alpha}^-$ be the image in ${\mathbb{P}}(V_{\alpha})$ of the sum of the weight spaces $(V_{\alpha})_{\lambda}$ for $\lambda\in\Lambda_{\alpha}\smallsetminus\{ \chi_{\alpha}\} $.
1. Given $\varepsilon>0$ with $d(x_{\alpha}^+,X_{\alpha}^-)\geq 2\varepsilon$, there exists $R_{\alpha}>0$ such that for any $z\in Z^+$ with $\langle\alpha,\mu(z)\rangle\geq R_{\alpha}$, the element $\rho_{\alpha}(z)$ is $\varepsilon$-proximal in ${\mathbb{P}}(V_{\alpha})$ with $\big(x_{\rho_{\alpha}(z)}^+,X_{\rho_{\alpha}(z)}^-\big)=(x_{\alpha}^+,X_{\alpha}^-)$.
2. We have $\rho_{\alpha}(N_{\alpha}^-)(x_{\alpha}^+)\cap X_{\alpha}^-=\emptyset$.
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1. It is sufficient to see that every restricted weight of $(\rho_{\alpha},V_{\alpha})$ except $\chi_{\alpha}$ belongs to $\chi_{\alpha}-\alpha-{\mathbb{N}}\Delta$. Consider the subgroup $W_{\alpha}$ of $W$ generated by the reflections $s_{\beta} : x\mapsto x-\langle x,\check{\beta}\rangle\,\beta$ for $\beta\in\Delta\smallsetminus\{ \alpha\} $. It acts transitively on the root subsystem of $\Phi$ generated by $\Delta\smallsetminus\{ \alpha\} $, and it fixes $\chi_{\alpha}$ since $\chi_{\alpha}$ is a multiple of $\omega_{\alpha}$ and $\langle\omega_{\alpha},\check{\beta}\rangle =0$ for all $\beta\in\Delta\smallsetminus\{ \alpha\} $. Therefore, for every weight $\lambda\in\chi_{\alpha}-{\mathbb{N}}(\Delta\smallsetminus\{ \alpha\} )$ there exists $w\in W_{\alpha}$ such that $w\cdot\lambda\in\chi_{\alpha}+{\mathbb{N}}\Delta$, which implies that $\lambda=\chi_{\alpha}$.
2. For $n\in N_{\alpha}^-$, the identity element $1\in G$ belongs to the closure of the conjugacy class $\{ znz^{-1},\ z\in Z\} $, hence $x_{\alpha}^+$ belongs to the closure of the orbit $\rho_{\alpha}(Zn)(x_{\alpha}^+)$ in ${\mathbb{P}}(V_{\alpha})$. But $X_{\alpha}^-$ is closed in ${\mathbb{P}}(V_{\alpha})$, stable under $Z$, and does not contain $x_{\alpha}^+$.
The point $x_{\alpha}^+\in{\mathbb{P}}(V_{\alpha})$ is fixed by $P_{\alpha}$. Moreover, $\rho_{\alpha}(N_{\alpha}^-)(x_{\alpha}^+)\cap X_{\alpha}^-=\emptyset$ by Lemma \[remarques V\_alpha\], hence there exists $\varepsilon>0$ such that $$d\big(\rho_{\alpha}(\mathcal{C}_{\alpha} P_{\alpha})(x_{\alpha}^+),X_{\alpha}^-\big)\geq 2\varepsilon.$$ Let $R_{\alpha}$ be given by Lemma \[remarques V\_alpha\] and let $r_{\alpha}=r_{\varepsilon}/\log q$, where $r_{\varepsilon}>0$ is given by Lemma \[produits d’elements proximaux decales\] and $q=e$ if ${\mathbf{k}}$ is Archimedean, $q$ is the cardinal of the residue field of $\mathcal{O}$ otherwise. Let $g_1,\ldots,g_n\in G$ satisfy $\langle\alpha,\mu(g_i)\rangle\geq R_{\alpha}$ and $\ell_{g_i}k_{g_{i+1}}\in\mathcal{C}_{\alpha} P_{\alpha}$ for all $i$. By Lemma \[remarques V\_alpha\], $\rho_{\alpha}(z_{g_i})$ is $\varepsilon$-proximal in ${\mathbb{P}}(V_{\alpha})$ with $x_{\rho_{\alpha}(z_{g_i})}^+=x_{\alpha}^+$ and $X_{\rho_{\alpha}(z_{g_i})}^-=X_{\alpha}^-$. Moreover, it induces a homothety of ratio $\Vert\rho_{\alpha}(z_{g_i})\Vert_{\alpha}$ on the line $x_{\alpha}^+$. By Lemma \[produits d’elements proximaux decales\], $$q^{-nr_{\alpha}}\cdot\prod_{i=1}^n \Vert\rho_{\alpha}(z_{g_i})\Vert_{\alpha}\ \leq\ \Vert\rho_{\alpha}(g_1\ldots g_n)\Vert_{\alpha}\ \leq\ \prod_{i=1}^n \Vert\rho_{\alpha}(z_{g_i})\Vert_{\alpha}.$$ Using (\[norme des representations and Cartan projection, cas reel\]) and (\[norme des representations and Cartan projection, cas ultrametrique\]), we get $$\Big\langle\chi_{\alpha},\sum_{i=1}^n \mu(g_i)\Big\rangle - nr_{\alpha}\ \leq\ \langle\chi_{\alpha},\mu(g_1\ldots g_n)\rangle\ \leq\ \Big\langle\chi_{\alpha},\sum_{i=1}^n \mu(g_i)\Big\rangle.\qedhere$$
Transverse products {#Partie produit transverse}
===================
In this section we explain how, under the assumptions of Theorem \[varphi ne change pas beaucoup mu\], Proposition \[mu d’un produit transverse selon alpha\] applies to the elements $\gamma\in\nolinebreak\Gamma$ and their images $\varphi(\gamma)$ under a small deformation $\varphi\in{\mathrm{Hom}}(\Gamma,G)$. We use Guichard’s idea [@gui] of writing every element $\gamma\in\Gamma$ as a “transverse product” $\gamma_0\ldots\gamma_n$ of elements of a fixed finite subset $F$ of $\Gamma$.
Transversality in $L$ {#Transversalite en rang un}
---------------------
Let ${\mathbf{k}}$ be a local field and $\mathbf{L}$ a connected reductive algebraic ${\mathbf{k}}$-group of ${\mathbf{k}}$-rank $1$. Fix a Cartan decomposition $L=K_LA_L^+K_L$ or $L=K_LZ_L^+K_L$, where $K_L$ is a maximal compact subgroup of $L$, where $\mathbf{A_L}$ is a maximal ${\mathbf{k}}$-split ${\mathbf{k}}$-torus of $\mathbf{L}$, and where $\mathbf{Z_L}$ is the centralizer of $\mathbf{A_L}$ in $\mathbf{L}$. Let $\mu_L : L\rightarrow E_L^+$ denote the corresponding Cartan projection, where $E_L=Y(\mathbf{A_L})\otimes_{{\mathbb{Z}}}{\mathbb{R}}$. Since $\mathbf{L}$ has ${\mathbf{k}}$-rank $1$, the vector space $E_L$ is a line, and any isomorphism from $E_L$ to ${\mathbb{R}}$ gives a Cartan projection $\mu_L^{{\mathbb{R}}} : L\rightarrow{\mathbb{R}}$.
If $\mathbf{L}$ has semisimple ${\mathbf{k}}$-rank 1, then $\mu_L^{{\mathbb{R}}}$ takes only nonnegative or only nonpositive values. We denote by $\alpha_L$ the indivisible positive restricted root of $\mathbf{A_L}$ in $\mathbf{L}$, by $\mathbf{P_L}=\mathbf{P}_{\alpha_L}$ the proper parabolic subgroup of $\mathbf{L}$ associated with $\alpha_L$, and by $\mathbf{N_L^-}=\mathbf{U}_{-\alpha_L}$ the unipotent subgroup associated with $-\alpha_L$.
If $\mathbf{L}$ has semisimple ${\mathbf{k}}$-rank $0$, then $\mathbf{A_L}$ is central in $\mathbf{L}$, hence $\mathbf{Z_L}=\mathbf{L}$. In this case $\mu_L^{{\mathbb{R}}}$ is a group homomorphism from $L$ to ${\mathbb{R}}$, thus taking both positive and negative values. We set $\mathbf{P_L}=\mathbf{Z_L}=\mathbf{L}$ and $\mathbf{N_L^-}=\{ 1\} $.
For the reader’s convenience, we give a proof of the following result, which is due to Guichard in the real semisimple case ([@gui], Lem. 7 & 9). We consider the more general situation of a reductive algebraic group over a local field.
\[produit d’elements transverses\] Let ${\mathbf{k}}$ be a local field, $L$ the set of ${\mathbf{k}}$-points of a connected reductive algebraic ${\mathbf{k}}$-group of ${\mathbf{k}}$-rank $1$, and $\mu_L^{{\mathbb{R}}} : L\rightarrow{\mathbb{R}}$ a Cartan projection. If ${\mathbf{k}}={\mathbb{R}}$ or ${\mathbb{C}}$, let $\Gamma$ be a convex cocompact subgroup of $L$; if ${\mathbf{k}}$ is non-Archimedean, let $\Gamma$ be any finitely generated discrete subgroup of $L$. Then there exist $D>0$ and a compact subset $\mathcal{C}_L$ of $N_L^-$ such that for $R\geq D$, any $\gamma\in\Gamma$ may be written as $\gamma=\gamma_0\ldots\gamma_n$, where
1. $|\mu_L^{{\mathbb{R}}}(\gamma_0)|\leq R+D$ and $R-D\leq |\mu_L^{{\mathbb{R}}}(\gamma_i)|\leq R+D$ for all $1\leq i\leq n$,
2. $\mu_L^{{\mathbb{R}}}(\gamma_1),\ldots,\mu_L^{{\mathbb{R}}}(\gamma_n)$ are all $\geq 0$ or all $\leq 0$,
3. $\ell_{\gamma_i} k_{\gamma_{i+1}} \in \mathcal{C}_L P_L$ for all $1\leq i\leq n-1$.
To prove Proposition \[produit d’elements transverses\] we use the following lemma, which translates the transversality condition (3) in terms of $\mu_L^{{\mathbb{R}}}$.
\[equivalence transversalite\] Under the assumptions of Proposition \[produit d’elements transverses\], there exists $D_0\geq\nolinebreak 0$ with the following property: given any $D\geq D_0$, there is a compact subset $\mathcal{C}_L$ of $N_L^-$ such that for $k\in K_L$, if $$|\mu_L^{{\mathbb{R}}}(z_1kz_2)|\geq |\mu_L^{{\mathbb{R}}}(z_1)|+|\mu_L^{{\mathbb{R}}}(z_2)|-D$$ for some $z_1,z_2\in Z_L^+$ with $|\mu_L^{{\mathbb{R}}}(z_1)|,|\mu_L^{{\mathbb{R}}}(z_2)|\geq D$, then $k\in\mathcal{C}_L P_L$.
Note that Proposition \[mu d’un produit transverse selon alpha\] implies some kind of converse to Lemma \[equivalence transversalite\]: for any compact subset $\mathcal{C}_L$ of $N_L^-$, there exists $D\geq 0$ such that for all $k\in K_L\cap\mathcal{C}_LP_L$ and all $z_1,z_2\in Z^+$, $$|\mu_L^{{\mathbb{R}}}(z_1kz_2)|\geq |\mu_L^{{\mathbb{R}}}(z_1)|+|\mu_L^{{\mathbb{R}}}(z_2)|-D.$$
We may assume that $\mathbf{L}$ has semisimple ${\mathbf{k}}$-rank $1$. Then $L/P_L$ is the disjoint union of $N_L^-\cdot P_L$ and $\{ w\cdot P_L\} $, where $w$ denotes the nontrivial element of the (restricted) Weyl group of $L$. It is therefore sufficient to prove the existence of a neighborhood $\mathcal{U}$ of $w\cdot P_L$ in $L/P_L$ such that for all $k\in K_L$ with $k\cdot P_L\in\mathcal{U}$ and all $z_1,z_2\in Z_L^+$ with $\mu_L^{{\mathbb{R}}}(z_1),\mu_L^{{\mathbb{R}}}(z_2)\geq D$, we have $$|\mu_L^{{\mathbb{R}}}(z_1kz_2)| < |\mu_L^{{\mathbb{R}}}(z_1)| + |\mu_L^{{\mathbb{R}}}(z_2)| - D.$$ Let $X_L$ denote either the Riemannian symmetric space or the Bruhat-Tits tree of $L$, depending on whether ${\mathbf{k}}$ is Archimedean or not. The space $X_L$ is Gromov-hyperbolic and we may identify $L/P_L$ with the boundary at infinity $\partial X_L$ of $X_L$, *i.e.*, with the set of equivalence classes $[\mathcal{R}]$ of geodesic half-lines $\mathcal{R} : [0,+\infty[\rightarrow X_L$ for the equivalence relation “to stay at bounded distance”. The point $P_L\in L/P_L$ (resp. $w\cdot P_L\in L/P_L$) corresponds to the equivalence class $[\mathcal{R}^+]$ (resp. $[\mathcal{R}^-]$) of the geodesic half-line $\mathcal{R}^+ : [0,+\infty[\rightarrow X$ (resp. $\mathcal{R}^- : [0,+\infty[\rightarrow X$) whose image is $Z_L^+\cdot x_0$ (resp. $(w\cdot Z_L^+)\cdot x_0$). Let $d$ be the distance on $X_L$ and $x_0$ the point of $X_L$ whose stabilizer is $K_L$. By (\[prelim mu distance, reel\]) and (\[prelim mu distance, ultrametrique\]), we may assume that $|\mu_L^{{\mathbb{R}}}(g)|=d(x_0,g\cdot x_0)$ for all $g\in L$. By the “shadow lemma” (see [@bou], Lem. 1.6.2, for instance), there is a constant $D_0>0$ such that the open sets $$\mathcal{U}_t = \Big\{ [\mathcal{R}],\quad \mathcal{R}(0)=x_0\ \mathrm{and}\ d\big(\mathcal{R}(t),\mathcal{R}^-(t)\big)<D_0\Big\} ,$$ for $t\in [0,+\infty[ $, form a basis of neighborhoods of $[\mathcal{R}^-]$ in $\partial X_L$. Fix $D\geq D_0$. For all $k\in K_L$ and $z_1,z_2\in Z_L^+$ with $t_1:=\mu_L^{{\mathbb{R}}}(z_1)\geq D$ and $t_2:=\mu_L^{{\mathbb{R}}}(z_2)\geq D$, we have $$\begin{aligned}
|\mu_L^{{\mathbb{R}}}(z_1kz_2)| & = & d(x_0,z_1kz_2\cdot x_0)\\
& = & d(z_1^{-1}\cdot x_0,kz_2\cdot x_0)\\
& = & d\big(\mathcal{R}^-(t_1),k\cdot\mathcal{R}^+(t_2)\big)\\
& \leq & d\big(\mathcal{R}^-(t_1),\mathcal{R}^-(D)\big) + d\big(\mathcal{R}^-(D),k\cdot\mathcal{R}^+(D)\big)\\
& & \quad\ +\, d\big(k\cdot\mathcal{R}^+(D),k\cdot\mathcal{R}^+(t_2)\big)\\
& = & t_1 - D + d\big(\mathcal{R}^-(D),k\cdot\mathcal{R}^+(D)\big) + t_2 - D\\
& = & |\mu_L^{{\mathbb{R}}}(z_1)| + |\mu_L^{{\mathbb{R}}}(z_2)| - 2D + d\big(\mathcal{R}^-(D),k\cdot\mathcal{R}^+(D)\big).\end{aligned}$$ Therefore, if $[k\cdot\mathcal{R}^+]\in\mathcal{U}_D$ then $|\mu_L^{{\mathbb{R}}}(z_1kz_2)|<|\mu_L^{{\mathbb{R}}}(z_1)|+|\mu_L^{{\mathbb{R}}}(z_2)|-D$. This completes the proof of Lemma \[equivalence transversalite\].
As in the proof of Lemma \[equivalence transversalite\], let $X_L$ denote either the Riemannian symmetric space or the Bruhat-Tits tree of $L$, depending on whether ${\mathbf{k}}$ is Archimedean or not. Let $d$ be the distance on $X_L$ and $x_0$ the point of $X_L$ whose stabilizer is $K_L$. By (\[prelim mu distance, reel\]) and (\[prelim mu distance, ultrametrique\]), we may assume that $|\mu_L^{{\mathbb{R}}}(g)|=d(g\cdot x_0,x_0)$ for all $g\in L$. Let $X'_L$ denote the convex hull of the limit set of $\Gamma$ in $X_L$. It is a closed subset of $X_L$ on which $\Gamma$ acts cocompactly: indeed, if ${\mathbf{k}}={\mathbb{R}}$ or ${\mathbb{C}}$ this is the convex cocompacity assumption; if ${\mathbf{k}}$ is non-Archimedean it follows from [@bas], Prop. 7.9. Fix a compact fundamental domain ${\mathcal{D}}$ of $X'_L$ for the action of $\Gamma$, and fix $x'_0\in{\mathrm{Int}}({\mathcal{D}})$. Let $\mathrm{d}_{{\mathcal{D}}}$ be the diameter of ${\mathcal{D}}$ and $D_0$ the constant given by Lemma \[equivalence transversalite\]. Let $$D = \max\big(D_0,6\,\mathrm{d}_{{\mathcal{D}}}+6\,d(x_0,x'_0)\big) > 0$$ and let $\mathcal{C}_L$ be the corresponding compact subset of $N_L^-$ given by Lemma \[equivalence transversalite\]. We claim that $D$ and $\mathcal{C}_L$ satisfy the conclusions of Proposition \[produit d’elements transverses\]. Indeed, let $R\geq D$. Fix $\gamma\in\Gamma$ and let $I$ be the geodesic segment of $X'_L$ with endpoints $x'_0$ and $\gamma^{-1}\cdot x'_0$. Let $n\in{\mathbb{N}}$ such that $$nR\, \leq\, d(x'_0,\gamma^{-1}\cdot x'_0)\, <\, (n+1)R.$$ For all $1\leq i\leq n$, let $x'_i\in I$ satisfy $d(x'_i,x'_0)=iR$. We have $x'_i\in\lambda_i\cdot{\mathcal{D}}$ for some $\lambda_i\in\Gamma$. Let $\gamma_0=\gamma\lambda_n\in\Gamma$ and $\gamma_i=\lambda_{n-i+1}^{-1}\lambda_{n-i}\in\Gamma$ for $i\geq 1$ (where $\lambda_0=1$), so that $\gamma=\gamma_0\ldots\gamma_n$. For all $1\leq i\leq n$, $$\begin{aligned}
\big||\mu_L^{{\mathbb{R}}}(\gamma_i)| - d(x'_{n-i},x'_{n-i+1})\big| & = & \big|d(\lambda_{n-i}\cdot x_0,\lambda_{n-i+1}\cdot x_0) - d(x'_{n-i},x'_{n-i+1})\big|\\
& \leq & d(\lambda_{n-i}\cdot x_0,\lambda_{n-i}\cdot x'_0) + d(\lambda_{n-i}\cdot x'_0,x'_{n-i})\\
& & +\, d(x'_{n-i+1},\lambda_{n-i+1}\cdot x'_0) + d(\lambda_{n-i+1}\cdot x'_0,\lambda_{n-i+1}\cdot x_0)\\
& \leq & 2\,\mathrm{d}_{{\mathcal{D}}} + 2\,d(x_0,x'_0).\end{aligned}$$ Since $d(x'_{n-i},x'_{n-i+1})=R$, we have $\big||\mu_L^{{\mathbb{R}}}(\gamma_i)|-R\big| \leq 2\,\mathrm{d}_{{\mathcal{D}}} + 2\,d(x_0,x'_0)$. Similarly, $$\big||\mu_L^{{\mathbb{R}}}(\gamma_0)| - d(x'_n,\gamma^{-1}\cdot x'_0)\big| \leq 2\,\mathrm{d}_{{\mathcal{D}}} + 2\,d(x_0,x'_0),$$ hence $|\mu_L^{{\mathbb{R}}}(\gamma_0)|\leq R+2\,\mathrm{d}_{{\mathcal{D}}}+2\,d(x_0,x'_0)$. For $1\leq i\leq n-1$, the same reasoning shows that $$\begin{aligned}
\label{comparaison entre mu(gamma_i gamma_i+1) et mu(gamma_i) + mu(gamma_i+1)}
|\mu_L^{{\mathbb{R}}}(\gamma_i\gamma_{i+1})| & \geq & d(x_{n-i-1},x_{n-i+1}) - 2\,\mathrm{d}_{{\mathcal{D}}} - 2\,d(x_0,x'_0)\nonumber\\
& = & 2R - 2\,\mathrm{d}_{{\mathcal{D}}} - 2\,d(x_0,x'_0)\nonumber\\
& \geq & |\mu_L^{{\mathbb{R}}}(\gamma_i)| + |\mu_L^{{\mathbb{R}}}(\gamma_{i+1})| - 6\,\mathrm{d}_{{\mathcal{D}}} - 6\,d(x_0,x'_0)\nonumber\\
& \geq & |\mu_L^{{\mathbb{R}}}(\gamma_i)| + |\mu_L^{{\mathbb{R}}}(\gamma_{i+1})| - D.\end{aligned}$$ By Lemma \[equivalence transversalite\], we have $\ell_{\gamma_i}k_{\gamma_{i+1}}\in \mathcal{C}_L P_L$ for all $1\leq i\leq n-1$.
We claim that $\mu_L^{{\mathbb{R}}}(\gamma_1),\ldots,\mu_L^{{\mathbb{R}}}(\gamma_n)\in{\mathbb{R}}$ all have the same sign. Indeed, we may assume that $\mathbf{L}$ has semisimple ${\mathbf{k}}$-rank $0$, in which case $\mu_L^{{\mathbb{R}}} : L\rightarrow{\mathbb{R}}$ is a group homomorphism. If $\mu_L^{{\mathbb{R}}}(\gamma_i)$ and $\mu_L^{{\mathbb{R}}}(\gamma_{i+1})$ had different signs for some $1\leq i\leq\nolinebreak n-1$, then (\[comparaison entre mu(gamma\_i gamma\_i+1) et mu(gamma\_i) + mu(gamma\_i+1)\]) would imply that $$\min\big(|\mu_L^{{\mathbb{R}}}(\gamma_i)|,|\mu_L^{{\mathbb{R}}}(\gamma_{i+1})|\big) \leq \frac{D}{2},$$ which would contradict the fact that $$\begin{aligned}
|\mu_L^{{\mathbb{R}}}(\gamma_i)|,|\mu_L^{{\mathbb{R}}}(\gamma_{i+1})| & \geq & R-2\,\mathrm{d}_{{\mathcal{D}}}-2\,d(x_0,x'_0)\\
& \geq & D-2\,\mathrm{d}_{{\mathcal{D}}}-2\,d(x_0,x'_0)\ >\ \frac{D}{2}.\qedhere\end{aligned}$$
Transversality in $G$ {#Transversalite en rang superieur}
---------------------
Let ${\mathbf{k}}$ be a local field, $\mathbf{G}$ a connected reductive algebraic ${\mathbf{k}}$-group, and $\mathbf{L}$ a closed connected reductive subgroup of $\mathbf{G}$ of ${\mathbf{k}}$-rank $1$. Fix a Cartan decomposition $G=KA^+K$ or $G=KZ^+K$.
\[decompositions de Cartan compatibles\] After conjugating $\mathbf{L}$ by some element of $G$, we may assume that $L$ admits a Cartan decomposition $L=K_LA_L^+K_L$ or $L=K_LZ_L^+K_L$ with $K_L\subset K$, with $\mathbf{A_L}\subset\mathbf{A}$, and with $A_L^+\cap A^+$ noncompact.
Indeed, $\mathbf{A_L}$ is contained in some maximal ${\mathbf{k}}$-split ${\mathbf{k}}$-torus of $\mathbf{G}$, and these tori are all conjugate over ${\mathbf{k}}$ ([@bot], Th. 4.21). Thus, after conjugating $\mathbf{L}$ by some element of $G$, we may assume that $\mathbf{A_L}\subset\mathbf{A}$. We now use a result proved by Mostow [@mos55] and Karpelevich [@kar] in the Archimedean case, and by Landvogt [@lan] in the non-Archimedean case: after conjugating $\mathbf{L}$ again by some element of $G$, we may assume that $K_L\subset K$. Finally, after conjugating $\mathbf{L}$ by some element of the Weyl group $W$, we may assume that $A_L^+\cap A^+$ is noncompact.
Assume that the conditions of Remark \[decompositions de Cartan compatibles\] are satisfied. The following lemma provides a link between Propositions \[mu d’un produit transverse selon alpha\] and \[produit d’elements transverses\]. We use the notation of Subsection \[Sous-groupes paraboliques maximaux\].
\[P\_L and P\_alpha\] If the restriction of $\alpha\in\Delta$ to $\mathbf{A_L}$ is nontrivial, then $P_L\subset P_{\alpha}$ and $N_L^- \subset N_{\alpha}^- P_{\alpha}$.
Fix $\alpha\in\Delta$ whose restriction to $\mathbf{A_L}$ is nontrivial, and let $a\in A_L^+\cap A^+$ such that $|\alpha(a)|>1$. Note that ${\mathfrak{g}}={\mathfrak{n}}_{\alpha}^-\oplus{\mathfrak{p}}_{\alpha}$ and ${\mathfrak{p}}_{\alpha}={\mathfrak{p}}_{\emptyset}\oplus{\mathfrak{n}}_{\alpha^c}^-$, where $$\begin{aligned}
{\mathfrak{n}}_{\alpha}^- & = & \bigoplus_{\beta\in\Phi^+} {\mathfrak{u}}_{-(\alpha+\beta)},\\
{\mathfrak{p}}_{\emptyset} & = & {\mathfrak{z}}\oplus \bigoplus_{\beta\in\Phi^+} {\mathfrak{u}}_{\beta},\\
\mathrm{and}\quad\quad {\mathfrak{n}}_{\alpha^c}^- & = & \bigoplus_{\beta\in{\mathbb{N}}(\Delta\smallsetminus\{ \alpha\} )} {\mathfrak{u}}_{-\beta}\end{aligned}$$ are all direct sums of eigenspaces of $\operatorname{Ad}(a)$, with eigenvalues of absolute value $<1$ on ${\mathfrak{n}}_{\alpha}^-$ and $\geq 1$ on ${\mathfrak{p}}_{\emptyset}$. Since ${\mathfrak{p}}_L$ is a sum of eigenspaces of $\operatorname{Ad}(a)$ for eigenvalues of absolute value $\geq 1$, we have ${\mathfrak{p}}_L\subset{\mathfrak{p}}_{\alpha}$. Given that $\mathbf{P_L}$ and $\mathbf{P}_{\alpha}$ are connected, this implies that $P_L\subset P_{\alpha}$. Since ${\mathfrak{n}}_L^-$ is a sum of eigenspaces of $\operatorname{Ad}(a)$ for eigenvalues of absolute value $<1$, we have ${\mathfrak{n}}_L^-\subset{\mathfrak{n}}_{\alpha}^-\oplus{\mathfrak{n}}_{\alpha^c}^-$. Note that $[{\mathfrak{n}}_{\alpha}^-,{\mathfrak{n}}_{\alpha^c}^-] \subset {\mathfrak{n}}_{\alpha}^-$, hence $N_{\alpha}^-$ is normalized by the group $N_{\alpha^c}^-$ generated by the groups $U_{-\beta}$ for $\beta\in{\mathbb{N}}(\Delta\smallsetminus\{ \alpha\} )$. This implies that $$N_L^- \,\subset\, N_{\alpha}^- N_{\alpha^c}^- \,\subset\, N_{\alpha}^- P_{\alpha}.\qedhere$$
Cartan projection and deformation {#Demonstration des theoremes}
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In this section we prove Theorem \[varphi ne change pas beaucoup mu\] using Propositions \[mu d’un produit transverse selon alpha\] and \[produit d’elements transverses\]. By the triangular inequality, it is sufficient to prove the following proposition.
\[mu d’un produit et somme des mu\] Let ${\mathbf{k}}$ be a local field, $G$ the set of ${\mathbf{k}}$-points of a connected reductive algebraic ${\mathbf{k}}$-group $\mathbf{G}$, and $L$ the set of ${\mathbf{k}}$-points of a closed reductive subgroup $\mathbf{L}$ of $\mathbf{G}$ of ${\mathbf{k}}$-rank $1$. Fix a Cartan projection $\mu : G\rightarrow E^+$ and a norm $\Vert\cdot\Vert$ on $E$. If ${\mathbf{k}}={\mathbb{R}}$ or ${\mathbb{C}}$, let $\Gamma$ be a convex cocompact subgroup of $L$; if ${\mathbf{k}}$ is non-Archimedean, let $\Gamma$ be any finitely generated discrete subgroup of $L$. Then for any $\varepsilon>0$, there exist a finite subset $F_{\varepsilon}$ of $\Gamma$, a neighborhood $\mathcal{U}_{\varepsilon}\subset{\mathrm{Hom}}(\Gamma,G)$ of the natural inclusion, and a constant $C_{\varepsilon}\geq 0$ such that any $\gamma\in\Gamma$ may be written as $\gamma=\gamma_0\ldots\gamma_n$ for some $\gamma_0,\ldots,\gamma_n\in F_{\varepsilon}$ with
1. $n \leq \varepsilon\,\Vert\mu(\gamma)\Vert + C_{\varepsilon}$,
2. $\Vert\mu(\varphi(\gamma_i))-\mu(\gamma_i)\Vert\leq 1$ for all $\varphi\in\mathcal{U}_{\varepsilon}$ and $0\leq i\leq n$,
3. for all $\varphi\in\mathcal{U}_{\varepsilon}$, $$\Big\Vert\mu(\varphi(\gamma)) - \sum_{i=0}^n \mu(\varphi(\gamma_i))\Big\Vert \leq \varepsilon \Vert\mu(\gamma)\Vert + C_{\varepsilon}.$$
The proof of Proposition \[mu d’un produit et somme des mu\] will be given in Subsection \[Demonstration de la proposition sur mu\].
Norms on $E$ and its subspaces
------------------------------
Under the assumptions of Proposition \[mu d’un produit et somme des mu\], let $G=KA^+K$ or $G=KZ^+K$ be the Cartan decomposition corresponding to $\mu$. By (\[inegalite fine pour mu\]), in order to prove Proposition \[mu d’un produit et somme des mu\], we may assume that $\mathbf{L}$ is connected and replace it by any conjugate by $G$. By Remark \[decompositions de Cartan compatibles\], we may assume that $L$ admits a Cartan decomposition $L=K_LA_L^+K_L$ or $L=K_LZ_L^+K_L$ with $K_L\subset K$, with $\mathbf{A_L}\subset\mathbf{A}$, and with $A_L^+\cap A^+$ noncompact. Let $\mu_L : L\rightarrow E_L^+$ be the corresponding Cartan projection. We naturally see $E_L$ as a line in $E$. If $\mathbf{L}$ has semisimple ${\mathbf{k}}$-rank $1$, then $E_L^+$ is a half-line in $E^+$; if $\mathbf{L}$ has semisimple ${\mathbf{k}}$-rank $0$, then $E_L^+=E_L$ is a line in $E$, intersecting $E^+$ in a half-line, and $$\mu(g) = E^+\cap W\cdot\mu_L(g)$$ for all $g\in L$. Since all norms on $E$ are equivalent, we may assume that $\Vert\cdot\Vert$ is the $W$-invariant Euclidean norm introduced in Section \[Preliminaires\]. By composing $\mu_L$ with some isomorphism from $E_L$ to ${\mathbb{R}}$, we get a Cartan projection $\mu_L^{{\mathbb{R}}} : L\rightarrow{\mathbb{R}}$ with $$|\mu_L^{{\mathbb{R}}}(g)|=\Vert\mu(g)\Vert$$ for all $g\in L$. For every $\alpha\in\Delta$ there are constants $t_{\alpha}^+,t_{\alpha}^-\geq 0$ such that $$\label{definition t_alpha}
\langle\alpha,\mu(g)\rangle =
\begin{cases}
t_{\alpha}^+\,|\mu_L^{{\mathbb{R}}}(g)| & \text{if $\mu_L^{{\mathbb{R}}}(g)\geq 0$,}\\
t_{\alpha}^-\,|\mu_L^{{\mathbb{R}}}(g)| & \text{if $\mu_L^{{\mathbb{R}}}(g)\leq 0$.}
\end{cases}$$ Let $\Delta_L = \{ \alpha\in\Delta,\ t_{\alpha}^{\pm}>0\} $ denote the set of simple roots of $\mathbf{A}$ in $\mathbf{G}$ whose restriction to $\mathbf{A_L}$ is nontrivial. Let $E_{\Delta_L}$ denote the subspace of $E$ spanned by the coroots $\check{\alpha}$ for $\alpha\in\Delta_L$, and let $\operatorname{pr}_{E_{\Delta_L}} : E\rightarrow E_{\Delta_L}$ denote the orthogonal projection on $E_{\Delta_L}$. Then $$|v|_{E_{\Delta_L}} = \Vert\operatorname{pr}_{E_{\Delta_L}}(v)\Vert$$ defines a seminorm $|\cdot|_{E_{\Delta_L}}$ on $E$. For $\alpha\in\Delta$, let $\chi_{\alpha}$ denote the highest weight of the representation $(\rho_{\alpha},V_{\alpha})$ of $\mathbf{G}$ introduced in Subsection \[Representations de G\]. Recall that $\langle\chi_{\alpha},\check{\alpha}\rangle\neq 0$ and $\langle\chi_{\alpha},\check{\beta}\rangle= 0$ for all $\beta\in\nolinebreak\Delta\smallsetminus\nolinebreak\{ \alpha\} $, hence $\{ \langle\chi_{\alpha},\cdot\rangle,\ \alpha\in\Delta_L\} $ is a basis of the dual of $E_{\Delta_L}$. Thus the function $$v \longmapsto \sum_{\alpha\in\Delta_L} |\langle\chi_{\alpha},v\rangle|$$ is a norm on $E_{\Delta_L}$. Since all norms on $E_{\Delta_L}$ are equivalent, there exists $c\geq 1$ such that $$\label{normes equivalentes}
c^{-1}\cdot\sum_{\alpha\in\Delta_L} |\langle\chi_{\alpha},v\rangle|\ \leq\ |v|_{E_{\Delta_L}}\ \leq\ c\cdot\sum_{\alpha\in\Delta_L} |\langle\chi_{\alpha},v\rangle|$$ for all $v\in E$.
Norm of the projection on $E_{\Delta_L}$
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The main step in the proof of Proposition \[mu d’un produit et somme des mu\] consists of the following proposition, which gives an upper bound for the seminorm $|\cdot|_{E_{\Delta_L}}$.
\[lemme majoration E\_Delta\_L\] Under the assumptions of Proposition \[mu d’un produit et somme des mu\], for any $\delta>0$ there exist a finite subset $F'_{\delta}$ of $\Gamma$, a neighborhood $\mathcal{U}'_{\delta}\subset{\mathrm{Hom}}(\Gamma,G)$ of the natural inclusion, and a constant $C'_{\delta}\geq 0$ such that any $\gamma\in\Gamma$ may be written as $\gamma=\gamma_0\ldots\gamma_n$ for some $\gamma_0,\ldots,\gamma_n\in F'_{\delta}$ with
1. $n\leq\delta\sum_{i=0}^n \Vert\mu(\gamma_i)\Vert$,
2. $\sum_{i=1}^n \Vert\mu(\gamma_i)\Vert = \Vert\sum_{i=1}^n \mu(\gamma_i)\Vert$,
3. $\Vert\mu(\varphi(\gamma_i))-\mu(\gamma_i)\Vert\leq 1$ for all $\varphi\in\mathcal{U}'_{\delta}$ and $0\leq i\leq n$,
4. for all $\varphi\in\mathcal{U}'_{\delta}$, $$\Big|\mu(\varphi(\gamma)) - \sum_{i=0}^n \mu(\varphi(\gamma_i))\Big|_{E_{\Delta_L}} \leq \delta\,\Big(\sum_{i=0}^n \Vert\mu(\gamma_i)\Vert\Big) + C'_{\delta}.$$
To prove Proposition \[lemme majoration E\_Delta\_L\], we use Propositions \[mu d’un produit transverse selon alpha\] and \[produit d’elements transverses\], together with Lemma \[P\_L and P\_alpha\].
Let $D>0$ be the constant and $\mathcal{C}_L$ the compact subset of $N_L^-$ given by Proposition \[produit d’elements transverses\]. By Lemma \[P\_L and P\_alpha\], for any $\alpha\in\Delta_L$, the set $\mathcal{C}_L$ is contained in ${\mathrm{Int}}(\mathcal{C}_{\alpha})P_{\alpha}$ for some compact subset $\mathcal{C}_{\alpha}$ of $N_{\alpha}^-$, where ${\mathrm{Int}}(\mathcal{C}_{\alpha})$ denotes the interior of $\mathcal{C}_{\alpha}$. Let $r_{\alpha},R_{\alpha}>0$ be the corresponding constants given by Proposition \[mu d’un produit transverse selon alpha\]. Fix $\delta>0$ and choose $R>D$ large enough so that $\frac{1}{R-D} \leq \delta$ and $\min(t_{\alpha}^+,t_{\alpha}^-)(R-D) - 1\ \geq\ R_{\alpha}$ for all $\alpha\in\Delta_L$, where $t_{\alpha}^+$ and $t_{\alpha}^-$ are defined by (\[definition t\_alpha\]). Let $F'_{\delta}$ be the set of elements $\gamma\in\Gamma$ such that $|\mu_L^{{\mathbb{R}}}(\gamma)|\leq R+D$, and $F''_{\delta}$ the subset of elements $\gamma\in F'_{\delta}$ such that $|\mu_L^{{\mathbb{R}}}(\gamma)|\geq R-D$. Note that $F'_{\delta}$ et $F''_{\delta}$ are finite since $\mu_L^{{\mathbb{R}}}$ is a proper map and $\Gamma$ is discrete in $L$. Let $\mathcal{U}'_{\delta}\subset{\mathrm{Hom}}(\Gamma,G)$ be the neighborhood of the natural inclusion whose elements $\varphi$ satisfy the following two conditions:
- $\Vert\mu(\varphi(\gamma))-\mu(\gamma)\Vert\leq 1$ and $|\langle\alpha,\mu(\varphi(\gamma))-\mu(\gamma)\rangle|\leq 1$ for all $\gamma\in F'_{\delta}$ and all $\alpha\in\Delta_L$,
- $\ell_{\varphi(\gamma)}k_{\varphi(\gamma')}\in \mathcal{C}_{\alpha} P_{\alpha}$ for all $\gamma,\gamma'\in F''_{\delta}$ with $\ell_{\gamma}k_{\gamma'}\in \mathcal{C}_L P_L$ and all $\alpha\in\nolinebreak\Delta_L$,
where for $g\in G$ we write $g = k_g z_g \ell_g$ with $k_g,\ell_g\in K$ and $z_g\in Z^+$. We claim that $F'_{\delta}$ and $\mathcal{U}'_{\delta}$ satisfy the conclusions of Proposition \[lemme majoration E\_Delta\_L\] for some constant $C'_{\delta}$. Indeed, let $\gamma\in\Gamma$. By Proposition \[produit d’elements transverses\], we may write $\gamma=\gamma_0\ldots\gamma_n$ for some elements $\gamma_0\in F'_{\delta}$ and $\gamma_1,\ldots,\gamma_n\in F''_{\delta}$ such that
- $\ell_{\gamma_i} k_{\gamma_{i+1}} \in \mathcal{C}_L P_L$ for all $1\leq i\leq n-1$,
- $\mu_L^{{\mathbb{R}}}(\gamma_1),\ldots,\mu_L^{{\mathbb{R}}}(\gamma_n)$ are all $\geq 0$ or all $\leq 0$.
This last condition implies that $\mu_L^{{\mathbb{R}}}(\gamma_1),\ldots,\mu_L^{{\mathbb{R}}}(\gamma_n)$ all belong to the same half-line in $E^+$, hence $$\sum_{i=1}^n \Vert\mu(\gamma_i)\Vert = \Big\Vert\sum_{i=1}^n \mu(\gamma_i)\Big\Vert.$$ Moreover, since $\gamma_1,\ldots,\gamma_n\in F''_{\delta}$, we have $$n \,\leq\, \frac{1}{R-D}\cdot\sum_{i=0}^n |\mu_L^{{\mathbb{R}}}(\gamma_i)| \,\leq\, \delta\,\sum_{i=0}^n \Vert\mu(\gamma_i)\Vert.$$ Let $\varphi\in\mathcal{U}'_{\delta}$. According to (\[normes equivalentes\]), in order to prove Condition (4) it is sufficient to bound $$\bigg|\Big\langle\chi_{\alpha},\mu(\varphi(\gamma)) - \sum_{i=0}^n \mu(\varphi(\gamma_i))\Big\rangle\bigg|$$ for all $\alpha\in\Delta_L$. For $\alpha\in\Delta_L$ and $1\leq i\leq n$ we have $$\begin{aligned}
\langle\alpha,\mu(\varphi(\gamma_i))\rangle & \geq & \langle\alpha,\mu(\gamma_i)\rangle - 1\\
& \geq & \min(t_{\alpha}^+,t_{\alpha}^-)\,|\mu_L^{{\mathbb{R}}}(\gamma_i)| - 1\\
& \geq & \min(t_{\alpha}^+,t_{\alpha}^-)(R-D) - 1\ \geq\ R_{\alpha}\end{aligned}$$ and $\ell_{\varphi(\gamma_i)} k_{\varphi(\gamma_{i+1})} \in \mathcal{C}_{\alpha} P_{\alpha}$. Proposition \[mu d’un produit transverse selon alpha\] thus implies that $$\begin{aligned}
\bigg|\Big\langle\chi_{\alpha},\mu(\varphi(\gamma_1\ldots\gamma_n)) - \sum_{i=1}^n \mu(\varphi(\gamma_i))\Big\rangle\bigg| & \leq & nr_{\alpha}\\
& \leq & \frac{r_{\alpha}}{R-D}\,\sum_{i=0}^n \Vert\mu(\gamma_i)\Vert.\end{aligned}$$ On the other hand, by (\[normes equivalentes\]) and (\[inegalite fine pour mu\]), $$\begin{aligned}
\big|\big\langle\chi_{\alpha},\mu(\varphi(\gamma))-\mu(\varphi(\gamma_1\ldots\gamma_n))\big\rangle\big|
& \leq & c\,\Vert\mu(\varphi(\gamma_0))\Vert\\
& \leq & c \cdot \max_{f\in F'_{\delta}} \big(\Vert\mu(f)\Vert + 1\big).\end{aligned}$$ By the triangular inequality, we finally get $$\bigg|\Big\langle\chi_{\alpha},\mu(\varphi(\gamma)) - \sum_{i=0}^n \mu(\varphi(\gamma_i))\Big\rangle\bigg| \leq \frac{r_{\alpha}}{R-D}\,\Big(\sum_{i=0}^n \Vert\mu(\gamma_i)\Vert\Big) + C'_{\delta}$$ with $$C'_{\delta} = c \cdot \max_{f\in F'_{\delta}} \big(\Vert\mu(f)\Vert + |\langle\chi_{\alpha},\mu(f)\rangle| + 2\big).$$ By (\[normes equivalentes\]), this implies Condition (4) whenever $$\sum_{\alpha\in\Delta_L} \frac{c\,r_{\alpha}}{R-D}\ \leq\ \delta,$$ which holds for $R$ large enough.
Proof of Proposition \[mu d’un produit et somme des mu\] {#Demonstration de la proposition sur mu}
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Proposition \[mu d’un produit et somme des mu\] follows from Proposition \[lemme majoration E\_Delta\_L\] and from the following general observation.
\[remarque formelle\] Let $(E,\Vert\cdot\Vert)$ be a Euclidean space, with corresponding scalar product $\langle\cdot,\cdot\rangle$, and $E_1$ a subspace of $E$. For any $x\in E$, let $|x|_1=\Vert\operatorname{pr}_1(x)\Vert$, where $\operatorname{pr}_1 : E\rightarrow E_1$ denotes the orthogonal projection on $E_1$. For any $\delta>0$, fix $C''_{\delta}\geq 0$ and let $\mathcal{I}_{\delta}$ be the set of pairs $(x,x')\in E^2$ such that
1. $x\in E_1$,
2. $|x'-x|_1\leq 2\delta\,\Vert x\Vert+C''_{\delta}$,
3. $\Vert x'\Vert\leq (1+\delta)\,\Vert x\Vert+C''_{\delta}$.
Then $$\sup_{(x,x')\in\mathcal{I}_{\delta},\ x\neq 0} \frac{\Vert x'-x\Vert - 4C''_{\delta}/\delta}{\Vert x\Vert} \longrightarrow_{\delta\rightarrow 0} 0.$$
Let $\delta\in ]0,1]$ and $(x,x')\in\mathcal{I}_{\delta}$. If $\Vert x\Vert\leq C''_{\delta}/\delta$, then $$\Vert x'-x\Vert \leq \Vert x'\Vert + \Vert x\Vert \leq (2+\delta)\,\Vert x\Vert + C''_{\delta} \leq \frac{4C''_{\delta}}{\delta}.$$ If $\Vert x\Vert\geq C''_{\delta}/\delta$, then $$\Vert x'\Vert \leq (1+2\delta)\,\Vert x\Vert$$ and $$\langle x,x'\rangle = \langle x,\operatorname{pr}_1(x')\rangle \,\geq\, \Vert x\Vert^2 - \Vert x\Vert\,|x-x'|_1 \,\geq\, (1-3\delta)\,\Vert x\Vert^2.$$ To conclude, note that for $y\in E$ with $\Vert y\Vert=1$, the diameter of the set $$\big\{ y'\in E,\quad 1-3\delta \,\leq\, \langle y,y'\rangle \,\leq\, \Vert y'\Vert \,\leq\, 1+2\delta\big\}$$ uniformly tends to $0$ with $\delta$.
Fix $\varepsilon\in ]0,1]$. For $\delta\in ]0,1]$, let $F'_{\delta}$, $\mathcal{U}'_{\delta}$, and $C'_{\delta}$ be given by Proposition \[lemme majoration E\_Delta\_L\]. Let $\gamma\in\Gamma$. By Proposition \[lemme majoration E\_Delta\_L\], we may write $\gamma=\gamma_0\ldots\gamma_n$ for some $\gamma_0,\ldots,\gamma_n\in F'_{\delta}$ such that
1. $n\leq\delta\sum_{i=0}^n \Vert\mu(\gamma_i)\Vert$,
2. $\sum_{i=1}^n \Vert\mu(\gamma_i)\Vert = \Vert\sum_{i=1}^n \mu(\gamma_i)\Vert$,
3. $\Vert\mu(\varphi(\gamma_i))-\mu(\gamma_i)\Vert\leq 1$ for all $\varphi\in\mathcal{U}'_{\delta}$ and $0\leq i\leq n$,
4. for all $\varphi\in\mathcal{U}'_{\delta}$, $$\Big|\mu(\varphi(\gamma)) - \sum_{i=0}^n \mu(\varphi(\gamma_i))\Big|_{E_{\Delta_L}} \leq \delta\,\Big(\sum_{i=0}^n \Vert\mu(\gamma_i)\Vert\Big) + C'_{\delta}.$$
Conditions (1), (3), and (4), together with (\[inegalite triangulaire pour mu\]) and the triangular inequality, imply that for all $\varphi\in\mathcal{U}'_{\delta}$, $$\begin{aligned}
\Big|\mu(\varphi(\gamma)) - \sum_{i=0}^n \mu(\gamma_i)\Big|_{E_{\Delta_L}} & \leq & \Big|\mu(\varphi(\gamma)) - \sum_{i=0}^n \mu(\varphi(\gamma_i))\Big|_{E_{\Delta_L}} + \sum_{i=0}^n \Vert\mu(\varphi(\gamma_i))-\mu(\gamma_i)\Vert\\
& \leq & 2\delta\,\Big(\sum_{i=0}^n \Vert\mu(\gamma_i)\Vert\Big) + C'_{\delta} + 1\end{aligned}$$ and $$\Vert\mu(\varphi(\gamma))\Vert \,\leq\, \sum_{i=0}^n \Vert\mu(\varphi(\gamma_i))\Vert \,\leq\, (1+\delta)\,\Big(\sum_{i=0}^n \Vert\mu(\gamma_i)\Vert\Big) + 1.$$ Moreover, Condition (2) implies that $$\label{norme de la somme des mu(gamma_i)}
\sum_{i=0}^n \Vert\mu(\gamma_i)\Vert \,\leq\, \Big\Vert\sum_{i=0}^n \mu(\gamma_i)\Big\Vert + 2\,\max_{g\in F'_{\delta}} \Vert\mu(g)\Vert.$$ Therefore, for $\varphi\in\mathcal{U}'_{\delta}$, Lemma \[remarque formelle\] applies to $$\begin{aligned}
E_1 & = & E_{\Delta_L},\\
C''_{\delta} & = & C'_{\delta} + 1 + 4\max_{g\in F'_{\delta}} \Vert\mu(g)\Vert,\\
x & = & \sum_{i=0}^n \mu(\gamma_i),\\
x' & = & \mu(\varphi(\gamma)):\end{aligned}$$ we obtain that if $\delta$ is small enough, then $$\label{mu(varphi(gamma)) et somme des mu(gamma_i)}
\Big\Vert\mu(\varphi(\gamma)) - \sum_{i=0}^n \mu(\gamma_i)\Big\Vert \,\leq\, \frac{\varepsilon}{4}\ \Big\Vert\sum_{i=0}^n \mu(\gamma_i)\Big\Vert + \frac{4C''_{\delta}}{\delta}$$ for all $\varphi\in\mathcal{U}'_{\delta}$. Now Conditions (1) and (3), together with the triangular inequality and (\[norme de la somme des mu(gamma\_i)\]), imply that $$\begin{aligned}
\Big\Vert\Big(\sum_{i=0}^n \mu(\varphi(\gamma_i))\Big) - \Big(\sum_{i=0}^n \mu(\gamma_i)\Big)\Big\Vert & \leq & \delta\,\Big(\sum_{i=0}^n \Vert\mu(\gamma_i)\Vert\Big) + 1,\\
& \leq & \delta\,\Big\Vert\sum_{i=0}^n \mu(\gamma_i)\Big\Vert + 2\delta\,\max_{g\in F'_{\delta}} \Vert\mu(g)\Vert + 1.\end{aligned}$$ Therefore, if $\delta$ is small enough, then $$\Big\Vert\mu(\varphi(\gamma)) - \sum_{i=0}^n \mu(\varphi(\gamma_i))\Big\Vert \,\leq\, \frac{\varepsilon}{2}\ \Big\Vert\sum_{i=0}^n \mu(\gamma_i)\Big\Vert + C'''_{\delta}$$ for all $\varphi\in\mathcal{U}'_{\delta}$, where $$C'''_{\delta} = \frac{4C''_{\delta}}{\delta} + 2\delta\,\max_{g\in F'_{\delta}} \Vert\mu(g)\Vert + 1.$$ In particular, taking $\varphi$ to be the natural inclusion of $\Gamma$ in $G$, we get $$\label{mu(gamma) et somme des mu(gamma_i)}
\Big\Vert\sum_{i=0}^n \mu(\gamma_i)\Big\Vert \,\leq\, \frac{1}{1-\frac{\varepsilon}{2}}\,\big(\Vert\mu(\gamma)\Vert + C'''_{\delta}\big) \,\leq\, 2\,\Vert\mu(\gamma)\Vert + 2C'''_{\delta},$$ hence $$\Big\Vert\mu(\varphi(\gamma)) - \sum_{i=0}^n \mu(\varphi(\gamma_i))\Big\Vert \leq \varepsilon\,\Vert\mu(\gamma)\Vert + 3C'''_{\delta}$$ for all $\varphi\in\mathcal{U}'_{\delta}$ whenever $\delta$ is small enough. Finally, Condition (1), together with (\[norme de la somme des mu(gamma\_i)\]) and (\[mu(gamma) et somme des mu(gamma\_i)\]), implies that $$n \,\leq\, 2\delta\,\Vert\mu(\gamma)\Vert + 3C'''_{\delta}.$$ Thus the triple $(F_{\varepsilon},\mathcal{U}_{\varepsilon},C_{\varepsilon})=(F'_{\delta},\mathcal{U}'_{\delta},3C'''_{\delta})$ satisfies the conclusions of Proposition \[mu d’un produit et somme des mu\] for $\delta$ small enough.
Properness and deformation {#Proprete and deformations}
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Let us now briefly explain how to deduce Theorems \[proprete, groupes de Lie\] and \[proprete, groupes algebriques\] from Theorem \[varphi ne change pas beaucoup mu\].
Theorem \[proprete, groupes algebriques\] follows from Theorem \[varphi ne change pas beaucoup mu\] and from the *properness criterion* of Benoist ([@ben96], Cor. 5.2) and Kobayashi ([@kob96], Th. 1.1). Under the assumptions of Theorem \[proprete, groupes algebriques\], this criterion states that a subgroup $\Gamma$ of $G$ acts properly on $G/H$ if and only if the set $\mu(\Gamma)\cap(\mu(H)+\mathcal{C})$ is bounded for any compact subset $\mathcal{C}$ of $E$. This condition means that the set $\mu(\Gamma)$ “gets away from $\mu(H)$ at infinity”.
We may assume that $\mathbf{G}$, $\mathbf{H}$, and $\mathbf{L}$ are all connected. Let $G=KA^+K$ or $G=KZ^+K$ be a Cartan decomposition of $G$, and let $\mu : G\rightarrow E^+$ be the corresponding Cartan projection. Endow $E$ with a $W$-invariant norm $\Vert\cdot\Vert$ as in Section \[Preliminaires\]. By Remark \[decompositions de Cartan compatibles\], we may assume that $L$ admits a Cartan decomposition $L=K_LA_L^+K_L$ or $L=K_LZ_L^+K_L$ with $K_L\subset K$ and $\mathbf{A_L}\subset\mathbf{A}$. If $\mu_L : L\rightarrow E_L^+$ denotes the corresponding Cartan projection, then $E_L$ is naturally seen as a line in $E$ and $$\mu(\ell) = E^+\cap (W\cdot\mu_L(\ell))$$ for all $\ell\in L$. Thus $\mu(L)$ is contained in the union $U_L$ of two half-lines of $E^+$. Using Remark \[decompositions de Cartan compatibles\] again, there is an element $g\in G$ such that $gHg^{-1}$ admits a Cartan decomposition $gHg^{-1}=K_HA_H^+K_H$ or $gHg^{-1}=K_HZ_H^+K_H$ with $K_H\subset K$ and $\mathbf{A_H}\subset\mathbf{A}$. The set $\mu(gHg^{-1})$ is contained in a finite union $U_H$ of subspaces of $E$ intersected with $E^+$, parametrized by the Weyl group $W$. By (\[inegalite fine pour mu\]), the Hausdorff distance between $\mu(gHg^{-1})$ and $\mu(H)$ is $\leq 2\,\Vert\mu(g)\Vert$. Therefore $U_L\cap U_H=\{ 0\} $ by the properness criterion, and there is a constant $\varepsilon>0$ such that $$d(\mu(\ell),\mu(H))\geq 2\varepsilon\,\Vert\mu(\ell)\Vert-2\,\Vert\mu(g)\Vert$$ for all $\ell\in L$. By Theorem \[varphi ne change pas beaucoup mu\], there is a neighborhood $\mathcal{U}_{\varepsilon}\subset{\mathrm{Hom}}(\Gamma,G)$ of the natural inclusion and a constant $C_{\varepsilon}\geq 0$ such that $$\Vert\mu(\varphi(\gamma))-\mu(\gamma)\Vert \leq \varepsilon\,\Vert\mu(\gamma)\Vert + C_{\varepsilon}$$ for all $\varphi\in\mathcal{U}_{\varepsilon}$ and $\gamma\in\Gamma$. Fix $\varphi\in\mathcal{U}_{\varepsilon}$. For all $\gamma\in\Gamma$, $$\begin{aligned}
d(\mu(\varphi(\gamma)),\mu(H)) & \geq & d(\mu(\gamma),\mu(H)) - \Vert\mu(\varphi(\gamma))-\mu(\gamma)\Vert\\
& \geq & \varepsilon\,\Vert\mu(\gamma)\Vert - C_{\varepsilon} - 2\,\Vert\mu(g)\Vert.\end{aligned}$$ Therefore, using the fact that $\Gamma$ is discrete in $G$ and $\mu$ is a proper map, we get that $\mu(\varphi(\Gamma))\cap(\mu(H)+\mathcal{C})$ is finite for any compact subset $\mathcal{C}$ of $E$. By the properness criterion, this implies that $\varphi(\Gamma)$ acts properly on $G/H$. It also implies that $\varphi(\Gamma)$ is discrete in $G$ and that the kernel of $\varphi$ is finite. Since $\Gamma$ is torsion-free, $\varphi$ is injective.
By Theorem \[proprete, groupes algebriques\], there is a neighborhood $\mathcal{U}\subset\linebreak{\mathrm{Hom}}(\Gamma,G)$ of the natural inclusion such that any $\varphi\in\mathcal{U}$ is injective and $\varphi(\Gamma)$ is discrete in $G$, acting properly discontinuously on $G/H$. Since $\varphi\in\mathcal{U}$ is injective, $\varphi(\Gamma)$ has the same cohomological dimension as $\Gamma$. We conclude using the fact, due to Kobayashi ([@kob89], Cor. 5.5), that a torsion-free discrete subgroup of $G$ acts cocompactly sur $G/H$ if and only if its cohomological dimension is $d(G)-d(H)$, where $d(G)$ (resp. $d(H)$) denotes the dimension of the symmetric space of $G$ (resp. of $H$).
Application to the compact quotients of ${\mathrm{SO}}(2n,2)/{\mathrm{U}}(n,1)$ {#quotients compacts Zariski-denses}
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Fix an integer $n\geq 1$. Note that ${\mathrm{U}}(n,1)$ naturally embeds in ${\mathrm{SO}}(2n,2)$ by identifying the Hermitian form $|z_1|^2+\ldots+|z_n|^2-|z_{n+1}|^2$ on ${\mathbb{C}}^{n+1}$ with the quadratic form $x_1^2+\ldots+x_{2n}^2-x_{2n+1}^2-x_{2n+2}^2$ on ${\mathbb{R}}^{2n+2}$. As Kulkarni [@kul] pointed out, ${\mathrm{U}}(n,1)$, seen as a subgroup of ${\mathrm{SO}}(2n,2)$, acts transitively on the anti-de Sitter space $$\begin{aligned}
\mathrm{AdS}^{2n+1} & = & \big\{ (x_1,\ldots,x_{2n+2})\in{\mathbb{R}}^{2n+2},\ x_1^2 + \ldots + x_{2n}^2 - x_{2n+1}^2 - x_{2n+2}^2 = -1\big\} \\
& \simeq & {\mathrm{SO}}(2n,2)/{\mathrm{SO}}(2n,1).\end{aligned}$$ The stabilizer of $(0,\ldots,0,1)$ is the compact subgroup ${\mathrm{U}}(n)$, hence $\mathrm{AdS}^{2n+1}$ identifies with ${\mathrm{U}}(n,1)/{\mathrm{U}}(n)$ and the action of ${\mathrm{U}}(n,1)$ on ${\mathrm{SO}}(2n,2)/{\mathrm{SO}}(2n,1)$ is proper. By duality, the action of ${\mathrm{SO}}(2n,1)$ on ${\mathrm{SO}}(2n,2)/{\mathrm{U}}(n,1)$ is proper and transitive. In particular, any uniform lattice $\Gamma$ of ${\mathrm{SO}}(2n,1)$ provides a standard compact quotient $\Gamma\backslash{\mathrm{SO}}(2n,2)/{\mathrm{U}}(n,1)$ of ${\mathrm{SO}}(2n,2)/{\mathrm{U}}(n,1)$.
Corollary \[quotients compacts de SO(2n,2)/SU(n,1)\] follows from Theorem \[proprete, groupes de Lie\] and from the existence of Zariski-dense deformations in ${\mathrm{SO}}(m,2)$ of certain uniform lattices of ${\mathrm{SO}}(m,1)$. Such deformations can be obtained by a bending construction due to Johnson and Millson. This construction is presented in [@jm] for deformations in ${\mathrm{SO}}(m+1,1)$ or in ${\mathrm{PGL}}_{m+1}({\mathbb{R}})$, but not in ${\mathrm{SO}}(m,2)$. For the reader’s convenience, we shall describe Johnson and Millson’s construction in the latter case, and check that the deformations obtained is this way are indeed Zariski-dense in ${\mathrm{SO}}(m,2)$.
From now on we use Gothic letters to denote the Lie algebras of real Lie groups (*e.g.* ${\mathfrak{g}}$ for $G$).
Uniform arithmetic lattices of ${\mathrm{SO}}(m,1)$
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Fix $m\geq 2$. The uniform lattices of ${\mathrm{SO}}(m,1)$ considered by Johnson and Millson are obtained in the following classical way. Fix a square-free integer $r\geq 2$ and identify ${\mathrm{SO}}(m,1)$ with the special orthogonal group of the quadratic form $$x_1^2+\ldots+x_m^2-\sqrt{r}x_{m+1}^2$$ on ${\mathbb{R}}^{m+1}$. Let $\mathcal{O}_r$ denote the ring of integers of the quadratic field ${\mathbb{Q}}(\sqrt{r})$. The group $\Gamma={\mathrm{SO}}(m,1)\cap{\mathrm{M}}_{m+1}(\mathcal{O}_r)$ is a uniform lattice in ${\mathrm{SO}}(m,1)$ (see [@bor63] for instance). For any ideal $I$ of $\mathcal{O}_r$, the congruence subgroup $\Gamma\cap (1+{\mathrm{M}}_{m+1}(I))$ has finite index in $\Gamma$, hence is a uniform lattice in ${\mathrm{SO}}(m,1)$. By [@mr], after replacing $\Gamma$ by such a congruence subgroup, we may assume that it is torsion-free. Then $M=\Gamma\backslash\mathbb{H}^m$ is a $m$-dimensional compact hyperbolic manifold whose fundamental group identifies with $\Gamma$. By [@jm], Lem. 7.1 & Th. 7.2, after possibly replacing $\Gamma$ again by some congruence subgroup, we may assume that $N=\Gamma_0\backslash{\mathbb{H}}^{m-1}$ is a connected, orientable, totally geodesic hypersurface of $M$, where $$\Gamma_0=\Gamma\cap{\mathrm{SO}}(m-1,1)$$ and where $${\mathbb{H}}^{m-1} \simeq \big\{ (x_2,\ldots,x_{m+1})\in{\mathbb{R}}^m,\ x_2^2+\ldots+x_m^2-\sqrt{r}x_{m+1}^2=-1\ \mathrm{and}\ x_{m+1}>0\big\}$$ is embedded in $${\mathbb{H}}^m \simeq \{ (x_1,\ldots,x_{m+1})\in{\mathbb{R}}^{m+1},\ x_1^2+\ldots+x_m^2-\sqrt{r}x_{m+1}^2=-1\ \mathrm{and}\ x_{m+1}>0\}$$ in the natural way. Since the centralizer of $\Gamma_0$ in ${\mathrm{SO}}(m,2)$ contains a subgroup isomorphic to ${\mathrm{SO}}(1,1)\simeq{\mathbb{R}}^{\ast}$, the idea of the bending construction is to deform $\Gamma$ “along this centralizer”, as we shall now explain.
Deformations in the separating case
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Assume that $N$ separates $M$ into two components $M_1$ and $M_2$, and let $\Gamma_1$ (resp. $\Gamma_2$) denote the fundamental group of $M_1$ (resp. of $M_2$). By van Kampen’s theorem, $\Gamma$ is the amalgamated product $\Gamma_1\ast_{\Gamma_0}\Gamma_2$. Fix an element $Y\in{\mathfrak{so}}(m,2)\smallsetminus{\mathfrak{so}}(m,1)$ that belongs to the Lie algebra of the centralizer of $\Gamma_0$ in ${\mathrm{SO}}(m,2)$. Following Johnson and Millson, we consider the deformations of $\Gamma$ in ${\mathrm{SO}}(m,2)$ that are given, for $t\in{\mathbb{R}}$, by $$\varphi_t(\gamma) =
\begin{cases}
\ \ \,\,\,\,\,\gamma & \text{for $\gamma\in\Gamma_1$,}\\
e^{tY}\gamma e^{-tY} & \text{for $\gamma\in\Gamma_2$.}
\end{cases}$$ Note that $\varphi_t : \Gamma\rightarrow{\mathrm{SO}}(m,2)$ is well defined since $e^{tY}$ centralizes $\Gamma_0$. Moreover, it is injective and $\varphi_t(\Gamma)$ is discrete in ${\mathrm{SO}}(m,2)$. We now check Zariski-density.
\[lemme cas separant\] For $t\neq 0$ small enough, $\varphi_t(\Gamma)$ is Zariski-dense in ${\mathrm{SO}}(m,2)$.
We need the following remark.
\[remarque algebres de Lie\] For $m\geq 2$, the only Lie subalgebra of ${\mathfrak{so}}(m,2)$ that strictly contains ${\mathfrak{so}}(m,1)$ is ${\mathfrak{so}}(m,2)$.
Indeed, ${\mathfrak{so}}(m,2)$ decomposes uniquely into a direct sum ${\mathfrak{so}}(m,1)\oplus\nolinebreak{\mathbb{R}}^{m+1}$ of irreducible ${\mathrm{SO}}(m,1)$-modules, where ${\mathrm{SO}}(m,1)$ acts on ${\mathfrak{so}}(m,1)$ (resp. on ${\mathbb{R}}^{m+1}$) by the adjoint (resp. natural) action.
Recall that ${\mathrm{SO}}(m,2)$ is Zariski-connected. Therefore, in order to prove that $\varphi_t(\Gamma)$ is Zariski-dense in ${\mathrm{SO}}(m,2)$, it is sufficient to prove that the Lie algebra of $\overline{\varphi_t(\Gamma)}$ is ${\mathfrak{so}}(m,2)$, where $\overline{\varphi_t(\Gamma)}$ denotes the Zariski closure of $\varphi_t(\Gamma)$ in ${\mathrm{SO}}(m,2)$.
By [@jm], Lem. 5.9, the groups $\Gamma_1$ and $\Gamma_2$ are Zariski-dense in ${\mathrm{SO}}(m,1)$. By [@jm], Cor. 5.3, and [@ser77], § I.5.2, Cor. 1, they naturally embed in $\Gamma$. Therefore $\overline{\varphi_t(\Gamma)}$ contains both ${\mathrm{SO}}(m,1)$ and $e^{tY}{\mathrm{SO}}(m,1) e^{-tY}$, and the Lie algebra of $\overline{\varphi_t(\Gamma)}$ contains both ${\mathfrak{so}}(m,1)$ and the Lie algebra of $e^{tY}{\mathrm{SO}}(m,1) e^{-tY}$. By Remark \[remarque algebres de Lie\], in order to prove that $\varphi_t(\Gamma)$ is Zariski-dense in ${\mathrm{SO}}(m,2)$, it is sufficient to prove that the Lie algebra of $e^{tY}{\mathrm{SO}}(m,1) e^{-tY}$ is not ${\mathfrak{so}}(m,1)$.
But if the Lie algebra of $e^{tY}{\mathrm{SO}}(m,1) e^{-tY}$ were ${\mathfrak{so}}(m,1)$, then we would have $e^{tY}{\mathrm{SO}}(m,1)^{\circ}e^{tY}={\mathrm{SO}}(m,1)^{\circ}$, *i.e.*, $e^{tY}$ would belong to the normalizer $N_{{\mathrm{SO}}(m,2)}({\mathrm{SO}}(m,1)^{\circ})$ of the identity component ${\mathrm{SO}}(m,1)^{\circ}$ of ${\mathrm{SO}}(m,1)$. Recall that the exponential map induces a diffeomorphism between a neighborhood $\mathcal{U}$ of $0$ in ${\mathfrak{so}}(m,2)$ and a neighborhood $\mathcal{V}$ of $1$ in ${\mathrm{SO}}(m,2)$, which itself induces a one-to-one correspondence between $\mathcal{U}\cap{\mathfrak{n}}_{{\mathfrak{so}}(m,2)}({\mathfrak{so}}(m,1))$ and $\mathcal{V}\cap N_{{\mathrm{SO}}(m,2)}({\mathrm{SO}}(m,1)^{\circ})$. Therefore, if we had $e^{tY}\in N_{{\mathrm{SO}}(m,2)}({\mathrm{SO}}(m,1)^{\circ})$ for some $t\neq 0$ small enough, then we would have $$Y \in {\mathfrak{n}}_{{\mathfrak{so}}(m,2)}({\mathfrak{so}}(m,1)) = \{ X\in{\mathfrak{so}}(m,2),\ \operatorname{ad}(X)({\mathfrak{so}}(m,1))={\mathfrak{so}}(m,1)\} .$$ But Remark \[remarque algebres de Lie\] implies that ${\mathfrak{n}}_{{\mathfrak{so}}(m,2)}({\mathfrak{so}}(m,1))$ is equal to ${\mathfrak{so}}(m,1)$, since it contains ${\mathfrak{so}}(m,1)$ and is different from ${\mathfrak{so}}(m,2)$. Thus we would have $Y\in{\mathfrak{so}}(m,1)$, which would contradict our choice of $Y$.
Deformations in the nonseparating case
--------------------------------------
We now assume that $S=M\smallsetminus N$ is connected. Let $j_1:\Gamma_0\rightarrow\pi_1(S)$ and $j_2:\Gamma_0\rightarrow\pi_1(S)$ denote the inclusions in $\pi_1(S)$ of the fundamental groups of the two sides of $N$. The group $\Gamma$ is a HNN extension of $\pi_1(S)$, *i.e.*, it is generated by $\pi_1(S)$ and by some element $\nu\in\Gamma$ such that $$\nu\,j_1(\gamma)\,\nu^{-1} = j_2(\gamma)$$ for all $\gamma\in\Gamma_0$. Fix an element $Y\in{\mathfrak{so}}(m,2)\smallsetminus{\mathfrak{so}}(m,1)$ that belongs to the Lie algebra of the centralizer of $j_1(\Gamma_0)$ in ${\mathrm{SO}}(m,2)$. Following Johnson and Millson, we consider the deformations of $\Gamma$ in ${\mathrm{SO}}(m,2)$ that are given, for $t\in{\mathbb{R}}$, by $$\left \{
\begin{array}{c @{\ =\ } l l}
\varphi_t(\gamma) & \ \,\gamma & \mathrm{for}\ \gamma\in\pi_1(S),\\
\varphi_t(\nu) & \nu e^{tY}.
\end{array}
\right.$$ Note that $\varphi_t : \Gamma\rightarrow{\mathrm{SO}}(m,2)$ is well defined since $e^{tY}$ centralizes $j_1(\Gamma_0)$. Moreover, it is injective and $\varphi_t(\Gamma)$ is discrete in ${\mathrm{SO}}(m,2)$.
For $t\neq 0$ small enough, $\varphi_t(\Gamma)$ is Zariski-dense in ${\mathrm{SO}}(m,2)$.
Let $\overline{\varphi_t(\Gamma)}$ denote the Zariski closure of $\varphi_t(\Gamma)$ in ${\mathrm{SO}}(m,2)$. By [@jm], Lem. 5.9, the group $\pi_1(S)$ is Zariski-dense in ${\mathrm{SO}}(m,1)$, hence $\overline{\varphi_t(\Gamma)}$ contains both ${\mathrm{SO}}(m,1)$ and $\nu e^{tY}$. But $\nu\in{\mathrm{SO}}(m,1)$, hence $e^{tY}\in\overline{\varphi_t(\Gamma)}$. Therefore $\overline{\varphi_t(\Gamma)}$ contains both ${\mathrm{SO}}(m,1)$ and $e^{tY}{\mathrm{SO}}(m,1)e^{-tY}$, and we may conclude as in the proof of Lemma \[lemme cas separant\].
[\[BoT\]]{}
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|
---
abstract: 'Astronomy depends on ever increasing computing power. Processor clock-rates have plateaued, and increased performance is now appearing in the form of additional processor cores on a single chip. This poses significant challenges to the astronomy software community. Graphics Processing Units (GPUs), now capable of general-purpose computation, exemplify both the difficult learning-curve and the significant speedups exhibited by massively-parallel hardware architectures. We present a generalised approach to tackling this paradigm shift, based on the analysis of algorithms. We describe a small collection of foundation algorithms relevant to astronomy and explain how they may be used to ease the transition to massively-parallel computing architectures. We demonstrate the effectiveness of our approach by applying it to four well-known astronomy problems: Högbom <span style="font-variant:small-caps;">clean</span>, inverse ray-shooting for gravitational lensing, pulsar dedispersion and volume rendering. Algorithms with well-defined memory access patterns and high arithmetic intensity stand to receive the greatest performance boost from massively-parallel architectures, while those that involve a significant amount of decision-making may struggle to take advantage of the available processing power.'
bibliography:
- 'abbrevs.bib'
- 'benbarsdell.bib'
title: Analysing Astronomy Algorithms for GPUs and Beyond
---
\[firstpage\]
methods: data analysis – gravitational lensing: micro – pulsars: general
Introduction
============
Computing resources are a fundamental tool in astronomy: they are used to acquire and reduce observational data, simulate astrophysical processes, and analyse and visualise the results. Advances in the field of astronomy have depended heavily on the increase in computing power that has followed Moore’s Law [@moore1965] since the mid 1960s; indeed, many contemporary astronomy survey projects and astrophysics simulations would simply not be possible without the evolution Moore predicted.
Until recently, increased computing power was delivered in direct proportion to the increase in central processing unit (CPU) clock rates. Astronomy software executed more and more rapidly with each new hardware release, without any further programming work. But around 2005, the advance in clock rates ceased, and manufacturers turned to increasing the [*instantaneous*]{} processing capacity of their CPUs by including additional processing cores in a single silicon chip package. Today’s mainstream multi-core CPUs typically have between 2 and 8 processing cores; these are routinely deployed in large-scale compute clusters.
![Clock-rate versus core-count phase space of Moore’s Law binned every two years for CPUs (circles) and GPUs (diamonds). There is a general trend for performance to increase from bottom left to top right.[]{data-label="fig:MulticoreMooresLaw"}](fig1){width="8.5cm"}
Fig. \[fig:MulticoreMooresLaw\] places the mainstream CPUs from the last $\sim 20$ years in the clock rate versus core-count phase space. In this space, the evolution of CPUs turns a ‘corner’ around 2005 when clock rates plateaued and multi-core processors emerged. Lying above today’s fastest multi-core CPUs in Fig. \[fig:MulticoreMooresLaw\] though, are the contemporary graphics processing units (GPUs), boasting hundreds of cores (‘[ *many-core*]{}’) and $\sim 1$ Ghz clock rates. GPUs are already useful in their own right—providing $\sim 30$ times the raw computation speed of CPUs—but perhaps more interestingly, they represent the likely evolution of CPUs. GPUs demonstrate how computing power can continue to follow Moore’s Law in an era of zero (or even negative) growth in clock rates.
The plateau in processor clock rates is problematic for astronomy software composed of [*sequential*]{} codes, wherein instructions are executed one after the other. Such codes derive no direct performance benefit from the presence of multiple processing cores, and their performance will languish for as long as processor clock rates remain steady at $\sim3$–4 GHz. Astronomy software must be (re-)written to take advantage of many-core processors.
Astronomers are already cogniscent of this issue. Shared- and distributed-memory multi-core CPU machines have been exploited using the well-known OpenMP and MPI programming models[^1]. In addition, a number of researchers have adapted, written and/or re-written classic astronomy codes for the GPU architecture in the last $\sim3$ years, and gained performance improvements ranging from factors of a few to factors of several hundred. Some highlights include N-body (e.g., @HamadaEtal2009), radio-telescope signal correlation (e.g., @WaythEtal2009), adaptive mesh refinement (e.g., @SchiveEtal2010), galaxy spectral energy distribution [@JonssonPrimack2009] and gravitational microlensing [@ThompsonEtal2010] codes.
Inevitably, a section of the astronomy community will continue with an [*ad hoc*]{} approach to the adaptation of software from single-core to many-core architectures. In this paper, we demonstrate that there is a significant difference between current computing techniques and those required to efficiently utilise new hardware architectures such as many-core processors, as exemplified by GPUs. These techniques will be unfamiliar to most astronomers and will pose a challenge in terms of keeping our discipline at the forefront of computational science. We present a practical, effective and simple methodology for creating astronomy software whose performance scales well to present and future many-core architectures. Our methodology is grounded in the classical computer science field of algorithm analysis.
In Section \[sec:ourmethodology\] we introduce the key concepts in algorithm analysis, with particular focus on the context of many-core architectures. We present four foundation algorithms, and characterise them as we outline our algorithm analysis methodology. In Section \[sec:AstronomyAlgorithms\] we demonstrate the proposed methodology by applying it to four well-known astronomy problems, which we break down into their constituent foundation algorithms. We validate our analysis of these problems against [*ad hoc*]{} many-core implementations as available in the literature and discuss the implications of our approach for the future of computing in astronomy in Section \[sec:discussion\].
A Strategic Approach: Algorithm Analysis {#sec:ourmethodology}
========================================
Algorithm analysis, pioneered by Donald Knuth (see, e.g., @Knuth1998), is a fundamental component of computer science – a discipline that is more about how to solve problems than the actual implementation in code. In this work, we are not interested in the specifics (i.e., syntax) of implementing a given astronomy algorithm with a particular programming language or library (e.g., CUDA, OpenCL, Thrust) on a chosen computing architecture (e.g., GPU, Cell, FPGA). As @Harris2007 notes, algorithm-level optimisations are much more important with respect to overall performance on many-core hardware (specifically GPUs) than implementation optimisations, and should be made first. We will return to the issue of implementation in future work.
Here we present an approach to tackling the transition to many-core hardware based on the analysis of algorithms. The purpose of this analysis is to determine the potential of a given algorithm for a many-core architecture *before* any code is written. This provides essential information about the optimal approach as well as the return on investment one might expect for the effort of (re-)implementing a particular algorithm. Our methodology was in part inspired by the work of @Harris2005.
Work in a similar vein has also been undertaken by @AsanovicEtal2006 [@AsanovicEtal2009] who classified parallel algorithms into 12 groups, referring to them as ‘dwarfs’. While insightful and opportune, these dwarfs consider a wide range of parallel architectures, cover all areas of computation (including several that are not of great relevance to astronomy) and are limited as a resource by the coarse nature of the classification. In contrast, the approach presented here is tailored to the parallelism offered by many-core processor architectures, contains algorithms that appear frequently within astronomy computations, and provides a fine-grained level of detail. Furthermore, our approach considers the fundamental concerns raised by many-core architectures at a level of abstraction that avoids dealing with hardware or software-specific details and terminology. This is in contrast to the work by @CheEtal2008, who presented a useful but highly-targeted summary of general-purpose programming on the NVIDIA GPU architecture.
For these reasons this work will serve as a valuable and practical resource for those wishing to analyse the expected performance of particular astronomy algorithms on current and future many-core architectures.
For a given astronomy problem, our methodology is as follows:
1. Outline each step in the problem.
2. Identify steps that resemble known algorithms (see below).
1. Outlined steps may need to be further decomposed into sub-steps before a known counterpart is recognised. Such composite steps may later be added to the collection of known algorithms.
3. For each identified algorithm, refer to its pre-existing analysis.
1. Where a particular step does not appear to match any known algorithm, refer to a relevant analysis methodology to analyse the step as a custom algorithm (see Sections \[sec:CriticalIssues\], \[sec:complexity\_analysis\] and \[sec:analysis\]). The newly-analysed algorithm can then be added to the collection for future reference.
4. Once analysis results have been obtained for each step, apply a global analysis to the algorithm to obtain a complete picture of its behaviour (see Section \[sec:macro\_analysis\]).
Here we present a small collection of foundation algorithms[^2] that appear in computational astronomy problems. This is motivated by the fact that complex algorithms may be *composed* from simpler ones. We propose that *algorithm composition* provides an excellent approach to turning the multi-core corner. Here we focus on its application to algorithm analysis; in future work we will show how it may also be applied to implementation methodologies. The algorithms are described below using a *vector* data structure. This is a data structure like a Fortran or C array representing a contiguous block of memory and providing constant-time random access to individual elements[^3]. We use the notation $v[i]$ to represent the $i^{\rm th}$ element of a vector $v$.
**Transform:** Returns a vector containing the result of the application of a specified function to every individual element of an input vector. $${\rm out}[i] = f({\rm in}[i])$$ Functions of more than one variable may also be applied to multiple input vectors. Scaling the brightness of an image (defined as a vector of pixels) is an example of a transform operation.
**Reduce:** Returns the sum of every element in a vector. $${\rm out} = \sum_i {\rm in}[i]$$ Reductions may be generalised to use any associative binary operator, e.g., product, min, max etc. Calculating image noise is a common application of the reduce algorithm.
**Gather:** Retrieves values from an input vector according to a specified index mapping and writes them to an output vector. $${\rm out}[i] = {\rm in}[{\rm map}[i]]$$ Reading a shifted or transformed subregion of an image is a common example of a gather operation.
**Interact:** For each element $i$ of an input vector, in$_1$, sums the interaction between $i$ and each element $j$ in a second input vector, in$_2$. $${\rm out}[i] = \sum_j f({\rm in}_1[i], {\rm in}_2[j])$$ where $f$ is a given interaction function. The best-known application of this algorithm in astronomy is the computation of forces in a direct N-body simulation, where both input vectors represent the system’s particles and the interaction function calculates the gravitational force between two particles.
These four algorithms were chosen from experience with a number of computational astronomy problems. The transform, reduce and gather operations may be referred to as ‘atoms’ in the sense that they are indivisible operations. While the interact algorithm is technically a composition of transforms and reductions, it will be analysed as if it too was an atom, enabling rapid analysis of problems that use the interact algorithm without the need for further decomposition.
We now describe a number of algorithm analysis techniques that we have found to be relevant to massively-parallel architectures. These techniques should be applied to the individual algorithms that comprise a complete problem in order to gain a detailed understanding of their behaviour.
Principle characteristics {#sec:CriticalIssues}
-------------------------
Many-core architectures exhibit a number of characteristics that can impact strongly on the performance of an algorithm. Here we summarise four of the most important issues that must be considered.
**Massive parallelism:** To fully utilise massively-parallel architectures, algorithms must exhibit a high level of parallel *granularity*, i.e., the number of required operations that may be performed simultaneously must be large and scalable. *Data-parallel* algorithms, which divide their *data* between parallel processors rather than (or in addition to) their *tasks*, exhibit parallelism that scales with the size of their input data, making them ideal candidates for massively-parallel architectures. However, performance may suffer when these algorithms are executed on sets of input data that are small relative to the number of processors in a particular many-core architecture[^4].
**Memory access patterns:** Many-core architectures contain very high bandwidth main memory[^5] in order to ‘feed’ the large number of parallel processing units. However, high latency (i.e., memory transfer startup) costs mean that performance depends strongly on the *pattern* in which memory is accessed. In general, maintaining ‘locality of reference’ (i.e., neighbouring threads accessing similar locations in memory) is vital to achieving good performance[^6]. Fig. \[fig:memory\_access\_patterns\] illustrates different levels of locality of reference.
![Representative memory access patterns indicating varying levels of locality of reference. Contiguous memory access is the optimal case for many-core architectures. Patterns with high locality will generally achieve good performance; those with low locality may incur severe performance penalties.[]{data-label="fig:memory_access_patterns"}](fig2){width="8cm"}
Collisions between threads trying to read the same location in memory can also be costly, and write-collisions must be treated using expensive atomic operations in order to avoid conflicts between threads.
**Branching:** Current many-core architectures rely on single instruction multiple data (SIMD) hardware. This means that neighbouring threads that wish to execute different instructions must wait for each other to complete the divergent code section before execution can continue in parallel (see Fig. \[fig:branching\]). For this reason, algorithms that involve significant branching between different threads may suffer severe performance degradation. Similar to the effects of memory access locality, performance will in general depend on the locality of branching, i.e., the number of different code-paths taken by a group of neighbouring threads.
![A schematic view of divergent execution within a SIMD architecture. Lines indicate the flow of instructions; white diamonds indicate branch points, where the code paths of neighbouring threads diverge. The statements on the left indicate typical corresponding source code. White space between branch points indicates a thread waiting for its neighbours to complete a divergent code section.[]{data-label="fig:branching"}](fig3){width="8cm"}
**Arithmetic intensity:** Executing arithmetic instructions is generally much faster than accessing memory on current many-core hardware. Algorithms performing few arithmetic operations per memory access may become memory-bandwidth-bound; i.e., their speed becomes limited by the rate at which memory can be accessed, rather than the rate at which arithmetic instructions can be processed. Memory bandwidths in many-core architectures are typically significantly higher than in CPUs, meaning that even bandwidth-bound algorithms may exhibit strong performance; however, they will not be able to take full advantage of the available computing power. In some cases, it may be beneficial to re-work an algorithm entirely in order to increase its arithmetic intensity, even at the cost of performing more numerical work in total.
For the arithmetic intensities presented in this paper, we assume an idealised cache model in which only the first memory read of a particular piece of data is included in the count; subsequent or parallel reads of the same data are assumed to be made from a cache, and are not counted. The ability to achieve this behaviour in practice will depend strongly on the memory access pattern (specifically the locality of memory accesses).
\[tbl:analysis\_of\_simple\_algorithms\] Transform Reduction Gather Interact
------------------------------------------ ------------------ ------------------------ ------------------ ------------------------------------------- -- -- -- --
Work $\mathcal{O}(N)$ $\mathcal{O}(N)$ $\mathcal{O}(N)$ $\mathcal{O}(NM)$
Depth $\mathcal{O}(1)$ $\mathcal{O}(\log N)$ $\mathcal{O}(1)$ $\mathcal{O}(M)$ or $\mathcal{O}(\log M)$
Memory access locality Contiguous Contiguous Variable Contiguous
Arithmetic intensity $1:1:\alpha$ $1:\frac{1}{N}:\alpha$ $1:1:0$ $1+\frac{M}{N}:1:2M\alpha$
Complexity analysis {#sec:complexity_analysis}
-------------------
The complexity of an algorithm is a formal measure of its execution time given a certain size of input. It is often used as a means of comparing the speeds of two different algorithms that compute the same (or a similar) result. Such comparisons are critical to understanding the relative contributions of different parts of a composite algorithm and identifying bottle-necks.
Computational complexity is typically expressed as the total run-time, $T$, of an algorithm as a function of the input size, $N$, using ‘Big O’ notation. Thus $T(N) = \mathcal{O}(N)$ means a run-time that is proportional to the input size $N$. An algorithm with complexity of $T(N) =
\mathcal{O}(N^2)$ will take four times as long to run after a doubling of its input size.
While the complexity measure is traditionally used for algorithms running on serial processors, it can be generalised to analyse parallel algorithms. One method is to introduce a second parameter: $P$, the number of processors. The run-time is then expressed as a function of both $N$ and $P$. For example, an algorithm with a parallel complexity of $T(N, P) = \mathcal{O}(\frac{N}{P})$ will run $P$ times faster on $P$ processors than on a single processor for a given input size; i.e., it exhibits perfect parallel scaling. More complex algorithms may incur overheads when run in parallel, e.g., those requiring communication between processors. In these cases, the parallel complexity will depend on the specifics of the target hardware architecture.
An alternative way to express parallel complexity is using the *work*, $W$, and *depth*, $D$, metrics first introduced formally by @Blelloch1996. Here, work measures the total number of computational operations performed by an algorithm (or, equivalently, the run-time on a single processor), while depth measures the longest sequence of sequentially-dependent operations (or, equivalently, the run-time on an infinite number of processors). The depth metric is a measure of the amount of inherent parallelism in the algorithm. A perfectly parallel algorithm has work complexity of $W(N) = \mathcal{O}(N)$ and depth complexity of $D(N) = \mathcal{O}(1)$, meaning all but a constant number of operations may be performed in parallel. An algorithm with $W = \mathcal{O}(N)$ and $D = \mathcal{O}(\log{N})$ is highly parallel, but contains some serial dependencies between operations that scale as a function of the input size. Parallel algorithms with work complexities equal to those of their serial counterparts are said to be ‘work efficient’; those that further exhibit low depth complexities are considered to be efficient parallel algorithms. The benefit of the work/depth metrics over the parallel run-time is that they have no dependence on the particular parallel architecture on which the algorithm is executed, i.e., they measure properties inherent to the algorithm.
A final consideration regarding parallel algorithms is Amdahl’s law [@Amdahl1967], which states that the maximum possible speedup over a serial algorithm is limited by the fraction of the parallel algorithm that cannot be (or simply *is not*) parallelised. Assuming an infinite number of available processors, the run-time of the parallel part of the algorithm will reduce to a constant, while the serial part will continue to scale with the size of the input. In terms of the work/depth metrics, the depth of the algorithm represents the fraction that cannot be parallelised, and the maximum theoretical speedup is given by $S_\mathrm{max} \approx
\frac{W}{D}$. Note the implication that the maximum speedup is actually a function of the input size. Increasing the problem size in addition to the number of processors allows the speedup to scale more effectively.
Analysis results {#sec:analysis}
----------------
We have applied the techniques discussed in Sections \[sec:CriticalIssues\] and \[sec:complexity\_analysis\] to the four foundation algorithms introduced at the beginning of Section \[sec:ourmethodology\]. We use the following metrics:
- **Work and depth:** The complexity metrics as described in Section \[sec:complexity\_analysis\].
- **Memory access locality:** The nature of the memory access patterns as discussed in Section \[sec:CriticalIssues\].
- **Arithmetic intensity:** Defined by the triple ratio $r:w:f$ representing the number of read, write and function evalation operations respectively that the algorithm performs (normalised to the input size). The symbol $\alpha$ is used, where applicable, to represent the internal arithmetic intensity of the function given to the algorithm.
The results are presented in Table \[tbl:analysis\_of\_simple\_algorithms\]. Note that this analysis is based on the most-efficient known parallel version of each algorithm.
Global analysis {#sec:macro_analysis}
---------------
Once local analysis results have been obtained for each step of a problem, it is necessary to put them together and perform a global analysis. Our methodology is as follows:
1. Determine the components of the algorithm where most of the computational work lies by comparing work complexities. Components with similar work complexities should receive similar attention with respect to parallelisation in order to avoid leaving behind bottle-necks as a result of Amdahl’s Law.
2. Consider the amount of inherent parallelism in each algorithm
by observing its theoretical speedup $S_{\rm max} \approx \frac{W}{D}$.
3. Use the theoretical arithmetic intensity of each algorithm to determine the likelihood of it being limited by memory bandwidth rather than instruction throughput. The theoretical *global* arithmetic intensity may be obtained by comparing the total amount of input and output data to the total amount of arithmetic work to be done in the problem.
4. Assess the memory access patterns of each algorithm to identify the potential to achieve peak arithmetic intensity[^7].
5. If particular components exhibit poor properties, consider alternative algorithms.
6. Once a set of component algorithms with good theoretical performance has been obtained, the algorithm decomposition should provide a good starting point for an implementation.
Application to Astronomy Algorithms {#sec:AstronomyAlgorithms}
===================================
We now apply our methodology from Section \[sec:ourmethodology\] to four typical astronomy computations. In each case, we demonstrate how to identify the steps in an outline of the problem as foundation algorithms from our collection described at the beginning of Section \[sec:ourmethodology\]. We then use this knowledge to study the exact nature of the available parallelism and determine the problem’s overall suitability for many-core architectures. We note that we have deliberately chosen simple versions of the problems in order to maximise clarity and brevity in illustrating the principles of our algorithm analysis methodology.
Inverse ray-shooting gravitational lensing {#sec:rayshooting}
------------------------------------------
**Introduction:** Inverse ray-shooting is a numerical technique used in gravitational microlensing. Light rays are projected backwards (i.e., from the observer) through an ensemble of lenses and on to a source-plane pixel grid. The number of rays that fall into each pixel gives an indication of the magnification at that spatial position relative to the case where there was no microlensing. In cosmological scenarios, the resultant maps are used to study brightness variations in light curves of lensed quasars, providing constraints on the physical size of the accretion disk and broad line emission regions.
The two main approaches to ray-shooting are based on either the direct calculation of the gravitational deflection by each lens [@KayserEtal1986; @SchneiderWeiss1986; @SchneiderWeiss1987] or the use of a tree hierarchy of psuedo-lenses [@Wambsganss1990; @Wambsganss1999]. Here, we consider the direct method.
**Outline:** The ray-shooting algorithm is easily divided into a number of distinct steps:
1. Obtain a collection of lenses according to a desired distribution, where each lens has position and mass.
2. Generate a collection of rays according to a uniform distribution within a specified 2D region, where each ray is defined by its position.
3. For each ray, calculate and sum its deflection due to each lens.
4. Add each ray’s calculated deflection to its initial position to obtain its deflected position.
5. Calculate the index of the pixel that each ray falls into.
6. Count the number of rays that fall into each pixel.
7. Output the list of pixels as the magnification map.
**Analysis:** To begin the analysis, we interpret the above outline as follows:
- Steps (i) and (ii) may be considered *transform* operations that initialise the vectors of lenses and rays.
- Step (iii) is an example of the *interact* algorithm, where the inputs are the vectors of rays and lenses and the interaction function calculates the deflection of a ray due to the gravitational potential around a lens mass.
- Steps (iv) and (v) apply further transforms to the collection of rays.
- Step (vi) involves the generation of a histogram. As we have not already identified this algorithm in Section \[sec:ourmethodology\], it will be necessary to analyse this step as a unique algorithm.
According to this analysis, three basic algorithms comprise the complete technique: transform, interact and histogram generation. Referring to Table \[tbl:analysis\_of\_simple\_algorithms\], we see that, in the context of a lensing simulation using $N_{\rm rays}$ rays and $N_{\rm lenses}$ lenses, the amount of work performed by the transform and interact algorithms will be $W =
\mathcal{O}(N_{\rm rays}) + \mathcal{O}(N_{\rm lenses})$ and $W =
\mathcal{O}(N_{\rm rays} N_{\rm lenses})$ respectively.
We now analyse the histogram step. Considering first a serial algorithm for generating a histogram, where each point is considered in turn and the count in its corresponding bin is incremented, we find the work complexity to be $W = \mathcal{O}(N_{\rm rays})$. Without further analysis, we compare this to those of the other component algorithms. The serial histogram and the transform operations each perform similar work. The interact algorithm on the other hand must, as we have seen, perform work proportional to $N_{\rm
rays} \times N_{\rm lenses}$. For large $N_{\rm lenses}$ (e.g., as occurs in cosmological microlensing simulations, where $N_{\rm lenses} > 10^4$) this step will dominate the total work. Assuming the number of lenses is scaled with the amount of parallel hardware, the interact step will also dominate the total run-time.
Given the dominance of the interact step, we now choose to ignore the effects of the other steps in the problem. It should be noted, however, that in contrast to cosmological microlensing, planetary microlensing models contain only a few lenses. In this case, the work performed by the interact step will be similar to that of the other steps, and thus the use of a serial histogram algorithm alongside parallel versions of all other steps would result in a severe performance bottle-neck. Several parallel histogram algorithms exist, but a discussion of them is beyond the scope of this work.
Returning to the analysis of the interact algorithm, we again refer to Table \[tbl:analysis\_of\_simple\_algorithms\]. Its worst-case depth complexity indicates a maximum speedup of $S \approx W =
\mathcal{O}(N_{\rm rays})$, i.e., parallel speedup scaling perfectly up to the number of rays. The arithmetic intensity of the algorithm scales as $N_{\rm lenses}$ and will thus be very high. Contiguous memory accesses indicate strong potential to achieve this high arithmetic intensity. We conclude that direct inverse ray-shooting for cosmological microlensing is an ideal candidate for an efficient implementation on a many-core architecture.
Högbom CLEAN
------------
**Introduction:** Raw (‘dirty’) images produced by radio interferometers exhibit unwanted artefacts as the result of the incomplete sampling of the visibility plane. These artefacts can inhibit image analysis and should ideally be removed by deconvolution. Several different techniques have been developed to ‘clean’ these images. For a review, see @Briggs1995. Here we analyse the image-based algorithm first described by @Hogbom1974. We note that the algorithm by @Clark1980 is now the more popular choice in the astronomy community, but point out that it is essentially an approximation to Högbom’s algorithm that provides increased performance at the cost of reduced accuracy.
The algorithm involves iteratively finding the brightest point in the ‘dirty image’ and subtracting from the dirty image an image of the beam centred on and scaled by this brightest point. The procedure continues until the brightest point in the image falls below a prescribed threshold. While the iterative procedure must be performed sequentially, the computations within each iteration step are performed independently for every pixel of the images, suggesting a substantial level of parallelism. The output of the algorithm is a series of ‘clean components’, which may be used to reconstruct a cleaned image.
**Outline:** The algorithm may be divided into the following steps:
1. Obtain the beam image.
2. Obtain the image to be cleaned.
3. Find the brightest point, $b$, the standard deviation, $\sigma$, and the mean, $\mu$, of the image.
4. If the brightness of $b$ is less than a prescribed threshold (e.g., $|b-\mu|<3\sigma$), go to step (ix).
5. Scale the beam image by a fraction (referred to as the ‘loop gain’) of the brightness of $b$.
6. Shift the beam image to centre it over $b$.
7. Subtract the scaled, shifted beam image from the input image to produce a partially-cleaned image.
8. Repeat from step (iii).
9. Output the ‘clean components’.
**Analysis:** We decompose the outline of the Högbom <span style="font-variant:small-caps;">clean</span> algorithm as follows:
- Steps (i) and (ii) are simple data-loading operations, and may be thought of as transforms.
- Step (iii) involves a number of reduce operations over the pixels in the dirty image.
- Step (v) is a transform operation, where each pixel in the beam is multiplied by a scale factor.
- Step (vi) may be achieved in two ways, either by directly reading an offset subset of the beam pixels, or by switching to the Fourier domain and exploiting the shift theorem. Here we will only consider the former option, which we identify as a gather operation.
- Step (vii) is a transform operation over pixels in the dirty image.
We thus identify three basic algorithms in Högbom <span style="font-variant:small-caps;">clean</span>: *transform*, *reduce* and *gather*. Table \[tbl:analysis\_of\_simple\_algorithms\] shows that the work performed by each of these algorithms will be comparable (assuming the input and beam images are of similar pixel resolutions). This suggests that any acceleration should be applied equally to *all* of the steps in order to avoid the creation of bottle-necks.
The depth complexities of each algorithm indicate a limiting speed-up of $S_{\rm max} \approx \mathcal{O}(\frac{N_{\rm pxls}}{\log{N_{\rm pxls}}})$ during the reduce operations. While not quite ideal, this is still a good result. Further, the algorithms do not exhibit high arithmetic intensity (the calculations involving only a few subtractions and multiplies) and are thus likely to be bandwidth-bound. This will dominate any effect the limiting speed-up may have.
The efficiency with which the algorithm will use the available memory bandwidth will depend on the memory access patterns. The transform and reduce algorithms both make contiguous memory accesses, and will thus achieve peak bandwidth. The gather operation in step (vi), where the beam image is shifted to centre it on a point in the input image, will access memory in an offset but contiguous 2-dimensional block. This 2D locality suggests the potential to achieve near-peak memory throughput.
We conclude that the Högbom <span style="font-variant:small-caps;">clean</span> algorithm represents a good candidate for implementation on many-core hardware, but will likely be bound by the available memory bandwidth rather than arithmetic computing performance.
Volume rendering
----------------
**Introduction:** There are a number of sources of volume data in astronomy, including spectral cubes from radio telescopes and integral field units, as well as simulations using adaptive mesh refinement and smoothed particle hydrodynamics techniques. Visualising these data in physically-meaningful ways is important as an analysis tool, but even small volumes (e.g., $256^3$) require large amounts of computing power to render, particularly when real-time interactivity is desired.
Several methods exist for rendering volume data; here we analyse a direct (or *brute-force*) ray-casting algorithm [@Levoy1990]. While similarities exist between ray-shooting for microlensing (Section \[sec:rayshooting\]) and the volume rendering technique we describe here, they are fundamentally different algorithms.
**Outline:** The algorithm may be divided into the following steps:
1. Obtain the input data cube.
2. Create a 2D grid of output pixels to be displayed.
3. Generate a corresponding grid of *rays*, where each is defined by a position (initially the centre of the corresponding pixel), a direction (defined by the viewing transformation) and a colour (initially black).
4. Project each ray a small distance (the *step size*) along its direction.
5. Determine which voxel each ray now resides in.
6. Retrieve the colour of the voxel from the data volume.
7. Use a specified *transfer function* to combine the voxel colour with the current ray colour.
8. Repeat from step (iv) until all rays exit the data volume.
9. Output the final ray colours as the rendered image.
**Analysis:** We interpret the steps in the above outline as follows:
- Steps (ii) to (v) and (vii) are all transform operations.
- Step (vi) is a gather operation.
All steps perform work scaling with the number of output pixels, $N_{\rm
pxls}$, indicating there are no algorithmic bottle-necks and thus acceleration should be applied to the whole algorithm equally.
Given that the number of output pixels is likely to be large and scalable, we should expect the transforms and the gather, with their $\mathcal{O}(1)$ depth complexities, to parallelise perfectly on many-core hardware.
The outer loop of the algorithm, which marches rays through the volume until they leave its bounds, involves some branching as different rays traverse thicker or thinner parts of the arbitrarily-oriented cube. This will have a negative impact on the performance of the algorithm on a SIMD architecture like a GPU. However, if rays are ordered in such a way as to maintain 2D locality between their positions, neighbouring threads will traverse similar depths through the data cube, resulting in little divergence in their branch paths and thus good performance on SIMD architectures.
The arithmetic intensity of each of the steps will typically be low (common transfer functions can be as simple as taking the average or maximum), while the complete algorithm requires $\mathcal{O}(N_{\rm pxls}N_d)$ memory reads, $\mathcal{O}(N_{\rm pxls})$ memory writes and $\mathcal{O}(N_{\rm pxls}N_d)$ function evaluations for an input data volume of side length $N_d$. This global arithmetic intensity of $N_d:1:N_d\alpha$ indicates the algorithm is likely to remain bandwidth-bound.
The use of bandwidth will depend primarily on the memory access patterns in the gather step (the transform operations perform ideal contiguous memory accesses). During each iteration of the algorithm, the rays will access an arbitrarily oriented plane of voxels within the data volume. Such a pattern exhibits 3D spatial locality, presenting an opportunity to cache the memory reads effectively and thus obtain near-peak bandwidth.
We conclude that the direct ray-casting volume rendering algorithm is a good candidate for efficient implementation on many-core hardware, although, in the absence of transfer functions with significant arithmetic intensity, the algorithm is likely to remain limited by the available memory bandwidth.
Pulsar time-series dedispersion {#sec:dedispersion}
-------------------------------
**Introduction:** Radio-telescopes observing pulsars produce time-series data containing the pulse signal. Due to its passage through the interstellar medium, the pulse signature gets delayed as a function of frequency, resulting in a ‘dispersing’ of the data. The signal can be ‘dedispersed’ by assuming a frequency-dependent delay before summing the signals at each frequency. The data are dedispersed using a number of trial dispersion measures (DMs), from which the true DM of the signal is measured.
There are two principle dedispersion algorithms used in the literature: the direct algorithm and the tree algorithm [@Taylor1974]. Here we consider the direct method, which simply involves delaying and summing time series for a range of DMs. The calculation for each DM is entirely independent, presenting an immediate opportunity for parallelisation. Further, each sample in the time series is operated-on individually, hinting at additional fine-grained parallelism.
**Outline:** Here we describe the key steps of the algorithm:
1. Obtain a set of input time series, one per frequency channel.
2. If necessary, transpose the input data to place it into channel-major order.
3. Impose a time delay on each channel by offsetting its starting location by the number of samples corresponding to the delay. The delay introduced into each channel is a quadratic function of its frequency and a linear function of the dispersion measure.
4. Sum aligned samples across every channel to produce a single accumulated time series.
5. Output the result and repeat (potentially in parallel) from step (iii) for each desired trial DM.
**Analysis:** We interpret the above outline of the direct dedispersion algorithm as follows:
- Step (ii) involves transposing the data, which is a form of *gather*.
- Step (iii) may be considered a set of *gather* operations that shift the reading location of samples in each channel by an offset.
- Step (iv) involves the summation of many time series. This is a nested operation, and may be interpreted as either a *transform*, where the operation is to sum the time sample in each channel, or a *reduce*, where the operation is to sum whole time series.
The algorithm therefore involves gather operations in addition to nested transforms and reductions. For data consisting of $N_s$ samples for each of $N_c$ channels, each step of the computation operates on all $\mathcal{O}(N_s
N_c)$ total samples. Acceleration should thus be applied equally to all parts of the algorithm.
According to the depth complexity listed in Table \[tbl:analysis\_of\_simple\_algorithms\], the gather operation will parallelise perfectly. The nested transform and reduce calculation may be parallelised in three possible ways: a) by parallelising the transform, where $N_s$ parallel threads each compute the sum of a single time sample over every channel sequentially; b) by parallelising the reduce, where $N_c$ parallel threads cooperate to sum each time sample in turn; or c) by parallelising both the transform and the reduce, where $N_s \times N_c$ parallel threads cooperate to complete the entire computation in parallel.
Analysing these three options, we see that they have depth complexities of $\mathcal{O}(N_c)$, $\mathcal{O}(N_s\log{N_c})$ and $\mathcal{O}(\log{N_c})$ respectively. Option (c) would appear to provide the greatest speedup; however, it relies on using significantly more parallel processors than the other options. It will in fact only be the better choice in the case where the number of available parallel processors is much greater than $N_s$. For hardware with fewer than $N_s$ parallel processors, option (a) will likely prove the better choice, as it is expected to scale perfectly up to $N_s$ parallel threads, as opposed to the less efficient scaling of option (c). In practice, the number of time samples $N_s$ will generally far exceed the number of parallel processors, and thus the algorithm can be expected to exhibit excellent parallel scaling using option (a).
Turning now to the arithmetic intensity, we observe that the computation of a single trial DM involves only an addition for each of the $N_s \times N_c$ total samples. This suggests the algorithm will be limited by memory bandwidth. However, this does not take into account the fact that we wish to compute many trial dispersion measures. The computation of $N_{\rm DM}$ trial DMs still requires only $\mathcal{O}(N_s \times N_c)$ memory reads and writes, but performs $N_{\rm DM} \times N_s \times N_c$ addition operations. The theoretical global arithmetic intensity is therefore $1:1:N_{\rm DM}$. Given a typical number of trial DMs of $\mathcal{O}(100)$, we conclude that the algorithm could, in theory at least, make efficient use of all available arithmetic processing power.
The ability to achieve such a high arithmetic intensity will depend on the ability to keep data in fast memory for the duration of many arithmetic calculations (i.e., the ability to efficiently cache the data). This in turn will depend on the memory access patterns. We note that in general, similar trial DMs will need to access similar areas of memory; i.e., the problem exhibits some locality of reference. The exact memory access pattern is non-trivial though, and a discussion of these details is outside the scope of this work.
We conclude that the pulsar dedispersion algorithm would likely perform to a high efficiency on a many-core architecture. While it is apparent that some locality of reference exists within the algorithm’s memory accesses, optimal arithmetic intensity is unlikely to be observed without a thorough and problem-specific analysis of the memory access patterns.
Discussion {#sec:discussion}
==========
The direct inverse ray-shooting method has been implemented on a GPU by @ThompsonEtal2010. They simulated systems with up to $10^9$ lenses. Using a single GPU, they parallelised the interaction step of the problem and obtained a speedup of $\mathcal{O}(100\times)$ relative to a single CPU core – a result consistent with the relative peak floating-point performance of the two processing units[^8]. These results validate our conclusion that the inverse ray-shooting algorithm is very well suited to many-core architectures like GPUs.
Our conclusions regarding the pulsar dedispersion algorithm are validated by a preliminary GPU implementation we have written. With only a simplistic approach to memory caching, we have recorded a speedup of $15\times$ over an efficient multi-core CPU code. This result is in line with the relative peak memory bandwidth of the two architectures, supporting the conclusions of Section \[sec:dedispersion\] that, without a detailed investigation into the memory access patterns, the problem will remain bandwidth-bound.
Some astronomy problems are well-suited to a many-core architecture, others are not. It is important to know how to distinguish between these. In the astronomy community, the majority of work with many-core hardware to date has focused on the implementation or porting of specific codes perhaps best classified as ‘low-hanging fruit’. Not surprisingly, these codes have achieved significant speed-ups, in line with the raw performance benefits offered by their target hardware.
A more generalised use of ‘novel’ computing architectures was undertaken by @BrunnerEtal2007, who, as a case study, implemented the two-point angular correlation function for cosmological galaxy clustering on two different FPGA architectures[^9]. While they successfully communicated the advantages offered by these new technologies, their focus on implementation details for their FPGA hardware inhibits the ability to generalise their findings to other architectures.
It is interesting to note that previous work has in fact identified a number of common concerns with respect to GPU implementations of astronomy algorithms. For example, the issues of optimal use of the memory hierarchy and underuse of available hardware for small particle counts have been discussed in the context of the direct N-body problem (e.g., @BellemanEtal2008). These concerns essentially correspond to a combination of what we have referred to as memory access patterns, arithmetic intensity and massive parallelism. While originally being discussed as implementation issues specific to particular choices of software and hardware, our abstractions re-cast them at the algorithm level, and allow us to consider their impact across a variety of problems and hardware architectures.
Using algorithm analysis techniques, we now have a basis for understanding which astronomy algorithms will benefit most from many-core processors. Those with well-defined memory access patterns and high arithmetic intensity stand to receive the greatest performance boost, while problems that involve a significant amount of decision-making may struggle to take advantage of the available processing power.
For some astronomy problems, it may be important to look beyond the techniques currently in use, as these will have been developed (and optimised) with traditional CPU architectures in mind. Avenues of research could include, for instance, using higher-order numerical schemes [@NitadoriMakino2008] or choosing simplicity over efficiency by using brute-force methods (Bate et al. submitted). Some algorithms, such as histogram generation, do not have a single obvious parallel implementation, and may require problem-specific input during the analysis process.
In this work, we have discussed the future of astronomy computation, highlighting the change to many-core processing that is likely to occur in CPUs.
The shift in commodity hardware from serial to parallel processing units will fundamentally change the landscape of computing. While the market is already populated with multi-core chips, it is likely that chip designs will undergo further significant changes in the coming years. We believe that for astronomy, a generalised methodology based on the analysis of algorithms is a prudent approach to confronting these changes – one that will continue to be applicable across the range of hardware architectures likely to appear in the coming years: CPUs, GPUs and beyond.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Amr Hassan and Matthew Bailes for useful discussions regarding this paper, and the reviewer Gilles Civario for helpful suggestions.
\[lastpage\]
[^1]: While the specifics of parallel CPU systems lie outside the scope of this paper, we note that many of the algorithm analysis techniques we describe lend themselves equally well to these architectures.
[^2]: Note that for these algorithms we have used naming conventions that are familiar to us but are by no means unique in the literature.
[^3]: Here we use *constant-time* in the algorithmic sense, i.e., constant with respect to the size of the input data. In this context we are not concerned with hardware-specific performance factors.
[^4]: Note also that oversubscription of threads to processors is often a requirement for good performance in many-core architectures. For example, an NVIDIA GT200-class GPU may be under-utilised with an allocation of fewer than $\sim 10^4$ parallel threads, corresponding to an oversubscription rate of around $50\times$.
[^5]: Memory bandwidths on current GPUs are $\mathcal{O}(100$GB/s$)$.
[^6]: Locality of reference also affects performance on traditional CPU architectures, but to a lesser extent than on GPUs.
[^7]: Studying the memory access patterns will also help to identify the optimal caching strategy if this level of optimisation is desired.
[^8]: We note that @ThompsonEtal2010 did not use the CPU’s Streaming SIMD Extensions, which have the potential to provide a speed increase of up to $4\times$. However, our conclusion regarding the efficiency of the algorithm on the GPU remains unchanged by this fact.
[^9]: Field Programmable Gate Arrays are another hardware architecture exhibiting significant fine-grained parallelism, but their specific details lie outside the scope of this paper.
|
---
abstract: 'In an extension of the Standard Model with a scalar color octet, the possibility of the strongly first-order electroweak phase transition is studied, by examining the finite-temperature effective Higgs potential at the one-loop level. It is found that there are wide regions in the parameter space that allow the strongly first-order electroweak phase transition, where the Higgs boson mass is larger than the experimental lower bound of 115 GeV, and the masses of the scalar color octet is around 200 GeV. The parameter regions may be explored at the LHC with respect to the electroweak phase transition.'
author:
- |
S. W. Ham$^{(1,2)}$[^1], Seong-A Shim$^{(3)}$[^2], and S. K. Oh$^{(4)}$[^3]\
\
[*(1) School of Physics, KIAS, Seoul 130-722, Korea*]{}\
[*(2) Korea Institute of Science and Technology Information*]{}\
[*Daejeon, 305-806, Korea*]{}\
[*(3) Department of Mathematics, Sungshin Women’s University*]{}\
[*Seoul 136-742, Korea*]{}\
[*(4) Department of Physics, Konkuk University, Seoul 143-701, Korea*]{}\
\
title: Electroweak phase transition in an extension of the Standard Model with scalar color octet
---
Introduction
============
The observed baryon asymmetry of the universe, or the excess of matter over anti matter, is a challenging problem for any theoretical model to be phenomenologically realistic. Several decades ago, Sakharov pointed out that theoretical models can generate the baryon asymmetry dynamically if they satisfy three essential conditions: the violation of baryon number conservation, the violation of both C and CP, and the deviation from thermal equilibrium \[1\]. It is well known that, in order to ensure sufficient deviation from thermal equilibrium, the electroweak phase transition (EWPT) should be first order, and its strength should be strong, since otherwise the baryon asymmetry generated during the phase transition subsequently would disappear \[2-10\]. As the universe cools down, the shape of the potential of the scalar field that is responsible for the electroweak symmetry breaking has two degenerate minima, where one of them is the (false) vacuum of the symmetric state and the other is the (true) vacuum of the broken state, at the critical temperature. The first-order electroweak phase transition takes place from the false vacuum to the true vacuum. In general, the first-order electroweak phase transition is regarded as strong if the vacuum expectation value (VEV) of the scalar field at the true vacuum is larger than the critical temperature.
The Standard Model (SM) is certainly the most successful theory so far for electroweak interactions, yet it is found, however, that the SM faces severe difficulty to satisfy the Sakharov conditions. First, the complex phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix cannot produce large enough CP violation to generate the baryon asymmetry. Further, for the present experimental lower bound on the mass of the SM Higgs boson, the strength of the first-order EWPT in the SM is too weak. Consequently, the possibility of a strongly first-order EWPT may be studied in the models modified or extended the SM in order to explain the baryon asymmetry of the universe.
In the literature, a number of models alternative to the SM have been investigated in this context. These models are phenomenologically well motivated, if not necessarily motivated by the baryon asymmetry. We are interested in studying the possibility of a strongly first-order EWPT in these models. Among them is an extension of the SM with an additional scalar color octet. Popov, Povarov, and Smirnov have studied it within the context of Pati-Salam unification \[11\]. Manohar and Wise have studied the general structure of this model, and examined new impact on the Higgs phenomenology and flavor physics \[12\]. Recently, other authors have also investigated the implications of the scalar color octet at the CERN Large Hadron Collider (LHC) \[13-19\].
It is also noticed that the presence of the scalar color octet causes additional sources of CP violation beyond the CKM matrix in the SM \[12\]. Since a sufficient CP violation is required by one of the Sakharov conditions for generating the baryon asymmetry, this model is in a better position than the SM in this respect. We are thus interested in whether this model also allows a strongly first-order EWPT.
By studying the finite temperature effective Higgs potential at the one-loop level, we find that there are parameter regions in this model where the EWPT is strongly first order to generate the desired baryon asymmetry. In the parameter regions, the Higgs boson mass can be as large as 197 GeV and the masses of the scalar color octet are as large as about 250 GeV. Thus, these parameter regions may be explored at the LHC.
The Model
=========
In Ref. \[12\], Manohar and Wise have considered the generalization of the SM with the most general scalar sector, with a natural suppression of flavor changing neutral currents. Explicitly, they have considered the case of an additional scalar color octet, from the point of view of the LHC phenomenology. Let us briefly describe the model.
The scalar sector of this model consists of $H$, the usual SM Higgs doublet, and $S^\alpha$ ($\alpha = 1, \cdot, 8$), the scalar color octet. The SM Higgs doublet is defined as $H^T = (H^+, H^0)$, where $H^+$ and $H^0$ are the charged and neutral Higgs fields, respectively. The scalar color octet are defined as $$\begin{aligned}
S^\alpha= \bigg(
\begin{array}{c}
S^{+\alpha} \\
S^{0\alpha}
\end{array} \bigg)
= \bigg (
\begin{array}{c}
S^{\alpha}_C \\
{\displaystyle S_R^{\alpha} + i S_I^{\alpha} \over \displaystyle \sqrt{2} }
\end{array}
\bigg ) \ ,\end{aligned}$$ where $S_C^\alpha$ is the charged scalar color octet and $S_R^{\alpha}$ and $S_I^{\alpha}$ are respectively the real and the complex components of the neutral scalar color octet.
Under $SU(3) \times SU(2) \times U(1)$, the SM Higgs field transforms as $({\bf 1}, {\bf 2})_{1/2}$ and the scalar color octet as $({\bf 8}, {\bf 2})_{1/2}$. The Yukawa coupling of the SM Higgs boson to the SM quarks is given as $${\cal L} = - g_{ij}^U {\bar u}_{R i} Q_{L j} H
- g_{ij}^D {\bar d}_{R i} Q_{L j} H^{\dagger} + {\rm H.c.} \ ,$$ where $i$ and $j$ are flavor indices ($i,j = 1,2,3$), $g_{ij}^U$ and $g_{ij}^D$ are the Yukawa coupling coefficients, $Q_{Lj}$ are the quark doublets, and $u_{Ri}$ and $d_{Ri}$ are the quark singlets. The Yukawa couplings of the scalar color octet to the SM quarks are given as $${\cal L} = - \eta_U g_{ij}^U {\bar u}_{R i} T^\alpha Q_{L j} S^\alpha
- \eta_D g_{ij}^D {\bar d}_{R i} T^\alpha Q_{L j} S^{\alpha \dagger} + {\rm H.c.} \ ,$$ where $\eta_U$ and $\eta_D$ are generally complex constants, $\alpha$ is the color index ($\alpha = 1, \cdot, 8$), $T^\alpha$ are the $SU(3)$ generators with the normalization condition of ${\rm Tr} (T^\alpha T^\beta) = \delta^{\alpha\beta}/2$.
In terms of the usual SM Higgs doublet and the scalar color octet, the most general form of the scalar potential at the tree level at zero temperature is given as \[12\] $$\begin{aligned}
V_0 &=& -\mu^2 H^{\dagger i} H_i + \lambda \left(H^{\dagger i} H_i \right)^2 \cr
&& + 2 m_S^2 {\rm Tr} S^{\dagger i} S_i
+ \lambda_1 H^{\dagger i} H_i {\rm Tr} S^{\dagger j} S_j
+ \lambda_2 H^{\dagger i} H_j {\rm Tr} S^{\dagger j} S_i \cr
&& + \Bigl[ \lambda_3 H^{\dagger i} H^{\dagger j} {\rm Tr} S_i S_j
+ \lambda_4 H^{\dagger i} {\rm Tr} S^{\dagger j} S_j S_i
+ \lambda_5 H^{\dagger i} {\rm Tr} S^{\dagger j} S_i S_j + {\rm H.c.} \Bigr] \cr
&&+ \lambda_6 {\rm Tr} S^{\dagger i} S_i S^{\dagger j} S_j
+ \lambda_7 {\rm Tr} S^{\dagger i} S_j S^{\dagger j} S_i
+ \lambda_8 {\rm Tr} S^{\dagger i} S_i {\rm Tr} S^{\dagger j} S_j \cr
&& + \lambda_9 {\rm Tr} S^{\dagger i} S_j {\rm Tr} S^{\dagger j} S_i
+\lambda_{10} {\rm Tr} S_i S_j {\rm Tr} S^{\dagger i} S^{\dagger j}
+ \lambda_{11} {\rm Tr} S_i S_j S^{\dagger j} S^{\dagger i} \ ,\end{aligned}$$ where $i, j$ are the $SU(2)$ indices, traces are over the color $SU(3)$ indices. $S = S^\alpha T^\alpha$, $m^2_S$ is the mass parameter for the scalar color octet, $\lambda$ is the quartic coupling coefficient of the SM Higgs field, and ${\lambda}_i$ ($i=$ from 1 to 11) are the quartic coupling coefficients of the scalar color octet. Note that the phases of $\eta_U$, $\eta_D$, $\lambda_4$ or $\lambda_5$ might be additional sources of CP violation beyond the complex phase in the CKM matrix.
In this model, since the color symmetry is intact, the scalar color octet would not develop any VEV. Thus, the electroweak symmetry breaking is triggered by the Higgs doublet alone, and the EWPT is determined by the shape of the Higgs potential. We assume that the real component of the neutral Higgs field, ${\rm Re}H^0 $, develops the VEV. Let us introduce the physical Higgs boson $\phi$ as $\phi = {\rm Re}H^0 /\sqrt{2}$. The tree-level zero-temperature Higgs potential may then be written as $$V_0 (\phi, 0) = -{1\over 2} \mu^2 \phi^2 + {1\over 4} \lambda \phi^4 \ .$$ At the tree level at zero temperature, we would have $\langle \phi \rangle = v_0$, where $v_0 = 246$ GeV, the tree-level VEV. The tree-level masses of the gauge bosons $W$, $Z$, and top quark $t$, respectively, are given as $m_W = g_2 v_0 /2$, $m_Z = \sqrt{g_1^2 + g_2^2} v_0 /2$, and $m_t = h_t v_0 /\sqrt{2}$. The tree-level masses of the Higgs boson and Goldstone boson are given by $m_{\phi} = \sqrt{2 \lambda} v_0$ and $m_G = \sqrt{\lambda} v_0$, respectively. Also, the tree-level masses for $S^{\alpha}_C$, $S_R^{\alpha}$, and $S_I^{\alpha}$ are given respectively as $$\begin{aligned}
& & m^2_{S_C} = m_S^2+\lambda_1{ v^2_0 \over 4} \ , \cr
& & m^2_{S_R} = m_S^2+\left(\lambda_1+\lambda_2+2\lambda_3\right){v^2_0 \over 4} \ , \cr
& & m^2_{S_I} = m_S^2+\left(\lambda_1+\lambda_2-2\lambda_3\right){v^2_0 \over 4} \ .\end{aligned}$$
At the one-loop level, the zero-temperature Higgs potential is given by including the one-loop contributions, which is calculated, by using the effective potential method \[20\], as $$\Delta V_1(\phi, 0) = 2 B v^2_0 \phi^2 - {3 \over 2} B \phi^4 +
B \log \bigg ( { \phi^2 \over v^2_0 } \bigg ) \phi^4 \ ,$$ where $$B = {3 \over 64 \pi^2 v^4_0} \bigg ( 2 m_W^4 + m_Z^4 - 4 m_t^4
+ {16 \over 3} m_{S_C}^4 + {8 \over 3} m_{S_R}^4 + {8 \over 3} m_{S_I}^4
+ {1 \over 3} m_{\phi}^4 + m_G^4 \bigg ) \ .$$ In the radiative corrections, we include the loops of $W$ boson, $Z$ boson, top quark, the Higgs boson, the Goldstone boson, the charged scalar color octet, and the neutral scalar color octet. The full one-loop zero-temperature Higgs potential is therefore given as $$V_1(\phi, 0) = V_0 (\phi, 0) + \Delta V_1 (\phi, 0) \ .$$
Now, the finite-temperature contribution at the one-loop level to the Higgs potential is given as \[21\] $$\begin{aligned}
V_1 (\phi, T) & = &
\sum_{l = B, F} {n_l T^4 \over 2 \pi^2}
\int_0^{\infty} dx \ x^2 \ \log
\bigg [1 \pm \exp{ \bigg ( - \sqrt {x^2 + {m_l^2 (\phi)/T^2 }} \bigg ) } \bigg ] ,\end{aligned}$$ where the negative sign is for bosons ($B$) and the positive sign for fermions ($F$). $m_l (\phi)$ is the field-dependent tree-level mass of the participating $l$-th particle, and $n_W = 6$, $n_Z = 3$, $n_t = - 12$, $n_{\phi} = 1$, $n_G = 3$, and $n_{S_C} = 2 n_{S_R} = 2 n_{S_I} = 16$ for the degrees of freedom for each particle, including the color factor of 8. The full one-loop finite-temperature Higgs potential is therefore given as $$V (\phi, T) = V_1 (\phi, 0) + V_1 (\phi, T) \ .$$ We note that the one-loop corrected VEV of the Higgs field at zero temperature is given by the minimum condition $${d V_1(\phi, 0) \over d \phi} = 0 \ ,$$ and the one-loop corrected mass of the Higgs boson is given by $$m^2_H = \left. {d^2 V_1(\phi, 0) \over d \phi^2} \right |_{\phi = v} \ .$$
For qualitative discussions on the EWPT, we take the high temperature approximation of $V_1(\phi, T)$. It is known that in the SM the high temperature approximation is consistent with the exact numerical integration within 5 % at temperature $T$ for $m_F/T < 1.6$ and $m_B/T < 2.2$, where $m_F$ and $m_B$ are the mass of the relevant fermion and boson, respectively. We assume that a similar level of accuracy may be expected in our case. Explicitly, $V_1(\phi, T)$ is given in the high temperature approximation as $$V_1^H (\phi, T) \simeq (D T^2 - E) \phi^2 - F T \phi^3 + G \phi^4 ,$$ where $$\begin{aligned}
&& D = {1 \over 24 v^2} (\sum_B n_B m_B^2 + 6 m_t^2 ) \ , \cr
&& E = {m_H^2 \over 4} - {1 \over 32 \pi^2 v^2} ( \sum_{l=B,F} n_l m_l^4) \ , \cr
&& F = {1 \over 12 \pi v^3} (\sum_B n_B m_B^3) \ , \cr
&& G = {m_H^2 \over 8 v^2}
- {1 \over 64 \pi^2 v^4} \left [\sum_{l=B,F} n_l \log {m_l^2 \over a_l T^2} \right ] \ ,\end{aligned}$$ with $\log (a_F) = 1.14$, $\log (a_B) = 3.91$, and $n_W = 6$, $n_Z = 3$, $n_t = - 12$, $n_{\phi} = 1$, $n_G = 3$, and $n_{S_C} = 2 n_{S_R} = 2 n_{S_I} = 16$ for the degrees of freedom for each particle, where the color factor of 8 is taken into account.
From this formula for $V_1(\phi, T)$ in the high-temperature approximation, one may notice that $V_1(\phi, T) \simeq D T^2 \phi^2$ at very high temperature, and that $V_1(\phi, T) \simeq -E\phi^2 +G \phi^4$ at very low temperature. Therefore, the term proportional to $F$ is crucial at intermediate temperature for the EWPT. We note that contributions from the scalar color octet loops are present in $V_1(\phi, T)$. In particular, the strength of the first-order EWPT is enhanced by the term proportional to $F$ due to the scalar color octet contributions. If the contributions of the scalar color octet are neglected, $V_1 (\phi, T)$ would contain the contributions of the weak gauge bosons and top quark alone, thus would become exactly equivalent to the SM Higgs potential. In this case, the EWPT in this model would also become either weakly first order or higher order. Consequently, the contributions from the loops of the scalar color octet are important for this model to realize the strongly first-order EWPT. For the numerical analysis in the following section, however, we perform the exact numerical integration of $V_1(\phi, T)$.
Numerical Analysis
==================
Let us first examine if the relevant parameters of the Higgs potential diverge at high energy scale. In order to do so, we consider the renormaliztion group (RG) equations for them. We know that $V_1(\phi, 0)$ contains three parameters $\lambda_1$, $\lambda_2$, and $\lambda_3$, through the contributions of the scalar color octet, as well as the SM parameters: $g_1$, $g_2$, and $g_3$, which are the $U(1)$, $SU(2)$, and $SU(3)$ gauge coupling constants, respectively, and $h_t$, which is the Yukawa coupling coefficient of top quark. These parameters are generally renormalizable, and thus should satisfy the RG equations.
Explicitly, the RG equation for the quartic coupling coefficient of the SM Higgs field, $\lambda$, is given as \[15\] $$16\pi^2 \frac{d \lambda}{dt} = 24 \lambda^2 + 48 \lambda_1^2 + 16 \lambda_2^2 + 16\lambda_3^2
-(3 g_1^2+ 9 g_2^2 - 12 h_t^2)\lambda + \frac{3}{8}g_1^4 + \frac{3}{4} g_1^2 g_2^2
+ \frac{9}{8} g_2^4 - 6 h_t^4 \ ,$$ where $t = \log{\mu}$, with $\mu$ being the running mass, and the RG equations for the quartic coupling coefficients of the scalar color octet, $\lambda_1$, $\lambda_2$, and $\lambda_3$, are given as \[15\] $$\begin{aligned}
16\pi^2 \frac{d\lambda_1}{dt} &=& 16 \lambda_1^2 + 8 \lambda \lambda_1
- (\frac{3}{2}g_1^2+ \frac{9}{2} g_2^{2} - 6 h_t^2)\lambda_1
- (\frac{3}{2}g_1^2 + \frac{9}{2} g_2^2 + 18 g_3^{2}) \lambda_1 \cr
&&\mbox{} + \frac{3}{8} g_1^4 + \frac{3}{8} g_2^4
+ \frac{3}{4} g_1^2 g_2^2 \ , \cr
16\pi^2 \frac{d \lambda_2}{dt} & = & 8 \lambda_2^2 + 16 \lambda \lambda_2
- (\frac{3}{2}g_1^2 + \frac{9}{2} g_2^2 - 6 h_t^2) \lambda_2
- (\frac{3}{2} g_1^2 + \frac{9}{2} g_2^2 + 18 g_3^2) \lambda_2 \ , \cr
16\pi^2 \frac{d \lambda_3}{dt} & = & 8 \lambda_3^2 + 16 \lambda \lambda_3
- (\frac{3}{2}g_1^2 + \frac{9}{2} g_2^{2} - 6 h_t^2)\lambda_3
- (\frac{3}{2}g_1^2 + \frac{9}{2} g_2^2 + 18 g_3^{2}) \lambda_3 \ ,\end{aligned}$$ where terms proportional to $\lambda_i$ ($i=4, \cdots,11$) are neglected, and the contributions of $\eta_U$, $\eta_D$ and the bottom quark sector are ignored. There are also RG equations for $g_1$, $g_2$, $g_3$, and $h_t$, which are not shown. These RG equations exhibit the mixing terms between the SM Higgs fields and the scalar color octet. These mixing effects may diverge at very low energies.
We calculate numerically the above RG equations, using the Runge-Kutta method. It is found that the three SM gauge coupling coefficients, $g_1$, $g_2$, and $g_3$, do not have any Landau poles for the whole region of the running mass up to the Planck scale. Also, the quartic coupling coefficient $\lambda$ and the Yukawa coupling coefficient $h_t$ are seen to increase much more slowly than the quartic coefficients of the scalar color octet, $\lambda_i$ ($i = 1,2,3$), as the running mass increases. Thus, we concentrate on the existence of the Landau poles of the quartic coupling coefficients of the scalar color octet.
For the sake of simplicity, we set hereafter $\lambda_3 = 0$, thus $m_{S_R} = m_{S_I}$, and neglect $\lambda_i$ ($i=4, \cdots,11$). Thus, we are left with $\lambda_1$ and $\lambda_2$, that account for the scalar color octet contributions.
We assign an initial value for both $\lambda_1$ and $\lambda_2$ at the electroweak scale, and then let them evolve from the electroweak scale $10^2$ GeV to the unification scale over $10^{12}$ GeV, through their respective RG equations. If any one of them diverges in between the two scales, we set the Landau poles for both of them. In this way, we examine the Landau poles for $\lambda_1$ and $\lambda_2$, from 0 to $4\pi$, where $4\pi$ is set by the perturbative boundary value of the quartic coupling coefficients. We assume that the masses of the scalar color octet are larger than 200 GeV. We take $h_t(m_Z) = 1$ and the SM gauge coupling coefficients at $Z$ boson mass scale.
Our result for $m_H = 120$ GeV is shown in Fig. 1, where a curve of the Landau poles is established. For given initial value of $\lambda_1 = \lambda_2$ at $\mu =10^2$ GeV, the curve shows the value of $\mu$ beyond which $\lambda_1$ or $\lambda_2$ becomes divergent. Thus, the area of Fig. 1 is divided by a boundary of the Landau poles. The lower region of Fig. 1 is free of divergence, whereas the upper region is nonpertabative.
One may notice that if we start with a larger initial value for $\lambda_1$ and $\lambda_2$, the Landau pole occurs at a smaller running mass If the initial value for $\lambda_1$ and $\lambda_2$ is smaller than 0.4, the RG equations show no divergence for the whole range of the running mass from $10^2$ GeV to $10^{12}$ GeV. We note that, for the initial value of $\lambda_1 = \lambda_2 = 1$, the Landau pole occurs when the running mass is a few TeV.
Another result for $m_H = 200$ GeV is shown in Fig. 2, where the values of other parameters are the same as Fig. 1. In Fig. 2, for the same initial value of $\lambda_1 = \lambda_2$ at the electroweak scale, it may be observed that the Landau pole occurs at a comparatively smaller running mass than Fig. 1, if the initial value is taken between about 0.1 and 1.
Now, we study the EWPT in this model. For a given set of parameter values, we examine the shape of $V(\phi,T)$, by varying $T$. If the Higgs potential exhibits the typical shape for the first-order EWPT, with two degenerate minima and a potential barrier between them, at a certain temperature, we define the temperature as $T_c$, the critical temperature. We then calculate the distance between the two degenerate minima, which is defined as $v_c$, the critical VEV, and determine the strength of the first-order EWPT. $v_c/T_c$. In this way, we examine the parameter space of this model for the possibility of the strongly first-order EWPT.
A result is shown in Fig. 3, where $m_H = 120$, $\lambda_1 = \lambda_2 = 1$, and $m_S = 160$ GeV. These parameter values yield $m_{S_C} = 201$ GeV and $m_{S_R} = m_{S_I} = 236$ GeV. We find that the Higgs potential has the shape for the first-order EWPT at the temperature $T_c = 99.7$ GeV. The corresponding critical VEV is calculated to be $v_c = 210$ GeV. Hence, the strongly first-order EWPT, since $v_c/T_c = 2.1$.
In Fig. 4, we show another result for a different set of parameter values, $m_H = 120$, $\lambda_1 = \lambda_2 = 1$, and $m_S = 200$ GeV, where the value of $m_S$ is changed. The masses of the scalar color octet are obtained as $m_{S_C} = 235$ GeV and $m_{S_R} = m_{S_I} = 265$ GeV. The critical temperature for these parameter values is $T_c = 62.1$ GeV and the corresponding critical VEV is $v_c = 241$ GeV. Thus, for these parameter values, too, the EWPT is strongly first-order, since $v_c/T_c = 3.8$. The difference between Fig. 3 and Fig. 4 may be attributed to the change in $m_S$.
We now change the parameter value of $m_H$. For $\lambda_1 = \lambda_2 = 1$, and $m_S = 160$ GeV, we determine the critical temperature and the corresponding critical VEV for $m_H > 115$ GeV, and calculate the strength of the first-order EWPT. We find that for $m_H$ up to 163 GeV, the first-order EWPT is strong enough, in other words, $v_c/T_c > 1$. The result is shown in Fig. 5, where $v_c/T_c$ is plotted as a function of $m_H$, as a solid curve.
Also, for $\lambda_1 = \lambda_2 = 1$, and $m_S = 200$ GeV, we do the same calculation by changing $m_H$. We find that the EWPT may be strongly first-order, for $115 < m_H < 193$ GeV. The result is shown in the same Fig. 5, as a dashed curve. Therefore, Fig. 5 tells that the EWPT in this model may be strongly first-order for $115 < m_H < 163$ GeV, $\lambda_1 = \lambda_2 = 1$, and $m_S = 160$ GeV, as well as for $ 115 < m_H < 193$ GeV, $\lambda_1 = \lambda_2 = 1$, and $m_S = 200$ GeV. Note that the lower bound on $m_H$ is set by the present Higgs search result, not by numerical analysis.
We also examine other regions in the parameter space of this model. Let us set $\lambda_1 = \lambda_2 = 0.05$. At this small value, $\lambda_1$ and $\lambda_2$ are free of the Landau poles up to $\mu = 10^{12}$ GeV, as the results of the RG equations show. With this value, we repeat the numerical analysis. The results are shown in Figs. 6, 7, and 8. Let us briefly describe them.
In Fig. 6, the shape of the Higgs potential at $T_c = 113.5$ GeV is shown, for $m_H = 120$ GeV, and $m_S = 200$ GeV. The critical VEV is obtained as $v_c = 183$ GeV, and the strength of the first-order EWPT is 1.6. The masses of the scalar color octet are calculated as $m_{S_C} = 201$ GeV and $m_{S_R} = m_{S_I} = 203$ GeV.
In Fig. 7, $m_S$ is changed to 250 GeV, while other parameter values are fixed. The critical temperature is $T_c = 60$ GeV, the critical VEV is $v_c = 246$ GeV, the strength of the first-order EWPT is 4.0, and the scalar color octet masses are $m_{S_C} = 251$ GeV and $m_{S_R} = m_{S_I} = 253$ GeV, for $m_H = 120$ GeV, and $m_S = 250$ GeV.
In Fig. 8, the strength of the first-order EWPT is plotted as a function of $m_H$. The solid curve is obtained for $m_S = 200$ GeV, and the dashed curve for $m_S = 250$ GeV. These curves show that this model allows the strongly first-order EWPT for $115 < m_H < 146$ GeV, $\lambda_1 = \lambda_2 = 0.05$, and $m_S = 200$ GeV, as well as for $ 115 < m_H < 197$ GeV, $\lambda_1 = \lambda_2 = 0.05$, and $m_S = 250$ GeV.
Since the sizable quartic couplings to the Higgs, $\lambda_1$ and $\lambda_2$, are crucial for allowing for a strong first-order phase transition, we plot in Fig. 9 the strength of the first-order EWPT versus $\lambda_1/\lambda_2$ for some values of $\lambda_2$: $\lambda_2 = 0.05$, $\lambda_2 = 0.1$, and $\lambda_2 = 0.5$. The values of the other free parameters are the same as in Fig. 6. As an illustration, we obtain that the strength of the phase transition is $v_c/T_c=1.63$ for $\lambda_1=0.05$ and $\lambda_2=0.1$ whereas $v_c/T_c=1.66$ for $\lambda_1=0.1$ and $\lambda_2=0.05$.
One may notice in Fig. 9 that the strength of the phase transition increases as $\lambda_1/\lambda_2$ increases. For given $\lambda_2$, $v_c/T_c$ increases as $\lambda_1$ increases. Also, for given $\lambda_1$, $v_c/T_c$ increases as $\lambda_2$ increases. However, comparing the three curves in Fig. 9, one may induce that the increasing rate of $v_c/T_c$ depends much strongly on $\lambda_1$ than $\lambda_2$. This is mainly due to the fact that, as one may see in Eq. 6, $m_{S_C}$ does not depend on $\lambda_2$.
Recently, there are some extensions of the SM, in which discussions on the electroweak phase transitions are presented \[22-24\]. The model with a number of additional Higgs siglets, Ref.\[24\], may have similar effects on the EWPT due to the couplings between Higgs siglets to the SM Higgs doublet. However, we note that we have the charged scalar color octet in the present model as well as neutral Higgs scalar boson, whereas there is no scalar color octet in the models with additional Higgs singlets. Thus, the search for the charged scalar color octet would be helpful to distinguish the present model from the model with additional Higgs singlets. Also, by examining the Higgs productions via the the scalar color octet loop through the gluon fusion process at the LHC may provide the distinctions among various models.
Conclusions
===========
The extension of the SM with scalar color octet is significantly different from the SM with respect to the EWPT. In order to activate the strongly first-order EWPT, the SM requires a very light Higgs boson, well below the experimental lower bound 114.4 GeV. In other words, the strongly first-order EWPT is practically not allowed in the SM. The existence of the scalar color octet in the SM improves the situation considerably, since the strongly first-order EWPT is possible for $m_H > 115$ GeV, as our numerical analysis shows. The thermal loop contributions to the Higgs potential at the one-loop level, given by the scalar color octet, may play quite remarkable role on the strength of the EWPT. In other words, the first-order EWPT might become stronger due to the thermal contributions by the scalar color octet.
Our numerical analysis suggests that there are wide regions in the parameter space of this model where the strongly first-order EWPT is allowed. The allowed parameter regions are established where the mass of the Higgs boson may be consistent with the present experimental lower bound ($m_H >115$ GeV), the masses of the scalar color octet are within the reach of the forthcoming LHC ($m_S \simeq 200$ GeV), and the quartic coupling coefficients for the scalar color octet are free of Landau poles.
We would like to note that we simplify our calculations by neglecting the quartic coupling coefficients $\lambda_i$ ($i = 3, \cdots, 11$), and $\eta_U$ and $\eta_D$ in the RG equations. It is known that the presence of these parameters would impose a stricter bound on the running mass coming from the perturbative-theoretic considerations such that the Landau poles would appear at a lower running mass. We note these simplifications are consistent with the parameter region we consider.
On the other hand, we find that the contributions of Higgs boson loops and the Goldstone boson loops at the one-loop level are negligibly small in the present parameter region. This is mainly because the contributions due to the Higgs boson loops are smaller than the contributions due to the scalar color octet, in particular when the Higgs boson mass is smaller than 200 GeV and the masses of the scalar color octet are larger than 200 GeV.
Summarizing, we establish the possibility of a strongly first-order EWPT, for the electroweak baryogenesis, in the extension of the SM with scalar color octet.
Acknowledgments {#acknowledgments .unnumbered}
===============
S. W. Ham thanks S. Baek, P. Ko, and Chul Kim for valuable comments. He would like to acknowledge the support from KISTI under “The Strategic Supercomputing Support Program (No. KSC-2008-S01-0011)” with Dr. Kihyeon Cho as the technical supporter. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0086961).
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[^1]: s.w.ham@hotmail.com
[^2]: shims@sungshin.ac.kr
[^3]: sunkun@konkuk.ac.kr
|
---
abstract: 'We prove new exponents for the energy version of the Erdős-Szemerédi sum-product conjecture, raised by Balog and Wooley. They match the previously established milestone values for the standard formulation of the question, both for general fields and the special case of real or complex numbers, and appear to be the best ones attainable within the currently available technology. Further results are obtained about multiplicative energies of additive shifts and a strengthened energy version of the “few sums, many products" inequality of Elekes and Ruzsa. The latter inequality enables us to obtain a minor improvement of the state-of the art sum-product exponent over the reals due to Konyagin and the second author, up to $\frac{4}{3}+\frac{1}{1509}$. An application of energy estimates to an instance of arithmetic growth in prime residue fields is presented.'
address:
- 'Misha Rudnev, Department of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom'
- 'Ilya D. Shkredov, Steklov Mathematical Institute, Division of Number Theory, ul. Gubkina, 8, Moscow, 119991; IITP RAS, Bolshoy Karetny per. 19, Moscow, 127994 and MIPT, Institutskii per. 9, Dolgoprudnii, 141701'
- 'Sophie Stevens, Department of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom'
author:
- Misha Rudnev
- 'Ilya D. Shkredov'
- Sophie Stevens
title: 'On the energy variant of the sum-product conjecture'
---
Preface
=======
In this paper we show that the milestone results in the current sum-product theory literature allow for a pure energy formulation involving both addition and multiplication. Previous inequalities of sum-product type involve either $|A\cdot A|$ and $|A+A|$, or $\E^+(A)$ and $|A\cdot A|$, or $\E^*(A)$ and $|A+A|$. The energy-energy formulation was raised by Balog and Wooley [@BW]; its key feature being that the sum-product conjecture cannot hold in its maximum strength in the energy formulation. Owing to an example in [@BW], which we will shortly retell, the cardinality formulation as Conjecture \[esc\] below should be weakened to the energy version as Conjecture \[esc1\]. The latter suggests a somewhat uncomfortably-looking fractional exponent.
We advance largely on the technical level. The well-established tool in additive combinatorics for passing from energy-type results to the existence of subsets with the desired structure is the Balog-Szemerédi-Gowers theorem. This has been done, in particular, by Balog and Wooley, as well as much earlier work on the relation between geometric incidences and algebraic growth, such as [@BKT], [@GT]. But for someone concerned with quantitative values of the resulting exponents, the use of the Balog-Szemrerédi-Gowers theorem usually comes at a price.
It was first observed that the Balog-Szemerédi-Gowers theorem can be avoided by Konyagin and the second author [@KS2] who succeeded in significantly strengthening the main estimate of [@BW]. But the final exponent in the resulting energy-energy inequality in [@KS2] has nonetheless stopped short of its cardinality-cardinality predecessor due to Elekes [@E].
In this paper we remove this gap and prove a variety of energy-energy sum-product type inequalities, which have the same exponents as their cardinality prototypes, up to logarithmic factors. The first of our key inequalities dealing with real/complex numbers is , which is the energy-energy analogue of the classical sum-product $5/4$ result by Elekes [@E], which we record here as . The two energies appearing in this inequality can be replaced by energies of two different additive shifts of a set (which is useful for applications), see Theorem \[t:BW\_mult\]. The second inequality is given by , which, in a certain sharp regime, is the energy version of the “few sums, many products" inequality by Elekes and Ruzsa [@ER]. As a matter of fact has better exponents than its prototype, see the discussion before the formulation of the corresponding Theorem \[t:fsmp\].
We also furnish the general field variants of our energy inequalities, valid in particular in positive characteristic, with an inevitable constraint on how small the set in question should be. They are usually just slightly weaker than for the particular case of the real or complex field. This combines the pruning techniques developed in this paper with the use of the incidence theorem of the first author from [@misha] and its further development in [@RRS], [@AMRS].
We strongly feel (although we make no attempt to substantiate this claim), that these energy exponents are the best ones attainable within the currently available technology. For the real or complex field there are better sum-product exponents, based on the foundational work by Solymosi [@S1], [@Sol05]. But these appear to necessarily involve cardinality of at least one counterpart, rather than the two energies. For general fields, there are no better ones so far. It appears that further significant progress towards the sum-product conjecture challenges one to break these energy barriers.
Finally, in the main body of the paper we discuss some applications of our energy-energy sum-product type inequalities: some in passing and one at length.
Introduction and main results
=============================
Let $\F$ be a field, with the multiplicative group $\F^*$ and let $A\subset \F$ be a finite set. The sum set, product set and quotient set of $A$ are defined respectively as $$\begin{aligned}
&A+A:=\{a+b:a,b\in A\},\\
&A\cdot A:=\{a\cdot b:a,b\in A\},\\
&A/A:=\{a/b ~:~ a,b\in{A},\,b\neq0\}.\end{aligned}$$ Sum, etc. sets involving different sets, say $A+B$ are defined in a similar way. For $x\in \F^*$, we write simply $xA$ for $\{x\}A$ and $x+A$ or $A+x$ for $\{x\}+A$.
It has long been observed that unless $A$ is in some sense close to a coset of a subfield of $\F$, additive and multiplicative structure find it hard to coexist : it is easy to minimise one of $|A+A|$ or $ |A\cdot A|$ but then the other set becomes very large. The question was raised – originally in the context of integers – and the first quantitative result obtained by Erdős and Szemerédi [@ES], leading to the renowned conjecture:
\[esc\] Let $A\subset \mathbb{R}$. Then $$\max(|A + A|,\, |A \cdot A|)\, \gtrsim\, |A|^{2}.$$
To avoid trivialities we further assume the $0\not\in A$ and $|A|>1$, this will be implicit in all statements we make, as well as that the sets $A,B,C,\dots$ are finite.
As usual, we use the notation $|\cdot|$ for cardinalities of finite sets. The symbols $\ll$, $\gg,$ suppress absolute constants in inequalities, as do their respective equivalents $O$ and $\Omega$. Besides, $X=\Theta(Y)$ means that $X=O(Y)$ and $X=\Omega(Y)$.
In addition, e.g., in the above statement of Conjecture \[esc\], the symbols $\lesssim, \; \gtrsim, \; \sim$ are used to replace, respectively $O,\;\Omega,\;\Theta$ when the inequalities involved are weakened by a power of $\log|A|$. Thus $|A|$ is viewed as an asymptotic parameter. The suppressed constants are therefore independent of $|A|$.
If $\F$ has positive characteristic $p$, we only deal with a “small set” case, $|A|$ being at most some less-than-$1$ power of $p$, so $p$, always denoting the positive characteristic of $\F$, is regarded as an asymptotic parameter as well.
The sum-product conjecture remains open although current world records in [@KS2] for the reals and [@RRS], [@AMRS] in positive characteristic have edged nearer to the statement of the conjecture.
Fundamental to the study of the sum-product phenomenon is a $L^2$ quantity expressing the additivity or multiplicativity of a set, known as energy. The additive energy between sets $A$ and $B$ is defined as $$\E^{+}(A,B):=|\{(a_1,a_2;b_1,b_2)\in A^{2}\times B^2 :a_1+b_1=a_2+b_2\}|.$$ We write $\E^+(A,A)=\E^+(A)$; if this quantity is considerably larger than the trivial lower bound $|A|^2$ (and closer to the trivial upper bound $|A|^3$), we say $A$ has an additive structure.
The multiplicative energy $\E^\times(A,B), \,\E^\times(A)$ is defined similarly. The Cauchy-Schwarz inequality relates the energy to the Erdős-Szemerédi conjecture: $$\label{CSE} \E^{+}(A)|A+A|,~\E^{*}(A)|A\cdot A|\;\geq \;|A|^{4}.$$ Note that by rearranging the equation defining energy, say $A+A$ in can be replaced by $A-A$, even though the latter two sets may in principle differ quite a bit.
Geometrically, the multiplicative energy $\E^\times(A)$ equals the number of ordered pairs of points of the plane set $A\times A$, supported on lines through the origin (corresponding to ratios in $A/A$). The additive energy $\E^+(A)$ equals the number of ordered pairs of points of $A\times A$ on parallel lines with the slope plus or minus $1$.
To what quantitative extent do additive and multiplicative structure find it hard to coexist in a set? Recently Balog and Wooley [@BW] raised an interesting question as to what is the correct energy, that is $L^2$, formulation of Conjecture \[esc\]. If one believes that one of $|A+A|$ or $|A\cdot A|$ must be $\gtrsim|A|^2$ can both $\E^+(A)$ and $\E^\times(A)$ be nonetheless large? An easy example shows, yes: take $A$ as the union of two disjoint equal in size arithmetic and geometric progressions. Then both energies are $\Omega(|A|^3)$, that is, up to constants, as big as it gets. Moreover, any subset containing more than, say $51\%$ of $A$ would have both energies $\Omega(|A|^3)$. On the other hand, for either $\E^+,$ or $\E^\times$, one can find a subset containing every second member of $A$, where the corresponding energy is $\lesssim |A|^2$.
It is possible to intertwine an arithmetic and geometric progression in a smarter way to ensure that any subset of $A$ containing a positive proportion of its members, has both energies considerably in excess of $|A|^2$. Balog and Wooley constructed such an example over the integers. We take a moment to review it briefly, owing to its appeal.
Let $A$ be the union of $n$ disjoint dilates of the integer interval $I= [n^2,\ldots,2n^2)$ by factors $1,2,\ldots, 2^{n-1}$. Note that $A\times A$ contains the union of $n$ disjoint square grids $P=(I\times I) \,\cup\, (2I\times 2I)\,\cup\, \ldots\,\cup \,(2^{n-1}I \times 2^{n-1}I)$. Then $\E^+(A)\gg |A|^{7/3}$ (sum of additive energies of each grid in $P$) while $\E^\times(A)\gg |A|^{7/3}\log^s|A|,$ because of and the fact that $|A\cdot A| \sim n^5$. (See [@Fo] proving the explicit value of $s = 0.086...\,$.) If $A'\subset A$ has cardinality $\alpha|A|$, for $0<\alpha<1$, it is easy to estimate $\E^+(A')$ from below just by looking at intersections $A'\times A'$ with $P$; the minimum estimate is achieved when $A'\times A'$ intersects each square forming $P$ uniformly at $(\alpha n^2)^2$ points. For $\E^+(A')$ use and the obvious inclusion $A'\cdot A' \,\subseteq\, A\cdot A.$ Thus $$\E^+(A'), \,\E^\times(A') \gg \alpha^4 n^7 = \alpha^4 \Theta( |A|^{7/3} ).$$ Besides, $\E^\times(A')$ actually exceeds the right-hand side by a power of $\log|A|$. By slight manipulations with the number $n$ of dilates of $I$ versus its size $n^2$, one can easily ensure that the logarithmic factor in $|A|$ is present in the estimate for both energies of $A'$.
The Balog–Wooley example shows that even though multiplicative and additive structure may not conjecturally coexist in any $A$ in the strong sense of Conjecture \[esc\], they cannot be completely divorced in the $L^2$ sense even by taking reasonably small subsets. On the other hand, we do not believe that the standard construction arsenal offers a stronger one than described above. It seems likely that the following claim is true.
\[esc1\] Let $A\subset \mathbb{R}$. There exists $A'\subseteq A$, such that $|A'|\geq |A|/2$, and $$\min [ \E^+(A'),\,\E^\times(A') ] \lesssim |A|^{7/3}.$$
Note that trivially we cannot expect to destroy multiplicative (respectively additive) structure by taking a subset, unless the latter is very thin, as is the case if, for example, $A$ is a geometric (respectively arithmetic) progression.
Balog and Wooley formulated their results in terms of the decomposition of $A$ as follows.
Let $A\subset \mathbb R$ be a set and $\d = 2/33$. Then there are two disjoint subsets $B$ and $C$ of $A$ such that $A = B\sqcup C$ and $$\max\{ \E^{+} (B), \E^{\times} (C)\} \lesssim |A|^{3-\delta}.$$
\[t:BW\]
The sum-product phenomenon is not restricted specifically to reals. Its study in prime residue fields $\F_p$ was initiated by Bourgain, Katz, and Tao [@BKT]. If the field $\F$ has positive characteristic $p>0$, we consider the case when $A$ is suitably small in terms of $p$. We do not know of any evidence that Conjectures \[esc\], \[esc1\] may be false if, say $|A|<p^{1/3}$. To this end [@BW] contains a positive characteristic version of Theorem \[t:BW\], with a smaller value of $\delta=4/101,$ subject to the constraint roughly $|A|<p^{101/161}.$\[rmk\]
The two main ingredients of the argument in [@BW] were the two following geometric incidence theorems and additive combinatorics. The known sum-product results over the real or complex field are somewhat stronger than in fields of positive characteristic largely due to order properties of reals, which so far have been indispensable for proofs of the celebrated Szemerédi-Trotter theorem in the plane. See [@SzT], [@To] for the original proof for reals and subsequent extension to the complex field.
\[Szemerédi-Trotter Theorem\]\[t:ST\] The number of incidences between a set of $m$ lines and $n$ points in $\mathbb C^2$ is $O[(mn)^{2/3} + m + n]$.
For arbitrary fields the first author proved a weaker geometric incidence theorem in $\F^3$.
\[[@misha]\] \[t:MR\] The number of incidences between a set of $m$ planes and $n\leq m$ points in $\F^3$ is $O[m (n^{1/2} +k )]$, where $k$ is the maximum number of collinear points, under an additional constraint $n\leq p^2$ in positive characteristic.
Thus real or complex numbers will be the special case in the sequel, and since $\mathbb R$ is not special for the ensuing discussion versus $\mathbb C$, we formulate the corresponding results in terms of the latter field.
It was shown in [@AMRS] that Theorem \[t:MR\] implies a weaker version of the Szemerédi-Trotter theorem for a general $\F$, with the main term $m^{3/4}n^{2/3}$ if the point set is a Cartesian product $A\times B$, with $|B|\le |A|\leq p^{2/3}$ in positive characteristic. This was recently improved by the third author and de Zeeuw and generalised to arbitrary point sets as follows.
\[[@SZ]\]\[t:SzT\_Fp\] Consider a set of $m$ lines in $\F^2$.
\(i) Let $A\times B\subset \F^2$ be a set of $n$ points, with $n^{1/2}<m<n^{3/2}$ and the constraint $mn^2<p^4$ in positive characteristic. The number of incidences between the above sets of points and lines is $O(m^{3/4}n^{5/8}).$
\(ii) For any set of $n$ points in $\F^2$, with $n^{7/8}<m<n^{8/7}$ and $m^{13}n^{-2} < p^{15}$, the number of incidences with the set of $m$ lines is $O[(mn)^{11/15}].$
Incidence theorems have been widely used in arithmetic combinatorics. Elekes [@E] realised that Theorem \[t:ST\] applies to sum-product type problems and proved the following estimate towards Conjecture \[esc\]: $$\label{ele}
\max( |A+A|,\;|A\cdot A| )\gg |A|^{1+\delta},~~\delta=1/4.$$
Roche-Newton and the first two authors [@RRS] applied Theorem \[t:MR\] in a similar vein and proved, for any field $\F$, that $$\label{us}
\max( |A+A|,\;|A\cdot A| )\gg |A|^{1+\delta},~~\delta=1/5.$$ under an additional constraint $|A|\leq p^{5/8}$ in positive characteristic.
There has been a series of improvements of the estimate in the real and complex case, started by Solymosi [@S1], and currently up to $\gtrsim |A|^{\frac{4}{3}+\frac{5}{9813}}$ in the right-hand side by Konyagin and the second author [@KS2]. All such improvements of (see, e.g., the references in [@KS2]) used crucially the order properties of the reals (the arguments would usually generalise to $\mathbb C$, see for example [@KR]) and benefited by repeated applications of the Szemerédi-Trotter theorem involving the sets of sums or products themselves. Without order in (a subfield of) $\F$, the sum-product estimate is the best one known.
Our main result is the following theorem, which implies, up to factors of $\log|A|$, the latter two estimates in the context of Conjecture \[esc1\]. We establish the following.
\[mainth\] Let $A\subset \F$.There exists $A'\subseteq A$, such that $|A'|\geq |A|/2$, and $$\min [ \E^+(A'),\,\E^\times(A') ] \lesssim |A|^{3-\delta},$$ where $\delta=1/4$ in the special case $\F=\mathbb C$ and $\delta = 1/5$ for any $\F$, with an additional constraint $|A|\leq p^{5/8}$ in positive characteristic.
Theorem \[mainth\] is an immediate consequence of the forthcoming, and stronger, Theorem \[t:BW\_R\], which is an improvement of Theorem \[t:BW\]. The values of $\delta$ we establish match those in the estimates , . Thus, our arguments emphasise the geometric (and reduce the additive) combinatorics content of the proof: passing from the estimates , to their weaker $L^2$ formulation in Theorem \[mainth\] only incurs logarithmic factors in $|A|$.
Since these estimates are, of course, partial apropos of Conjecture \[esc1\], we have not troubled ourselves with calculating the exact powers of $\log|A|$. On the other hand, we do not expect that, modulo these factors, the estimates in question can be improved within today’s state of the art toolkit. Once again, all the improvements of the Elekes estimate in the real/complex case appear to relate (multiplicative or additive) energy to the size of the counterpart (respectively product or sum) set and do not work in the energy-energy sense, cf. the title of the breakthrough paper [@Sol05] by Solymosi.
These improvements, in particular were enabled by the idea of the second author and Schoen to use the [*third,*]{} rather the [*second*]{} moment, or cubic (and higher order) energy of the convolution function arising in the description of sum or product sets. See, e.g., [@SS]. This opportunity is inherent in the numerical values of the exponents arising in the Szemerédi-Trotter theorem. It does not appear to be granted by the weaker Theorem \[t:MR\]. Nor does it seem to be at hand if one pursues energy-energy estimates. This is why, we believe, Theorem \[mainth\] marks a certain milestone, and to improve its exponents, which are $\delta=1/4$ for $\mathbb F=\mathbb C$ and $\delta =1/5$ otherwise, one needs a conceptual innovation, whether this is about the reals or a general $\F$.
On the technical level we do much better than Theorem \[t:BW\] by avoiding the use of the Balog-Szemerédi-Gowers theorem. The latter, presented as Theorem \[t:BSzG\_Schoen\] below, has been a standard arithmetic combinatorial tool for passing from large energy bounds to subsets with small doubling, in particular Balog and Wooley used it to prove Theorem \[t:BW\]. Unfortunately, on the quantitative level applying the Balog-Szemerédi-Gowers theorem is usually quite wasteful. Konyagin and the second author found a way to avoid it in the context of Theorem \[t:BW\], where they proved $\delta=1/5$ over the reals [@KS2]. In the proof of the key result of this paper, the forthcoming Theorem \[t:BW\_R\], we follow the main line of the argument in Section 4 of [@KS2], making it somewhat stronger, which yields what we believe is the strongest result, within reach of today’s machinery, that is $\delta=1/4$ for real and complex numbers.
We develop the argument in the context of a general field $\F$, where we use Theorem \[t:MR\], while Theorem \[t:ST\] applies to $\F=\mathbb C$ as a special case. This enables us to prove $\delta=1/5$ for a general $\F$, matching its value in the sum-product estimate , while $\delta=1/4$ for $\F=\mathbb C$, matches the Elekes estimate .
As far as applications are concerned, we are interested in quantitative arithmetic growth estimates. By arithmetic growth we mean, for an integer $n\geq 2$, having a function $f: \F^n\to \F$, such that for any $A\subset \F$, sufficiently small in terms of $p$ in the positive characteristic case, the cardinality of the range of $f$, restricted to $A^n$, exceeds $|A|$ by orders of magnitude. See [@AMRS] and the references contained therein for some quantitative estimates for $n=2,3,4$ over general fields and general discussion.
It appears that our energy method enables one to obtain stronger quantitative growth estimates, for they often result in a relation binding energies of two different types, say $\E^\times(A)$ and $\E^+(A)$ for some putative set $A$. Our results, see the forthcoming Corollary \[t:BW\_C\], provide upper bounds for the product of the two energies if one passes to a pair of large subsets. Previously available estimates would bind, say multiplicative energy and the sum set, see e.g., [@Sol05]. Passing from $\E^+(A)$ to the sum set, aiming basically to invert the first inequality in would invoke a quantitatively costly application of the Balog-Szemerédi-Gowers theorem. Our method enables one to bypass this.
There is a well established connection between arithmetic growth and incidence geometry estimates, in both directions. However exploring this connection from the former towards the latter estimates would invariably invoke the Balog-Szemerédi-Gowers theorem. We limit the references to the well-known works of Bourgain, Katz and Tao [@BKT] and Green and Tao [@GT]; see Section 6 in both papers. It turns out, however, that the Balog-Szemerédi-Gowers theorem can be avoided. In the context of incidence estimates, namely Theorem \[t:SzT\_Fp\] (ii), this was achieved in [@SZ]. In the same vein, we challenge an interested reader to embark on reducing – and strengthening – the fairly lengthy proof of Proposition 6.6 in [@GT] to a much shorter energy argument, avoiding the Balog-Szemerédi-Gowers theorem in the vein of the forthcoming Theorem \[t:BW\_mult\].
We presently limit the number of applications considered in detail to one, concerning the prime residue field $\F_p$. Given $f: \F^n_p\to \F_p$, what is the lower bound on $|A|$, such that for any $A\subset \F_p$, the range of $f$, restricted to $A^n$, takes up a positive proportion of the field $\F_p$? For many such $f$ one can relatively easily prove the threshold $|A|=\Omega(p^{2/3}),$ via character sums or often just linear algebra methods that work well for relatively large sets with respect to $p$. See e.g [@CEIK]. However, these techniques usually fail to work for smaller $A$. To this effect, the challenge is to reduce the threshold $|A|=\Omega(p^{2/3})$ for some $f$.
Petridis [@Petridis] proved recently that if $|A| \geq p^{5/8}$, then the cardinalities of the sets $(A+ A)\cdot (A + A), \,(A+A) / (A+A)$ are both $\Omega(p)$, having incorporated the so-called “generic projections” argument from [@BKT] and Theorem \[t:MR\] into a rather involved argument. Generic projections is an easy pigeonholing argument showing that the sets $(A-A) / (A-A)$, as well as $\{\frac{ac-bd}{a-d}:\,a,b,c,d\in A,\,a\neq d \}$ are both equal to $\F_p$ as long as $|A| >p^{1/2}$.
As a matter of fact, Petridis establishes a stronger $L^2$ claim that the number of solutions of the equation $$(a+b)(c+d) = (a'+b')(c'+d'):\,a,\ldots,d'\in A$$ is bounded as $O(|A|^8/p)$, that is up to a constant the expected number, as long as $|A|>p^{5/8}$. Such a bound appears to be out of reach by methods of [@BKT] even regarding the set $(A-A) / (A-A)$ if one rearranges the latter equation as fractions and replaces the plus signs by minuses.
In this paper we establish the following.
\[thm:had\] Let $A\subseteq \F_p, $ with $|A| \gg p^{25/42}\log^K |A|$, for some absolute constant $K$. There are disjoint $B,C\subset A$, each of cardinality $\geq |A|/3$, such that number of solutions to the equation $$\frac{ab - c}{a - d}=\frac{a'b' - c'}{a' - d'}:\,a,b,a',b' \in B;\,c,d,c',d' \in C \label{e:had}$$ is $O({\aa^8}/{p})$, and therefore $\left|\left\{
\frac{ab-c}{a-d}:a,b,c,d\in A
\right\}
\right|=\Omega(p)\,.$
The reader can verify that in the latter theorem all the minus signs can be replaced by plus signs as well.
Further results
---------------
Here we present a somewhat stronger formulation of Theorem \[mainth\], its analogue for multiplicative energies of additive shifts and the energy version of the Elekes-Ruzsa few sums, many products inequality. The latter is available only in the real/complex setting; over the reals it yields a minor improvement of the best known sum-product exponent, after being plugged into the argument recently developed by Konyagin and the second author. We also provide an auxiliary subsection which contains a suitably tailored version of the Balog-Szemerédi-Gowers theorem – which may be interesting in its own right – and some indication of what our results would look like if the Balog-Szemerédi-Gowers theorem had to be used.
### Sum-product decomposition and energy inequalities
\[t:BW\_R\] Let $A\subset \F$. There exist two disjoint subsets $B$ and $C$ of $A$, such that $A=B\sqcup C$, and $$\max [ \E^+(B),\,\E^\times(C) ] \lesssim |A|^{3-\delta},$$ where $\delta=1/4$ in the special case $\F=\mathbb C$ and $\delta = 1/5$ for any $\F$, with an additional constraint $|A|\leq p^{5/8}$ in positive characteristic.
Theorem \[t:BW\_R\] clearly implies Theorem \[mainth\]: one of $B,C$ has size $\geq|A|/2$.
As we have mentioned, one cannot expect both $B$ and $C$ in Theorem \[t:BW\_R\] to constitute a positive proportion of $A$. But this can be achieved by weakening the claim as follows, to be used in the proof of Theorem \[thm:had\].
\[t:BW\_C\] Let $A\subset \F$, with an additional constraint $|A|\leq p^{3/5}$ in positive characteristic. There exist two disjoint subsets $B$ and $C$ of $A$, each of cardinality $\geq |A|/3$, such that $$\label{formula1}
\E^{+} (B) \cdot \E^\times(C)^{3/2} \; \lesssim \;|A|^7.$$ In the latter estimate the additive and multiplicative energy can be swapped (for some other $B,C$). Besides, there exist two disjoint subsets $B$ and $C$ of $A$, each of cardinality $\Omega( |A|)$, such that $$\label{formula2} \E^+(B)\cdot \E^\times(C) \, \lesssim \, |A|^{28/5}.$$ Furthermore, if $\mathbb F=\mathbb C$, the estimate improves to $$\label{formula3}
\E^{+} (B) \cdot \E^\times(C) \; \lesssim \;|A|^{11/2}.$$
We will spell out the proof of Theorem \[t:BW\_R\], the key quantitative result, in all detail. This proof, furthermore, allows for a number of straightforward variations, which result from the fact established in the quoted literature. Some of these variations are left without detailed proofs, for they would repeat the main arguments more or less line by line.
### Multiplicative energy of translates
In [@Shkredov_R[A]] the second author considers a slightly more general context than usual sum–products setting. The proof of Theorem \[t:BW\_R\] combined with the arguments of [@KS2], enables one to establish a variant of Theorem \[t:BW\_R\] as follows.
Let $A\subset \mathbb C$ be a set, $\a \neq 0$, and $\d = 1/4$. Then there are two disjoint subsets $B$ and $C$ of $A$ such that $A = B\sqcup C$ and $$\label{f:BW_mult1}
\max\{ \E^{\times} (B), \E^{\times} (\a+C)\} \lesssim |A|^{3-\d} \,.$$ Further, there are disjoint subsets $B'$ and $C'$ of $A$ such that $A = B'\sqcup C'$ and $$\label{f:BW_mult2}
\max\{ \E^{+} (B'), \E^{+} (1/ C')\} \lesssim |A|^{3-\d} \,.$$ \[t:BW\_mult\]
We will further present a proof of the following consequence of Theorem \[t:BW\_mult\], which improves a result from [@s_E_k].
Let $A\subset \mathbb C$ be a set, and let $$R[A] := \left\{ \frac{a_1-a}{a_2-a} ~:~ a,a_1,a_2 \in A,\, a_2 \neq a \right\} \,.$$ Then there are two sets $R', R'' \subseteq R[A]$, $|R'|, |R''| \ge |R[A]|/2$ such that $\E^\times (R') \lesssim |R'|^{3-1/4}$ and $\E^+ (R'') \lesssim |R''|^{3-1/4}$. \[c:R\_energy\]
Note that the set $R[A]$ is the set of finite pinned cross-ratios, generated by the projective set $A\cup\{\infty\}$, defined by quadruples $(a,a_1,a_2,\infty)$.
Naturally, an analogue of Theorem \[t:BW\_mult\] over a general field $\F$, following from the proof of Theorem \[t:BW\_R\] also exists. It is established by combining the arguments of the proof of Theorem \[t:BW\_R\] and the proof of Proposition 2 in [@KS2][^1].
Let $A\subset \F$. If $\F$ has positive characteristic $p$, suppose $|A|\leq p^{5/8}.$ Let $\a \in \F^*$ and $\d =1/5$. Then there are two disjoint subsets $B$ and $C$ of $A$ such that $A = B\sqcup C$ and $$\max\{ \E^{\times} (B), \E^{\times} (\a+C)\} \lesssim |A|^{3-\d} \,.$$ \[t:BW\_mult\_Fp\]
We cannot obtain an equally strong analogue for the second statement of Theorem \[t:BW\_mult\] about the set of the reciprocals. Extending the bound (\[f:BW\_mult2\]) to general fields would require incidence results for hyperbolae rather than affine objects. However, a weaker result can most likely be derived on the basis of Proposition 14 in [@AMRS].
In addition, Corollary \[t:BW\_C\] also applies if one replaces the two energies appearing therein by multiplicative energies of two distinct translates of $A$.
### Few Sums, Many Products
Our approach also allows for the energy generalisation of the well-known result of Elekes and Ruzsa, from the paper [@ER], whose title we have used for this subsection. Namely, for $A\subset \mathbb R$ (as well as of $\mathbb C$) one has $$|A+A|^4|AA|\gtrsim |A|^6$$ In fact, we strengthen the above result to an energy-energy inequality, which is “morally” equivalent to $|A+A|^3|AA|\gtrsim |A|^5.$ We present the result as an energy inequality and remark that for general fields we do not have an analogue that would be stronger than in Corollary \[t:BW\_C\].
\[t:fsmp\] Let $A\subset \mathbb C$. There exist two disjoint subsets $B$ and $C$ of $A$, each of cardinality $\geq |A|/3$, such that $$\label{formula4}
\E^{\times} (B) \cdot \E^+(C)^{3} \; \lesssim \;|A|^{11}.$$
However, since the exponents $1,3$ in the above estimate are quite far from one another, estimate beats the non-optimal estimate only if the additive anergy is sufficiently large. Besides, the two energies cannot be swapped in Theorem \[t:fsmp\].
### The sum-product estimate over $\mathbb R$
Theorem \[t:fsmp\] yields a minor improvement of the stat-of-the-art sum-product exponent over the reals. Konyagin and the second author [@KS1], [@KS2] set a new world record towards Conjecture \[esc\], having shown that for a finite set $A$ of reals, $$\label{ksr}
\max(|A+A|,\,|A\cdot A|)\;\gtrsim\; |A|^{\frac{4}{3}+\frac{5}{9813}}.$$ This improved the previous best-known exponent $\frac{4}{3}$ obtained some ten years earlier as a result of a graceful and renowned construction by Solymosi [@Sol05], which only relies on the order properties of reals and does not use the Szemerédi-Trotter theorem. Within the arguments in [@KS1], [@KS2], the margin by which the value $\frac{4}{3}$ can be beaten depends on the best known estimates apropos of two issues, which can be described as “few products, many sums” and “few sums, many products”. The current approach to both issues that furnishes sufficiently strong estimates is largely based on the Szemerédi-Trotter theorem. Dealing with the “few sums, many products” side of the coin has been much more successful; this was first done by Elekes and Ruzsa [@ER]. Its counterpart proves to be much harder; it is referred by some authors as the [*weak Erdős-Szemerédi conjecture*]{}, with the best known estimate stated as [@KS2 Theorem 12], originating in [@s_sumsets].
Konyagin and the second author proved an energy version of the estimate of Elekes and Ruzsa [@KS1 Theorem 9]; it is slightly weaker than estimate above herein. The following improvement of the sum-product inequality comes after a calculation if one replaces the estimate of [@KS1 Theorem 9] with a variant of estimate . This improves the estimates of [@KS2 Lemma 18] and if one chases through the ensuing [@KS2 Proof of Theorem 3], the result becomes as follows.
\[objeli\] For a finite set $A\subset \mathbb R$, one has $$\max(|A+A|,\,|A\cdot A|)\;\gtrsim\; |A|^{\frac{4}{3}+\frac{1}{1509}}.$$
### Balog-Szemerédi-Gowers Theorem
We present some auxiliary results in this short section as a weaker, but arguably less technical alternative to the forthcoming key Propositions \[l:BW\_FF\], \[l:BW\_R\]. In contrast, this section is about the Balog–Szemerédi–Gowers theorem [@Gow_m], which our main proofs avoid. We take advantage of the opportunity to present a small but potentially useful modification of one result from Schoen’s paper [@S_BSzG]. For modern forms of the Balog-Szemerédi-Gowers theorem see, e.g., [@BouGar09] and [@S_BSzG].
It is easy to see that the following statement implies the original Balog–Szemerédi–Gowers theorem.
Let $(G,+)$ be an abelian group. Let $A\subseteq G$ be a set, $K\ge 1$ be a real number, and $k\ge 2$ be an integer. Suppose that $\E^+ (A) \ge |A|^3 / K$. Then there are sets $A_* \subseteq A$, $P \subseteq A-A$ such that $|A_*| \ge |A|/(8kK)$, $|P| \le 8kK|A|$ and for any $a_1,\dots, a_k \in A_*$ one has $$\label{f:BSzG_Schoen}
|A \cap (P+a_1) \cap \dots \cap (P+a_k)| \ge \frac{|A|}{4K} \,.$$ \[t:BSzG\_Schoen\]
Theorem \[t:BSzG\_Schoen\] allows, e.g., for the following analogue of the forthcoming key Prorosition \[l:BW\_R\], which basically stand for the Balog-Wooley decomposition. The estimates are weaker but proofs are simpler.
Let $A \subset \mathbb C$ be a set. Then there is $A_1 \subseteq A$ such that $|A_1| \gtrsim \E^\times (A) |A|^{-2}$ and $$\label{f:sp_E_E_1'}
(\E^{+} (A_1))^2 (\E^\times (A))^9 \lesssim |A|^{32} \,.$$ \[t:sp\_E\_E’\]
The special case $\mathbb F=\mathbb C$ in the above formulation indicates the use of the Szemerédi-Trotter theorem in the proof below. We challenge an interested reader to formulate a general field analogue, replacing the use of Theorem \[t:ST\] by Theorem \[t:SzT\_Fp\].
Proof of Theorem \[t:BW\_R\]
============================
We start out with two intermediate results towards the estimate of Theorem \[t:BW\_R\]: one will later result in $\delta=1/5$ for a general $\F$ and the other in $\delta=1/4$ for $\F=\mathbb C$ in Theorem \[t:BW\_R\].
\[l:BW\_FF\] Let $A\subset \mathbb{F}$, with $|A|^6\lesssim p^2 \E^\times(A)$ in positive characteristic. Then there is a set $A_{1}\subseteq A$ such that $|A_1|\gtrsim \sqrt{\E^\times(A)/|A|}$ and $$\label{f:ksdf}
\E^{+}(A_1)\lesssim |A_1|^{11/2}|A|^{3/2}(\E^\times(A))^{-3/2}\,.$$ The energies $\E^\times,\E^+$ in the above statement can be swapped (for some other $A_1$).
Using the pigeonhole principle, we choose a dyadic group $P$ of ratios from $A/A$, with approximately some $t$ realisations, which supports at least a fraction of $\frac{1}{\log_2|A|}$ of $\E^\times(A)$. More precisely, there is a set $P\subseteq A/A$ and an integer $t$, such that $|A|^2/(2|A/A|)\leq t\leq |A|$, $t^2|P|\sim \E^\times(A)$, and $t<|A\cap xA|\leq 2t$ for any $x\in P$. That is, each line through the origin in $\F^2$, with a slope in $P$ supports about $t$ points of $A\times A$.
Let $S\subseteq A\times A$ be the set of points supported on these lines with slopes in $P$; so $|P|t\leq|S| <2 |P|t$. Let $\pi_x : S\mapsto A$ be the projection of points of $S$ to the $x$-axis: $\pi_x(s_x,s_y)=s_x$. The projection $\pi_y$ is similarly defined as the projection to the set of ordinates.
Consider the set $A_x$ of abscissae of $S$, that is $A_x=\pi_x(S)$. By another dyadic pigeonhole argument, we find a set $A'\subseteq A_x$ of popular abscissae for $S$. There exists $A'\subseteq A_x$, and a number $q'$ such that for every $x'\in A'$, the vertical line through $x'$ supports approximately $q'$ points of $S$ (more precisely $q'<|S\cap \{x=x'\}|\leq 2q'$), and $|A'|q'\sim |S|$. Observe that $q'\leq \min(|A|,|P|)$, but in the sequel we need an analogue of the slightly stronger inequality $q'\lesssim |A'|$, which is not necessarily true.
So if $q'\leq |A'|$ then we set $A_1=A'$ and $q=q'$. Otherwise we do another dyadic pigeonholing, now by ordinates. I.e., we consider the plane set $S'=S\cap \pi_x^{-1}(A')$ and find a number $q''$ and a set $A''\subseteq \pi_y(S')$ such that $q''<|S'\cap \{y=y''\}|\leq 2q''$ for all $y''\in A''$. In other words, $A''$ is the set of popular ordinates of the set $S'$ (which in turn is almost as big as $S$): the horizontal line through each $y\in A''$ contains about $q''$ points of $S'$.
We have $$q'|A'|\sim q''|A''|$$ and since clearly $q''\leq |A'|$, plus since we assume $q'>|A'|$, we must have $|A''|\gtrsim q'>|A'|\geq q''$. We conclude that $q''< q' \lesssim |A''|$ and set $A_1=A''$ and $q=q''$.
Hence now $|A_1|\gtrsim q$, and so we have $$\label{a1bd} |A_1|^2\gtrsim q|A_1|\sim |S|\geq |P|t=\frac{|P|t^2}{t}\sim \frac{\E^\times(A)}{t}\geq \frac{\E^\times(A)}{|A|}\,.$$
Besides, since $|P|t \le |S| \lesssim |A_1|^2$, and $t^2\sim \E^\times(A)/|P|$ we conclude that $$\label{need}
|P|\lesssim |A_1|^4/\E^\times(A).$$
Without loss of generality let us regard $A_1=A'$, that is the set of popular abscissae, rather than ordinates. We then have by construction of $A'$ a set, each member of which can be represented at least $q$ times as a ratio from $A/P$: $$\begin{aligned}
\label{trick}
\nonumber \E^+(A_1)&=|\{(a,a',b,b')\in A_1^4: a+b=a'+b'\}|\\
&\leq q^{-2}|\{(a,a',p_*,p'_*,\a,\a')\in A_1^2\times P^2\times A^2: a+\a/p_* =a'+\a'/p'_* \}|\end{aligned}$$ Note that by symmetry of $A\times A$, $P=P^{-1}$, so we can (but do not have to – this is only a gesture towards the fact that there is no difference as to whether $A_1$ has been taken as $A'$ or $A''$) replace division by $p$ by multiplication.
Consider then the family of $m=|A_1||P||A|$ planes, with equations $a+p_* x=y+\a'z:\;(a,p_*,\a')\in A_1\times P\times A$ and the same number of points $(x,y,z)\in P\times A_1\times A$. Applying Theorem \[t:MR\] we claim that, in terms of the contribution of the main term $m^{3/2}$ in the estimate of the Theorem $$\label{alm}\E^+(A_1)\ll (|A||P||A_1|)^{3/2}q^{-2},$$ to be fully justified shortly.
Indeed, the maximum number of collinear points $k$ in the estimate of Theorem \[t:MR\] is bounded by $\max(|A|,|P|)$. If the maximum equals $|P|$ then to drop the $km$ term in the estimate of Theorem \[t:MR\] it therefore suffices to show that $|P|\lesssim |A||A_1|$. This is true, since we have established that $|P|\lesssim |A_1|^2$. Now if $\max(|A|,|P|) = |A|$ then we need to check $|A| \lesssim |A_1| |P|$. But $|P| \gtrsim \E^\times (A) / |A_1|^2$ and hence everything follows from a trivial bound $\E^\times (A) \ge |A|^2 \ge |A_1| |A|$.
In positive characteristic Theorem \[t:MR\] is applicable when $ |A_1||P||A|\leq p^2$. By and the trivial bound $|A_1|\leq|A|$, this will be true given that $|A|^6\lesssim p^2\E^\times(A)$. To strengthen the latter constraint to $|A|^6\leq p^2\E^\times(A)$, as claimed, we proceed as follows.
Suppose, $|A|^6\leq p^2\E^\times(A)$. Partition the set of points $\mathcal P\times A_1\times A$ in $\lesssim 1$ pieces $\{\mathcal P_i\}$ (say, by partitioning $A$), whose size differs by at most an absolute constant factor, and such that each $|\mathcal P_i|\leq p^2$. The number of solutions of the equation in the second line of is the sum, over $1\leq i\lesssim 1$, of the number of incidences between the above $m$ planes and the point set $\mathcal P_i$. By Theorem \[t:MR\] it is $O(m^{3/2})$, for each $i$.
Thus the summation over $i\lesssim 1$ results only in the change of the power of $\log|A|$ hidden in the incidence estimate , that is if $|A|^6\leq p^2\E^\times(A)$ the estimate is true, with a different factor of $\log|A|$ hidden in the $\lesssim$ symbol.
We pass from formula to by setting $q \sim |P|t/|A_1|$, $\E^\times(A)\sim |P|t^2$ and using to bound the remaining $\sqrt{|P|}$ in the numerator.
We have established the formula as it is. The fact that energies can be swapped follow by taking the above set $P$ as a subset of $A-A,$ rather than of $A/A$. and repeat the argument. The only modification is that the equivalent of will now deal with $\E^\times(A_1)$, followed by the equivalent of , as worked out explicitly in the proof of Proposition 1 in [@AMRS].
This completes the proof of Proposition \[l:BW\_FF\].
In the special case of the real or complex field we have a slightly stronger result via the Szemerédi-Trotter theorem.
\[l:BW\_R\] Let $A\subset \mathbb{C}$ be a set. Then there is $A_{1}\subseteq A$ such that $|A_1|^2\gtrsim \E^\times(A)|A|^{-1}$ and $$\E^{+}(A_1)\E^\times(A)\lesssim {|A_1|}^{9/2}|A| \,.\label{f:stuff}$$ The energies $\E^\times,\E^+$ in the above statement can be swapped (for some other $A_1$).
One repeats the pigeonholing arguments in the proof of Proposition \[l:BW\_FF\] with the same notations $t,P,q,A_1$ up to and inclusive of the estimate . Without loss of generality we assume that $A_1$ is the set of popular abscissae $A'$ in the construction of Proposition \[l:BW\_FF\].
Using the notation of [@KS1], we observe that $$A_1\subset \text{Sym}_q(A,P):=\{x:|A\cap xP^{-1}|\geq q\}\,,$$ which means that, by construction, each member of $A_1$ can be represented at least $q$ times as a ratio from $A/P$ (or a product $AP$, since $P=P^{-1}$).
Using Lemma 13 and Corollary 11 of [@KS2] we conclude[^2] that $$\label{shen} \E^+(A_1) \ll d_*(A)^{1/2} |A_1|^{5/2},$$ where the parameter (see [@KS2]) $$d_*(A_1) \le \frac{|A|^2|P|^2}{q^3 |A_1|}.$$ Using the relations $|A_1|q\sim |P|t\lesssim |A_1|^2$ from the proof of Proposition \[l:BW\_FF\] we obtain $$d_*(A_1) \sim \frac{|A|^2|A_1|^2}{|P|t^3} \sim\frac{|A|^2|A_1|^2|P|t}{{\E^\times (A)}^2}\sim\frac{|A|^2|A_1|^4}{{\E^\times (A)}^2}\,.$$
Substituting the latter estimate into completes the proof of Proposition \[l:BW\_R\]. The fact that the energies $\E^\times,\E^+$ can be swapped follows taking $P\subseteq A-A$ instead of $A/A$ and repeating the argument. See also Theorem 20 in [@KS2].
\[rrem\] The same proof can be easily modified to an application of Theorem \[t:SzT\_Fp\] here instead of Theorem \[t:ST\]. The latter theorem would also enable one to obtain decomposition estimates involving higher energies, along the lines of those obtained by the second author [@s_E_k] over the reals. We note, however, that the proofs of Propositions \[l:BW\_FF\], \[l:BW\_R\] use the symmetry between the $x$ and $y$-axes and are not applicable to the quantities $\mathrm{D}^{+}, \mathrm{D}^\times$, studied by the second author in [@s_E_k].
To conclude the proof of Theorem \[t:BW\_R\], the above intermediate results are iterated via a simple lemma: see, e.g., [@TaoVubook].
\[1/4ineq\] Let $A_1,\dots,A_n$ be subsets of an abelian group. Then $$\left(\E^{+} \left(\bigcup_{i=1}^n A_i\right)\right)^{1/4} \le \sum_{i=1}^n (\E^{+} (A_i))^{1/4}.$$
Observe that Propositions \[l:BW\_FF\], \[l:BW\_R\] imply a weaker version of Theorem \[t:BW\_R\], where one replaces $\max(B,C)$ with $\min(B,C)$.
### Conclusion of the proof of Theorem \[t:BW\_R\] and proofs of Corollary \[t:BW\_C\]
Suppose $\E^\times(A)\lesssim |A|^{3-\delta}$ or there is nothing to prove: we can always take a small $B$ such that $|B|^3< |A|^{3-\delta}$ and $C=A\setminus B$.
Heuristically, we use Proposition \[l:BW\_FF\] for a general $\F$ and Proposition \[l:BW\_R\] in the special case $\F=\mathbb R$ or $\mathbb C$ to pull out from $A$, one by one, subsets $A_1$ with a small additive energy and stop once the energy of the remainder $C$ of $A$ will become smaller than $|A|^{3-\delta}.$ This is bound to happen if $C$ becomes sufficiently small relative to $|A|$. The above lemmas are going to guarantee that $\E^+(B=A\setminus C)\lesssim |A|^{3-\delta}$.
Formally, let $M\geq 1$ be a parameter which we choose later, assume that $\E^\times(A)\leq |A|^{3}/M$ and $|A|^3\leq p^2 / M$ in positive characteristic.
We construct a decreasing sequence of sets $C_1=A \supseteq C_2 \supseteq \dots \supseteq C_k$ and an increasing sequence of sets $B_0 = \emptyset \subseteq B_1 \subseteq \dots \subseteq B_{k-1} \subseteq A$ such that for any $j=1,2,\dots, k$ the sets $C_j$ and $B_{j-1}$ are disjoint and moreover $A = C_j \sqcup B_{j-1}$. If at some step $j$ we have $\E^{\times} (C_j) \leq |A|^3 / M$, we stop and set $C=C_j$, $B = B_{j-1}$, and $k=j-1$. Else, we have $\E^{\times} (C_j) > |A|^3 / M$. We apply Proposition \[l:BW\_FF\] to the set $C_j$, finding the subset $D_j$ of $C_j$ such that $$|D_j|^2 \gtrsim \frac{\E^\times(C_j)}{|C_j|}>\frac{|A|^3}{M|C_j|}\geq\frac{|A|^2}{M}\label{hm}$$ and $$\label{dest}
\E^{+} (D_j) \lesssim \frac{|D_j|^{11/2} |C_j|^{3/2}}{(\E^\times(C_j))^{3/2}}\leq\frac{M^{3/2}|D_j|^{11/2}|C_j|^{3/2}}{|A|^{9/2}}<|D_j|^{11/2}M^{3/2}|A|^{-3} \,.$$ After that we put $C_{j+1} = C_j \setminus D_j$, $B_j = B_{j-1} \sqcup D_j$ and repeat the procedure. In view of the uniform lower bound on $|D_j|$, the process will terminate, as $|C_j|$ decreases, after say $k$ iterations, when $\E(C=C_{k+1})\leq |A|^3/M$. We set $B=B_k = \bigsqcup_{j=1}^k D_j$ and $C=C_{k+1}$, so $ A=B\sqcup C$.
Trivially $|B| =\sum_{j=1}^{k}|D_j|\leq |A|$.\
Then, using Lemma \[1/4ineq\] and the bound of Proposition \[l:BW\_FF\] we get $$\begin{aligned}
\label{formula}
\E^{+} (B) &\lesssim M^{3/2} |A|^{-3} \left( \sum_{j=1}^{k} |D_j|^{11/8}\right)^4\\ \nonumber
&\leq M^{3/2}|A|^{-3}\max_j\{|D_j|\}^{3/2}\left(\sum_{j=1}^k|D_j|\right)^4\\ \nonumber
&\leq M^{3/2}|A|^{5/2}
\,.\end{aligned}$$ Optimising over $M$, with $\E^{*} (C) \leq |A|^3/M$, that is choosing $M=|A|^{1/5}$, we obtain the result for a general $F$. In particular, the constraint in terms of $p$ in positive characteristic boils down to $|A|^6<|A|^{3-1/5}p^2$, so $|A|\leq p^{5/8}.$
In the special case of $\F=\mathbb C$ the analogue of the estimates and comes from using , rather than . Namely, we have $$\E^{+} (D_j) \lesssim \frac{|D_j|^{9/2} |C_j|}{\E^\times(C_j)}\leq \frac{ M |D_j|^{9/2}} {|A|^{2} }\,$$ and $$\E^{+} (B) \; \lesssim \; M |A|^{-2} \left( \sum_{j=1}^{k} |D_j|^{9/8}\right)^4
\;\leq \; M |A|^{5/2}.$$ Optimising with $\E^{*} (C) \leq |A|^3/M$, yields $M=|A|^{1/4}$. This proves Theorem \[t:BW\_R\].
Observe that if $A$ has a subset $C$ with $|C|\geq 2|A|/3$ and $\E^\times(C)\leq |C|^{8/3} $, there is nothing to prove: just take $B$ as half of $C$, with the trivial bound $\E^+(B)\leq |A|^3$ and rename $C$ as the other half.
Otherwise we repeat the argument in the preceding proof, finishing it at the first instance when either $\E^\times(C=C_{k+1}) <|A|^{8/3}$ or $|B=B_k|>|A|/3$. Without loss of generality we can assume that in addition to the estimates and we have, say $|D_j|<|A|/100$ for every $j$ (by partitioning $D_j$ if necessary). Since the sequence of sets $\{C_j\}$ is decreasing, we have $\E^\times(C) \leq \E^\times(C_j),\;\forall j\leq k+1.$ Hence, in view of , the calculation becomes
$$\E^{+} (B) \E^\times(C)^{3/2} \; \lesssim \; |A|^{3/2} \left( \sum_{j=1}^{k} |D_j|^{11/8}\right)^4 \;\leq |A|^7,$$ thus proving .
Note that Proposition \[l:BW\_FF\] does indeed apply on each step in positive characteristic, for the condition $|A|^6 \leq p^2\E^\times(C)$ is satisfied, since we have assumed $\E^\times(C)\geq |A|^{8/3}$ and $|A|\leq p^{3/5}.$
To derive of the Corollary from we observe that according to the above proofs one can swap the two energies in . We do this for the set $B$. Namely $B$ gets partitioned into $B'$ and $C'$, each of size at least $|B|/3$, such that $$\E^+(B')^{3/2} \E^\times(C') \lesssim |A|^7.$$ Multiplying the latter two estimates we obtain $$(\E^\times(B) \E^+(B') )^{3/2} \cdot (\E^+(C)\E^\times(C') )\lesssim |A|^{14}.$$ If $\E^\times(B) \E^+(B') \leq \E^+(C)\E^\times(C')$, we get $$(\E^\times(B) \E^+(B'))^{5/2} \lesssim |A|^{14},$$ (and one can replace $B$ by $C'$, disjoint from $B'$) otherwise we get the same inequality involving the disjoint $C$ and $C'$.
Renaming the two subsets in question as $B$ and $C$ finishes the proof, the price we’ve paid is just that $B$ and $C$ no longer partition $A$.
Proof of Theorem \[t:fsmp\] and Corollary \[objeli\]
====================================================
We give a detailed sketch of the proof of Theorem \[t:fsmp\], which largely repeats our previous arguments, the key benefit being derived from using an estimate from [@MR-NS].
[Sketch of proof]{}
We invoke the additive version of the construction in the proof of Proposition \[l:BW\_FF\], with the same notations, to derive a version of the proposition, given by the forthcoming estimate . $P$ is now a dyadic group of popular sums, with approximately (that is up to a constant factor) $t$ realisations each, that supports $|P|t^2 \gtrsim \E^+$ of additive energy. $S$ is the corresponding subset of $A\times A$. $A_1$ is the set of popular abscissae (ordinates) for $S$, with approximately $q$ realisations, and $q\lesssim |A_1|$. Moreover, $|S|t\lesssim |A_1|^2$ and $|A_1|\gtrsim \sqrt{\E^+(A)/|A|}.$
Assuming that $A$ does not contain zero, consider the multiplicative energy equation $$a/b = a'/b' : \,a,b,a',b'\in A_1.$$
For the left-hand side there are approximately $q$ choices to add some $c\in A$ in the numerator and some $d\in A$ in the denominator to replace it with
$$\frac{(a+c) - c}{ (b+d) - d} = \frac{ s-c }{r-d},$$ where $s,r \in P$. Thus $\E^\times(A_1)$ is bounded by a constant, times $q^{-4}$, times the number of solutions of $$\frac{s-c}{r-d} = \frac{s'-c'}{r'-d'} \neq 0,\infty: \;(s,r,s',r')\in P^4,\; (c,d,c',d')\in A^4.$$ The latter equation has been studied, in particular in the paper of Murphy, Roche-Newton, and the second author [@MR-NS] (see Lemma 2.5 therein) which proves the upper bond $\lesssim |P|^3|A|^3$ for the number of solutions.
Hence, once $|A_1|^2\gtrsim q|A_1| \gtrsim |P|t$ we have $$\label{vers}
\E^\times(A_1) \lesssim q^{-4} |P|^3 |A|^3 \lesssim \frac{|P|^3 |A|^3} {|P|^4t^4/|A_1|^4} = |A_1|^4|A|^3\frac{|P|^2t^2}{|P|^3t^6}\lesssim \frac{ |A_1|^{8}|A|^3}{{\E^+(A)}^3}.$$
A straightforward adaptation of the iterative argument in the first two passages of the proof of Corollary \[t:BW\_C\] to the latter estimate completes the proof.
Proof of Corollary \[objeli\]
-----------------------------
Observe that inequality implies, by Cauchy-Schwarz, that $$\label{subs}
|A_1\cdot A_1|, \;|A_1/A_1| \gtrsim \frac{{\E^+(A)}^3}{ |A_1|^{4}|A|^3}\geq \frac{{\E^+(A)}^3}{ |A|^7},$$ hence the same lower bound for the supersets $|A\cdot A|, \;|A/A|$. In [@KS1 Theorem 9] a weaker estimate was established: $$|A\cdot A|, \;|A/A| \gtrsim \frac{{\E^+(A)}^4}{|A|^{10}}.$$ The latter estimate was used to obtain [@KS1 Lemma 12, estimates (34), (36)], restated as [@KS2 Lemma 18, estimates (22), (24)]. If one uses instead, this improves the term $L^{-16}$ in these estimates to $L^{-12}$. Recalculating [@KS2 Proof of Theorem 3 from (26) on] yields the new sum-product exponent $\frac{4}{3}+\frac{1}{1509}$ as claimed.
Proof of Corollary \[c:R\_energy\]
==================================
Put $R=R[A]$, $R^* = R \setminus \{0\}$, and $\d =1/4$. Using Theorem \[t:BW\_mult\], we find $B,C \subseteq R$ such that $R = B\sqcup C$ and $$\max\{ \E^{\times} (B), \E^{\times} (C-1)\} \lesssim |R|^{3-\d} \,.$$ If $|B|\ge |R|/2$ then we are done. Suppose not. Then $|C| \ge |R|/2$ and in view of formula $R=1-R$, see [@Shkredov_R[A]], we obtain that $C' := 1-C \subseteq R$, $|C'| = |C| \ge |R|/2$ and $$\E^{\times} (C') = \E^{\times} (1-C) = \E^\times (C-1) \lesssim |R|^{3-\d} \,.$$ So, putting $R'$ equals $B$ or $C'$, we obtain the result. To find the set $R''$ note that $(R^*)^{-1} = R^*$ and use the second part of Theorem \[t:BW\_mult\].
The same proof allows us to find a subset $A'_s$ of the set $A_s \cup (-A_s)$, $A_s = A\cap (A+s)$, $s\in A-A$, $A \subset \mathbb C$ of cardinality $|A_s|/2$ such that $\E^\times (A'_s) \lesssim |A'_s|^{3-1/4}$. This question is a dual of one which appeared in [@KS1], [@KS2]. The same result holds for some multiplicative analogue of the sets $A_s$, namely, $A^*_s = A\cap (s/A)$, $s\in AA$.
Proof of Theorem \[thm:had\]
============================
We now turn to the proof of Theorem \[thm:had\]. We shall apply Corollary \[t:BW\_C\] to the set $A$: it applies when $|A|\leq p^{3/5}$, which we may assume, passing to a subset if $A$ is too big.
So, if $|A|\leq p^{3/5}$, Corollary \[t:BW\_C\] gives us two positive proportion disjoint subsets $B,C$ of $A$, whose energies satisfy the estimate and whereon we consider equation .
Let us denote $$Q= \left|
\left\{\frac{ab-c}{a-d}:\;a,b,c,d\in A, a\neq d\right\}\right|.$$
By Cauchy-Schwarz $$\begin{aligned}
\label{e:hadcs}
\Theta(|A|^8) = |B|^4 |C|^4 & \leq \; Q \left(\sum_{x\in \mathbb{F}_{p}}\left|\left\{x=\frac{ab-c}{a-d}: \,a,b\in B;\,c,d\in C\right\}\right|\right)^2 .\end{aligned}$$
Let us isolate the case $x=0$. This means $ab=c$ and $a'b'=c'$, so the trivial bound for this is $|A|^6$.
We can then denote $$\mathcal E= \sum_{x\in \mathbb{F}^*_{p}}\left|\left\{x=\frac{ab-c}{a-d}: \,a,b\in B;\,c,d\in C\right\}\right|^2$$ and assuming $|A|>p^{1/2}$ rewrite as $$|A|^8\ll Q\mathcal E,
\label{easy}$$ for trivially $Q\leq p$. We further aim to find the upper bound on $\mathcal E$.
Rearranging and applying Cauchy-Schwarz once more yields $$\label{int}
\mathcal E = \sum_{x\neq 0}|\{a(b-x)=c-dx\}|^2\leq \sum_{x\neq 0}\E^\times(B,x+B)\E^+(C,xC)\,.$$ We evaluate these energy terms using an argument of [@Petridis], modifying it to suit our needs. Although [@Petridis] considers the term containing $\E^+(C,xC)$ only, the analysis applies almost verbatim to the quantity $\E^\times(B,x+B)$ as well.
In the following lemmata, $A$ denotes a dummy set, to be replaced, respectively, by $C$ and $B$. We first quote a well known fact, which has been in the literature since “generic projections” in [@BKT], recorded as Lemma 3 in [@Petridis].
\[ensum\] Let $A\subseteq \F_p$ with $|A|>p^{1/2}$. Then $$\sum_{x\in \F_p^*} \E^+(A,xA), \; \sum_{x\in \F_p^*} \E^\times(A,x+A) \ll |A|^4.$$
Heuristically the above statement means that knowing, say $a,b,a',b'$ in the equation $a(b-x)=a'(b'-x)$ generically defines $x$, the constraint $|A|>p^{1/2}$ taking care of degeneracies. So the latter $\ll$ estimate is, in fact, an asymptotic identity.
Besides, by Cauchy-Schwarz, for every $x\in \F_p^*$ we have $$\E^+(A,xA),\; \E^\times(A,x+A) \geq |A|^4/p.\label{trb}$$ Indeed, we simply use $p$ as the upper bound for, say $|A+xA|$.
Combining the formula – where we set $\E^\times(B,x+B)= (\E^\times(B,x+B)-|B|^4/p) + |B|^4/p$ and similarly for $\E^+(C,xC)$ – with Lemma \[ensum\] we have, assuming $|A|>p^{1/2}$ and using the Hölder inequality we obtain: $$\begin{aligned}
\label{long}
\mathcal E&\ll \frac{|A|^8}{p} + \sum_{x\neq 0}\left(\E^+(C,xC)-\frac{|C|^4}{p}\right) \left(\E^\times(B,x+B)-\frac{|B|^4}{p}\right)\\ \nonumber
& \leq \frac{|A|^8}{p} +\left( \sum_{x\neq 0}\left(\E^+(C,xC)-\frac{|C|^4}{p}\right)^{5/3}\right)^{3/5} \left(\sum_{x\neq 0}\left(\E^\times(B,x+B)-\frac{|B|^4}{p}\right)^{5/2}\right)^{2/5}.\end{aligned}$$ The choice of exponents in the Hölder inequality has been made to conform with the estimate in the sequel.
The main part of the argument is the following proposition, an analogue of Proposition 8 in [@Petridis].
\[p:had\] Let $A\subseteq \mathbb{F}_p$, with $p^{1/2}<|A|\leq p^{2/3}$ and $s\in (0,3)$. Then $$\sum_{x\neq 0}\left(\E^+(A,xA)-\frac{|A|^{4}}{p}\right)^{1+s}=O\left(p^{1-\frac{1}{3}s}\E^+(A)^{\frac{2}{3}s}|A|^{2+\frac{4}{3}s}\right)$$ and $$\sum_{x\neq 0}\left(\E^\times(A,x+A)-\frac{|A|^{4}}{p}\right)^{1+s}=O\left(p^{1-\frac{1}{3}s}\E^\times(A)^{\frac{2}{3}s}|A|^{2+\frac{4}{3}s}\right).$$
Proposition \[p:had\] relies on the following lemma, the analogue of Theorem 2 in [@Petridis], which follows from Theorem \[t:MR\].
\[inccount\] Let $A\subseteq \mathbb{F}_p,$ $X\subseteq \mathbb{F}_p^*$. Suppose $|X|=O(|A|^2)$ and $|A|^2|X|=O(p^2)$. Then $$\sum_{x\in X}\E^+(A,xA) \ll \E^+(A)^{1/2} |A|^{3/2}|X|^{3/4}$$ and $$\sum_{x\in X}\E^\times(A,x+A) \ll \E^\times(A)^{1/2}|A|^{3/2}|X|^{3/4} \,.$$
To keep the exposition more self–contained we sketch the proof of Lemma \[inccount\], dealing with its second estimate: the first one was the result of Theorem 2 in [@Petridis], and the two proofs are in essence identical.
(Sketch of proof.) Let $Y=\sum_{x\in X}\E^\times(A,x+A)$, that is the number of solutions of the equation $$a(x+b)=c(x+d): \;x\in X,\,a,b,c,d\in A.$$ It is easy to see that the condition $|X|=O(|A|^2)$ implies that the number of trivial solutions of the last equation (e.g. $a=c=0$ and $x,b,d$ are any or $x=-b$, $c=0$ and $a,d$ are arbitrary) is negligible. Rearranging and applying Cauchy-Schwarz we get $Y\ll (\E^\times(A)Z)^{1/2}$, where $Z$ is the number of solutions of the equation $$\frac{x+b}{x+d} = \frac{x'+b'}{x'+d'}:\;x,x'\in X, b,d,b',d' \in A.$$ The number of nontrivial solutions of the latter equation can be estimated by using Theorem 19 in [@AMRS], which is a particular case of Theorem \[t:MR\]. This was done, in particular, in the proof of Corollary 8 of the latter paper. We nonetheless briefly show how.
Consider a family of planes in $\F_p^3$, with equations, in coordinates $(x_1,x_2,x_3)$ not to be confused with $x,x'\in X$, as follows: $$\frac{1}{x+d} x_1 - x_2 - b'x_3 + \frac{x}{x+d} = 0,$$ and their incidences with points $(x_1,x_2,x_3) = (b, \frac{x'}{x'+d'}, \frac{1}{x'+d'})$. It is easy to verify, see [@AMRS], that there are $m=|X||A|^2$ planes and points. The constraints of Lemma \[inccount\] ensure that the main term $m^{3/2}$ dominates in the estimate Theorem \[t:MR\] and the theorem is applicable. Thereupon, the second estimate of Lemma \[inccount\] follows.
Using methods from [@MR-NS] (see the proofs of Lemmas 2.3, 2.4) one can prove another variant of Lemma \[inccount\], namely, for any $A\subseteq \mathbb{F}_p$ and $X\subseteq \mathbb{F}_p^*$ with $|A| \leq p^{2/3}$ the following holds $$\sum_{x\in X}\E^+(A,xA) \ll |A|^{13/4} |X|^{1/2} \,,$$ and $$\sum_{x\in X}\E^\times(A,x+A) \ll |A|^{13/4}|X|^{1/2} \,.$$
Lemma \[inccount\] can be restated in the standard way, similar to the well-known restatement of the Szemerédi-Trotter theorem, Theorem \[t:ST\] here, as $O\left(\frac{n^2}{\tau^3} + \frac{n}{\tau}\right)$ as the upper bound on the number of $\tau$-rich lines. See Lemma 4 of [@Petridis]. We quote an auxiliary Lemma from \[BKT\] which appears (and is proved) explicitly in [@Petridis] as Lemma 3.
\[BKT\] For every set $X \subseteq \F_p^*$ we have the following inequality $$\sum_{x\in X} \left(\E^+(A, xA) -\frac{ |A|^4}{p}\right),\, \sum_{x\in X} \left(\E^\times(A, x+A) -\frac{ |A|^4}{p}\right) \; \leq \; p|A|^2\,.$$
\[proplemma4\] Let $A\subseteq \mathbb{F}_{p}$, $|A|> p^{1/2}$ and $1\leq K\leq p\frac{\E^+(A)}{2|A|^{4}}$.\
Then the number of $x\in \mathbb{F}_p^*$ such that $\E^+(A,xA)>\frac{\E^+(A)}{K}$ is $O\left(K^{4}\frac{|A|^{6}}{\E^+(A)^{2}}\right)$. A similar bound holds for the number of $x\in \mathbb{F}_p^*$ such that $\E^+(A,x+A)>\frac{\E^\times(A)}{K}$.
Note that $K$ is well defined, by . Let $X$ be the set of $x$ in question.\
Firstly we show that $|A|^{2}|X|=O(p^{2})$. From Lemma \[BKT\] we know that $$p|A|^2\geq \sum_{x\in X}\left(\E^+(A,xA)-\frac{|A|^4}{p}\right)\geq\sum_{x\in X}\left(\frac{\E^+(A)}{K}-\frac{|A|^4}{p}\right)\,.$$ Hence we have $$p|A|^2\geq |X|\left(\frac{\E^+(A)}{K}-\frac{|A|^4}{p}\right),$$ and because of our choice of $K$, the first term dominates. So $|X|\leq \frac{2p |A|^2 K}{\E^+(A)}$; substitution yields $|A|^2|X|=O(p^2)$ and $|X| = O(|A|^2)$.\
Next, we use Lemma \[inccount\] to obtain $$\sum_{x\in X}\E^+(A,xA)=O(\E^+(A)^{1/2}|A|^{3/2}|X|^{3/4})\,.$$ All that remains is to evaluate the following: $$\frac{|X|\E^+(A)}{K}\leq \sum_{x\in X}\E^+(A,xA)=O(\E^+(A)^{1/2}|A|^{3/2}|X|^{3/4}).$$ One rearranges to obtain the desired bound on $|X|$.
The multiplicative energy case is identical.
Thus we pass from having to satisfy the conditions of Lemma \[inccount\] to meeting those of Lemma \[proplemma4\]. We now prove Proposition \[p:had\]; as the proofs for both statements are the same, we prove the first one only.
Note that by Cauchy-Schwarz, for every $x\neq 0$, we have $\E^+ (A,xA)\leq \E^+(A).$
Set $M=(p \E^+ (A) / 8|A|^4)^{1/3}$, so $1/2\leq M\leq p \E^+ (A) /(2|A|^4)$ by . Since also $p^{1/2}<|A|\leq p^{2/3}$, for any $1\leq K \leq M$ the conditions of Lemma \[proplemma4\] are satisfied.
For ‘small’ energies when $\E^+ (A,xA)\leq \E^+(A)/M$, we rely on which ensures no sign alterations and proceed with a trivial inequality and Lemma \[BKT\] to obtain: $$\begin{aligned}
\sum\left( \E^+(A,xA)-\frac{\aa^4}{p}\right)^{1+s} & \leq \sum\left(\frac{\E^+(A)}{M}\right)^s\left( \E^+(A,xA)-\frac{\aa^4}{p}\right) \\ &
\leq p\aa^2\left(\frac{\E^+(A)}{M}\right)^s, \end{aligned}$$ where both sums are taken over the set $\{x\neq 0: \E^+(A,xA)\leq \E^+(A)/M\}$. Note that if $M<1$ then the inequalities $\E^+(A,xA)-\frac{\aa^4}{p} \leq \E^+ (A,xA)\leq \E^+(A)/M$ hold by trivial reasons.
For ‘large’ $\E^+(A,xA)$ we use Lemma \[proplemma4\] and a dyadic argument. Let $$X_i:=\{x: 2^i \E^+(A)/M < \E^+(A,xA)\leq \min(2^{i+1}\E^+(A)/M, \E^+(A)\}.$$
For $0\leq i\ll \log M$, set $2^{-i}M=K_i$, so $1\leq K_i \leq M.$
By Lemma we know that $|X_{i}|=O\left(2^{-4i}\frac{M^4|A|^{6}}{\E^{+}(A)^{2}}\right)$. So $$\begin{aligned}
\sum_{\substack{\E^+(A,xA)>\E^+(A)/M}}\left(\E^+(A,xA)-\frac{|A|^{4}}{p}\right)^{1+s}&\leq \; \sum_i\sum_{x\in X_{i}}\E^+(A,xA)^{1+s}\\
&\ll\; \sum_i|X_i|\E^+(A)^{1+s}2^{(1+s)i} M^{-(1+s)} \\ & \ll \frac{ M^{3-s} |A|^{6}}{\E^+(A)^{1-s}}.\end{aligned}$$ Here we have used that $s\in (0,3)$ to sum the geometric progression in $i$, so the constant hidden in the last inequality depends on $s$. In view of the choice of $M$, independent of $s$, the latter two estimates match, concluding the proof of Proposition \[p:had\].
To conclude the proof of Theorem \[thm:had\] we observe that by Proposition \[p:had\], applied to the estimate we have $$\label{almost}
\mathcal E \; \ll \; |A|^8/p + p^{2/3} |A|^{10/3} [ \E^+( C) \E^\times (B)^{3/2} ] ^{4/15},$$ given that $|A|\leq p^{3/5}$. Applying of Corollary \[t:BW\_C\] (with the notations $B$ and $C$ reversed) it is immediate to conclude that the first term in the estimate dominates if $|A|\geq p^{25/42} \log^K|A|$, where $K$ depends only on the power of $\log|A|$ hidden in the estimate . Note that $25/42<3/5$, so the upper bound on $|A|$ to make Corollary \[t:BW\_C\] applicable has been satisfied.
For such $A$ the number of solutions of equation is $O(|A|^8/p)$, which completes the proof of Theorem \[thm:had\].
Proof of Theorem \[t:BSzG\_Schoen\] and ensuing statements {#sec:further}
==========================================================
We prove Theorem \[t:BSzG\_Schoen\].
[**Proof.**]{} Let $\E= \E^+(A) \geq \aa^3/K$. Put $A_s = A\cap (A-s)$. We have $$\label{f:energy_BSzG}
\E = \sum_{s\in G} |A_s|^2 = \sum_s \sum_{x,y} A_s (x) A_s (y) = \sum_s \sum_{x,y} A(x) A(y) A(x+s) A(y+s) \,.$$ Let $\eps \in (0,1)$ be a real number which we will choose later. Let us put $$P = P_\eps = \left\{ s ~:~ |A_s| \ge \frac{\eps \aa}{2K} \right\} \,.$$ Clearly, the set $P_\eps$ is symmetric for any $\eps$ and $|P_\eps| \le 2K \eps^{-1} |A|$. As $$\sum_s\, \sum_{x-y\notin P_\eps} A_s (x) A_s (y) = \sum_{x-y\notin P_\eps} A(x) A(y) |A_{x-y}|
<
\frac{\eps \aa^3}{2K}
$$ then by combining the last estimate with (\[f:energy\_BSzG\]), we get $$\sum_{s~:~ |A_s| \ge \aa/2K}\, \sum_{x-y\in P_\eps} A_s (x) A_s (y)
-
\eps^{-1} \sum_{s~:~ |A_s| \ge \aa/2K}\, \sum_{x-y\notin P_\eps} A_s (x) A_s (y)
> 0 \,.$$ It follows that there is $s$ with $|A_s| \ge \aa/(2K)$ satisfying, $$\eps \sum_{x-y\in P_\eps} A_s (x) A_s (y) > \sum_{x-y\notin P_\eps} A_s (x) A_s (y) \,.$$ In other words $$\label{tmp:17.07.2014_1}
\sum_{x-y\in P_\eps} A_s (x) A_s (y) > (1-\eps) |A_s|^2 \,.$$ Now let us consider a non–oriented graph (with loops) $\mathcal{G}=(\mathcal{V},\mathcal{E})$ with the vertex set equal to $A_s$ such that its vertices $x,y$ are connected iff $x-y\in P_\eps$. It is easy to see from (\[tmp:17.07.2014\_1\]) that $|\mathcal{E}| > (1-\eps) |A_s|^2$. Put $$A_* = \{ v \in V ~:~ \deg v \ge (1-2\eps) |\mathcal{V}| \} \subseteq A_s \,.$$ Inequality (\[tmp:17.07.2014\_1\]) implies $|A_*| \ge \eps |\mathcal{V}| \ge \eps |A|/(2K)$. Moreover for any vertices $a_1, \dots, a_k \in A_*$ there are at least $(1-2k\eps)|\mathcal{V}|$ common neighbours $x\in \mathcal{V}$; i.e. vertices $x$ such that $(a_1,x), \dots, (a_k,x) \in \mathcal{E}$. By putting $\eps = 1/(4k)$, we obtain the result. $\hfill\Box$
Theorem \[t:BSzG\_Schoen\] enables one to achieve Balog-Wooley type decomposition results, dealing with the energies directly.
(Sketch of proof of Proposition \[t:sp\_E\_E’\]) Set $\E^\times (A) = |A|^3 /K$. Using Theorem \[t:BSzG\_Schoen\] in multiplicative form with $k=2$, we find a set $A_* \subseteq A$, $|A_*| \gg |A| /K$ and a set $P\subseteq A/A$, $|P| \ll K|A|$ such that for any $a,b \in A_*$ the following holds $|A\cap aP \cap bP| \gg |A|/K$. Thus $\E^{+} (A_*)$ is bounded from above by $$O\left( (|A| /K)^{-2} |\{ a(p^{-1}_1 + p^{-1}_2) = b(p^{-1}_3 + p^{-1}_4) ~:~ p_1,p_2,p_3,p_4 \in P,\, a,b \in A \}|\right) \,.$$ Using a consequence of the Szemerédi–Trotter theorem, see e.g. Lemma 2.5 from [@MR-NS], we get $$\E^{+} (A_*) \lesssim (|A| /K)^{-2} \cdot (\E^\times (A))^{1/2} |P|^3
\ll
K^5 |A| (\E^\times (A))^{1/2} \,.$$ This completes the proof.
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[^1]: The proof merely requires replacing the equation in the proof of the forthcoming Proposition \[l:BW\_FF\] by the equation corresponding to the energy $\E^\times(\a+A_1)$, rather than $\E^+(A_1)$, where $A_1\subseteq A$ is constructed throughout the proof of Proposition \[l:BW\_FF\]. One proceeds by applying Theorem \[t:MR\] to the latter equation in essentially the same way it is done in the proof of Proposition \[l:BW\_FF\]; the actual application of the theorem can be copied from the proof of Proposition 2 of [@AMRS].
[^2]: For a reader not willing to consult [@KS2] we sketch the argument that goes back to the paper of Elekes [@E] and is similar to the one in the conclusion of the proof of the Proposition \[l:BW\_FF\]. Instead of Theorem \[t:MR\] one uses the Szemerédi-Trotter theorem to estimate the cardinality of the set $S_\tau$ of sums $s=a+b$ in $A_1+A_1$ with approximately $1\leq\tau\leq |A_1|$ realisations. One can rewrite $s=a+ \a p,$ with $\a\in A$, $p\in P^{-1}=P$, and therefore there are at least $q\tau$ incidences between the point set $A\times S_\tau$ and a set of $|P||A_1|$ lines. Theorem \[t:ST\] then gives the upper bound on $|S_\tau|$; recycling this into an energy estimate is a standard exercise. \[ftn\]
|
---
author:
- 'Christian M. Fromm'
- Manel Perucho
- Eduardo Ros
- Tuomas Savolainen
- 'J. Anton Zensus'
date: ' Draft 1.0: '
title: On the location of the supermassive black hole in CTA102
---
[Relativistic jets in active galactic nuclei represent one of the most powerful phenomena in the Universe. They form in the surroundings of the supermassive black holes as a by-product of accretion onto the central black hole in active galaxies. The flow in the jets propagates at velocities close to the speed of light. The distance between the first part of the jet that is visible in radio images (core) and the black hole is still a matter of debate.]{} [Only very-long-baseline interferometry observations resolve the innermost compact regions of the radio jet. Those can access the jet base, and combining data at different wavelenghts, address the physical parameters of the outflow from its emission.]{} [We have performed an accurate analysis of the frequency-dependent shift of the VLBI core location for a multi-wavelength set of images of the blazar including data from 6cm down to 3mm.]{} [The measure of the position of the central black hole, with mass $\sim10^{8.93}\,M_\odot$, in the blazar reveals a distance of $\sim 8\times10^4$ gravitational radii to the 86 GHz core, in agreement with similar measures obtained for other blazars and distant radio galaxies, and in contrast with recent results for the case of nearby radio galaxies, which show distances between the black hole and the radio core that can be two orders of magnitude smaller.]{}
Introduction
============
The blazar has a redshift $z=1.037$ (Schmidt [@sch65]; a luminosity distance of 6.7Gpc, and an image scale of [8.11]{}pc/milliarcsec) and hosts a supermassive black hole (SMBH) of $10^{8.93}\,M_\odot$ (Zamaninasab et al. [@zam14]). The source was one of the first to show strong variability in the radio after its discovery in the 1960s. Its radio morphology shows a strong core with a southward jet on pc-scales, and two lobes in north-west and south-east directions (e.g., Stanghellini et al. [@sta98], Fromm et al. [@fro13a]).
The radio core of a jet is defined as its first observed surface at a given frequency out of the black hole neighbourhood. The position at which it appears at each frequency may be different if the core corresponds to the first region that is optically thin to radiation at this frequency, the lower frequencies appearing downstream of the higher ones, as the jet flow becomes diluter and more transparent (Marcaide & Shapiro [@mar84], Lobanov [@lob98]). On the contrary, if the core corresponds to a strong reconfinement shock, the core at the highest frequencies should converge at the position of the shock. A recent study (Hada et al. [@had11], Marscher [@mar11]) on M87 suggested that the core corresponds to the first optically thin surface and that the 43GHz surface is at 14 gravitational radii ($R_{\rm s}$) from the central black hole. On the contrary, this distance has been suggested to be of $10^4 - 10^5$ gravitational radii in the case of blazars (Marscher et al. [@mar08; @mar10]) and more distant radio galaxies than M87 such as 3C 111 and 3C 120 (Marscher et al. [@mar02], Chatterjee et al. [@chat09; @chat11]).
Fundamental differences between radio galaxies and blazar jets have been invoked to explain the difference (Hada et al. [@had11]; Marscher et al. [@mar11]). In blazars, the jet is observed at a very small angle to the line of sight, which favors the observation of the fast flow due to the relativistic Doppler boosting of the radiation (e.g., Zensus [@zen97]). In radio galaxies, the jet is typically observed at a larger angle, and the faster flow could be missed due to de-boosting of the radiation. The observed jet in the two types of source could correspond to the slower, outer layers of the jet, originated in the accretion disk surrounding the central black hole in the case of radio galaxies, and to the faster spine of the jet generated closer the the black hole, respectively. Therefore, the physical properties of the observed radio core in the two types of objects could be different.
In this letter, we use a 86 GHz VLBA observations made before the start of the flare as observed at 220 GHz (2005.6, Fromm et al. [@fro11]) together with observations down to 5 GHz to measure the distance from the black-hole to the radio-core at 86 GHz. This observation made during a quiescent state of the jet provides an opportunity to measure the core shift in a “clean” setting without the potential confusing effects of the flaring component. We expect changes in the opacity of the region during the flare, if it is followed by the propagation of a component, thus changing the measured distances following the core-shift method (Lobanov [@lob98]). Moreover, Fromm et al. ([@fro13b]) showed that the core-shift vectors on the plane of the sky presented a very irregular behaviour during the 2006 flare. The comparison of the steady state with these epochs would require a detailed understanding of the non-radial motions at the core, which is out of the scope of this paper. We refer the reader to recent works that show this behaviour, known as *jet wobbling*, in other sources (Agudo et al. [@ag07; @ag12], Molina et al. [@mol14]). This letter is structured as follows: In Section 2 we present the relevant data for this work; in Section 3 we present our results; Section 4 is devoted to a brief discussion on possible effects that may have an influence on the result, and in Section 5 we summarize our work.
VLBA observations and data analysis
===================================
{width="99.00000%"}
We observed the jet in the blazar using the Very Long Baseline Array (VLBA) at different frequencies, ranging from 5GHz to 86GHz. The results from these observations were presented in Fromm et al. ([@fro11; @fro13a; @fro13b]), but for the case of the 86 GHz data, which is presented in Fig. \[fig:coreshiftallfreq\]. Those observations cover two years around a major flare in the source in 2006 (Fromm et al. [@fro11]). The 86 GHz epoch that we use here (2005.4) is the only one out of eight observations that yields high enough signal-to-noise ratio in the extended jet emission to allow alignment of the 86 GHz image with the lower frequencies. In addition, the strong flares observed in many radio-sources are usually related with the later detection of a region of enhanced emission that travels along the jet and can be followed by fitting the interferometric data with a Gaussian function (usually called *component*) at each epoch. These injected components can be related to an increase of the number of particles injected (Perucho et al. [@per08]), which can affect the opacity in the region and change the relative position of the core at different frequencies (Kovalev et al. [@kov08]).
Figure \[fig:coreshiftallfreq\] shows the core region at all frequencies for epoch 2005.4. We aligned the images of the jet at different frequencies at each epoch using a cross-correlation method based on the optically thin jet regions (Croke & Gabuzda [@cro08]; Fromm et al. [@fro13b]). This analysis revealed a two-dimensional shift of the core that can only be explained by non-axial (pattern or flow) motion of the emitting region (Fromm et al. [@fro13b]; see also, e.g., Agudo et al. [@ag07], [@ag12], Perucho et al. [@per12]).
\[tab:coreshift\]
\[fig:totalcoreshift\] {width="0.99\columnwidth"}
Figure \[fig:totalcoreshift\] displays the total core-shifts, computed by integrating the two-dimensional path through all the intermediate frequencies. Our results for the first epoch show that the core does not converge to zero at our highest frequency, i.e., measurable shift between the 86GHz and 43GHz core. This result is consistent with the 86GHz core still corresponding to the ($\tau$=1)-surface.
The fit of the relative position of the core at the different frequencies with respect to the largest ($r_\mathrm{core}\propto \nu^{-1/k_r}$, Lobanov [@lob98]) results in a value ($k_r=1.0\pm0.1$) compatible with the expected in the case of a conical jet in which the energies of the non-thermal particles and the magnetic field are in equipartition, close to the minimum value of those energies required to explain the observed radio flux. This supports the interpretation of the jet in becoming transparent to the different frequencies as it expands.
The single-dish data from the 2006 radio flare allow us to follow the evolution of the source luminosity with time at different frequencies. The first increase in flux density associated to the flare was detected at a turnover frequency of 222$\pm$99GHz in 2005.6 (Fromm et al. [@fro11]). Subtracting the spectrum of the source previous to the flare allows to follow the evolution of the injected flow related to the flare. By doing this, we could identify extra flux at frequencies larger than 100GHz before the main radio flare, i.e., before the radio feature could be observed out of the radio core at high frequencies. We can assign this extra flux to the injected flux beyond errors and claim that it is optically thin at those frequencies before it is at 43GHz or 86GHz. This represents independent evidence of the relation of radio-cores to optical depth in this source. Our conclusion is that core-shift analysis performed at higher frequencies than allowed by present techniques would result in non-convergence of the core at a given position. We have to note that our observations indicate that a standing feature, possibly a re-confinement shock lying 0.1mas away from the core at 43GHz. This leaves room for the coincidence, within errors, of the core and a re-confinement shock in other sources.
Results: black hole relative location.
======================================
Using the results of the power law fit to the obtained core-shifts and a viewing angle of $\vartheta=2.6^\circ$ (Fromm et al. [@fro13b]), we can derive the distance of the black hole to the radio core at 86GHz (see Fig. 2) Following the approach of Hada et al. ([@had11]) the distance to the black hole is $(7.0\pm3.2)$pc, which is equivalent to $(8.5\pm3.9)\times10^4$ gravitational radii. In order to validate our result we compute the distance to the black hole using the projected core-shift along the average P.A. of $90^\circ$ since the calculation along the 2D path could lead to an over-estimate of the distance. The distance obtained using the projected core-shifts is $(6.4\pm5.5)$pc corresponding to $(7.8\pm6.7)\times10^4$ gravitational radii similar to the one using the core-shifts along the 2D path.
This distance to the black hole that we obtain fits nicely to the results obtained by Kutkin et al. ([@kut14]) and Zamaninasab et al. ([@zam13]) for the blazar 3C454.3 ($r_{\mathrm{core},43\,\mathrm{GHz}}\sim9\,\mathrm{pc}$). It is also comparable with the predictions for blazar and quasar jets (Marscher et al. [@mar08; @mar10]) and distant radiogalaxies (Marscher et al. [@mar02], Chatterjee et al. [@chat09; @chat11]). On the contrary, it is much larger than in the case of the radio galaxy M87 (Hada et al. [@had11]; Marscher et al. [@mar11]).
The reason for this difference could then well be a matter of resolution and that observing M87 at the distance of those other radiogalaxies would bring the core to $10^{(4-5)}\,\mathrm{R_s}$. In this context, HST-1 (at 1 arcsec from the core) could be observed as the radio-core or within it of M87, with the particularity that it can be identified with a recollimation shock. Another relevant aspect is the viewing angle: Sources observed at small viewing angles pile up all the emission from the compact, bright regions. Therefore, if the jet brightness is high up to the radio-core, it may coincide with the last bright, projected surface, which obscures all the regions between the $\tau$=1 surface and this last surface. This effect could again be avoided by increasing resolution.
A relevant limitation of this kind of measures is set by the uncertainties in the alignment of the highest frequency image, especially if no extended emission to align properly with lower frequencies is available.
Despite all the difficulties associated with this calculation (see the discussion), we can claim that the separation of the radio core at tens of GHz to the black hole in CTA 102 is of the order of parsecs, implying a separation of $10^{(4-5)}\,\mathrm{R_s}$, on the basis of the values obtained for the 22 GHz (see Table 4 in Fromm et al. [@fro13b]), 43 GHz and 86 GHz radio-cores.
Discussion
==========
Core-shifts and the 2006 flare in CTA 102
-----------------------------------------
It is difficult to derive conclusions from the influence of the passage of the component through the core region due to the large errors obtained in the calculation of the core-shifts for the affected epochs. However, a general trend that we observe by performing the same kind of analysis for the remaining epochs is that the exponent of the core-shift with frequency decreases when a perturbation crosses the core region (see Table 5 in Fromm et al. [@fro13b]). Actually, the crossing of the 43 GHz core by another component at the beginning of 2007 (Fromm et al. [@fro13a]) seems to cause the same effect as the 2006 flare (see Table 5 in Fromm et al. [@fro13b]). This effect can be assigned to changes in the opacity as the perturbation propagates: As the opacity increases, the high-frequency core positions are dragged downstream, and the relative core-shifts are reduced. This is reflected in the decrease of $k_r$.
Core-shifts and jet wobbling
----------------------------
Fromm et al. ([@fro13b]) showed that the core-shift direction may also change in the plane of the sky. This effect could be due to *jet wobbling*, which has been observed in other sources (e.g., Agudo et al. [@ag07], [@ag12]). We would like to point out that these changes in the relative positions of the cores on the plane of the sky could result in changes in the measured projected projected core-shifts. In addition, this effect is independent from the presence of a perturbation, although a perturbation can introduce further changes. Both effects seem to be acting on the jet in CTA 102, which makes it very difficult to disentangle their relative role.
However, even the large apparent changes in direction of the core-shift vectors observed at the two first epochs presented in Fromm et al. ([@fro13b], one during the steady state and another one at the first stages of the evolution of the flare) do not translate into significant changes in the value of $k_r$ obtained for both epochs ($1.0\pm0.1$ versus $0.8\pm0.3$).
Nevertheless, the changes of the relative positions on the plane of the sky should have a negligible effect, save errors, on the calculation of the relative location of the black hole, because of the deprojection done to obtain it (Lobanov [@lob98]).
Summary
=======
We present here the first 86 GHz map of CTA 102 within the series of observations around the 2006 flare. Unfortunately the other monitoring images at this frequency show poor quality. The map presented here corresponds to the first of those epochs, prior to the triggering of the flare. Making use of the simultaneous observations of the source from 5 GHz to 86 GHz and considering that the core is unaffected in this epoch by the propagation of any component, we have measured the core-shifts during what we expect to be a steady-state like situation in CTA 102. As a result, we obtain a slope for the core-shifts ($k_{\rm r}=1.0\pm0.1$) that is compatible with a conical jet in adiabatic expansion. This result allows us to compute the relative position of the 86 GHz core to the black hole, which results to be $\simeq 7\,{\rm pc}$ or $8.5\times10^4$ gravitational radii.
We have discussed the possible effect of perturbations and changes on the directions of the core-shift vector and the estimates of the distance between the radio-core and the black hole. Despite the uncertainties, the same kind of calculations were performed for all epochs using the 22 GHz and 43 GHz cores as reference and provided values for the distance between them and the black-hole of the order of several parsecs in all cases, confirming our result (Fromm et al. [@fro13b]). Future work should include the validation of our results and possibly confirm the discussed effects on them.
We acknowledge L. Fuhrmann for careful reading and for useful comments and suggestions to the manuscript. We thank the anonymous referee for comments and criticism that helped to improve this manuscript. C.M.F. was supported for this research through a stipend from the International Max Planck Research School (IMPRS) for Astronomy and Astrophysics at the Universities of Bonn and Cologne. Part of this work was supported by the COST Action MP0905 ‘Black Holes in a violent Universe’. E.R. acknowledges partial support from MINECO grants AYA2009-13036-C02-02 and AYA2012-38491-C02-01 as well as Generalitat Valenciana grant PROMETEO/2009/104. M.P. acknowledges financial support from MINECO grants AYA2010-21322-C03-01, AYA2010-21097-C03-01, and CONSOLIDER2007-00050. This work is based on observations with the VLBA, which is operated by the NRAO, a facility of the NSF under cooperative agreement by Associated Universities Inc. This research made use of data from the MOJAVE database that is maintained by the MOJAVE team (Lister et al. [@lis09a]).
Agudo, I., Bach, U., Krichbaum, T.P., et al. 2007, 476, L17 Agudo, I., Marscher, A.P., Jorstad, S.G., Gómez, J.-L., Perucho, M., Piner, B.G., Rioja, M., Dodson, R. 2012, 747, 63 Blandford, R. D., & Königl A. 1979, 232, 34 Chatterjee, R., Marscher, A.P., Jorstad, S.G., et al. 2009, 704, 1689 Chatterjee, R., Marscher, A.P., Jorstad, S.G., et al. 2011, 734, 43 Croke, S.M., & Gabuzda D.C. 2008, 386, 619 Fromm, C.M., Perucho, M., Ros, E., et al. 2011, 531, A95 Fromm, C.M., Ros, E., Perucho, M., et al. 2013a, 551, A32 Fromm, C.M., Ros, E., Perucho, M., et al. 2013b, 557, A105 Hada, K., Doi, A., Kino, M., et al. 2011, 477, 185 Kovalev, Y.Y., Lobanov, A.P., Pushkarev, A.B., & Zensus, J.A. 2008, 483, 759 , A. M., [Sokolovsky]{}, K. V., [Lisakov]{}, M. M., et al. 2014, 437, 3396 Lister, M.L., Aller, H.D., Aller, M.F., et al. 2009, 137, 3718 Lobanov, A.P. 1998, 330, 79 Marcaide, J.M., & Shapiro, I.I. 1984, 276, 56 Marscher, A.P., Jorstad, S.G., Gómez, J.-L., Aller, M.F., Teräsranta, H., Lister, M.L., Stirling, A-M. 2002, 417, 625 Marscher, A.P., Jorstad, S.G., D’Arcangelo, F.D., et al. 2008, 452, 966 Marscher, A.P., Jorstad, S.G., Larionov, V.M., et al. 2010, 710, L126 Marscher, A.P. 2011, 477, 164 Molina, S.N., Agudo, I., Gómez, J.-L., Krichbaum, T.P., Martí-Vidal, I., Roy, A.L. 2014, 566, 26 McNaron-Brown, K., Johnson, W.N., Jung, G.V., et al. 1995, 541, 575 Perucho, M., Agudo, I., Gómez, J.L., et al. 2008, 483, L29 Perucho, M., Kovalev, Y.Y., Lobanov, A.P., Hardee, P.E., & Agudo, I. 2012, 749, 55 Schmidt, M. 1965 141, 1295–1300 Stanghellini, C., O’Dea, C.P., Dallacasa, D., et al. 1998, 131, 303 Zamaninasab, M., Savolainen, T., Clausen-Brown, E., Hovatta, T., Lister, M.L., Krichbaum, T.P., Kovalev, Y.Y., Pushkarev, A.B. 2013, 436, 3341 Zamaninasab, M., Clausen-Brown, E., Savolainen, T., Tchekhovskoy, A. 2014, 510, 126 Zensus, J.A. 1997, 35, 607
|
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author:
- |
Atsuo T. Okazaki$^1$, Christopher M. P. Russell$^{1}$\
$^1$ Faculty of Engineering, Hokkai-Gakuen University, Toyohira-ku, Sapporo 062-8605, Japan\
[*E-mail(ATO): okazaki@lst.hokkai-s-u.ac.jp*]{}
title: ' 3D Dynamical Modeling of Wind Accretion in Cyg X-3 '
---
Introduction
============
Cyg X-3 is a unique laboratory for high energy astrophysics. It is a very bright, high mass X-ray binary consisting of a Wolf-Rayet star and a compact object, which is most likely a black hole, in a circular 4.8 h orbit. At present, it is the only microquasar from which variable GeV gamma-ray emission has been detected.
Cyg X-3 exhibits a 4.8 h modulation in multi-wavebands, which should provide important clues for the accretion/ejection mechnism. The X-ray modulation is basically explained by the absorption/scattering in the dense Wolf-Rayet wind, but the origin of the asymmetry (slow rise, fast decay”) remains an open question (Zdziarski et al. 2012; Vilhu & Hannikainen 2013).
Fast orbital motion and slow acceleration of the wind (due to a lack of hydrogen) means that the Coriolis force could play an important role to shape the accretion pattern. Therefore, constructing a 3D dynamic model of wind accretion, taking account of the orbital motion, is important to understand the high energy activity of Cyg X-3.
Numerical Setup
===============
We use a 3D SPH code to simulate the Wolf-Rayet wind around the compact object of Cyg X-3. The code uses the variable smoothing length and individual time-steps. The standard values of artificial viscosity parameters are adopted, i.e., $\alpha=1$ and $\beta=2$. In one simulation shown below, the wind is launched spherically symmetrically and the optically-thin radiative cooling is taken into account, while in the other simulations, isothermal wind particles are ejected only in a narrow range of azimuthal and vertical angles toward the compact object, in order to optimize the resolution and computational efficiency. In these simulations, the Wolf-Rayet wind is assumed to have a velocity distribution of the form $$v = v_\infty (1-r/R_{\mathrm WR})^\beta,$$ where $v_\infty$ is the terminal speed of the wind and $R_{\mathrm WR}$ is the radius of the Wolf-Rayet star. The stellar and wind paraemters adopted are as follows: $M_{\mathrm WR} = 10.3\,M_\odot$, $R_{\mathrm WR} = 6.1\,R_\odot$, $M_{\mathrm X} = 2.4\,M_\odot$, $\dot{M}_{\mathrm WR} = 6.5 \times 10^{-6}\,M_\odot\;\mathrm{yr}^{-1}$, and $v_\infty = 1,700\,\mathrm{km\;s}^{-1}$ (Zdziarski et al. 2013). The binary orbit is set in the $x$-$y$ plane. In what follows, Phase 0 corresponds to the superior conjunction.
Numerical Results
=================
Figure \[fig1\] shows the global structure of the $\beta=2$ wind in the orbital plane. We notice that the Wolf-Rayet wind, which is launched spherically symetrically, has a large-scale asymmetry around the compact object: significantly denser gas flow is seen on the rear side of the compact object than on the front side. This is because for $\beta = 2$ the wind speed near the compact object is comparable to the orbital speed of the compact object.
Table \[table1\] compares the accretion rates in isothermal simulations for $\beta=$1, 2, and 3 with the correspronding Bondi-Hoyle-Lyttleton (BHL) accretion rate. The simulated accretion rate is comparable with the BHL accretion rate for $\beta=1$, but is significantly lower for $\beta \ge 2$. As a result, the increase in the simulated accretion rate with increasing $\beta$ is weaker than in the BHL rate. This is because the flow around the compact object has a larger density gradient for larger $\beta$, while the BHL accretion rate is for a flow with uniform density distribution.
[cccc]{}\
$\beta$ & simulation & simulated $\dot{M}$ & BHL $\dot{M}$\
&& ($10^{18}\,\mathrm{g\;s}^{-1}$) & ($10^{18}\,\mathrm{g\;s}^{-1}$)\
1 & isothermal & 1.6 & 1.7\
2 & isothermal & 2.8 & 4.1\
2 & radiative cooling & 2.5 & 4.1\
3 & isothermal & 4.0 & 8.6\
\[table1\]
Finally, Figure \[fig2\] shows simulated modulation in mass column and resulting X-ray light curves for a $\beta=2$ wind with optically thin, radiative cooling. Moderate modulation is seen for small to intermediate inclination angles, while for high inclination angles, absorption/scattering by the accretion wake (phase 0.75) becomes remarkable. Unfortunately, the current model light curves do not reproduce the observed light-curve asymmetry. They are, however, computed for the first time based on the 3D dynamical simulation, where the effect of the orbital motion is also taken into account. We will continue our effort to better model the wind structure and absorption/scattering by the wind.
\
We thank Shunji Kitamoto for helpful comments on the X-ray behavior of Cyg X-3. This work was partially supported by the JSPS Grants-in-Aid for Scientific Research (C) (24540235).
References {#references .unnumbered}
==========
Zdziarski, A.A. et al. 2012 MNRAS., 426, 1031
Zdziarski, A.A. et al. 2013 MNRASL., 429, L104
Vilhu, O. & Hannikainen, D.C. 2013 A&A., 550, A48
\[last\]
|
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abstract: 'Transfer learning aims to reduce the amount of data required to excel at a new task by re-using the knowledge acquired from learning other related tasks. This paper proposes a novel transfer learning scenario, which distills robust phonetic features from grounding models that are trained to tell whether a pair of image and speech are semantically correlated, without using any textual transcripts. As semantics of speech are largely determined by its lexical content, grounding models learn to preserve phonetic information while disregarding uncorrelated factors, such as speaker and channel. To study the properties of features distilled from different layers, we use them as input separately to train multiple speech recognition models. Empirical results demonstrate that layers closer to input retain more phonetic information, while following layers exhibit greater invariance to domain shift. Moreover, while most previous studies include training data for speech recognition for feature extractor training, our grounding models are not trained on any of those data, indicating more universal applicability to new domains.'
address: |
Computer Science and Artificial Intelligence Laboratory\
Massachusetts Institute of Technology\
Cambridge, MA, USA
bibliography:
- 'main.bib'
title: 'Transfer Learning from Audio-Visual Grounding to Speech Recognition'
---
**Index Terms**: transfer learning, audio-visual grounding, multi-modal learning, semantic supervision, speech recognition
Introduction
============
Robustness of automatic speech recognition (ASR) systems is essential to generalization of using speech as interfaces for human computer interaction. Thanks to the strong modeling capacity of neural networks, recent studies [@amodei2016deep; @chiu2018state; @jaitly2013vocal; @cui2015data; @ko2015audio; @kim2017generation; @hsu2017unsupervised; @hsu2018unsupervised; @hosseini2018multi; @sun2018training] have demonstrated that by providing supervised examples as abundant and diverse as possible, such models can learn to extract domain invariant features and recognize linguistic units jointly. However, without additional treatment, good performance and robustness may not be achieved when labeled data are very limited in quantities or not available in all domains [@hermansky1994rasta]. One way to ease the burden of ASR systems is by providing better features, which are more invariant to nuisance factors while containing linguistic information ready for use (e.g., linear separability w.r.t. phonemes). Such features can be hand-crafted by leveraging prior knowledge [@hermansky1994rasta; @kingsbury1997recognizing; @kim2016power; @fredes2017locally], or they can be learned in a data-driven fashion. Furthermore, this learning can take place jointly with ASR [@sun2017unsupervised], or separately with some tasks that have aligned objectives [@hsu2017learning; @hsu2018extracting; @chung2019unsupervised].
Learning features from some source tasks that can benefit the target task is a common realization of transfer learning [@pan_transfer_learning_survey_2009]. In this work, we propose a novel inductive transfer learning scenario [@pan_transfer_learning_survey_2009], which utilizes speech features learned from audio-visual grounding for speech recognition. Audio-visual grounding [@harwath2016unsupervised] is a task which aims to distinguish whether a spoken caption is semantically associated with an image or not, and vice versa, without using any textual transcripts. Deep audio-visual embedding network (DAVEnet) [@harwath2018jointly] is a two-branched convolutional neural network model for this task, which learns to encode images and spoken captions into a shared embedding space that reflects semantic similarity. To successfully learn a semantic representation for speech, the model has to recognize its lexical content, which in turns requires identifying phonetic content. Therefore, one can expect intermediate layers of the speech branch in DAVEnet models to function as lexical or phonetic unit detectors. Furthermore, since non-linguistic aspects of speech, such as speaker, are not correlated with semantics, these information may be discarded, resulting in the intermediate outputs from the model being invariant to domain shift.
We conduct a series of ASR experiments probing properties of the features distilled from DAVEnet models at different layers. Results indicate higher in-domain accuracy using features closer to input, and better robustness to domain shift using features from latter layers. In addition, we also study how the choice of DAVEnet architectures and grounding performance affects the performance of distilled feature extractors. In summary, our contributions are three-fold:
To the best of our knowledge, this is the first work connecting audio-visual grounding with speech recognition.
Our empirical study verifies that the distilled feature extractors not only contain sufficient information for recognizing phonemes, but better remove nuisance information.
Moreover, the grounding models are trained on a different dataset from that used for ASR, indicating more general applicability of the distilled features.
Learning Spoken Languages through Audio-Visual Grounding
========================================================
In this section, we describe in detail the source task as well as the DAVEnet model, and then review several analysis studies which lay the foundation for our work.
![Graphical illustration of audio-visual grounding model training (left), ResDAVEnet architecture (center), and feature distillation pipeline for speech recognition (right).[]{data-label="fig:model"}](diagram.pdf){width="\linewidth"}
Audio-Visual Grounding
----------------------
Inspired by the fact that humans learn to speak before being able to read or write, audio-visual grounding of speech is a proxy task proposed in [@harwath2016unsupervised] that aims to examine the capability of computational models to learn a language using only semantic-level supervision. To simulate such a learning scenario, a model has access to images and their spoken captions during training. The goal of the model is to learn a semantic representation for each caption and each image, such that representations of semantically correlated utterances and images are similar to each other, while those from irrelevant pairs are dissimilar. Performance is evaluated using a cross-modality retrieval task: given a spoken sentence, a model is asked to rank a list of 1,000 images according to semantic relevance, with only one image being the correct answer, and vice versa. Recall@10 averaged over the retrieval tasks in both directions is used for evaluation.
Deep Audio-Visual Embedding Network (DAVEnet)
---------------------------------------------
DAVEnet is a convolutional neural network (CNN) for audio-visual grounding proposed in [@harwath2016unsupervised; @harwath2017learning; @harwath2018jointly], which consists of two branches: $f$ for speech and $g$ for image, as depicted in Figure \[fig:model\]. Each branch has a sequence of strided convolutional blocks, followed by a global mean-pooling layer to produce a fixed dimensional representation. The model is trained to minimize a triplet loss [@hoffer2015deep; @jansen2018unsupervised]: given a similarity measure $S(\cdot, \cdot)$, paired speech and image, $x_s$ and $x_i$, along with one imposter instance from each modality, $\tilde{x}_s$ and $\tilde{x}_i$, the loss enforces $S(f(x_s), g(x_i))$ to exceed both $S(f(x_s), g(\tilde{x}_i))$ and $S(f(\tilde{x}_s), g(x_i))$ by a predefined margin. Following [@harwath2019towards], imposter instances are drawn using a mixture of uniform sampling and within-batch semi-hard negative mining [@jansen2018unsupervised]. $S(z_1, z_2) = \langle z_1, z_2 \rangle $ is used here.
In our experiments, we make use of two DAVEnet model variants. The first is identical to the model used in [@harwath2018jointly], which uses an audio network comprised of 5 convolutional layers and the VGG16 architecture for the image network. The second model, ResDAVEnet, is based upon deep residual networks [@resnet]. The image network makes use of the ResNet50 architecture, while the audio network is based on strided 1-D convolutions with residual connections. The first layer of the ResDAVEnet audio model is comprised of 128 convolutional units each spanning all frequency channels but only one temporal frame, with a temporal shift of 1 frame. This is followed by a ReLU and a BatchNorm layer. The remainder of the network is a sequence of 4 residual stacks with channel dimensions 128, 256, 512, and 1024. Each residual stack is comprised of a sequence of two basic residual blocks (as described in [@resnet]) which share the same overall channel dimension, with 2-D 3x3 kernels replaced with 1-D kernels of length 9. Additionally, the first residual block in each layer in each stack is applied with a stride of two frames, resulting in an effective temporal downsampling ratio of $2^4$ for the entire network, as shown in Figure \[fig:model\] (center).
Emergence of Multi-Level Speech Unit Detectors
----------------------------------------------
Recent work [@drexler2017analysis; @harwath2017learning; @harwath2019towards] on analyzing DAVEnets have shown that, despite the fact that phoneme and word labels are never explicitly provided, such detectors automatically emerge within these models. Phoneme-like detectors reside in layers closer to the input [@drexler2017analysis; @harwath2019towards], while semantic word detectors reside in layers closer to the output [@harwath2017learning]. Such findings echo with the recent discovery in the computer vision community [@zhou2014learning; @zhou2015object] that in a trained scene classifier, layers closer to sensory input appear to be low-level pattern (e.g., shape, edge, and color) detectors, while object detectors emerge at later layers. This behavior can be mainly attributed to the compositionality of the prediction target as well as the inductive bias we impose in the model architecture. Just as a scene can often be determined by the objects that are present, the semantics of a spoken sentence is determined by the sequence of words, each of which in turn is determined by phoneme sequences. Prediction of semantic objects from a spoken sentence can therefore be regarded as a bottom-up process, which iteratively composes higher-level concepts from lower-level ones with the hierarchical convolution operations in CNNs.
Transfer Learning to Speech Recognition
=======================================
Distilling Robust Feature Extractors for ASR
--------------------------------------------
Both DAVEnet variants are trained on the Places Audio Caption dataset (PlacesAudCap) [@harwath2018jointly], derived from the Places205 scene classification dataset [@zhou2014learning]. PlacesAudCap is composed of over 400K image and unscripted spoken caption pairs collected from 2,954 speakers via Amazon Mechanical Turk, which sums up to over 1,000 hours. For the audio-visual grounding task, both models use 40-dimensional log Mel filterbank (FBank) features with 10ms shift and 25ms analysis window as input, and achieve R@10 of 0.629 and 0.720, respectively.
As a natural result of large-scale crowd-sourcing, this dataset exhibits great diversity not only in textual content, but also in speaker, background noise, and microphone channels. For both semantic grounding and speech recognition, these non-textual factors are nuisances to the target, and therefore would eventually be removed from the internal representations learned by the networks trained for these two tasks. Having been exposed to a vast amount of nuisance factors, we hypothesize that the audio branch of DAVEnet models would also learn domain invariant phonetic representations at later layers, which can be subsequently utilized for robust speech recognition. From now on, we denote features extracted from the $k$-th layer of model $M$ with $M$-L$k$, for example, ResDAVEnet-L2.
To account for the different frame rates at different layers in DAVEnet models, when extracting outputs from a layer with a down-sampling rate $r$ compared to the speech inputs, we repeat each step $r$ times for simplicity, as shown in Figure \[fig:model\] (right).
Evaluating Transfer Learning Performance
----------------------------------------
To evaluate transfer learning performance, we consider three criteria:
inclusion of phonetic content,
exclusion of nuisance factors, and
transferrability across datasets.
The first two are evaluated using a protocol similar to [@hsu2018extracting], where an ASR model is trained on a set of domains, and evaluated on both in-domain and out-of-domain speech (relative to the training data). Performance on in-domain data characterizes an upper bound for the amount of phonetic information that can be inferred from the input. The performance gap between in-domain and out-of-domain data quantifies the invariance of the features to nuisance factors: the smaller this gap, the more invariant the features are. To test the third criteria, instead of training the source task on a dataset that includes speech used for the target task, a separate dataset collected through a different process (i.e., PlacesAudCap) is used. We emphasize here that this is a more practical setting to consider than training one feature extractor for each target task.
Related Work
============
Transfer learning has a long history in the field of machine learning [@pan_transfer_learning_survey_2009]. More recently, deep neural network models have been shown to be extremely effective for learning representations of data with a high degree of re-usability across many different tasks and domains. Perhaps the most well-known example of this is the use of the ImageNet [@imagenet] image classification database to pre-train convolutional neural network models for other downstream computer vision tasks [@Razavian_2014; @faster_rcnn; @two_stream_cnns]. Other sub-fields have also developed similarly techniques. For example, in natural language processing, dense word vector models such as word2vec [@word2vec] and GloVe [@glove], or more advanced ones like ELMo [@elmo] and BERT [@bert] have quickly replaced one-hot word representations in many tasks and pushed the state-of-the-art forward on a variety of language understanding tasks. More recently, there is also an increasing interest in learning from multimodal data [@yeh2018unsupervised] and transfer learned representations from such tasks [@gupta2017aligned]
In the field of speech recognition, low-resource speech recognition is a scenario which heavily benefits from transfer learning, for example in the form of training on multilingual datasets [@ekapol_2016]. Other models capable of disentangling phonetic and domain information have recently been shown to learn acoustic features with a greater degree of domain invariance than traditional acoustic features [@hsu2017learning; @hsu2017unsupervised; @hsu2018extracting]. Another line of work has studied the use of the visual modality as a form of weak supervision using semantic information for acoustic modeling [@harwath2016unsupervised; @kamper_taslp19; @chrupala_2017], followed up with analysis on representations learned from such models [@drexler2017analysis; @alishahi_2017; @harwath2019towards]. In this paper, we build upon this prior work and quantify the degree to which these representations can be used to build robust ASR.
Experiments
===========
ASR Setup and Baselines
-----------------------
We consider TIMIT [@zue1990speech] and Aurora-4 [@pearce2002aurora] for training ASR systems to study robustness of the proposed method to speaker, channel, and noise. TIMIT contains 5.4 hours of 16kHz broadband recordings of read speech from 630 speakers, of which about 70% are male. Recordings from male speakers are used for training ASR systems, which are then tested on both genders. Aurora-4 is based on the Wall Street Journal (WSJ) corpus [@garofalo2007csr], containing recordings with microphone and noise variation. The set of conditions are divided into four groups: clean (A), noisy (B), channel (C), and noisy+channel (D). While recordings in A are recorded by one microphone in quiet environments, those in C are recorded with a different set of microphones than A. Recordings in B and D are created from A and C, respectively, with artificially added noises. Similar to [@hsu2018extracting], we use the clean set (A) for training ASR systems, and test on the four groups separately.
Kaldi [@povey2011kaldi] is used for training of initial HMM-GMM models, forced alignment, and decoding. The Microsoft Cognitive Toolkit (CNTK) [@seide2016cntk] is used for neural network-based acoustic model training. To simplify the pipeline and study only the effect of ASR input features, the same forced alignment derived from a HMM-GMM model trained on Mel-frequency cepstral coefficient (MFCC) features are used for all experiments, following the default recipe in Kaldi. A three-layer long short-term memory (LSTM) acoustic model with 1,024 memory cells and a 512-node linear projection at each layer is used [@sak2014long]. Training of LSTM acoustic models closely follows [@zhang2016highway], which minimizes a frame-level cross-entropy loss using stochastic gradient descent with a momentum of 0.9 starting from the second epoch. Initial learning rate is set to 0.2 per minibatch, and $L2$ regularization with a weight of $1e-6$ is used.
We consider two types of features to compare with our proposed method. The first one is FBank feature, which is the input to DAVEnet models and contains rich phonetic and domain information. The second one is the latent segment variable $z_1$ from a model called factorized hierarchical variational autoencoder (FHVAE) [@hsu2017learning]. FHVAE learns to encode sequence-level and segment-level information into separate latent variables without supervision by optimizing an evidence lower bound derived from a factorized graphical model, and has been shown effective for extracting domain invariant ASR features [@hsu2018extracting].
While previous work investigated usage of FHVAE for ASR by training FHVAE models on all domains of the target task (e.g., Aurora-4 with all four conditions) [@hsu2018extracting; @hsu2018unsupervised], we also evaluate FHVAE models trained on PlacesAudCap to test cross-dataset transferability, and on the subset of domains used for ASR training. We use FHVAE models with two LSTM layers, each with 256 cells, for both the encoders and decoder. A discriminative weight of $\alpha=10$ is applied for all models, and the scalable training algorithm proposed in [@hsu2018scalable] is used for training on PlacesAudCap dataset with a sequence batch size $K=5000$, because the original algorithm cannot handle large-scale datasets.
Main Results
------------
Tables \[tbl:a4\_res\] and \[tbl:timit\_res\] present the testing word error rates (WERs) on both in-domain and out-of-domain conditions for ASR systems trained with different features. *FE Train Set* denotes the data used for training feature extractors, and *A/I* following *Places* represents the audio and image portion of the PlacesAudCap dataset, respectively.
Starting with Table \[tbl:a4\_res\], we observe that FBank suffers from severe degradation in all out-of-domain conditions (B, C, and D), while FHVAE trained on all conditions of the Aurora-4 dataset achieves the best performance. However, when trained on *Places A*, improvement of FHVAE from FBank on out-of-domain data becomes less significant in the presence of additive noise, compared to the result in the purely channel-mismatched condition (C). Results of the proposed methods are shown in the second and the third section in Table \[tbl:a4\_res\]. While features from ResDAVEnet consistently outperforms FBank and FHVAE (Places A) for all layers, those from DAVEnet do not. We hypothesize that the much deeper architecture of ResDAVEnet at each layer (ResStack) enables better removal of nuisance factors and preserving of linguistic information compared to DAVEnet, which also reflects in the comparison of grounding performance as mentioned earlier.
It is also worth noting that, for both DAVEnet and ResDAVEnet models, performance in matched domain degrades when using latter layers, and except for ResDAVEnet-L1, all features are actually worse than the FBank baseline. This could indicate discarding of relevant phonetic information in the process of inferring higher-level semantic representation such as words. Table \[tbl:timit\_res\] demonstrates a similar trend as Table \[tbl:a4\_res\], where FHVAE trained on TIMIT dataset of all genders achieves the best out-of-domain WER, and ResDAVEnet-L2 is the best comparing to models trained on Places.
We also present qualitative visualizations in Figure \[fig:aurora4\_tsne\] using t-SNE [@tsne] comparing ResDAVEnet, FHVAE (Places A / Aurora4 All), and the baseline FBank feature. It can be observed from the first row that all three features contain phonetic information, as different phonetic manners are separated in each space. On the other hand, the project features of ResDAVEnet and FHVAE (Aurora4 All) are visually more environment invariant than those from the other two (for the FHVAE trained on PlacesAudCap, green and orange dots concentrate more at the center than red and blue dots). Such visualization correlates well with the performance of the various feature types in Tables \[tbl:a4\_res\].
To conclude, we learn that
despite being trained with exactly the same process, inductive bias introduced to model architectures (i.e., DAVEnet versus ResDAVEnet) still affects the properties of learned representations,
feature extractors distilled from ResDAVEnet models clearly preserve phonetic information while improving invariance to nuisance factors, and most importantly,
it achieves better cross-dataset transferrability compared to FHVAE and FBank features.
![Frame-level t-SNE projections for four different acoustic representations, color coded for phonetic manner class, speaker identity, and noise/environment type. Visually, the ResDAVEnet features encode the least amount of speaker and environment information.[]{data-label="fig:aurora4_tsne"}](aurora4_tsne_with_condition_4feat.png){width="\linewidth"}
Correlation with Source Task Performance
----------------------------------------
Finally, we study how the performance of the grounding task affects the transfer learning performance, conditioning on the same neural network architecture for the source task. We create two proper subsets of 200k and 80k paired image/audio captions, and train one ResDAVEnet model on each subset. R@10 of the retrieval task for the models trained with 80k, 200k, and 400k (original) are 0.343, 0.582, and 0.720, respectively.
Results are shown in Table \[tbl:a4\_wer\_frac\]. Except for the first layer, we can observe that WER decreases as the amount of source task training data increases. In fact, except for the out-of-domain conditions of the first layer, all layers improve in all conditions (full results not shown due to space limit). Discovery of such positive correlation affirms the relatedness of the two tasks and encourages collection of a larger dataset for building a general feature extractor based on semantic grounding tasks.
Concluding Discussion and Future Work
=====================================
In this paper, we present a successful example of transfer learning from a weakly supervised semantic grounding task to robust ASR. We achieve cross-dataset transferability, which is an important milestone toward building a generalized feature extractor to be used in many tasks and domains like BERT. In addition, along with the analysis in [@drexler2017analysis; @harwath2019towards], this work sheds light on using semantic level supervision to learn the compositional structure of a language. For future work, we would like to study methods for leveraging target task data, possibly through semi-supervised training or adaptation, in order to bridge the gap to FHVAE trained on those data. Furthermore, unlike FHVAE, it is unclear at which layer a ResDAVEnet model learns to maximally remove domain information. We would also like to explicitly force such disentanglement to occur at certain layers, which can possibly improve both the grounding performance and the robustness of distilled features.
|
---
abstract: |
The cyclic shift graph of a monoid is the graph whose vertices are the elements of the monoid and whose edges connect elements that are cyclic shift related. The Patience Sorting algorithm admits two generalizations to words, from which two kinds of monoids arise, the ${{\smash{\mathrm{rps}}}}$ monoid and the ${{\smash{\mathrm{lps}}}}$ (also known as Bell) monoid. Like other monoids arising from combinatorial objects such as the plactic and the sylvester, the connected components of the cyclic shift graph of the ${{\smash{\mathrm{rps}}}}$ monoid consists of elements that have the same number of each of its composing symbols. In this paper, with the aid of the computational tool SageMath, we study the diameter of the connected components from the cyclic shift graph of the ${{\smash{\mathrm{rps}}}}$ monoid.
Within the theory of monoids, the cyclic shift relation, among other relations, generalizes the relation of conjugacy for groups. We examine several of these relations for both the ${{\smash{\mathrm{rps}}}}$ and the ${{\smash{\mathrm{lps}}}}$ monoids.
address:
- 'Centro de Matemática e Aplicações, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal'
- 'Centro de Matemática e Aplicações and Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal'
- 'Departamento de Matemática and CEMAT-CIÊNCIAS, Faculdade de Ciências, Universidade de Lisboa, Lisboa 1749-016, Portugal.'
author:
- 'Alan J. Cain'
- António Malheiro
- 'Fábio M. Silva'
bibliography:
- '\\jobname.bib'
title: Conjugacy in Patience Sorting monoids
---
Introduction {#sec:introduction}
============
Patience Sorting has its origins in the works of Mallows [@Mallows62; @10.2307/2028347] and can be regarded as an insertion algorithm on standard words over a totally ordered alphabet ${\mathcal{A}}_n=\{1<2<\dots<n\}$, that is, words over ${\mathcal{A}}_n$ containing exactly one occurrence of each of the symbols from ${\mathcal{A}}_n$. As noticed by Burstein and Lankham [@BL2005], this algorithm can be viewed as a non-recursive version of Schensted’s insertion algorithm. This perspective suggests that a construction similar to the plactic monoid must also hold for this case. The plactic monoid can be constructed as the quotient of the free monoid over ${\mathcal{A}}$ (the infinite totally ordered alphabet of natural numbers), ${\mathcal{A}}^*$, by the congruence which relates words of ${\mathcal{A}}^*$ inserting to the same (semistandard) Young tableaux under Schensted’s insertion algorithm.
According to Aldous and Diaconis [@MR1694204] we can consider two generalizations of Patience Sorting to words, which we will call the right Patience Sorting insertion and the left Patience Sorting insertion ( and insertion, respectively, for short). Considering the alphabet ${\mathcal{A}}$, these generalizations lead to two distinct monoids, the monoid, denoted by ${{\smash{\mathrm{rps}}}}$, and the monoid (also known in the literature as the Bell monoid [@Maxime07]), denoted by ${{\smash{\mathrm{lps}}}}$, which are, respectively, the monoids given by the quotient of ${\mathcal{A}}^*$ by the congruence which relates words having the same insertion under the and insertion.
In a monoid $M$, two elements $u$ and $v$, are said to be related by a cyclic shift, denoted $u{\sim_{{\mathrm{p}}}}v$, if there exists $x,y\in M$ such that $u=xy$ and $v=yx$. In their seminal work concerning the plactic monoid [@MR646486], Lascoux and Schützenberger proved that any two elements in the plactic monoid, ${{\smash{\mathrm{plac}}}}$, having the same evaluation (that is, elements that contain the same number of each generating symbol) can be obtained one from the other by applying a finite sequence of cyclic shift relations. The same characterization is known to hold for other plactic-like monoids, such as the hypoplactic monoid [@1709.03974], the Chinese monoid [@MR1847182], the sylvester monoid [@MR2081336; @MR2142078], and the taiga monoid [@1709.03974]. In Section \[subsection:conjugacy\] we show that an analogous result holds for the monoids (of finite rank) and for the monoid of rank $1$, ${{\smash{\mathrm{lps}}}}_1$. Note that all these monoids are multihomogeneous, that is, they are defined by presentations where the two side of each defining relation contains the same number of each generator. Thus, the evaluation of an element of the monoid corresponds to the evaluation of some (and hence any) word that represents it.
The previous results can be rewritten in another form by considering what we will call as cyclic shift graph of a monoid $M$, denoted ${{\smash{\mathrm{K}}}}(M)$, which is the undirected graph whose vertices are the elements of $M$ and whose edges connect elements that differ by a cyclic shift. So, if $M={{\smash{\mathrm{plac}}}}$ or $M={{\smash{\mathrm{rps}}}}$, or their finite analogues, then the results mentioned in the previous paragraph can be restated as saying that the connected components of ${{\smash{\mathrm{K}}}}(M)$ consist of the elements of $M$ which have the same evaluation. Thus, it follows that the connected components of ${{\smash{\mathrm{K}}}}(M)$ are finite. With the aid of the computational tool SageMath we studied the diameter of the connected components of the cyclic shift graph ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$. In SageMath we wrote a program based on the insertion algorithm, which given a word of ${\mathcal{A}}^*$, outputs the connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ containing the element of ${{\smash{\mathrm{rps}}}}$ that corresponds to the evaluation of the inserted word .
Aiming to parallel the result obtained by Choffrut and Merca[ş]{} [@Choffrut2013], and refined by Cain and Malheiro [@1709.03974], concerning the maximal diameter of connected components of the cyclic shift graph of the plactic monoid of finite rank, we used the tools available in the SageMath library to construct tables containing the number of vertices and the diameter of connected components from ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}}_n)$. The experimental results obtained from these calculations lead us to establish some conjectures regarding diameters of specific connected components. In Section \[subsection:cyclic\_shift\], we show that some of these conjectures are in fact true. In particular we prove that the maximum diameter of a connected components of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}}_n)$, for $n\geq 3$, lies between $n-1$ and $2n-4$. We also draw some conclusions for the diameter of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}}_n)$ for particular elements of ${{\smash{\mathrm{rps}}}}_n$.
The cyclic shift relation previously defined generalizes the usual conjugacy relation for groups. That is, when considering groups, the cyclic shift relation is just the usual conjugacy relation. Since for monoids this relation is, in general, not transitive, it is natural to consider the transitive closure of ${\sim_{{\mathrm{p}}}}$, which we will henceforth denote by ${{\sim}_{{\mathrm{p}}}^*}$. (Note that ${{\sim}_{{\mathrm{p}}}^*}$-classes correspond to connected components of the cyclic shift graph.) We consider two other notions of conjugacy (see [@Araujo201493; @araujo2015four] for other conjugacy notions, their properties, and relations among them). The relation ${\sim_{{\mathrm{l}}}}$ on $M$, proposed by Lallement in [@MR530552], which can be defined as follows: given $u,v\in M$ $$u{\sim_{{\mathrm{l}}}}v\ \Leftrightarrow\ \exists g\in M\ ug=gv.$$ (There is a dual notion ${\sim_{{\mathrm{r}}}}$ relating elements for which $gu=vg$, instead.) As this relation is reflexive and transitive but, in general, not symmetric, in [@MR742135], Otto considered the equivalence relation ${\sim_{{\mathrm{o}}}}$ given by the intersection of ${\sim_{{\mathrm{l}}}}$ and ${\sim_{{\mathrm{r}}}}$.
All the mentioned relations are equal in the group case, and in any monoid, ${\sim_{{\mathrm{p}}}}\ \subseteq\ {{\sim}_{{\mathrm{p}}}^*}\ \subseteq\ {\sim_{{\mathrm{o}}}}\
\subseteq\ {\sim_{{\mathrm{l}}}}$ (cf. [@Araujo201493]). Denoting by ${\sim_{{\mathrm{ev}}}}$ the binary relation that pairs elements with the same evaluation, it is easy to see that for multihomogeneous monoids ${\sim_{{\mathrm{l}}}}\
\subseteq\ {\sim_{{\mathrm{ev}}}}$ (cf. [@cm_conjugacy Lemma 3.2]), and thus for all the above multihomogeneous monoids (plactic, hypoplactic, chinese, sylvester, taiga and ) we have ${{{\sim}_{{\mathrm{p}}}^*}} = {{\sim_{{\mathrm{o}}}}} = {{\sim_{{\mathrm{l}}}}} = {{\sim_{{\mathrm{ev}}}}}$. This property, is not a general property of multihomogeneous monoids, as it is known that in the stalactic monoid connected components of the cyclic shift graph are properly contained in ${\sim_{{\mathrm{ev}}}}$ [@1709.03974 Proposition 7.2]. In this paper we show that a similar situation occurs for monoids of rank greater than 1, since we will prove that ${{\sim_{{\mathrm{l}}}}} \subsetneq {{\sim_{{\mathrm{ev}}}}}$ in these cases.
Preliminaries and notation
==========================
In this section we introduce the fundamental notions that we will use along the paper. For more details regarding these concepts check for instance [@1706.06884], [@MR1905123], and [@howie1995fundamentals].
Words and presentations {#alphabetswords}
-----------------------
In this paper, we denote by ${\mathcal{A}}$ the infinite totally ordered alphabet $\{1<2<\dots \}$, that is, the set of natural numbers with the usual order viewed as an alphabet. For any $n\in\mathbb{N}$, the resriction of ${\mathcal{A}}$ to the first $n$ natural numbers is denoted by ${\mathcal{A}}_n$.
In general, if $\Sigma$ is an alphabet, then $\Sigma^+$ denotes the *free semigroup* over $\Sigma$, that is, the set of non-empty words over $\Sigma$, and if $\varepsilon$ denotes the empty word, then the *free monoid* over $\Sigma$ is $\Sigma^*=
\Sigma^+\cup \{\varepsilon\}$.
Next, we define several concepts that are directly related with the notion of word. Let $w\in{\mathcal{A}}^*$. Then:
- a word $u\in {\mathcal{A}}^*$, is said to be a *factor* of $w$ if there exist words $v_1,v_2\in {\mathcal{A}}^*$, such that $w=v_1uv_2$;
- for any symbol $a$ in ${\mathcal{A}}$, the number of occurrences of $a$ in $w$, is denoted by ${\left|w\right|}_{a}$;
- the *content of* $w$, is the set ${{\smash{\mathrm{cont}}}}(w)=\left\{{a}\in
{\mathcal{A}}: {\left|w\right|}_{a}\geq 1\right\}$;
- the *evaluation of* $w$, denoted by ${{{\mathrm{ev}}}\parens[]{w}}$, is the sequence of non-negative integers whose $a$-th term is ${\left|w\right|}_{a}$, for any $a\in {\mathcal{A}}$;
- the word is said to be *standard* if each symbol from ${\mathcal{A}}_n$, for a given $n$, occurs exactly once.
A *monoid presentation* is a pair $(\Sigma,
\mathcal{R})$, where $\Sigma$ is an alphabet and $\mathcal{R}\subseteq
\Sigma^*\times \Sigma^*$. We say that a monoid $M$ is *defined by a presentation* $(\Sigma,\mathcal{R})$ if $M\simeq \Sigma^*/\mathcal{R}^\#$, where $\mathcal{R}^\#$ is the smallest congruence containing $\mathcal{R}$ (see [@howie1995fundamentals Proposition 1.5.9] for a combinatorial description of the smallest congruence containing a relation).
A presentation is *multihomogeneous* if, for every relation $(w,w') \in\mathcal{R}$, we have ${{{\mathrm{ev}}}\parens[]{w}}={{{\mathrm{ev}}}\parens[]{w'}}$, in other words, if $w$ and $w'$ contain the same number of each of its composing symbols. Then, a monoid is multihomogeneous if there exists a multihomogeneous presentation defining the monoid.
PS. *tableaux* and insertion
-----------------------------
In this subsection we recall the basic concepts regarding patience sorting *tableaux*, and the insertion on such *tableaux*.
A *composition diagram* is a finite collection of boxes arranged in bottom-justified columns, where no order on the length of the columns is imposed. Let $\Sigma$ be a totally ordered alphabet. Then, an (resp. ) *tableau over* $\Sigma$ is a composition diagram with entries from $\Sigma$, so that the sequence of entries of the boxes in each column is strictly (resp., weakly) decreasing from top to bottom, and the sequence of entries of the boxes in the bottom row is weakly (resp., strictly) increasing from left to right. So, if $$\label{exmp2}
\ytableausetup
{aligntableaux=center, boxsize=1.25em}
R=\begin{ytableau}
\none &\none & 4 \\
4 & 5 & 3 \\
1 & 1 & 2
\end{ytableau}\ \text{ and }\ S=\begin{ytableau}
\none & 5 \\
4 & 4 \\
1 & 3 \\
1 & 2
\end{ytableau},$$ then $R$ is an *tableau*, and $S$ is an *tableau* both over ${\mathcal{A}}_n$, for $n \geq 5$. Henceforth, we shall often refer to an tableau or to an tableau simply as a PS. tableau, not distinguishing the cases whenever they can be dealt in a similar way.
The left and right Patience Sorting monoids can be given as the quotient of the free monoid ${\mathcal{A}}^*$ over the congruence which relates words that yield the same PS. tableau under a certain algorithm [@1706.06884 § 3.6]. This algorithm is presented in the following paragraph and merges in one the Algorithms 3.1 and 3.2 of [@1706.06884]. (Observe that we will use the notation ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}()$, ${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}()$ instead of, respectively, $\mathfrak{R}_\ell()$, $\mathfrak{R}_r()$ used in [@1706.06884].)
\[alg:PSinsertion\]
*Input:* A word $w$ over a totally ordered alphabet $\Sigma$.
*Output:* An *tableau* ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(w)$ (resp., *tableau* ${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(w)$).
*Method:*
1. If $w=\varepsilon$, output an empty *tableau* $\emptyset$. Otherwise:
2. $w=w_1\cdots w_n$, with $w_1,\ldots,w_n\in \Sigma$. Setting $$\begin{aligned}
{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(w_1)=\ytableausetup
{boxsize=1.1em, aligntableaux=center}\begin{ytableau}
w_1 \end{ytableau} ={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(w_1),
\end{aligned}$$ then, for each remaining symbol $w_j$ with $1<j\leq n$, denoting by $r_1\leq \dots\leq r_k$ (resp., $r_1< \dots< r_k$) the symbols in the bottom row of the *tableau* ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(w_1\cdots w_{j-1})$ (resp., ${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(w_1\cdots w_{j-1})$), proceed as follows:
- if $r_k\leq w_j$ (resp., $r_k < w_j$), insert $w_j$ in a new column to the right of $r_k$ in ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(w_1\cdots w_{j-1})$ (resp., ${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(
w_1\cdots w_{j-1})$);
- otherwise, if $m=\min\left\{i\in\{1,\ldots,
k\}:w_j< r_i\right\}$, (resp. $m=\min\left\{i\in\{1,\ldots,
k\}:w_j\leq r_i\right\}$) construct a new empty box on top of the column of ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(w_1\cdots w_{j-1})$ (resp. ${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(w_1\cdots w_{j-1})$) containing $r_m$. Then bump all the symbols of the column containing $r_m$ to the box above and insert $w_j$ in the box which has been cleared and previously contained the symbol $r_m$.
Output the resulting *tableau*.
Observe that the insertion of a given word $w=w_1\cdots w_n$ under Algorithm \[alg:PSinsertion\] is obtained through the insertion of each of its symbols, from left to right in the previously obtained tableaux (starting with the empty tableaux $\emptyset$). For instance, if $R$ is the tableau from Example \[exmp2\], and $u=4511432
\in{\mathcal{A}}^*_5$, then ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u)=R$ (see Figure \[figure:extended\_insertion\]). The reader can check that ${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(u)=S$.
$$\begin{aligned}
&\emptyset \xleftarrow[4]{\ \ \ }\
\ytableausetup
{mathmode, boxsize=1.3em, aligntableaux=center}
\begin{ytableau}
4
\end{ytableau}\
\xleftarrow[5]{\ \ \ }\
\begin{ytableau}
4 & 5
\end{ytableau}\
\xleftarrow[1]{\ \ \ }\
\begin{ytableau}
4 & \none\\
1 & 5
\end{ytableau}\
\xleftarrow[1]{\ \ \ }\
\begin{ytableau}
4 & 5\\
1 & 1
\end{ytableau}\\
&\xleftarrow[4]{\ \ \ }\
\begin{ytableau}
4 & 5 & \none \\
1 & 1 & 4
\end{ytableau}\
\xleftarrow[3]{\ \ \ }\
\begin{ytableau}
4 & 5 & 4 \\
1 & 1 & 3
\end{ytableau}
\xleftarrow[2]{\ \ \ }\
\begin{ytableau}
\none & \none & 4\\
4 & 5 & 3\\
1 & 1 & 2
\end{ytableau}={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u ).\
\end{aligned}$$
The Patience Sorting monoids
----------------------------
For each $\mathrm{x}\in \{{{\mathrm{l}}},{{\mathrm{r}}}\}$, we define a binary relation ${\equiv_{{\smash{\mathrm{xps}}}}}$ in ${\mathcal{A}}^*$ in the following way: given $u,v\in{\mathcal{A}}^*$, $$u{\equiv_{{\smash{\mathrm{xps}}}}}v\quad \textrm{iff}\quad {{\smash{\mathrm{P}}}_{\smash{\mathrm{xps}}}}(u)={{\smash{\mathrm{P}}}_{\smash{\mathrm{xps}}}}(v).$$ This relation is a congruence [@1706.06884 Proposition 3.21], and the quotient of ${\mathcal{A}}^*$ by ${\equiv_{{\smash{\mathrm{lps}}}}}$ is the so-called monoid, denoted ${{\smash{\mathrm{lps}}}}$, and the quotient of ${\mathcal{A}}^*$ by ${\equiv_{{\smash{\mathrm{rps}}}}}$ is the monoid which is denoted by ${{\smash{\mathrm{rps}}}}$. The rank-$n$ analogues of these monoids, denoted by ${{\smash{\mathrm{lps}}}}_n$ and ${{\smash{\mathrm{rps}}}}_n$, are obtained by restricting the alphabet and the relation to the set ${\mathcal{A}}_n^*$. Note that each equivalence class of these monoids is represented by a unique tableau, and hence we will identify elements of the monoid with their tableaux representation.
Words yielding the same PS. tableau (and hence in the same ${\equiv_{{\smash{\mathrm{xps}}}}}$-class) have necessarily the same content, and even the same evaluation. Thus, we can refer to the content and evaluation of an element of the monoid, and similarly to the content and evaluation of a tableau. Also, we shall refer to an element of ${{\smash{\mathrm{xps}}}}_n$ (or to its tableau representative) as *standard* if one (and hence any) of its words in the ${\equiv_{{\smash{\mathrm{xps}}}}}$-class has one occurrence of each of the symbols from ${\mathcal{A}}_n$.
As shown in [@1706.06884 § 3.6 & § 3.7], the left and right Patience Sorting monoids are defined by the multihomogeneous presentations $({\mathcal{A}}^*,{\mathcal{R}}_{{\smash{\mathrm{lps}}}})$ and $({\mathcal{A}}^*,{\mathcal{R}}_{{\smash{\mathrm{rps}}}})$, where $$\begin{aligned}
{\mathcal{R}}_{{\smash{\mathrm{lps}}}}&=\{\,(yux,yxu): m\in \mathbb{N},\, x,y,u_1,\ldots , u_m\in
{\mathcal{A}}, \\
&\qquad u=u_m\cdots u_1,\, x<y\leq
u_1< \cdots < u_m\,\}\end{aligned}$$ and $$\begin{aligned}
{\mathcal{R}}_{{\smash{\mathrm{rps}}}}&=\{\,(yux,yxu): m\in \mathbb{N},\, x,y,u_1,\ldots , u_m\in
{\mathcal{A}}, \\
&\qquad u=u_m\cdots u_1,\, x\leq y<
u_1\leq \cdots \leq u_m\,\}.\end{aligned}$$ Hence, the left and right Patience Sorting monoids, and their finite rank analogues, are multihomogeneous monoids.
We have seen how to obtain a PS. tableau from a word in ${\mathcal{A}}^*$. Now, we explain how to pass from PS. tableaux to words representing such diagrams. Given $\textrm{x}\in \{{{\mathrm{l}}},{{\mathrm{r}}}\}$ and an $x$PS. tableau $P$, the *column reading of* $P$ is the word obtained from reading the entries of the $x$PS. tableau $P$, column by column, from the leftmost to the rightmost, starting on the top of each column and ending on its bottom. For example, the column reading of the tableau $R$ in Example \[exmp2\] is $41\, 51\, 432$, while the column reading of the tableau $S$ is $411\,
5432$.
Combinatorics of cyclic shifts {#Subsection4.1}
==============================
\[subsection:cyclic\_shift\]
As noted in the introduction, the *cyclic shift graph of* a monoid $M$, ${{\smash{\mathrm{K}}}}(M)$, is the undirected graph with vertex set $M$, whose edges connect vertices that differ by a single cyclic shift. Since, ${{\smash{\mathrm{rps}}}}$ is a multihomogeneous monoid, we have ${{{\sim}_{{\mathrm{p}}}^*}}\subseteq
{{\sim_{{\mathrm{ev}}}}}$, and thus each connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ cannot contain elements with different evaluations and therefore they have finitely many vertices.
Our goal in this subsection is to study the diameter of the connected components from ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}}_n)$, which as we will show are bounded by a value that depends on the rank $n$. Note that in [@1709.03974 Example 3.1], the authors provide a finitely presented multihomogeneous monoid for which the connected components of the cyclic shift graph have unbounded diameter. Therefore, these are not particular cases of a more general result that holds for all multihomogeneous monoids.
The experimental results within this subsection were obtained with the aid of SageMath [@cocalc]. This computational tool allowed us to write a program for which: given an element of ${{\smash{\mathrm{rps}}}}_n$, provides the connected component from the cyclic shift graph of ${{\smash{\mathrm{rps}}}}_n$ containing that element.
The program starts by creating a vertex for each word from ${\mathcal{A}}^*_n$ that has the same evaluation as the given element from ${{\smash{\mathrm{rps}}}}_n$. Afterwards, it adds edges between the words that are cyclic shift related. Finally, by merging the vertices whose $x$PS. insertion is the same into a single vertex, it constructs the connected component of the cyclic shift graph of ${{\smash{\mathrm{rps}}}}_n$, ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}}_n)$, containing the given element from ${{\smash{\mathrm{rps}}}}_n$.
For instance in Figure \[fig:connected\_component\] we show the connected component of the cyclic shift graph of ${{\smash{\mathrm{rps}}}}_4$ containing the element ${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(1234)$ that can be seen to have diameter $4$.
(0.5,-3.5) – (9.5,6); (1)
[1]{} & [2]{} & [3]{} & [4]{}
; (2) \[right of=1\]
[2]{} & &\
[1]{} & [3]{} & [4]{}
; (3) \[above of=2\]
[3]{} & [4]{}\
[1]{} & [2]{}
; (4) \[below of=2\]
[4]{} & &\
[1]{} & [2]{} & [3]{}
; (5) \[right of=2\]
[3]{} &\
[2]{} &\
[1]{} & [4]{}
; (6) \[above of=5\]
[4]{} &\
[2]{} &\
[1]{} & [3]{}
; (7) \[above of=6\]
& [4]{} &\
[1]{} & [2]{} & [3]{}
; (8) \[below of=5\]
& [3]{} &\
[1]{} & [2]{} & [4]{}
; (9) \[right of=5\]
[2]{} & [4]{}\
[1]{} & [3]{}
; (10) \[above of=9\]
[4]{}\
[3]{}\
[2]{}\
[1]{}
; (11) \[above of=10\]
& [4]{}\
& [3]{}\
[1]{} & [2]{}
; (12) \[below of=9\]
[4]{} & [3]{}\
[1]{} & [2]{}
; (13) \[right of=9\]
& & [4]{}\
[1]{} & [2]{} & [3]{}
; (14) \[above of=13\]
[4]{} &\
[3]{} &\
[1]{} & [2]{}
; (15) \[below of=13\]
[3]{} & &\
[1]{} & [2]{} & [4]{}
;
\(1) edge node (2) (1) edge node (3) (1) edge node (4) (2) edge node (3) (2) edge node (4) (2) edge node (5) (2) edge node (6) (2) edge node (7) (2) edge node (8) (3) edge \[bend right=37pt\] node (4) (3) edge node (6) (3) edge node (7) (5) edge node (6) (5) edge node (8) (5) edge node (9) (5) edge node (10) (5) edge node (11) (5) edge node (12) (6) edge node (7) (6) edge \[bend right=30pt\] node (8) (8) edge node (9) (8) edge node (12) (9) edge node (10) (9) edge \[bend right=27pt\] node (11) (9) edge node (12) (9) edge node (13) (9) edge node (14) (9) edge node (15) (10) edge node (11) (13) edge node (14) (13) edge node (15) (14) edge \[bend left=38pt\] node (15) ;
The results of computer experimentation on the diameter of connected compontents is shown in Tables \[tab:std\_cyclic\_shift\_graph\] and \[tab:cyclic\_shift\_graph\]. In Table \[tab:std\_cyclic\_shift\_graph\] we present the diameter and number of vertices in the connected component of the cyclic shift graph of standard elements of lengths $1$ up to $9$, whereas in Table \[tab:cyclic\_shift\_graph\] the same information is presented but for some (non-standard) words of given fixed evaluations.
--- ------- ---- -------- --
1 1 0 $n-1$
2 2 1 $n-1$
3 5 2 $2n-4$
4 15 4 $2n-4$
5 52 6 $2n-4$
6 203 8 $2n-4$
7 877 10 $2n-4$
8 4140 12 $2n-4$
9 21147 14 $2n-4$
--- ------- ---- -------- --
: Examples of diameter and number of vertices in the connected component of the cyclic shift graph ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ for given evaluations of standard elements.
\[tab:std\_cyclic\_shift\_graph\]
The results in Table \[tab:std\_cyclic\_shift\_graph\] suggest the following:
\[conj:diameter\_std\] The diameter of a connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ containing a standard element of length $n\geq 3$ is $2n-4$.
Note that the connected components of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ and ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{lps}}}})$ coincide when restricted to standard elements.
The data gathered in both Table \[tab:std\_cyclic\_shift\_graph\] and Table \[tab:cyclic\_shift\_graph\] leads us to propose the following:
\[conj:diameter\_nonstd\] The diameter of a connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ containing an element with $n\geq 3$ symbols, with possible multiple appearences of each symbol, lies between $n-1$ and $2n-4$.
Evaluation
------------------- ------- ---- ------------
(5) 1 0 $n-1$
(5,3) 4 1 $n-1$
(4,1,4) 20 2 $n-1=2n-4$
(3,3,1,2) 75 3 $n-1=2n-5$
(1,2,4,2) 287 4 $n=2n-4$
(1,3,2,1,2) 656 5 $n=2n-5$
(2,1,1,2,3) 554 4 $n-1=2n-6$
(1,2,1,2,2) 711 6 $n+1=2n-4$
(1,1,1,3,1,2) 2409 7 $n+1=2n-5$
(1,1,2,2,1,2) 2840 6 $n=2n-6$
(1,2,1,1,2,2) 2373 8 $n+2=2n-4$
(1,1,1,1,2,1,2) 6499 9 $n+2=2n-5$
(1,1,1,2,1,1,2) 6078 8 $n+1=2n-6$
(1,1,1,1,1,2,2) 6768 10 $n+3=2n-4$
(1,1,1,1,1,2,1,1) 11695 11 $n+3=2n-5$
(1,1,1,1,2,1,1,1) 11224 10 $n+2=2n-6$
(1,1,1,1,1,1,2,1) 12002 12 $n+4=2n-4$
: Examples of diameter and number of vertices in the connected component of the cyclic shift graph ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ for given evaluations of non-standard elements.
\[tab:cyclic\_shift\_graph\]
One of the first results that was possible to obtain from the data was
\[prop:diameter12\] All elements of ${{\smash{\mathrm{rps}}}}$ containing two symbols, with the same evaluation, form a connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$. Furthermore, the component has diameter $1$.
As already noticed each connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ cannot contain elements with different evaluations. Let $u$ and $v$ be two elements of ${{\smash{\mathrm{rps}}}}$ with the same evaluation such that ${\left|{{\smash{\mathrm{cont}}}}(w)\right|}= 2$. Suppose without loss of generality that ${{\smash{\mathrm{cont}}}}(w)=\{1,2\}$. Then, these elements are of the form ${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^i1^j2^k)$, for some $i, k\in \mathbb{N}_0$ and $i+k,j\in\mathbb{N}$. So, $u={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^i1^j2^k)$ and $v={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^l1^n2^m)$ with $j=n$ and $i+k=l+m$. Therefore, $v={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^l1^j2^m)$ and we consider the following cases:
If $i\geq l$, then $k+i-l=m$. Setting $x={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{i-l})$ and $y={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^l1^j2^k)$, we have $$\begin{aligned}
&u={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{i-l}2^l1^j2^k)={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{i-l}) {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^l1^j2^k)=x y\,
\text{ and}\\
&v={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^l1^j2^k2^{i-l})={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^l1^j2^k) {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{i-l})=y x.
\end{aligned}$$
Otherwise, if $i<l$, then $m+l-i=k$. Setting $x={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{i}1^{j}2^{m})$ and $y={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{l-i})$, we get $$\begin{aligned}
&u={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{i}1^{j}2^{m}2^{l-i})={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{i}1^{j}2^{m})
{{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{l-i})=x y\, \text{ and}\\
&v={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{l-i}2^{i}1^{j}2^{m})={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{l-i})
{{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{i}1^{j}2^{m})=y x.
\end{aligned}$$ In both cases, $u{\sim_{{\mathrm{p}}}}v$. Therefore, the diameter of the connected component from ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ containing such elements is 1. The result follows.
In the following lemma we provide an upper bound for the diameter of the connected components from ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ of elements whose content is greater or equal to $3$, thus answering the upper bound part of Conjecture \[conj:diameter\_nonstd\].
By observing several connected components obtained with the program constructed with SageMath, we concluded that for any element $w\in {{\smash{\mathrm{rps}}}}$, with ${{\smash{\mathrm{cont}}}}(w)=\{1,\ldots,
n\}$ and $n\geq 3$, the element $${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left((n-1)^{{{\left|w\right|}}_{n-1}}
(n-2)^{{{\left|w\right|}}_{n-2}}\cdots 3^{{\left|w\right|}_3}
2^{{\left|w\right|}_2}1^{{\left|w\right|}_1}\ n^{{{\left|w\right|}}_{n}}\right)$$ plays a key role in the connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ which contains $w$. For instance, in Figure \[fig:connected\_component\], we see that the element $${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(3214)={\color{Grey}\begin{ytableau}
{\color{black}3} & \none\\
{\color{black}2} & \none\\
{\color{black}1} & {\color{black}4}
\end{ytableau}}$$ is in the center of the connected component. Using this insight we were able to prove the following result:
\[prop:diameter\_upper\_bound\] All elements of ${{\smash{\mathrm{rps}}}}$ containing $n\geq 3$ symbols, with the same evaluation, form a connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$. Furthermore, the component has diameter at most $2n-4$.
Let $w$ be an element of ${{\smash{\mathrm{rps}}}}$ with ${\left|{{\smash{\mathrm{cont}}}}(w)\right|}=n\geq 3$. Suppose without loss of generality that ${{\smash{\mathrm{cont}}}}(w)=\{1,\ldots, n\}$. Since each connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}}_n)$ cannot contain elements with different evaluations, to prove this result, it suffices to check that from $w$, by applying at most $n-2$ cyclic shift relations we can always obtain the element $$w'={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left((n-1)^{{\left|w\right|}_{n-1}} (n-2)^{
{\left|w\right|}_{n-2}}\cdots 2^{{\left|w\right|}_2}1^{{\left|w\right|}_1}
n^{{\left|w\right|}_{n}}\right)$$ of ${{\smash{\mathrm{rps}}}}$.
We will construct a path in ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}}_n)$ from $w$ to $w'$ of length at most $n-2$. We aim to find a sequence $w_0,w_1,\ldots,w_{n-2}$ of elements of ${{\smash{\mathrm{rps}}}}_n$ such that $w=w_0$, $w'=w_{n-2}$, and $w_i{\sim_{{\mathrm{p}}}}w_{i+1}$, for $i=0,\ldots, n-3$. The construction is inductive. First note that all the symbols $1$ occur in the bottom of the first column of $w$. If $w$ has only one column, then $w$ has column reading $n^{{\left|w\right|}_{n}}(n-1)^{{\left|w\right|}_{n-1}}
(n-2)^{{\left|w\right|}_{n-2}}\cdots 2^{{\left|w\right|}_2}1^{{\left|w\right|}_1}$ and applying one cyclic shift we get the intended result. Suppose $w$ has at least two columns. Let $k$ (necessarily $k\geq 2$) be the bottom symbol of the second column of $w$. Observe that any symbol $j$ less than $k$ must lie in the first column of $w$. Set $w=w_0=\dots =w_{k-1}$. We calculate the element $w_k$ from $w$ in the following way. Consider the column reading $ukv$, of $w$, for $u,v\in {\mathcal{A}}_n^*$, where $u$ is the prefix just up to before the first occurrence of a symbol $k$ occurring in the second column. Fix $w_k={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(kv) {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(u)$. Note that $w={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(u) {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(kv)$ and so $w{\sim_{{\mathrm{p}}}}w_k$.
Then, the first column of $w_k$ has column reading $k^{{\left|w\right|}_k}\dots 2^{{\left|w\right|}_2}1^{{\left|w\right|}_1}$, because all symbols in $v$ are greater or equal to $k$, and symbols in $u$ that are strictly less than $k$ appear in decreasing order. For $i\in\{k, \ldots, n-2\}$, let $w_{i}={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(u\right) {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left((i+1)v\right)$ where $u$ is the prefix of the column reading of $w_{i}$ up to just before the first occurrence of a symbol $i+1$ (in the second column) and let $w_{i+1}={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left((i+1)v\right) {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(u\right)$. Using this process we ensure that the first column of $w_{i+1}$ is precisely $${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left((i+1)^{{\left|w\right|}_{i+1}} i^{{\left|w\right|}_i}
\cdots 2^{{\left|w\right|}_2} 1^{{\left|w\right|}_1}\right).$$ The result follows by induction.
Regarding the lower bound of Conjecture \[conj:diameter\_nonstd\], we are only able to establish it for standard elements of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$. To prove such a result, we will use the notion of cocharge sequence for standard words over ${\mathcal{A}}$ and follow an approach similar to the one used in the case of the plactic monoid in [@1709.03974]. Note that it will be sufficient to prove the result for standard words over the alphabet ${\mathcal{A}}_n$.
For any standard word $w$ over ${\mathcal{A}}_n$, the *cocharge labels* from the symbols of $w$ are calculated as follows:
- draw a circle, place a point $*$ somewhere on its circumference, and, starting from $*$, write $w$ anticlockwise around the circle;
- let the cocharge label of the symbol $1$ be $0$;
- iteratively, suppose the cocharge label of the symbol $a$ from $w$ is $k$, then proceed clockwise from the symbol $a$ to the symbol $a+1$ and:
- if the symbol $a+1$ of $w$ is reached without passing the point $*$, then the cocharge label of $a+1$ is $k$;
- otherwise, if the symbol $a+1$ is reached after passing the point $*$, then the cocharge label of $a+1$ is $k+1$.
The *cocharge sequence* of a standard word $w$, ${{\smash{\mathrm{cochseq}}}}(w)$, is the sequence of the cocharge labels from the symbols of $w$, whose $a$-th term is the cocharge label from the symbol $a$ of $w$. So, it follows from the definition that if $w$ is a standard word over ${\mathcal{A}}_n$, then ${{\smash{\mathrm{cochseq}}}}(w)$ is a sequence of length $n$.
For example, the labelling of the standard word $w=4572631$ over ${\mathcal{A}}_7$, proceeds in the following way
(0,0) circle\[radius=5mm\];
(4:-4mm) arc\[radius=4mm,start angle=-170,end angle=-10\];
(-90:2mm);
(-18:13.5mm) arc\[radius=13mm,start angle=-10,end angle=-170\];
(-165:18mm) arc\[radius=18mm,start angle=-170,end angle=-10\];
i/in [0/\*,9/4,8/5,7/7,6/2,5/6,4/3,3/1]{} [ at ($ (90-\i*30:7.8mm) $) [$\ilabel$]{}; ]{}; i/in [9/2,8/2,5/2,7/3,3/0,6/1,4/1]{} [ at ($ (90-\i*30:11.2mm) $) [$\ilabel$]{}; ]{};
and thus ${{\smash{\mathrm{cochseq}}}}(w)=(0,1,1,2,2,2,3)$.
From the definition it also follows that the cocharge sequence is a weakly increasing sequence which starts at $0$ and such that each of the remaining terms is either equal to the previous term or greater by $1$.
\[standardinvariant\] For standard words $u,v$ over ${\mathcal{A}}_n$, if $u{\equiv_{{\smash{\mathrm{rps}}}}}v$, then ${{\smash{\mathrm{cochseq}}}}(u)={{\smash{\mathrm{cochseq}}}}(v)$.
It is enough to show that any two standard words over ${\mathcal{A}}_n$ such that one is obtained from the other by applying a relation from ${\mathcal{R}}_{{{\smash{\mathrm{rps}}}}_n}$ have the same cocharge sequence.
So, there is a factor $yu_i\cdots u_1x$ of one of the standard words, with $x,y,u_1,\ldots,u_i\in{\mathcal{A}}_n$ and $x<y< u_1<\cdots <u_i$, that is changed to the factor $yxu_i\cdots u_1$ of the other. Given any symbol $a\in {\mathcal{A}}_n\setminus\{1\}$, when applying such relation, the relative position between the symbols $a$ and $a-1$ is not changed. That is, if $a-1$ occurs to the right (resp. left) of $a$ in one of the standard words, then $a-1$ also occurs to the right (resp. left) of $a$ in the other. Thus, equal symbols of these standard words have the same cocharge label and therefore the cocharge sequence of these words is the same.
Given a standard element $u$ of ${{\smash{\mathrm{rps}}}}_n$ on $n$ generators, let ${{\smash{\mathrm{cochseq}}}}(u)$ be ${{\smash{\mathrm{cochseq}}}}(w)$ for any word $w\in {\mathcal{A}}^*_n$ such that ${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(w)=u$. Using the previous lemma we conclude that ${{\smash{\mathrm{cochseq}}}}(u)$ is well-defined.
\[prop:diameter\_lower\_bound\] The diameter of a connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}}_n)$, with $n\geq 2$, containing a standard element is at least $n-1$.
The case $n=2$ follows from Lemma \[prop:diameter12\]. Suppose $n\geq 3$.
From [@1709.03974 Lemma 2.2], we deduce that any two standard elements of ${{\smash{\mathrm{rps}}}}$ that differ by a single cyclic shift, have cocharge sequences whose corresponding terms differ by at most $1$.
The standard elements $u={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(1\cdot 2\cdots
n\right)$ and $v={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(n(n-1)\cdots 1\right)$ of ${{\smash{\mathrm{rps}}}}_n$ are in the same connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ by Lemma \[prop:diameter\_upper\_bound\]. Now, notice that ${{\smash{\mathrm{cochseq}}}}(u)=(0,0,\ldots,0)$ and that ${{\smash{\mathrm{cochseq}}}}(v)=(0,1,\ldots,n-1)$. Since the last term of both sequences differs by $n-1$, the standard elements $u$ and $v$ are at distance of at least $n-1$.
For instance, in Figure \[fig:connected\_component\], the distance between the elements $${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(1234)={\color{Grey}\begin{ytableau}
{\color{black}1} & {\color{black}2} & {\color{black}3} & {\color{black}4}
\end{ytableau}}\ \text{ and }\ {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(4321)={\color{Grey}\begin{ytableau}
{\color{black}4} \\
{\color{black}3} \\
{\color{black}2} \\
{\color{black}1}
\end{ytableau}}$$ in that connected component is precisely $3$, which is in accordance with the previous result.
Since for standard words the cyclic shif graphs ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{lps}}}})$ and ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ coincide, the previous result also give us a lower bound for connected components of standard words of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{lps}}}})$.
Combining Lemmata \[prop:diameter12\], \[prop:diameter\_upper\_bound\] and \[prop:diameter\_lower\_bound\] we get
1. Connected components of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ coincide with ${\sim_{{\mathrm{ev}}}}$-classes of ${{\smash{\mathrm{rps}}}}$.
2. The maximum diameter of a connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}}_n)$ is $n-1$, for $n=1,2$, and lies between $n-1$ and $2n-4$, for $n\geq 3$.
Other observations from computer experimental results lead us to conclude that the number of vertices in a given connected component is equal to the number of vertices in the connected component that has one more symbol $1$. This makes sense since the elements of the new connected component will be the elements of the former with an additional symbol 1 in the bottom of the first column.
Also, it seems that in a standard component, the addition of a new symbol $1$ leads to a connected component whose diameter can possibly decrease by 2 when compared with the original. In fact, we were able to establish the following result:
Let $w$ be an element of ${{\smash{\mathrm{rps}}}}$, with $n\geq 4$ symbols, such that the minimum symbol of $w$ has at least two occurences, and the second smallest symbol only occurs once. Then the diameter of the connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ containing $w$ is at most $2n-6$.
Without lost of generality, suppose that ${{\smash{\mathrm{cont}}}}(w)=\{1,\dots ,n\}$, with $n\geq 4$. The proof strategy is similar to the proof of Lemma \[prop:diameter\_upper\_bound\]. We aim to construct a path in ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ from $w$ to $$w'={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(1^{{\left|w\right|}_1}
(n-1)^{{\left|w\right|}_{n-1}}(n-2)^{{\left|w\right|}_{n-2}}\cdots 3^{{\left|w\right|}_3}2
n^{{\left|w\right|}_n}\right)$$ by applying at most $n-3$ cyclic shifts relations.
For an element $w$ of ${{\smash{\mathrm{rps}}}}$, under the given assumptions, we will distinguish particular readings of its tableau representation. For simplicity, we call these readings *delayed column readings*. Note that the symbol $1$ occurs more than once, and that all symbols $1$ appear on the bottom of the first column of such tableaux. If we proceed as in the column reading, but we read the symbol on the bottom of the first column (necessarily a symbol $1$) latter on, we obtain a delayed column reading. Following Algorithm \[alg:PSinsertion\], it is clear that all these words corresponding to delayed column readings also insert to the same element. For example, the element $S$ of has column reading $411\, 5432$ and has delayed column readings, $4151432$, $4154132$, $4154312$, and $4154321$.
If the tableau representation of $w$ has only one column, then it has the form $${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(
n^{{\left|w\right|}_n}(n-1)^{{\left|w\right|}_{n-1}}(n-2)^{{\left|w\right|}_{n-2}}\cdots
3^{{\left|w\right|}_3}2
1^{{\left|w\right|}_1}\right)$$ which is cyclic shift related to $${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(
(n-1)^{{\left|w\right|}_{n-1}}(n-2)^{{\left|w\right|}_{n-2}}\cdots
3^{{\left|w\right|}_3}2
1^{{\left|w\right|}_1}n^{{\left|w\right|}_n}\right)$$ which in turn has delayed column reading $$(n-1)^{{\left|w\right|}_{n-1}}(n-2)^{{\left|w\right|}_{n-2}}\cdots
3^{{\left|w\right|}_3}2
1^{{\left|w\right|}_1-1}n^{{\left|w\right|}_n}1.$$ By applying a cyclic shift we get the intended form since $$w'= {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(
1(n-1)^{{\left|w\right|}_{n-1}}(n-2)^{{\left|w\right|}_{n-2}}\cdots
3^{{\left|w\right|}_3}2
1^{{\left|w\right|}_1-1}n^{{\left|w\right|}_n}\right).$$
Otherwise, suppose first that the bottom symbol of the second column is $2$. Note that the symbol $3$ can appear in the first three columns of $w$, and if it appears in the third column, then its bottom symbol is a $3$. Consider the delayed column reading of $w$, $u13v$, where $u$ is the prefix up to before the first occurence of a symbol $3$ in the rightmost column where a symbol $3$ appears (necessarily on the first three columns). So, either $u$ or $v$ has the unique symbol $2$, and if $u$ or $v$ has the symbol $2$ then all symbols $3$ appear to its left. Let $w_3={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(13v) {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(u)$, and so $w{\sim_{{\mathrm{p}}}}w_3$, since $w={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(u) {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(13v)$. The first column of $w_3$ has column reading $1^{{\left|w\right|}_1}$ and the second column $3^{{\left|w\right|}_3}2$.
Now suppose the bottom symbol of the second column is $k>2$. Consider the delayed column reading of $w$, $u1kv$, where $u$ is the prefix up to before the first occurence of a symbol $k$ in the second column. Note that all symbols in $v$ are greater or equal to $k$, and symbols in $u$ that are strictly less than $k$ appear in decreasing order (from left to right). Let $w_k={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(1kv) {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(u)$, and so $w{\sim_{{\mathrm{p}}}}w_k$. The first column of $w_k$ has column reading $1^{{\left|w\right|}_1}$ and the second column $k^{{\left|w\right|}_{k}}\cdots
3^{{\left|w\right|}_3}2 $.
We will construct a path in ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}}_n)$ from $w_k$ to $w'$ of length at most $n-4$, by considering a sequence $w_k,\ldots,w_{n-1}$ of elements of ${{\smash{\mathrm{rps}}}}_n$, with $k\geq 3$, such that $w'=w_{n-1}$, and $w_i{\sim_{{\mathrm{p}}}}w_{i+1}$, for $i=k,\ldots, n-1$. For $i\in\{k, \ldots, n-2\}$, let $w_{i}={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(u\right) {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(1(i+1)v\right)$ where $u$ is the prefix of the delayed column reading $u1(i+1)v $ of $w_{i}$ up to just before the first occurrence of a symbol $i+1$ (on the third column) and let $w_{i+1}={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(1(i+1)v\right) {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(u\right)$. Note that all symbols in $v$ are greater or equal to $i+1$, and all symbols in $u$ that are strictly less than $i+1$ appear in decreasing order (from left to right). Thus the two first columns of $w_{i+1}$ have column readings $1^{{\left|w\right|}_1}$ and $(i+1)^{{\left|w\right|}_{i+1}} i^{{\left|w\right|}_i}\dots
3^{{\left|w\right|}_3}2$, respectively. The result follows by induction.
Conjugacy in the and monoids {#subsection:conjugacy}
==============================
Restating the results of Section \[subsection:cyclic\_shift\] in terms of the conjugacy relation ${\sim_{{\mathrm{p}}}}$ we have shown that in ${{\smash{\mathrm{rps}}}}_n$ we have ${{\sim_{{\mathrm{p}}}}}={{\sim_{{\mathrm{ev}}}}}$, for $n\in\{1,2\}$; and that ${{\sim_{{\mathrm{p}}}}}\subsetneq {{{\sim}_{{\mathrm{p}}}^*}}={{\sim_{{\mathrm{ev}}}}}$, for $n>2$. Thus, ${{{\sim}_{{\mathrm{p}}}^*}}={{\sim_{{\mathrm{ev}}}}}$ in the (infinite rank) right Patience Sorting monoid. In all cases, we deduce that any of the conjugacy relations ${{{\sim}_{{\mathrm{p}}}^*}}$, ${{\sim_{{\mathrm{o}}}}}$, and ${{\sim_{{\mathrm{l}}}}}$ coincides with ${{\sim_{{\mathrm{ev}}}}}$.
The case proves to be distinct from the case. In ${{\smash{\mathrm{lps}}}}_1$, it is immediate that ${{\sim_{{\mathrm{p}}}}}={{\sim_{{\mathrm{ev}}}}}$, but for $n\geq 2$, we will see that ${{\sim_{{\mathrm{p}}}}}\subsetneq
{{{\sim}_{{\mathrm{p}}}^*}}$ and ${{\sim_{{\mathrm{l}}}}}\subsetneq {{\sim_{{\mathrm{ev}}}}}$, in ${{\smash{\mathrm{lps}}}}_n$, and thus in ${{\smash{\mathrm{lps}}}}$. Whether the inclusion ${{{\sim}_{{\mathrm{p}}}^*}}\subseteq {{\sim_{{\mathrm{l}}}}}$ is strict or, in fact an equality, is left as an open question.
For any $n\geq 2$, in ${{\smash{\mathrm{lps}}}}_n$ we have ${{\sim_{{\mathrm{p}}}}}\subsetneq {{{\sim}_{{\mathrm{p}}}^*}}$.
From Lemma \[prop:diameter\_lower\_bound\], we deduce that ${{\sim_{{\mathrm{p}}}}}
\subsetneq {{{\sim}_{{\mathrm{p}}}^*}}$, for ${{\smash{\mathrm{lps}}}}_n$ with $n\geq 3$.
Regarding the ${{\smash{\mathrm{lps}}}}_2$ case, consider the elements ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21121)$ and ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21112)$ of ${{\smash{\mathrm{lps}}}}_2$. We have that $$\begin{aligned}
{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21121)&={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(211) {{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21){\sim_{{\mathrm{p}}}}{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21) {{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(211)={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21211)\\
&={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(22111)={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(2) {{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(2111){\sim_{{\mathrm{p}}}}{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(2111) {{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(2)\\
&={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21112),
\end{aligned}$$ and so ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21121)\ {{\sim}_{{\mathrm{p}}}^*}\ {{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21112)$ in ${{\smash{\mathrm{lps}}}}_2$. It is easy to check that ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21121){\nsim_{{\mathrm{p}}}}{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21112)$ in ${{\smash{\mathrm{lps}}}}_2$. Indeed, notice that the unique words $u$ and $v$ of ${\mathcal{A}}^*_2$ such that ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u)={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21121)$ and ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v)={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21112)$ are precisely, $21121$ and $21112$, respectively. Moreover, if ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21121)={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(st)$, for words $s,t\in{\mathcal{A}}_2^*$, then ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(ts)\neq {{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21112)$.
Resuming, we have a pair of elements of ${{\smash{\mathrm{lps}}}}_2$ which belong to ${{\sim}_{{\mathrm{p}}}^*}$ but not to ${\sim_{{\mathrm{p}}}}$.
In order to prove that ${{\sim_{{\mathrm{l}}}}}\subsetneq {{\sim_{{\mathrm{ev}}}}}$, in ${{\smash{\mathrm{lps}}}}_n$, we first prove two auxiliary results.
For any $k,n\in \mathbb{N}$ and $u,v\in {{\smash{\mathrm{lps}}}}_k$, if $n\geq k$, then: $$u{\sim_{{\mathrm{l}}}}v \text{ in } {{\smash{\mathrm{lps}}}}_n\ \Leftrightarrow\ u{\sim_{{\mathrm{l}}}}v \text{ in }
{{\smash{\mathrm{lps}}}}_k.$$
Let $u,v\in {{\smash{\mathrm{lps}}}}_k$ and $n\geq k$. Suppose that $u{\sim_{{\mathrm{l}}}}v$ in ${{\smash{\mathrm{lps}}}}_n$. Note that $u$ and $v$ have the same evaluation. There exists $g\in {{\smash{\mathrm{lps}}}}_n$ such that $u g= g v$. If $g$ is the identity then the result holds trivially. Assume that the tableau representation of $g$ has $j$ columns.
Since $u g= g v$, then $u^2g=u u g
= u g v= g v v=g v^2$. Using the same reasoning, it follows that for any $i\geq 1$, $u^i g=g v^i$. Note that if $a$ is the minimum symbol occuring in $u$, then $u^i$ has bottom row beginning (from left to right) with (at least) $i$ symbols $a$.
Suppose $g$ has a symbol greater than $k$. As ${{\smash{\mathrm{cont}}}}(u)\subseteq {\mathcal{A}}_k$, the symbols from ${g}$ that are greater or equal than $k$ have to be inserted in the tableau representation of $u^j$ to the right of the first $j$ columns. Now, in the tableau representation of $gv^i$, the symbols from $g$ are inserted into the first $j$ columns. This is a contradiction, since $u^ig=gv^i$. So all symbols from $g$ are less or equal than $k$, that is, $g\in{{\smash{\mathrm{lps}}}}_k$.
The converse direction of the lemma is obvious from the definition of ${\sim_{{\mathrm{l}}}}$.
Let $C_2=\left\{{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(1),{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21)\right\}$. As proved in [@1706.06884 Proposition 4.1], the submonoid of ${{\smash{\mathrm{lps}}}}_2$ generated by $C_2$, denoted $\langle C_2\rangle$, is free. Observe that the elements of $\langle C_2\rangle$ are precisely the elements of ${{\smash{\mathrm{lps}}}}_2$ whose tableau representation has bottom row filled with symbols $1$.
\[lem:left\_conjugacy\_Bell2\] For any $u,v\in \langle C_2\rangle$ and $n\geq 2$, $$u{\sim_{{\mathrm{l}}}}v \text{ in } {{\smash{\mathrm{lps}}}}_n\ \Leftrightarrow\ u{\sim_{{\mathrm{l}}}}v
\text{ in } \langle C_2\rangle.$$
Let $u,v\in \langle C_2\rangle$, $n\geq 2$ and suppose that $u{\sim_{{\mathrm{l}}}}v$ in ${{\smash{\mathrm{lps}}}}_n$. Suppose that $u\in\langle {{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21)\rangle$. Since $u{\sim_{{\mathrm{ev}}}}v$, then also $v\in
\langle{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21)\rangle$, and thus $u=v$. Therefore the result holds.
Suppose now that $u\notin \langle {{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21)\rangle$. Then at least one of the columns of the tableau representation of $u$ has height one and is filled with the symbol $1$. Note that the tableau representation of $v$ has the same number of columns of heigth two, and the same number of columns of heigth one (and each such box is filled with the symbol $1$).
Let $g\in {{\smash{\mathrm{lps}}}}_n$ be such that $u g= g v$. By the previous lemma we can assume $g\in{{\smash{\mathrm{lps}}}}_2$. If $g$ is the identity then the result holds trivially. Suppose that the tableau representation of $g$ has at least one column with height one filled with the symbol $2$. Attending to Algorithm \[alg:PSinsertion\] and since the bottom row of $u$ is filled with the symbol $1$, $ug$ is represented by a tableau that is composed by the columns of $u$ followed by the columns of $g$.
Now, the tableau representation of $gv$ has at least one less column. Indeed, consider the column reading of the tableau representation of $v$, which is a word from $\{1,21\}^*$, where at least one single symbol $1$ is used, that is, it does not belong to $\{21\}^*$. Applying Algorithm \[alg:PSinsertion\] we will first insert symbols from $g$, and get the tableau representation of $g$, followed by the insertion of the column reading from $v$. Now, each time a word $21$ is inserted we obtain a new column, but the first time a single symbol $1$ is inserted it will take place in the leftmost column of height one filled with the symbol $2$, becoming a column of heigth two and column reading $21$. Thus, the tableau representation of $gv$ cannot have the same number of columns as the tableau representation of $ug$. This is a contradiction. Therefore, the tableau representation of $g$ has bottom row filled with the symbol $1$, and hence $g\in\langle C_2\rangle$.
Since the converse direction is immediate, the result follows.
\[prop511\] For the monoid of rank $n$, with $n\geq 2$, we have $${\sim_{{\mathrm{l}}}}\ \subsetneq\
{\sim_{{\mathrm{ev}}}}.$$
In the free monoid of rank $2$ the relation ${{\sim}_{{\mathrm{p}}}^*}$ is equal to ${\sim_{{\mathrm{l}}}}$ [@lentin1967combinatorial Theorem 3], and it is properly contained in ${\sim_{{\mathrm{ev}}}}$ (For example, in ${\mathcal{A}}_2^*$, there are words with the same evaluation $2121$ and $2112$, for which $2121{\nsim_{{\mathrm{l}}}}2112$).
Consider the embedding $\eta:{\mathcal{A}}_2^*\to{{\smash{\mathrm{lps}}}}_n$ given by $1\mapsto
{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(1)$ and $2\mapsto {{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21)$. This map yields an isomorphism between ${\mathcal{A}}_2^*$ and the free submonoid of ${{\smash{\mathrm{lps}}}}_n$, $\langle C_2\rangle$. Using the example of the first paragraph and the isomorphism, we conclude that the elements ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(211211)$ and ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(211121)$ of $\langle C_2\rangle$ that have the same evaluation, satisfy ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(211211) {\nsim_{{\mathrm{l}}}}{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(211121)$ in $\langle
C_2\rangle$. By Lemma \[lem:left\_conjugacy\_Bell2\] we get ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(211211)
{\nsim_{{\mathrm{l}}}}{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(211121)$ in ${{\smash{\mathrm{lps}}}}_n$. The result follows.
Regarding the relation between ${{\sim}_{{\mathrm{p}}}^*}$ and ${\sim_{{\mathrm{l}}}}$ in the monoids of rank greater or equal than $3$ we leave the following:
In any multihomogeneous monoid the inclusion ${{\sim}_{{\mathrm{p}}}^*}\ \subseteq\ {\sim_{{\mathrm{l}}}}$ holds. For the monoid of rank $n$, ${{\smash{\mathrm{lps}}}}_n$, with $n\geq 3$, is the inclusion strict, or does the equality hold?
Considering this problem we were able to prove the following result:
Let $u,v$ be elements of ${{\smash{\mathrm{lps}}}}_n$ with exactly two symbols (with possible multiple occurrences) and $n\geq 2$. In ${{\smash{\mathrm{lps}}}}_n$, the following holds $$u\ {{\sim}_{{\mathrm{p}}}^*}\ v\ \Leftrightarrow\ u\ {\sim_{{\mathrm{l}}}}\ v.$$
Without lost of generality, assume that $u,v\in {{\smash{\mathrm{lps}}}}_2$ and that $u{\sim_{{\mathrm{l}}}}v$ in ${{\smash{\mathrm{lps}}}}_n$. Hence $u {\sim_{{\mathrm{ev}}}}v$ and thus for $a\in{\mathcal{A}}_2$, the number of symbols $a$ in $u$ and $v$ is the same.
As $u,v\in {{\smash{\mathrm{lps}}}}_2$, $u={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u'u'')$ and $v={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v'v'')$ where ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u'),{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v')
\in\langle C_2\rangle$, and ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''),{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v'')\in \langle{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(2)\rangle$. Note that $u{\sim_{{\mathrm{p}}}}{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''u')$ and $v{\sim_{{\mathrm{p}}}}{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v''v')$ in ${{\smash{\mathrm{lps}}}}_n$.
We consider two cases. If ${\left|u'u''\right|}_2\geq {\left|u'u''\right|}_1$, then ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''u')={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}\left((21)^i2^j\right)$ and ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v''v')
={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}\left((21)^k2^l\right)$ for some $i,j,k,l\in\mathbb{N}_0$. As ${\left|u''u'\right|}_a={\left|v''v'\right|}_a$ for all $a\in{\mathcal{A}}_2$, we deduce that $i=k$ and $i+j=k+l$, and thus it follows that $j=l$. So, we conclude that ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''u')={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v''v')$. Therefore $u{\sim_{{\mathrm{p}}}}{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''u')={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v''v'){\sim_{{\mathrm{p}}}}v$ and thus $u\ {{\sim}_{{\mathrm{p}}}^*}\ v$ in ${{\smash{\mathrm{lps}}}}_n$.
Now suppose that ${|u'u''|}_1>{|u'u''|}_2$. In this case ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''u'),
{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v''v')\in \langle C_2\rangle$. As in ${{\smash{\mathrm{lps}}}}_n$ ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''u') {\sim_{{\mathrm{p}}}}u$, $u{\sim_{{\mathrm{l}}}}v$, ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v''v'){\sim_{{\mathrm{p}}}}v$ and ${{\sim_{{\mathrm{p}}}}}\subseteq
{{\sim_{{\mathrm{l}}}}}$, it follows that ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''u'){\sim_{{\mathrm{l}}}}{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v''v')$ in ${{\smash{\mathrm{lps}}}}_n$, by the transitivity of ${\sim_{{\mathrm{l}}}}$. Hence, by Lemma \[lem:left\_conjugacy\_Bell2\], ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''u'){\sim_{{\mathrm{l}}}}{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v''v')$ in the free monoid $\langle C_2\rangle$. In a free monoid we have ${{{\sim}_{{\mathrm{p}}}^*}} ={{\sim_{{\mathrm{l}}}}}$ [@lentin1967combinatorial Theorem 3]. Therefore ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''u')\ {{\sim}_{{\mathrm{p}}}^*}\ {{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v''v')$ in $\langle C_2\rangle$. So, ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''u')\ {{\sim}_{{\mathrm{p}}}^*}\ {{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v''v')$ in ${{\smash{\mathrm{lps}}}}_n$. Combining this with fact that $u{\sim_{{\mathrm{p}}}}{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''u')$ and ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v''v'){\sim_{{\mathrm{p}}}}v$ in ${{\smash{\mathrm{lps}}}}_n$, it follows that $u\ {{\sim}_{{\mathrm{p}}}^*}\ v$ in ${{\smash{\mathrm{lps}}}}_n$.
In both cases $u\ {{\sim}_{{\mathrm{p}}}^*}\ v$ in ${{\smash{\mathrm{lps}}}}_n$ and the result follows.
|
---
abstract: 'Neural network training is commonly accelerated by using multiple synchronized workers to compute gradient updates in parallel. Asynchronous methods remove synchronization overheads and improve hardware utilization at the cost of introducing gradient delay, which impedes optimization and can lead to lower final model performance. We introduce Adaptive Braking (AB), a modification for momentum-based optimizers that mitigates the effects of gradient delay. AB dynamically scales the gradient based on the alignment of the gradient and the velocity. This can dampen oscillations along high curvature directions of the loss surface, stabilizing and accelerating asynchronous training. We show that applying AB on top of SGD with momentum enables training ResNets on CIFAR-10 and ImageNet-1k with delays exceeding 32 update steps with minimal drop in final test accuracy.'
bibliography:
- 'adabrake.bib'
---
Introduction
============
Computational workloads for training state-of-the-art deep learning models have grown rapidly in recent years [@amodei2018ai]. This growth has outpaced the growth in compute power available on individual accelerators. To keep training times manageable, these workloads are often distributed over a cluster of devices working in parallel. The most common form of distributed training is Distributed Synchronized SGD [@chen2016revisiting] which divides a mini-batch of samples between workers and accumulates the gradients from all workers before updating the model parameters. The workers are synchronized and can not start processing the next mini-batch until the weights have been updated which lowers hardware utilization.
@lian2015asynchronous [-@lian2015asynchronous] propose performing asynchronous weight updates to avoid the synchronization overhead. Asynchronous weight updates can improve hardware utilization at the cost of introducing gradient delay [@lian2015asynchronous]. The effects of gradient delay have been studied in several works. @yang2019pipemare [-@yang2019pipemare] show that delays cause unstable oscillations in the optimization trajectory lowering the maximum stable learning rate. @mitliagkas2016asynchrony [-@mitliagkas2016asynchrony] show that delays with a particular distribution can increase the effective momentum in the underlying optimization process. @giladi2019stabilitysedge [-@giladi2019stabilitysedge] and @kosson2020pipelined [-@kosson2020pipelined] suggest that gradient delay removes the benefits of momentum and that with delays momentum should only be used if modified. Without mitigation, gradient delay commonly results in slower optimization and worse final model performance [@chen2016revisiting]. Many methods have been proposed to improve convergence with gradient delay [@hakimi2019dana; @zhang2016staleness; @zheng2017asynchronous; @gaun2017delay; @rigazzi2019dc3sgd; @giladi2019stabilitysedge].
*Adaptive Braking* (AB) is a modification to the momentum update process that can greatly increase tolerance to delayed gradients with minimal compute overhead and no memory overhead. AB dynamically scales the gradient magnitude based on the angle between the gradient and velocity vectors, decreasing it for positive alignment (acute angle) and increasing it for negative alignment (obtuse angle). Intuitively AB can dampen oscillations along a single gradient component by reducing the velocity magnitude at every step. In the case of multiple components with different, constant, curvatures, the alignment of the gradient and velocity will be more strongly correlated with the high curvature components. This means that AB primarily dampens oscillations for the components with high curvature, stabilizing them without affecting the other components as much on average. This resembles the effect of higher order optimization methods which account for the loss landscape curvature.
In this work we focus on applying AB to Stochastic Gradient Descent with Momentum (SGDM). We show that training with SGDM+AB can improve asynchronous multi-worker training in multiple settings with no tuning of the single-worker hyperparameters. In particular, SGDM+AB enables training ResNet-20 on CIFAR-10 and ResNet-50 on ImageNet-1k with large delays $D \geq 32$ with minimal accuracy degradation. In our experiments we compare AB with several other mitigation methods showing that AB enables greater delay tolerance than other methods.
Algorithm
=========
Adaptive Braking (AB) is a general technique for momentum-based optimizers. It computes a gradient-velocity alignment score and uses it to scale the gradient. In this section we describe how AB is applied to SGDM.
The original SGDM update is: $$\begin{aligned}
{\mathbf}{v}_{t+1} &= m {\mathbf}{v}_t + {\mathbf}{g}_{t} \label{eq:sgd_v_updt} \\
{\mathbf}{w}_{t+1} &= {\mathbf}{w}_t - \eta {\mathbf}{v}_{t+1} \label{eq:sgd_w_updt}\end{aligned}$$
where ${\mathbf}{w}_t$ and ${\mathbf}{v}_t$ are the model weights and velocity at time $t$, $\eta$ is the learning rate, and $m$ is the momentum coefficient. The weight gradient applied at time $t$ is ${\mathbf}{g}_{t}$ which may have been computed with a delay, ${\mathbf}{g}_{t} = G({\mathbf}{w}_{t-D})$, where $G(\cdot)$ is the gradient function and $D$ is a random variable representing the system delay.
In and , each weight parameter is independent and can be processed separately. Adaptive Braking groups parameters so that it can compute a gradient-velocity alignment per group. By default we use a filter-wise grouping of parameters as described in Appendix \[sec:param-grouping\].
To apply Adaptive Braking, we compute the gradient scaling factor $\alpha^i_t$ based on the cosine similarity of the velocity and gradient vectors for parameter group $i$. The SGDM+AB update is:
$$\begin{aligned}
\alpha^i_{t} &= 1 - \rho \frac{\langle {\mathbf}{g}^i_{t}, {\mathbf}{v}^i_t \rangle}{\max(\|{\mathbf}{g}^i_{t}\| \|{\mathbf}{v}^i_t\|, \epsilon)} \label{eq:ab_alpha}\\
&\approx 1 - \rho \cos \angle \left( {\mathbf}{g}^i_{t}, {\mathbf}{v}^i_t \right) \nonumber \\
{\mathbf}{v}^i_{t+1}&= m {\mathbf}{v}^i_t + \alpha^i_t {\mathbf}{g}^i_{t} \label{eq:ab_v_updt} \\
{\mathbf}{w}^i_{t+1} &= {\mathbf}{w}^i_t - \eta {\mathbf}{v}^i_{t+1} \label{eq:ab_w_updt}\end{aligned}$$
where $\rho$ is a scalar hyperparameter we call the *braking coefficient* (Appendix \[sec:alpha\]), and $\epsilon$ is used for numerical stability. Different formulations of Adaptive Braking could substitute the cosine similarity with other distance functions.
Optimizing a Noisy Quadratic Model {#sec:nqm}
==================================
To gain insights into how AB can help optimization, we analyze its effect on convergence for a Noisy Quadratic Model (NQM). We adopt the setup that @zhang2019algorithmic [-@zhang2019algorithmic] used to model the effects of batch size in neural networks. The NQM allows us to explicitly control various aspects of the optimization such as the dimensionality, the amount of noise, the condition number, and the delay. We measure the quality of optimization trajectories by the number of optimization steps, $T$, required to reach a target loss. See Appendix \[sec:nqm\_setup\] for details about our setup.
-0.05in ![ This figure shows the number of optimization steps, $T$, required to reach the target loss on the NQM from Section \[sec:nqm\] for different hyperparameters. Each heatmap plots $T$ over different learning rates $\eta$ and momentum $m$. Black regions are unstable and white regions to not reach the target loss within the 500000 steps performed. The left column shows SGDM and the right column shows Adaptive Braking with $\rho=0.5$. The rows show different amounts of delay $D$ and noise $\sigma$. $T^*$ estimates the fastest trajectory based on the 1st percentile of $T$ over the colored region (to reduce the effects of noise and the choice of sampling grid). ](figures/heatmap_fix.pdf "fig:"){width="\linewidth"} -0.175in
-0.1in \[fig:heatmap\]
Figure \[fig:heatmap\] compares $T$ for SGDM with and without AB for different learning rates, momentum values, delay and noise. The first row shows the no-delay and no-noise case. In this case AB does not improve the speed and slightly decreases the highest stable learning rate. This happens because AB can magnify certain high-frequency oscillations, where $g$ and $v$ are almost always oppositely aligned, causing AB to effectively scale the learning rate by up to $1+\rho$. Appendix \[sec:nqm\_microstepping\] explores this effect further and shows how AB can be modified to avoid this.
-0.05in ![ AB can lower the steady state loss when optimizing a noisy quadratic model. The learning rate and momentum correspond to values that reach the target loss ($10^{-2}$) with AB but not SGDM in the second row of Figure \[fig:heatmap\]. **Left:** The total loss in each case and the contribution to the loss from the largest eigenvalue. The steady-state loss for the largest eigenvalues is lower with AB. **Right:** The relative energy decay ratio for each eigenvalue showing a greater dampening of high curvature components. []{data-label="fig:cq_dampening_noise"}](figures/cq_dampening_noise_small.pdf "fig:"){width="\linewidth"}
The results of adding noise in the no-delay case are shown in the second row of Figure \[fig:heatmap\]. In this case AB significantly speeds up the fastest trajectory and expands the region that reaches the target loss within the step budget. In the presence of noise, a constant learning rate trajectory will converge to an expected steady-state loss that depends on the hyperparameters and level of noise. @zhang2019algorithmic [-@zhang2019algorithmic] show that there is a trade-off with increasing the momentum and/or learning rate: it can improve the convergence rate (of the expectation) but magnifies the steady-state loss. The dampening effect of AB can reduce the steady-state loss, expanding the region that will converge within the time limit and unlocking the faster trajectories with larger step sizes. To measure the dampening effect of AB we can compare the energy after making an AB update ($E_{t+1}$) to what the energy would have been after making an SGDM update ($\hat{E}_{t+1})$ from each state (${\mathbf}{w}_t$, ${\mathbf}{v}_t$) along the AB optimization trajectory. The energy $E_t=\mathcal{L}({\mathbf}{w}_t) + \frac{1}{2}\eta \|{\mathbf}{v}_t\|^2$ accounts for both potential energy (the loss $\mathcal{L}({\mathbf}{w}_t)$) and kinetic energy $\frac{1}{2}\eta \|{\mathbf}{v}\|_t^2$ of an optimization state (see Appendix \[sec:nqm\_energy\]). The geometric mean of $E_{t+1}/\hat{E}_{t+1}$, which we call the *relative energy decay*, indicates how much faster AB dissipates energy compared to SGDM on average. The relative energy decay can be computed for each eigenvector to measure the dampening for different components. Figure \[fig:cq\_dampening\_noise\] shows that AB can lower the steady state-loss by dampening the large eigenvalue components.
The third and forth rows of Figure \[fig:heatmap\] show that AB can help with gradient delay. AB can expand the region of convergence and significantly reduce the time required to reach the target loss. Similar to @kosson2020pipelined [-@kosson2020pipelined], we note that standard momentum does not seem to help in the delay case but with AB there can be a significant benefit. Delays intuitively cause optimization to overshoot, introducing and amplifying oscillations. AB seems to help stabilize these oscillations improving convergence in the presence of gradient delay. Figure \[fig:cq\_dampening\_delay\] explores this effect. It shows that AB can dampen high curvature components stabilizing training with gradient delay.
-0.05in ![ AB can stabilize training with gradient delay. The learning rate and momentum correspond to values that reach the target loss with AB but are unstable for SGDM in the third row of Figure \[fig:heatmap\]. **Left:** The components corresponding to the three largest eigenvalues are unstable without AB. **Right:** AB dissipates energy in these components on average which stabilizes training. []{data-label="fig:cq_dampening_delay"}](figures/cq_dampening_delay_small.pdf "fig:"){width="\linewidth"}
Training Neural Networks
========================
To measure the effectiveness of AB for training neural networks with gradient delay, we simulate multi-worker ASGD. We do this on a single machine by storing a history of the master weights $[{\mathbf}{w}_t, {\mathbf}{w}_{t-1}, {\mathbf}{w}_{t-2} ... {\mathbf}{w}_{t-D}]$. We then use a chosen algorithm to compute the updated master weights ${\mathbf}{w}_{t+1}$ using the delayed gradient ${\mathbf}{g}_{t} = G({\mathbf}{w}_{t-D})$. In all experiments we use a constant delay $D$, which is representative of an ideal ASGD setting with $D+1$ workers and round robin scheduling. All experiments were implemented using the PyTorch framework [@pytorch2019], and executed on NVIDIA T4 or V100 GPUs.
The main metric we are interested in is the final test accuracy of our trained model compared to a zero-delay, single-worker SGDM baseline. This baseline represents the best possible convergence scenario albeit with no parallelism and no speedup. We evaluate the delay tolerance of algorithms by comparing how much the final test accuracy degrades when training with ASGD and different delays $D$. For consistency, we do not change the per-worker hyperparameters from the original SGDM baseline. We report experiments on two common image classification tasks: ResNet-20 trained on CIFAR-10 [@krizhevsky09learningmultiple] and ResNet-50 trained on ImageNet-1k [@krizhevsky2012imagenet]. Hyperparameter settings can be found in Appendix \[hsetting\].
In addition to Adaptive Braking, we also evaluate and compare against a variety of gradient delay mitigation strategies[^1]: Shifted Momentum (SM) [@giladi2019stabilitysedge], DANA [@hakimi2019dana], Delay-Compensation (DC) [@zheng2017asynchronous], and Staleness-Aware (SA) [@zhang2016staleness].
CIFAR-10
--------
In Figure \[fig: exp2-ab-compare\], we simulate asynchronous training of ResNet-20 on CIFAR-10. We evaluate SGDM combined with other delay mitigation strategies and compare them against SGDM+AB with $\rho=2$ (hyperparameter search shown in Appendix \[sec:cifar10-ext\]). We find that training with SGDM+AB leads to equivalent accuracy at small-to-moderate delays, and significantly outperforms the other mitigation strategies at large delays ($D = 128$). We also see more stability from run-to-run when compared to the other strategies.
![ResNet-20 + CIFAR-10 final test accuracy vs delay. AB provides greater delay tolerance than other mitigation strategies. Each line shows the median over five trials.[]{data-label="fig: exp2-ab-compare"}](figures/exp2-ab-compare.pdf "fig:"){width="\linewidth"} -0.1in
To gain further insight into AB’s effects, we measure key metrics $\alpha^i_t$ and $\|{\mathbf}{v}^i_t\|$ during CIFAR-10 training and discuss their implications in Appendices \[sec:alpha\] and \[sec:vel-norm\], respectively.
ImageNet-1k
-----------
In Figure \[fig:exp3-i1k-compare\], we simulate asynchronous training of ResNet-50 on ImageNet-1k with a delay of $D = 32$. We compare the vanilla SGDM optimizer to SGDM+AB with $\rho=2$. For our zero-delay baseline, in addition to using a single worker as in the CIFAR-10 experiments, we also include a more realistic Synchronous SGD (SSGD) setup with $D + 1 = 33$ workers. For the SSGD run we use a large batch size of $BS' = 32 * 33 = 1056$ and linearly-scaled learning rate $LR' = 0.00125 * 33 = 0.04125$.
![AB outperforms other delay mitigation strategies when training ResNet-50 on ImageNet with a delay of $D=32$.[]{data-label="fig:exp3-i1k-compare"}](figures/exp3-i1k-4pg.pdf "fig:"){width="\linewidth"} -0.1in
We confirm that training with vanilla SGDM and gradient delay leads to poor convergence at the start of training, and a final test accuracy degradation of -0.24% compared to the single-worker baseline. Using SGDM+AB leads to more stable convergence during early training; the test accuracy curve is closer to synchronous training. Overall, AB prevents final accuracy degradation for asynchronous training and even outperforms the single-worker baseline by +0.52%.
We also compare AB with other delay mitigation strategies in the same ASGD setting. We find that SGDM+AB outperforms the other algorithms in terms of final test accuracy. Among the other algorithms, SGDM+DANA performs the best, and following a similar trajectory to AB in the early stages of training. Final test accuracies for all methods are reported in Table \[tab:i1k\].
0.1in \[tab:i1k\]
Conclusion
==========
Adaptive Braking scales the gradient based on the alignment of the gradient and velocity. This is a non-linear operation that dampens oscillations along the high-curvature components of the loss surface without affecting the other components much on average. It is especially effective in the presence of gradient delay where it can stabilize components that would otherwise be unstable. We show that AB is competitive with state of the art methods for ASGD training.
The increased delay tolerance that AB provides could enable hardware speedups for both data-parallel distributed training as well as pipeline-parallel training with pipelined backpropagation [@Ptrowski1993PerformanceAO; @chen2012pipelined; @Harlap2018PipeDreamFA].
In this work we have focused on the SGDM optimizer, but future work could propose similar modifications to other optimizers such as Adam [@kingma2014adam].
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Joel Hestness, Vithursan Thangarasa, and Xin Wang for for their help and feedback that improved the manuscript.
Related Work
============
Asynchronous methods are used to improve compute utilization for neural network training but introduce gradient staleness. Gradients are stale because the gradient is computed using weight from $D$ time steps ago, ${\mathbf}{g}_t = G\left( {\mathbf}{w}_{t-D} \right)$. To mitigate this @giladi2019stabilitysedge [-@giladi2019stabilitysedge] propose adding delay to the velocity as well, ${\mathbf}{v}_{t-D}$. They do this by tracking an independent velocity for each worker and updating the master weights using the current worker’s velocity. Another class of mitigation strategies attempts to predict future weights $\hat{{\mathbf}{w}}_{t-D} \approx {\mathbf}{w}_t$ for use in the gradient computation, ${\mathbf}{g}_t = G\left( \hat{{\mathbf}{w}}_{t-D} \right)$. Most methods [@chen2018efficient; @hakimi2019dana; @kosson2020pipelined] use the velocity vector to estimate the future weights.
@zhang2016staleness [-@zhang2016staleness] propose Staleness-Aware (SA) and show that down-weighing the gradients based on the delay $D$ (gradient penalization) can improve asynchronous training. @kosson2020pipelined [-@kosson2020pipelined] characterize the impulse response of gradients in the optimization process and modify the delayed impulse response to match the non-delayed setting in a technique called Spike Compensation.
Delay Compensated ASGD [@zheng2017asynchronous] and its variants [@gaun2017delay; @rigazzi2019dc3sgd] estimate the gradient using the first two terms of the Taylor expansion of the delayed gradient function. Using the Taylor expansion of the delayed gradient function requires estimating the Hessian and storing the old weights. Applying DC-ASGD with a velocity approximation for the weight change is closely related to element-wise Adaptive Braking (See Appendix \[sec:ab-dc\]).
@odonoghue3982adaptive [-@odonoghue3982adaptive] use a method called Adaptive Restart (AR) to dampen oscillations and speed up optimization. Adaptive Restart resets the velocity, ${\mathbf}{v} = {\mathbf}{0}$, when ${\mathbf}{g}^T \cdot {\mathbf}{v} < 0$ which can be viewed as a measure of alignment. AB also measures alignment using cosine similarity but applies a continuous correction to ${\mathbf}{g}$ rather than a discrete reset of ${\mathbf}{v}$. This makes AB more applicable in a noisy optimization setting such as SGD. Periodically resetting the step direction is also used in nonlinear conjugate gradient optimization methods. Adaptive Braking can be seen as a form of nonlinear conjugate gradient optimization since the step direction accumulation is adaptively adjusted based on the current gradient. There are many variations of nonlinear conjugate gradient optimization but to the best of our knowledge, none of these forms are exactly equivalent to Adaptive Braking.
The rest of this section shows the algorithmic details of the methods we compare against in our experiments.
\[pseudocode\]
Asynchronous SGD (ASGD)
-----------------------
Always do: Receive parameters ${\mathbf}{w}_{t-D}$ from the master Compute gradient: ${\mathbf}{g}_{t;j} = G({\mathbf}{w}_{t-D})$ Send ${\mathbf}{g}_{t;j}$ to the master
For t = 1...T do: Receive gradient ${\mathbf}{g}_{t;j}$ from worker $j$ Update momentum: ${\mathbf}{v}_{t+1} = m {\mathbf}{v}_t+{\mathbf}{g}_{t;j}$ Update master’s weights: ${\mathbf}{w}_{t+1} = {\mathbf}{w}_t-\eta_t {\mathbf}{v}_{t+1}$ Send ${\mathbf}{w}_{t+1}$ to worker $j$
Staleness-Aware
---------------
Staleness-Aware divides the original learning rate by the delay of the current gradient in each update step.
Initialize an iteration array: $iter = [0]*N$ For t = 1...T do: Receive gradient ${\mathbf}{g}_{t;j}$ from worker $j$ Calculate worker $j$’s delay: $D_t = t-iter[j]$ Update momentum: ${\mathbf}{v}_{t+1} = m {\mathbf}{v}_t+{\mathbf}{g}_{t;j}$ Update master: ${\mathbf}{w}_{t+1} = {\mathbf}{w}_t-\frac{\eta_t}{D_t} {\mathbf}{v}_{t+1}$ Send ${\mathbf}{w}_{t+1}$ to worker $j$ Save current iteration: $iter[j] = t$
Shifted Momentum
----------------
Shifted Momentum assigns an independent velocity ${\mathbf}{v}_{t;j}$ to each worker $j$, and updates the master weights using the current worker’s velocity.
Always do: Receive parameters ${\mathbf}{w}_{t-D}$ from the master Compute gradient: ${\mathbf}{g}_{t;j} = G({\mathbf}{w}_{t-D})$ Update momentum ${\mathbf}{v}_{t+1;j} = m {\mathbf}{v}_{t;j}+{\mathbf}{g}_{t;j}$ Send ${\mathbf}{v}_{t+1;j}$ to the master
For t = 1...T do: Receive gradient ${\mathbf}{v}_{t+1;j}$ from worker $j$ Update master’s weights: ${\mathbf}{w}_{t+1} = {\mathbf}{w}_t - \eta_t {\mathbf}{v}_{t+1;j}$ Send ${\mathbf}{w}_{t+1}$ to worker $j$
DANA
----
DANA assigns an independent velocity $v_{t;j}$ to each worker $j$, and computes the gradient on estimated future weights.
Always do: Receive parameters $\hat{{\mathbf}{w}}_{t-D}$ from the master Compute gradient: ${\mathbf}{g}_{t;j} = G({\mathbf}{w}_{t-D})$ Update momentum: ${\mathbf}{v}_{t+1;j} = m {\mathbf}{v}_{t;j}+{\mathbf}{g}_{t;j}$ Send ${\mathbf}{v}_{t+1;j}$ to the master
For t = 1...T do: Receive gradient ${\mathbf}{v}_{t+1;j}$ from worker $j$ Update master’s weights: ${\mathbf}{w}_{t+1} = {\mathbf}{w}_t-\eta_t {\mathbf}{v}_{t+1;j}$ Estimates future weights: $\hat{{\mathbf}{w}}_{t+1} = {\mathbf}{w}_t-\eta_t m \sum_j {\mathbf}{v}_{t+1;j}$ Send $\hat{{\mathbf}{w}}_{t+1}$ to worker $j$
Delay-Compensated ASGD
----------------------
Delay-Compensated ASGD approximates the Hessian of the loss surface and corrects the delayed gradient based on the weight inconsistency.
For t = 1...T do: Receive gradient ${\mathbf}{g}_{t;j}$ from worker $j$ Compensate gradient: $\hat{{\mathbf}{g}}_{t;j} = {\mathbf}{g}_{t;j} + \nabla {\mathbf}{g}_{t;j} \cdot ({\mathbf}{w}_{t} - {\mathbf}{w}_{t-D}) $ Update momentum: ${\mathbf}{v}_{t+1} = m {\mathbf}{v}_t+\hat{{\mathbf}{g}}_{t;j}$ Update master’s weights: ${\mathbf}{w}_{t+1} = {\mathbf}{w}_t-\eta_t {\mathbf}{v}_{t+1}$ Send ${\mathbf}{w}_{t+1}$ to worker $j$
where $\nabla {\mathbf}{g}_{t;j}$ is approximated with $\lambda_t \cdot \text{diag}({\mathbf}{g}_{t;j} \odot {\mathbf}{g}_{t;j})$ and $\lambda_t$ is the variance control parameter, set using a moving-average as described in the original paper. We note that this algorithm is modified to work with SGDM.
Adaptive Braking
----------------
For t = 1...T do: Receive gradient ${\mathbf}{g}_{t;j}$ from worker $j$ For i in parameter groups do: Compute braking: $\alpha^i_{t} = 1 - \rho \cos \angle \left( {\mathbf}{g}^i_{t}, {\mathbf}{v}^i_t \right)$ Update momentum: ${\mathbf}{v}^i_{t+1} = m {\mathbf}{v}^i_t+\alpha^i_{t}{\mathbf}{g}^i_{t;j}$ Update master’s weights: ${\mathbf}{w}^i_{t+1} = {\mathbf}{w}^i_t-\eta_t {\mathbf}{v}^i_{t+1}$ Send ${\mathbf}{w}_{t+1}$ to worker $j$
Hyperparameter Settings {#hsetting}
=======================
The per-worker hyperparameter settings used in our neural network training experiments are listed in Table \[tab:hyperparameters\]. For CIFAR-10, we choose to use a small batch size of 32 rather than the standard setting of 128 to showcase a training setup with high momentum, which is where Adaptive Braking is most effective. For ImageNet-1k we use a per-worker batch size of 32 to reflect a common SSGD training setup with 8 GPUs and a total batch size of 256, and choose a momentum of 0.99 based on hyperparameter searches performed by @shallue2019measuring [-@shallue2019measuring]. For DC we use the adaptive form of the algorithm and adopt the original paper’s hyperparameters. The other mitigation strategies are hyperparameter-free.
-0.1in
0.1in \[tab:hyperparameters\]
CIFAR-10 Extended Results {#sec:cifar10-ext}
=========================
0.1in
Algorithm D=0 D=1 D=4 D=16 D=32 D=64 D=128
--------------------- ------------ ------------ ------------ ------------ ------------ ------------ ------------
SGDM 92.41% 92.34% 92.16% 90.41% 84.03% 10.09% 10.00%
SGDM+AB, $\rho=0.5$ 92.36% **92.61%** 92.16% 91.78% 89.22% 25.25% 45.11%
SGDM+AB, $\rho=1$ **92.61%** 92.51% 92.39% 92.18% 91.67% 89.82% 84.15%
SGDM+AB, $\rho=2$ 92.47% 92.43% 92.46% **92.27%** **91.99%** **91.21%** 89.98%
SGDM+AB, $\rho=3$ 92.44% 92.44% **92.54%** 91.87% 91.82% 90.69% 88.22%
SGDM+AB, $\rho=4$ 92.12% 91.98% 92.07% 91.57% 91.50% 90.87% **90.07%**
SGDM+AB, $\rho=5$ 92.12% 91.94% 92.01% 90.97% 90.41% 89.35% 87.06%
\[tab:cifar10-ext\]
In Figure \[fig: exp1-delay-tolerance\], we simulate asynchronous training of ResNet-20 on CIFAR-10 and measure the delay tolerance of SGDM with or without Adaptive Braking. Each experiment is repeated 5 times and the median final test accuracy is plotted.
We find that AB greatly improves the delay tolerance of SGDM. In particular, we can train asynchronously with a gradient delay of $D=32$ with only a -0.42% drop in test accuracy. Even at extreme settings with $D=128$, the degradation is only -2.43%, while vanilla SGDM fails to converge at all. We also find that the delay tolerance improves as $\rho$ is increased from $0.5$ to $2.0$.
![ResNet-20 + CIFAR-10 final test accuracy for different delays. Adaptive Braking improves the delay tolerance of SGDM when training in an ASGD setting.[]{data-label="fig: exp1-delay-tolerance"}](figures/exp1-delay-tolerance.pdf "fig:"){width="\linewidth"} -0.1in
In Table \[tab:cifar10-ext\] we list extended results with more settings of braking coefficient $\rho$. The results suggest that larger $\rho$ should be used for larger delays. The choice of $\rho = 2$ is the most consistent across $D = [0, 1, 4, 16, 32, 64, 128]$, performing best or second-best in almost all delay settings.
Parameter Grouping for AB {#sec:param-grouping}
=========================
The AB gradient scaling factor $\alpha^i_t$ is non-linear with respect to ${\mathbf}{g}^i_t$ and ${\mathbf}{v}^i_t$, and depends on the granularity with which the model parameters are grouped. We consider three levels of granularity for grouping:
- **Per tensor:** This is based on the default grouping of parameters into tensors in PyTorch. In this case each convolutional or linear layer has a weight tensor which contains all the multiplicative weights and optionally a bias which is a separate tensor. Normalization layers have their own bias and scaling tensors.
- **Per filter:** Here the weights of each neuron or filter are treated separately. The biases and other parameters such as those in the normalization layers are still grouped per tensor.
- **Per element:** Here each parameter is treated separately, and the scaling coefficient reduces to $\alpha_t^i = 1 - \rho \text{ sgn}(g_t^i \cdot v_t^i)$.
In Table \[tab:grouping\] we find that using a filter-wise grouping of parameters leads to the best performance for ResNet-20 trained on CIFAR-10, even when accounting for different optimal settings of $\rho$ for each grouping method. Therefore we use filter-wise Adaptive Braking for all of our experiments.
0.1in \[tab:grouping\]
AB with Weight Decay {#sec:ab-weight-decay}
====================
When weight decay is used with Adaptive Braking, we add the weight decay term to the velocity independently, and do not consider the weight decay to be part of the gradient when computing the cosine similarity:
$$\begin{aligned}
\alpha^i_{t} &= 1 - \rho \frac{\langle {\mathbf}{g}^i_{t}, {\mathbf}{v}^i_t \rangle}{\max(\|{\mathbf}{g}^i_{t}\| \|{\mathbf}{v}^i_t\|, \epsilon)} \\
&\approx 1 - \rho \cos \angle \left( {\mathbf}{g}^i_{t}, {\mathbf}{v}^i_t \right) \nonumber \\
{\mathbf}{v}^i_{t+1}&= m {\mathbf}{v}^i_t + \alpha^i_t {\mathbf}{g}^i_{t} + \lambda {\mathbf}{w}^i_t \\
{\mathbf}{w}^i_{t+1} &= {\mathbf}{w}^i_t - \eta {\mathbf}{v}^i_{t+1}\end{aligned}$$
This helps prevent $\alpha^i_t$ from being skewed by the weight decay term, which is correlated across steps.
AB compared with DC {#sec:ab-dc}
===================
Under a particular approximation, delay-compensated ASGD has a similar form to Adaptive Braking. DC attempts to correct the delayed gradient ${\mathbf}{g_t}$ by measuring the change in the master weights, and using a Hessian approximation to estimate the up-to-date gradient ${\mathbf}{\hat{g}_t}$ at the current master weights:
$$\begin{aligned}
{\mathbf}{g}_t &= G({\mathbf}{w}_{t-D}) \\
{\mathbf}{\hat{g}}_t &= G({\mathbf}{w}_t) \approx {\mathbf}{g}_t + \lambda {\mathbf}{H}_t \cdot ({\mathbf}{w}_t - {\mathbf}{w})_{t-D} \label{eq:delay_gd_2ord}\\
&\approx {\mathbf}{g}_t + \lambda ({\mathbf}{g}_t \cdot {\mathbf}{g}_t^T) \cdot ({\mathbf}{w}_t - {\mathbf}{w}_{t-D}) \label{eq:delay_gd_2ord_approx_H} \\
&\approx {\mathbf}{g}_t + \lambda \cdot \text{diag}({\mathbf}{g}_t \odot {\mathbf}{g}_t) \cdot ({\mathbf}{w}_t - {\mathbf}{w}_{t-D}) \label{eq:dc_diag}\\
&= {\mathbf}{g}_t + \lambda ({\mathbf}{g}_t \odot {\mathbf}{g}_t) \odot ({\mathbf}{w}_t - {\mathbf}{w}_{t-D}) \label{eq:dc}\end{aligned}$$
@zheng2017asynchronous [-@zheng2017asynchronous] use the Taylor Series expansion of the delayed gradient but truncate higher order terms . ${\mathbf}{H}_t$ is the hessian at time $t$, and $\lambda$ is a hyperparameter used to control the strength of the second-order correction. They then approximate ${\mathbf}{H}_t \approx {\mathbf}{g}_t \cdot {\mathbf}{g}_t^T \approx \text{diag}({\mathbf}{g}_t \odot {\mathbf}{g}_t)$ which makes the compensation method an element-wise operation. To arrive at AB, we maintain the outer-product form for the remainder of this section.
During SGDM training, the weight update at each step $t$ is $-\eta {\mathbf}{v_t}$. If we assume the velocity over the last $D$ steps is relatively unchanged, then we can approximate the total weight change from step $(t-D)$ to step $t$ as:
$$\begin{aligned}
{\mathbf}{w}_t - {\mathbf}{w}_{t-D} &= -\sum_{i=0}^{D-1} \eta {\mathbf}{v}_{t-i} \\
&\approx -\sum_{i=0}^{D-1} \eta {\mathbf}{v}_t \\
&= -\eta D {\mathbf}{v}_t\end{aligned}$$
Substituting this approximation into , we end up with a gradient scaling term that involves a dot product of the gradient and the velocity:
$$\begin{aligned}
\hat{{\mathbf}{g}}_t &\approx {\mathbf}{g}_t + \lambda ({\mathbf}{g}_t \cdot {\mathbf}{g}_t^T) \cdot (-\eta D {\mathbf}{v}_t) \\
&= {\mathbf}{g}_t (1 - \lambda \eta D ({\mathbf}{g}_t^T \cdot {\mathbf}{v}_t)) \\
&= {\mathbf}{g}_t (1 - \lambda' ({\mathbf}{g}_t^T \cdot {\mathbf}{v}_t))\end{aligned}$$
Finally, we can use an adaptive setting of $\lambda'$, normalizing by the magnitudes of the gradient and velocity at time $t$. At this point we are no longer approximating the master weight gradient ${\mathbf}{\hat{g}_t}$, so we adjust notation:
$$\begin{aligned}
\lambda'_t &= \frac{\lambda'_0}{\|{\mathbf}{g}_t\| \cdot \|{\mathbf}{v}_t\|} \\
{\mathbf}{g}'_t &= {\mathbf}{g}_t (1 - \lambda'_0 \cos \angle ({\mathbf}{g}_t, {\mathbf}{v}_t)) \label{eq:dc-approx}\end{aligned}$$
The gradient scaling term in is now exactly $\alpha_t = 1 - \rho \cos \angle ({\mathbf}{g}_t, {\mathbf}{v}_t)$ used for AB. Note that for AB the braking coefficient $\rho$ is chosen independently rather than set based on the learning rate $\eta$ and delay $D$.
Gradient scale $\alpha$ and braking coefficient $\rho$ {#sec:alpha}
======================================================
The strength of AB’s gradient scaling $\alpha^i_t$ depends on the choice of braking coefficient $\rho$. In general, the optimal setting of $\rho$ is task-dependent and can be optimized as a hyperparameter, but we find that values in $\rho \in [0.5, 2]$ work well across different delays and model architectures. Note that if $\rho$ is set larger than 1, it is possible for the gradient scaling at a particular step to be negative, but we find this to rarely happen in practice. We have also experimented with clamping $\alpha^i_t$ to be non-negative but do not see a significant effect on convergence, so for simplicity we do not perform clamping in the standard form of AB.
In Figure \[fig:exp4-alpha\], we measure the gradient scaling $\alpha^i_t$ applied by Adaptive Braking during ResNet-20 + CIFAR-10 training. At the start of training, successive gradients are well-aligned, so as expected $\alpha^i_t$ is less than one and AB scales down the gradients. In the later stages of training, successive gradients are not well aligned and $\alpha^i_t$ returns closer to 1, which would be equivalent to vanilla SGDM.
We also find that the gradient scaling rarely becomes negative, despite the fact that we are using a braking coefficient of $\rho=2$. This confirms that even though the potential range of the gradient scaling is $1 - \rho \leq \alpha^i_t \leq 1 + \rho$, the typical values seen during CNN training rarely reach such extreme values. This is probably due to the high dimensionality of the parameter groups, which leads the average cosine similarity to be closer to zero.
The norm of the gradient is also usually smaller than the norm of the velocity ($\tau^i_t < 1$), so even if $\alpha^i_t$ does become negative at an individual step, it will likely only reduce the velocity norm, not completely reverse the direction of optimization (gradient ascent). In extreme cases such as a loss plane with constant gradient, we would see oscillations if $\rho > 1$. But again, we do not see this behavior in practice when training CNNs.
![The average gradient scaling $\alpha^i_t$ increases throughout training. This plot measures $\alpha^i_t$ for four different convolutional layers in ResNet-20. The model is trained on CIFAR-10 with SGDM+AB, $\rho=2$, with a delay of $D=32$.[]{data-label="fig:exp4-alpha"}](figures/exp4-alpha.pdf "fig:"){width="\linewidth"} -0.1in
Velocity Norm and Gradient Velocity Ratio {#sec:vel-norm}
=========================================
AB tends to reduce or remove the growth in the velocity that happens if successive gradients are well aligned. This is a pervasive problem in delayed gradient training, especially at the very start of training where the first $\approx D$ gradient estimates are computed on the same initial weights. In Figure \[fig:exp4-vel-fine\], we plot a fine-grained view of $\|{\mathbf}{v}^i_t\|$ at the very start of ResNet-20 + CIFAR-10 training. We train with either SGDM (black) or SGDM+AB (red) and use a constant delay of $D=32$. Without AB, the velocity norm across many parameter groups explodes within the first few hundred steps. This means that the gradients are well aligned and sum constructively, leading to large $\|{\mathbf}{v}^i\|$. When AB is used, this initial blowup is greatly reduced.
![At the very start of training, SGDM+AB scales down similarly-aligned gradients, and prevents a blowup of the velocity norm that occurs with vanilla SGDM. Each plot measures $\|{\mathbf}{v}^i\|$ for a different group of convolutional layer weights ${\mathbf}{w}^i$ in ResNet-20. The y-axis is log scaled. The model is trained on CIFAR-10 with a delay of $D=32$.[]{data-label="fig:exp4-vel-fine"}](figures/exp4-vel-fine.pdf "fig:"){width="\linewidth"} -0.1in
As training continues, we notice that the average magnitude of $\|{\mathbf}{v}^i_t\|$ is not very different between SGDM and SGDM+AB, and can often be higher when using SGDM+AB (See Figure \[fig:exp4-vel\]). So even though Adaptive Braking is scaling down the gradient, and limiting the growth of the velocity, AB does not significantly decrease the average velocity norm $\|{\mathbf}{v}^i\|$. Instead, AB actually stabilizes the velocity and leads the optimizer to take larger steps in early training than it would with vanilla SGDM.
This suggests that replacing Adaptive Braking with a smaller learning rate would not produce the same benefits. Slowing down the growth of the velocity vector is not the same as reducing the magnitude of the weight updates. This idea is explored further in Appendix \[sec:unwrap\].
![The velocity norm $\|{\mathbf}{v}^i_t\|$ measured across a full training run, for four different convolutional layers in ResNet-20. During early training, the velocity norm is larger when using SGDM+AB than when using vanilla SGDM. The model is trained on CIFAR-10 with a delay of $D=32$.[]{data-label="fig:exp4-vel"}](figures/exp4-vel.pdf "fig:"){width="\linewidth"} -0.1in
To better measure the effect of AB across different layers and across the training schedule, we introduce Gradient Velocity Ratio (GVR), which measures the ratio of the gradient norm over the velocity norm for a group $i$ of parameters:
$$\begin{aligned}
\tau^i_t = \frac{\|{\mathbf}{g}^i_t\|}{\|{\mathbf}{v}^i_t\|} \label{eq:turn}\end{aligned}$$
We believe GVR is a good measure of a momentum-based optimizer’s ability to change its trajectory. We measure the GVR $\tau^i_t$ during training in Figure \[fig:exp4-turnability\], and find that using AB greatly increases GVR throughout training. This supports our theory that Adaptive Braking makes it easier to change the direction of the optimization trajectory during ASGD training.
![The GVR $\tau^i_t = \|{\mathbf}{g}^i_t\| / \|{\mathbf}{v}^i_t\|$ is higher when training with SGDM+AB than with vanilla SGDM. Each plot measures $\|{\mathbf}{v}^i\|$ for a different group of convolutional layer weights ${\mathbf}{w}^i$. The model is trained on CIFAR-10 with a delay of $D=32$.[]{data-label="fig:exp4-turnability"}](figures/exp4-gvr.pdf "fig:"){width="\linewidth"} -0.1in
Weight Update Direction
=======================
The instantaneous effect of AB on the SGDM weight update is to make the weight update more aligned with the gradient when the gradient and velocity directions disagree. This effect is illustrated in Figure \[fig:step\_direction\], where we compare the alignment of the gradient with the weight update for both SGDM and SGDM+AB, $\rho=2$. We plot measurements for a range of gradient-velocity ratio (GVR) values $\tau = [0.1, 0.2, 0.5, 0.8]$ which we find is typical in CNN training (See Appendix \[sec:vel-norm\] and Figure \[fig:exp4-turnability\]).
Note that a similar effect can be achieved for vanilla SGDM if we significantly reduce the momentum $m$: for instance if $m=0$ then the weight update is always perfectly aligned with the gradient. When training with delayed gradients and no mitigation, setting $m=0$ is a valid choice and can even be optimal when there is no gradient noise. However we find that in both the convex quadratic and neural network ASGD setting that we can achieve faster convergence if we use nonzero momentum combined with delay mitigation. This is why AB’s ability to reorient the weight update in the presence of momentum is valuable.
![Alignment of weight update with gradient, as a function of alignment of gradient and velocity. When the direction of ${\mathbf}{g}^i$ disagrees with the direction of ${\mathbf}{v}^i$, the SGDM + AB weight update is closer to ${\mathbf}{g}^i$.[]{data-label="fig:step_direction"}](figures/step_direction.pdf "fig:"){width="\linewidth"} -0.1in
AB Ablation Study {#sec:unwrap}
=================
Since AB scales the gradient during both the velocity update step and the weight update step, we can ask whether the delay tolerance of AB comes from just the instantaneous correction to the weight update, or from the long-term effect on the velocity. The two effects are made clear by looking at an unwrapped version of the SGDM+AB update equations:
$$\begin{aligned}
\label{eq:ab-vel-update}
{\mathbf}{v}_{t+1}^i &= m {\mathbf}{v}_{t}^i + \alpha_{t}^i {\mathbf}{g}_{t}^i \\
\label{eq:ab-weight-update}
{\mathbf}{w}_{t+1}^i &= {\mathbf}{w}_{t}^i - \eta \left( m {\mathbf}{v}_{t}^i + \alpha_{t}^i {\mathbf}{g}_{t}^i \right)\end{aligned}$$
For SGDM, the scaling factor $\alpha_t^i$ is always fixed to 1. For SGDM+AB, $\alpha_t^i$ is normally computed per-step as described by and applied in both equations. Alternatively, we can apply the scaling in only one equation or the other.
If we choose to apply gradient scaling only on the velocity update, we call this algorithm AB-vel-only, with update equations:
$$\begin{aligned}
\label{eq:ab-vel-only-1}
{\mathbf}{v}_{t+1}^i &= m {\mathbf}{v}_{t}^i + \alpha_{t}^i {\mathbf}{g}_{t}^i \\
\label{eq:ab-vel-only-2}
{\mathbf}{w}_{t+1}^i &= {\mathbf}{w}_{t}^i - \eta \left( m {\mathbf}{v}_{t}^i + {\mathbf}{g}_{t}^i \right)\end{aligned}$$
If we choose to apply gradient scaling only on the weight update, we call this algorithm AB-weight-only, with update equations:
$$\begin{aligned}
\label{eq:ab-step-only-1}
{\mathbf}{v}_{t+1}^i &= m {\mathbf}{v}_{t}^i + {\mathbf}{g}_{t}^i \\
\label{eq:ab-step-only-2}
{\mathbf}{w}_{t+1}^i &= {\mathbf}{w}_{t}^i - \eta \left( m {\mathbf}{v}_{t}^i + \alpha_{t}^i {\mathbf}{g}_{t}^i \right)\end{aligned}$$
In Figure \[fig:exp5-vel-step\], we measure the delay tolerance of SGDM with either AB-vel-only or AB-weight-only. We find that most of the delay tolerance of AB comes from scaling the gradient before updating the velocity. This supports the theory that balancing the velocity norm and dampening oscillations is crucial to mitigating delays.
![Scaling the gradient during the velocity update is more important for delay tolerance than scaling the gradient during the weight update.[]{data-label="fig:exp5-vel-step"}](figures/exp5-ab-variants.pdf "fig:"){width="\linewidth"} -0.1in
Extended Noisy Quadratic Model Analysis
=======================================
-0.05in {width="0.8\linewidth"} -0.175in
-0.1in \[fig:heatmap\_microstep\]
Problem Setup {#sec:nqm_setup}
-------------
We assume that the convex quadratic is centered and aligned with the axis. This can be done without a loss of generality since the optimizers considered are both translation and rotation-invariant[^2]. We write the loss as: $$\mathcal{L}({\mathbf}{w}) = \frac{1}{2} {\mathbf}{w}^T {\mathbf}{H} {\mathbf}{w} = \frac{1}{2} \sum_{k=1}^{N} \lambda_k w_k^2$$ where ${\mathbf}{w} = [w_1, ..., w_N]^T$ are the weights to be optimized and ${\mathbf}{H}=\textrm{diag}(\lambda_1, ..., \lambda_N)$ is the Hessian of the loss.
Following @zhang2019algorithmic [-@zhang2019algorithmic] we assume additive gradient noise with covariance equal to the Hessian of the loss. We also adopt their Hessian eigenvalue spectrum which is of the form $\{\frac{1}{j}\}_{j=1}^{N}$ with $N=10^4$ which they show can closely match certain neural networks. We write the gradient at timestep $t$ as: $${\mathbf}{g}_t = {\mathbf}{H} {\mathbf}{w}_{t-D} + \sigma \mathcal{N}({\mathbf}{0}, {\mathbf}{H})$$ where ${\mathbf}{w}_{t-D}$ are the weights with delay $D$, $\mathcal{N}({\mathbf}{0}, {\mathbf}{H})$ is the multivariate normal distribution noise with mean ${\mathbf}{0}$ and covariance matrix ${\mathbf}{H}$, and $\sigma$ scales the noise.
@zhang2019algorithmic [-@zhang2019algorithmic] use the noisy quadratic model to explore the effects of batch size (simulated by modifying the noise scale $\sigma$) with good predictive results for neural networks. Their focus is on linear optimizers which allows them to derive closed form solutions for the convergence. Since AB is non-linear and has a cross-feature dependency we explicitly carry out the optimization on the full quadratic and do not use any sort of binning of similar eigenvalues. The objective of the optimization is to bring the loss below the target loss $\varepsilon=0.01$ and the weights are initialized to ${\mathbf}{w}={\mathbf}{1}$. We measure the quality of trajectories with the number of steps, $T$, required to reach the target loss.
Energy Measure {#sec:nqm_energy}
--------------
In Section \[sec:nqm\] we explore the effect of AB on individual components. To do this effectively we introduce an energy measure to estimate the convergence of individual components and compare it between states. Using the loss for this is problematic because it oscillates and a low loss does not necessarily indicate convergence (if the velocity is large). In more realistic settings we can not easily determine what the components are and therefore can not compute component losses, apply different learning rates to different components or early stop individual components. We use a similar energy model as @hermans2018gradient [-@hermans2018gradient] that accounts for both the loss (potential energy) and velocity (kinetic energy). Our energy ($E$) is normalized with the learning rate ($\eta$) making it directly comparable with the loss: $$\begin{aligned}
E_t &= \mathcal{L}({\mathbf}{w}_t) + \frac{1}{2}\frac{\|{\mathbf}{w}_t-{\mathbf}{w}_{t-1}\|^2}{\eta} \\
&= \mathcal{L}({\mathbf}{w}_t) + \frac{1}{2}\eta \|{\mathbf}{v}_t\|^2\end{aligned}$$
Note that the energy upper bounds the loss so an energy of zero would mean that a component has fully converged. For an oscillating trajectory the energy is roughly equal to the loss at the extreme points where the velocity is approximately zero. Overall the energy can be viewed as roughly estimating the envelope of the loss for an oscillating component. This makes it easier to estimate convergence from a single state and compare the convergence of different states than using the loss directly.
For t = 1...T do: Compute gradient ${\mathbf}{g}$ ${\mathbf}{v} \gets m {\mathbf}{v}$ For i = 1...S do: $\alpha \gets 1 - \rho \frac{\langle {\mathbf}{g}, {\mathbf}{v} \rangle}{\max(\|{\mathbf}{g}\| \|{\mathbf}{v}\|, \epsilon)}$ ${\mathbf}{v} \gets {\mathbf}{v} + \frac{\alpha}{S}{\mathbf}{g}$ ${\mathbf}{w} \gets {\mathbf}{w}-\eta {\mathbf}{v}$
-0.05in {width="1.0\linewidth"} -0.175in
-0.1in \[fig:heatmap\_abew\]
Micro-stepping {#sec:nqm_microstepping}
--------------
Figure \[fig:heatmap\] shows that in the no-delay and no-noise case AB can slightly reduce the region of stability. This leads to sightly worse optimal trajectories. In Section \[sec:nqm\] we state that this happens because AB can magnify certain high frequency oscillations. With noise and delays this does not seem to be an issue, potentially because the baseline SGDM trajectories don’t converge to the target loss for hyperparameter settings where high frequency oscillations could occur.
As an example of AB magnifying oscillations, consider the case where AB is applied on a single component with curvature $\lambda$, learning rate $\frac{1}{\lambda} < \eta < \frac{2}{\lambda}$ and very small momentum value $m \approx 0$. This will result in a trajectory that overshoots the minimum at every step and ${\mathbf}{g}$ and ${\mathbf}{v}$ will always be oppositely aligned. This causes AB to apply a constant $\alpha=1+\rho$, effectively increasing the learning rate, potentially causing instability.
The issue arises from AB over-correcting the velocity when the gradient ${\mathbf}{g}_t$ and velocity ${\mathbf}{v}_t$ are oppositely aligned. This happens because AB scales the gradient based on the alignment of ${\mathbf}{v}_t$ and ${\mathbf}{g}_t$ without considering the resulting alignment of ${\mathbf}{g}_t$ and ${\mathbf}{v}_{t+1}$. In cases where $\|{\mathbf}{v}_t\|$ is small and ${\mathbf}{g}_t$ and ${\mathbf}{v}_{t+1}$ are oppositely aligned this can lead to larger $\|{\mathbf}{v}_{t+1}\|$. Various forms of clamping can help here, for example enforcing $\alpha \le 1$ but we have found that this can reduce the effectiveness of AB.
Another way is to change the velocity update to consider more than just the initial alignment of ${\mathbf}{g}_t$ and ${\mathbf}{v}_t$. We can divide the velocity update into $S$ “micro-steps", calculating a different $\alpha$ for each one as shown in Algorithm \[alg:AB\_microstepping\]. For large values of $S$ micro-stepping might have significant overhead but could help AB in the large batch size or low noise settings. Figure \[fig:heatmap\_microstep\] shows the effects of micro-stepping on the speed of convergence. It shows that with micro-stepping AB can tolerate higher learning rates than plain SGDM and slightly decreases the minimum steps needed to reach the target loss.
Parameter Grouping {#sec:nqm_grouping}
------------------
Adaptive Braking operates by computing an alignment score between the gradient and velocity for a group of parameters and then scaling the gradient based on the alignment. The performance of AB depends on the choice of groups. For neural networks we find that filter-wise grouping works well, see Appendix \[sec:param-grouping\]. In this section we explore the effect of grouping for convex quadratics, in particular we compare the global form (with a single group) to the element-wise form.
To decease compute requirements we use low dimensional models in this section. We use 32 components with a log-uniform eigenvalue spectrum from $10^{-4}$ to $1$ and a target loss of $\epsilon=10^{-5}$. Figure \[fig:heatmap\_abew\] shows the steps required to reach the target loss for different AB forms for a delay of $1$ and no noise. We can see that the global form of AB outperforms the baseline. The element-wise form works really well if the quadratic aligns with the axes. In this case it is really performing component-wise AB. This can speed up the convergence of all components that are sufficiently underdamped. For overdamped components this slows their convergence (by effectively lowering the learning rate). However, since all components are stabilized, higher learning rates can be used which at least partially compensates for this effect. Ideally we could apply AB selectively to the components that need to be dampened without affecting the other ones. Unfortunately we generally don’t know what the components are and element-wise AB does not necessarily outperform the global form of AB for a random alignment (see Figure \[fig:heatmap\_abew\]).
Overall there seems to be a trade-off in the group size. Each additional component in a group lowers the correlation of the scaling to the other components, weakening the dampening effect. Using a larger number of groups, with fewer components each, may give stronger correlations increasing the dampening effect. Ideally the most unstable components should fall in separate groups so they can be dampened effectively. It may also be important for components to be contained within a single group. If this is not the case, different coordinates of the gradient for a given component may be scaled differently. This effectively rotates the gradient, potentially causing it to interfere with the convergence of other components. This might be why element-wise AB generally doesn’t perform as well as using larger groups (when the loss is not aligned as is usually the case). The filter-wise grouping we use for neural networks (see Appendix \[sec:param-grouping\]) could strike a good balance between the number of groups and splitting components between groups.
[^1]: Algorithmic details can be found in Appendix \[pseudocode\].
[^2]: To make AB rotation-invariant we use a single group spanning all parameters. We investigate different groupings in Appendix \[sec:param-grouping\].
|
---
abstract: |
The local chromatic number of a graph was introduced in [@EFHKRS]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs.
We use an old topological result of Ky Fan [@kyfan] which generalizes the Borsuk-Ulam theorem. It implies the existence of a multicolored copy of the complete bipartite graph $K_{\lceil
t/2\rceil,\lfloor t/2\rfloor}$ in every proper coloring of many graphs whose chromatic number $t$ is determined via a topological argument. (This was in particular noted for Kneser graphs by Ky Fan [@kyfan2].) This yields a lower bound of $\lceil t/2\rceil+1$ for the local chromatic number of these graphs. We show this bound to be tight or almost tight in many cases.
As another consequence of the above we prove that the graphs considered here have equal circular and ordinary chromatic numbers if the latter is even. This partially proves a conjecture of Johnson, Holroyd, and Stahl and was independently attained by F. Meunier [@meunier]. We also show that odd chromatic Schrijver graphs behave differently, their circular chromatic number can be arbitrarily close to the other extreme.
author:
- |
[**Gábor Simonyi**]{}[^1] $\qquad$ [**Gábor Tardos**]{}[^2]\
\
Alfréd Rényi Institute of Mathematics,\
Hungarian Academy of Sciences,\
1364 Budapest, POB 127, Hungary\
\
[simonyi@renyi.hu]{} [tardos@renyi.hu]{}
title: 'Local chromatic number, Ky Fan’s theorem, and circular colorings'
---
Introduction
============
The local chromatic number of a graph is defined in [@EFHKRS] as the minimum number of colors that must appear within distance $1$ of a vertex. For the formal definition let $N(v)=N_G(v)$ denote the [*neighborhood*]{} of a vertex $v$ in a graph $G$, that is, $N(v)$ is the set of vertices $v$ is connected to.
\[defi:lochr\] [([@EFHKRS])]{} The [*local chromatic number*]{} $\psi(G)$ of a graph $G$ is $$\psi(G):=\min_c \max_{v\in V(G)} |\{c(u): u \in N(v)\}|+1,$$ where the minimum is taken over all proper colorings $c$ of $G$.
The $+1$ term comes traditionally from considering “closed neighborhoods” $N(v)\cup\{v\}$ and results in a simpler form of the relations with other coloring parameters.
While the local chromatic number of a graph $G$ obviously cannot be more than the chromatic number $\chi(G)$, somewhat surprisingly, it can be arbitrarily less, cf. [@EFHKRS], [@Fur]. On the other hand, it was shown in [@KPS] that $$\psi(G)\ge \chi_f(G)$$ holds for any graph $G$, where $\chi_f(G)$ denotes the fractional chromatic number of $G$. For the definition and basic properties of the fractional chromatic number we refer to the books [@SchU; @GR].
This suggests to investigate the local chromatic number of graphs for which the chromatic number and the fractional chromatic number are far apart. This is our main goal in this paper.
Prime examples of graphs with a large gap between the chromatic and the fractional chromatic number are Kneser graphs and Mycielski graphs, cf. [@SchU]. Other, closely related examples are provided by Schrijver graphs, that are vertex color-critical induced subgraphs of Kneser graphs, and many of the so-called generalized Mycielski graphs. In this introductory section we focus on Kneser graphs and Schrijver graphs, Mycielski graphs and generalized Mycielski graphs will be treated in detail in Subsection \[subsect:gmyc\].
We recall that the Kneser graph $KG(n,k)$ is defined for parameters $n\ge 2k$ as the graph with all $k$-subsets of an $n$-set as vertices where two such vertices are connected if they represent disjoint $k$-sets. It is a celebrated result of Lovász [@LLKn] (see also [@Bar; @Gre]) proving the earlier conjecture of Kneser, that $\chi(KG(n,k))=n-2k+2$. For the fractional chromatic number one has $\chi_f(KG(n,k))=n/k$ as easily follows from the vertex-transitivity of $KG(n,k)$ via the Erdős-Ko-Rado theorem, see [@SchU; @GR].
Bárány’s proof [@Bar] of the Lovász-Kneser theorem was generalized by Schrijver [@Schr] who found a fascinating family of subgraphs of Kneser graphs that are vertex-critical with respect to the chromatic number.
Let $[n]$ denote the set $\{1,2,\dots,n\}$.
[([@Schr])]{} The stable Kneser graph or [*Schrijver graph*]{} $SG(n,k)$ is defined as follows. $$\begin{aligned}
V(SG(n,k))&=&\{A\subseteq [n]: |A|=k,\forall i:\ \{i,i+1\}\nsubseteq
A\ \ \hbox{\rm and}\ \ \{1,n\}\nsubseteq A\}\\
E(SG(n,k))&=&\{\{A,B\}: A\cap B=\emptyset\}\end{aligned}$$
Thus $SG(n,k)$ is the subgraph induced by those vertices of $KG(n,k)$ that contain no neighboring elements in the cyclically arranged basic set $\{1,2,\dots,n\}$. These are sometimes called [*stable $k$-subsets*]{}. The result of Schrijver in [@Schr] is that $\chi(SG(n,k))=n-2k+2(=\chi(KG(n,k))$, but deleting any vertex of $SG(n,k)$ the chromatic number drops, i.e., $SG(n,k)$ is vertex-critical with respect to the chromatic number. Recently Talbot [@Tal] proved an Erdős-Ko-Rado type result, conjectured by Holroyd and Johnson [@HJ], which implies that the ratio of the number of vertices and the independence number in $SG(n,k)$ is $n/k$. This gives $n/k\leq
\chi_f(SG(n,k))$ and equality follows by $\chi_f(SG(n,k))\leq
\chi_f(KG(n,k))=n/k$. Notice that $SG(n,k)$ is not vertex-transitive in general. See more on Schrijver graphs in [@BjLo; @LihLiu; @Mat; @Zie].
Concerning the local chromatic number it was observed by several people [@ZF; @JK], that $\psi(KG(n,k))\ge
n-3k+3$ holds, since the neighborhood of any vertex in $KG(n,k)$ induces a $KG(n-k,k)$ with chromatic number $n-3k+2$. Thus for $n/k$ fixed but larger than $3$, $\psi(G)$ goes to infinity with $n$ and $k$. In fact, the results of [@EFHKRS] have a similar implication also for $2<n/k\leq 3.$ Namely, it follows from those results, that if a series of graphs $G_1,\dots, G_i,\dots$ is such that $\psi(G_i)$ is bounded, while $\chi(G_i)$ goes to infinity, then the number of colors to be used in colorings attaining the local chromatic number grows at least doubly exponentially in the chromatic number. However, Kneser graphs with $n/k$ fixed and $n$ (therefore also the chromatic number $n-2k+2$) going to infinity cannot satisfy this, since the total number of vertices grows simply exponentially in the chromatic number.
The estimates mentioned in the previous paragraph are elementary. On the other hand, all known proofs for $\chi(KG(n,k))\ge n-2k+2$ use topology or at least have a topological flavor (see [@LLKn; @Bar; @Gre; @MatCCA] to mention just a few such proofs). They use (or at least, are inspired by) the Borsuk-Ulam theorem.
In this paper we use a stronger topological result due to Ky Fan [@kyfan] to establish that all proper colorings of a $t$-chromatic Kneser, Schrijver or generalized Mycielski graph contain a multicolored copy of a balanced complete bipartite graph. This was noticed by Ky Fan for Kneser graphs [@kyfan2]. We also show that the implied lower bound of $\lceil
t/2\rceil+1$ on the local chromatic number is tight or almost tight for many Schrijver and generalized Mycielski graphs.
In the following section we summarize our main results in more detail.
Results {#sect:results}
=======
In this section we summarize our results without introducing the topological notions needed to state the results in their full generality. We will introduce the phrase that a graph $G$ is [*topologically $t$-chromatic*]{} meaning that $\chi(G)\ge t$ and this fact can be shown by a specific topological method, see Subsection \[subsect:bounds\]. Here we use this phrase only to emphasize the generality of the corresponding statements, but the reader can always substitute the phrase “a topologically $t$-chromatic graph” by “a $t$-chromatic Kneser graph” or “a $t$-chromatic Schrijver graph” or by “a generalized Mycielski graph of chromatic number $t$”.
Our general lower bound for the local chromatic number proven in Section \[sect:lowb\] is the following.
\[thm:lowb\] If $G$ is topologically $t$-chromatic for some $t\ge2$, then $$\psi(G)\ge\left\lceil t\over 2\right\rceil+1.$$
This result on the local chromatic number is the immediate consequence of the Zig-zag theorem in Subsection \[subsect:kyfzag\] that we state here in a somewhat weaker form:
\[thm:bip\] Let $G$ be a topologically $t$-chromatic graph and let $c$ be a proper coloring of $G$ with an arbitrary number of colors. Then there exists a complete bipartite subgraph $K_{\lceil{t\over 2}\rceil,\lfloor{t\over
2}\rfloor}$ of $G$ all vertices of which receive a different color in $c$.
We use Ky Fan’s generalization of the Borsuk-Ulam theorem [@kyfan] for the proof. The Zig-zag theorem was previously established for Kneser graphs by Ky Fan [@kyfan2].
We remark that János Körner [@JK] suggested to introduce a graph invariant $b(G)$ which is the size (number of points) of the largest completely multicolored complete bipartite graph that should appear in any proper coloring of graph $G$. It is obvious from the definition that this parameter is bounded from above by $\chi(G)$ and bounded from below by the local chromatic number $\psi(G)$. An obvious consequence of Theorem \[thm:bip\] is that if $G$ is topologically $t$-chromatic, then $b(G)\ge t$.
In Section \[sect:upb\] we show that Theorem \[thm:lowb\] is essentially tight for several Schrijver and generalized Mycielski graphs. In particular, this is always the case for a topologically $t$-chromatic graph that has a [*wide*]{} $t$-coloring as defined in Definition \[defi:wide\] in Subsection \[ss:wide\].
As the first application of our result on wide colorings we show, that if the chromatic number is fixed and odd, and the size of the Schrijver graph is large enough, then Theorem \[thm:lowb\] is exactly tight:
\[thm:upb\] If $t=n-2k+2>2$ is odd and $n\ge4t^2-7t$ then $$\psi(SG(n,k))=\left\lceil t\over 2\right\rceil+1.$$
See Remark 4 in Subsection \[subsect:schr\] for a relaxed bound on $n$. The proof of Theorem \[thm:upb\] is combinatorial. It will also show that the claimed value of $\psi(SG(n,k))$ can be attained with a coloring using $t+1$ colors and avoiding the appearance of a totally multicolored $K_{\lceil{t\over
2}\rceil,\lceil{t\over 2}\rceil}.$ To appreciate the latter property, cf. Theorem \[thm:bip\].
Since $SG(n,k)$ is an induced subgraph of $SG(n+1,k)$ Theorem \[thm:upb\] immediately implies that for every fixed even $t=n-2k+2$ and $n, k$ large enough $$\psi(SG(n,k))\in\left\{{t\over 2}+1,{t\over 2}+2\right\}.$$
The lower bound for the local chromatic number in Theorem \[thm:lowb\] is smaller than $t$ whenever $t\ge 4$ but Theorem \[thm:upb\] claims the existence of Schrijver graphs with smaller local than ordinary chromatic number only with chromatic number $5$ and up. In an upcoming paper [@up] we prove that the local chromatic number of all $4$-chromatic Kneser, Schrijver, or generalized Mycielski graphs is $4$. The reason is that all these graphs satisfy a somewhat stronger property, they are [*strongly*]{} topologically $4$-chromatic (see Definition \[defi:topres\]). We will, however, also show in [@up] that topologically $4$-chromatic graphs of local chromatic number $3$ do exist.
To demonstrate that requiring large $n$ and $k$ in Theorem \[thm:upb\] is crucial we prove the following statement.
\[prop:k2\] $\psi(SG(n,2))=n-2=\chi(SG(n,2))$ for every $n\ge 4$.
As a second application of wide colorings we prove in Subsection \[subsect:gmyc\] that Theorem \[thm:lowb\] is also tight for several generalized Mycielski graphs. These graphs will be denoted by $M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(K_2)$ where ${{\mbox{\boldmath$r$}}}=(r_1,\dots,r_d)$ is a vector of positive integers. See Subsection \[subsect:gmyc\] for the definition. Informally, $d$ is the number of iterations and $r_i$ is the number of “levels” in iteration $i$ of the generalized Mycielski construction. $M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(K_2)$ is proven to be $(d+2)$-chromatic “because of a topological reason” by Stiebitz [@Stieb]. This topological reason implies that these graphs are strongly topologically $(d+2)$-chromatic. Thus Theorem \[thm:lowb\] applies and gives the lower bound part of the following result.
\[thm:gmycspec7\] If ${{\mbox{\boldmath$r$}}}=(r_1,\ldots,r_d)$, $d$ is odd, and $r_i\ge 7$ for all $i$, then $$\psi(M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(K_2))=\left\lceil d\over2\right\rceil+2.$$
It will be shown in Theorem \[thm:gmycspec4\] that relaxing the $r_i\ge 7$ condition to $r_i\ge 4$ an only slightly weaker upper bound is still valid. As a counterpart we also show (see Proposition \[prop:myc2\] in Subsection \[subsect:gmyc\]) that for the ordinary Mycielski construction, which is the special case of ${{\mbox{\boldmath$r$}}}=(2,\dots,2)$, the local chromatic number behaves just like the chromatic number.
The Borsuk-Ulam Theorem in topology is known to be equivalent (see Lovász [@LLgomb]) to the validity of a tight lower bound on the chromatic number of graphs defined on the $n$-dimensional sphere, called Borsuk graphs. In Subsection \[subsect:ctopI\] we prove that the local chromatic number of Borsuk graphs behaves similarly as that of the graphs already mentioned above. In this subsection we also formulate a topological consequence of our results on the tightness of the result of Ky Fan [@kyfan]. We also give a direct proof for the same tightness result.
The circular chromatic number $\chi_c(G)$ of a graph $G$ was introduced by Vince [@Vin], see Definition \[defi:circ\] in Section \[sec:circ\]. It satisfies $\chi(G)-1<\chi_c(G)\le\chi(G)$. In Section \[sec:circ\] we prove the following result using the Zig-zag theorem.
\[thm:circ\] If $G$ is topologically $t$-chromatic and $t$ is even, then $\chi_c(G)\ge t$.
This theorem implies that $\chi_c(G)=\chi(G)$ if the chromatic number is even for Kneser graphs, Schrijver graphs, generalized Mycielski graphs, and certain Borsuk graphs. The result on Kneser and Schrijver graphs gives a partial solution of a conjecture by Johnson, Holroyd, and Stahl [@JHS] and a partial answer to a question of Hajiabolhassan and Zhu [@HZ]. These results were independently obtained by Meunier [@meunier]. The result on generalized Mycielski graphs answers a question of Chang, Huang, and Zhu [@CHZ].
We will also discuss the circular chromatic number of odd chromatic Borsuk and Schrijver graphs showing that they can be close to one less than the chromatic number. For generalized Mycielski graphs a similar result was proven by Lam, Lin, Gu, and Song [@LLGS], that we will also use.
Lower bound {#sect:lowb}
===========
Topological preliminaries {#toppre}
-------------------------
The following is a brief overview of some of the topological concepts we need. We refer to [@Bjhand; @Hat] and [@Mat] for basic concepts and also for a more detailed discussion of the notions and facts given below.
A [*$\mathbb{Z}_2$-space*]{} (or [*involution space*]{}) is a pair $(T,\nu)$ of a topological space $T$ and the involution $\nu:T\to T$, which is continuous and satisfies that $\nu^2$ is the identity map. The points $x\in T$ and $\nu(x)$ are called [*antipodal*]{}. The involution $\nu$ and the $\mathbb{Z}_2$-space $(T,\nu)$ are [*free*]{} if $\nu(x)\ne x$ for all points $x$ of $T$. If the involution is understood from the context we speak about $T$ rather than the pair $(T,\nu)$. This is the case, in particular, for the unit sphere $S^d$ in ${\mathbb R}^{d+1}$ with the involution given by the central reflection ${{\mbox{\boldmath$x$}}}\mapsto-{{\mbox{\boldmath$x$}}}$. A continuous map $f:S\to T$ between $\mathbb{Z}_2$-spaces $(S,\nu)$ and $(T,\pi)$ is a [*$\mathbb{Z}_2$-map*]{} (or an [*equivariant map*]{}) if it respects the respective involutions, that is $f\circ\nu=\pi\circ f$. If such a map exists we write $(S,\nu)\to(T,\pi)$. If $(S,\nu)\to(T,\pi)$ does not hold we write $(S,\nu)\not\to(T,\pi)$. If both $S\to T$ and $T\to S$ we call the $\mathbb Z_2$-spaces $S$ and $T$ $\mathbb
Z_2$-equivalent and write $S\leftrightarrow T$.
We try to avoid using homotopy equivalence and $\2$-homotopy equivalence (i.e., homotopy equivalence given by $\2$-maps), but we will have to use two simple observations. First, if the $\2$-spaces $S$ and $T$ are $\2$-homotopy equivalent, then $S\leftrightarrow T$. Second, if the space $S$ is homotopy equivalent to a sphere $S^h$ (this relation is between topological spaces, not $\2$-spaces), then for any involution $\nu$ we have $S^h\to(S,\nu)$.
The $\mathbb{Z}_2$-index of a $\mathbb{Z}_2$-space $(T,\nu)$ is defined (see e.g. [@MZ; @Mat]) as $${\rm ind}(T,\nu):=\min\{d\ge 0:(T,\nu)\to S^d\},$$ where ${\rm ind}(T,\nu)$ is set to be $\infty$ if $(T,\nu)\not\to S^d$ for all $d$.
The $\mathbb{Z}_2$-coindex of a $\mathbb{Z}_2$-space $(T,\nu)$ is defined as $${\rm coind}(T,\nu):=\max\{d\ge 0:S^d\to(T,\nu)\}.$$ If such a map exists for all $d$, then we set ${\rm coind}(T,\nu)=\infty$. Notice that if $(T,\nu)$ is not free, we have ${\rm
ind}(T,\nu)={\rm coind}(T,\nu)=\infty$.
Note that $S\to T$ implies $\ind(S)\le\ind(T)$ and $\coind(S)\le\coind(T)$. In particular, $\2$-equivalent spaces have equal index and also equal coindex.
The celebrated Borsuk-Ulam Theorem can be stated in many equivalent forms. Here we state three of them. For more equivalent versions and several proofs we refer to [@Mat]. Here (i) and (ii) are standard forms of the Borsuk-Ulam Theorem, while (iii) is clearly equivalent to (ii).
[**Borsuk-Ulam Theorem.**]{}
**
(i)
: (Lyusternik-Schnirel’man version) Let $d\ge0$ and let ${{\cal H}}$ be a collection of open (or closed) sets covering $S^d$ with no $H\in{{\cal H}}$ containing a pair of antipodal points. Then $|{{\cal H}}|\ge d+2$.
(ii)
: $S^{d+1}\not\to S^d$ for any $d\ge 0$.
(iii)
: For a $\mathbb{Z}_2$-space $T$ we have ${\rm ind}(T)\ge
{\rm coind}(T)$.
The suspension $\susp(S)$ of a topological space $S$ is defined as the factor of the space $S\times[-1,1]$ that identifies all the points in $S\times\{-1\}$ and identifies also the points in $S\times\{1\}$. If $S$ is a $\2$-space with the involution $\nu$, then the suspension $\susp(S)$ is also a $\2$-space with the involution $(x,t)\mapsto(\nu(x),-t)$. Any $\2$-map $f:S\to T$ naturally extends to a $\2$-map $\susp(f):\susp(S)\to\susp(T)$ given by $(x,t)\mapsto(f(x),t)$. We have $\susp(S^n)\cong S^{n+1}$ with a $\2$-homeomorphism. These observations show the well known inequalities below.
\[suspension\] For any $\2$-space $S$ $\ind(\susp(S))\le\ind(S)+1$ and $\coind(\susp(S))\ge\coind(S)+1$.
A(n abstract) simplicial complex $K$ is a non-empty, hereditary set system. That is, $F\in K$, $F'\subseteq F$ implies $F'\in K$ and we have $\emptyset\in K$. In this paper we consider only finite simplicial complexes. The non-empty sets in $K$ are called [*simplices*]{}. We call the set $V(K)=\{x:\{x\}\in K\}$ the set of [*vertices*]{} of $K$. In a [*geometric realization*]{} of $K$ a vertex $x$ corresponds to a point $||x||$ in a Euclidean space, a simplex $\sigma$ corresponds to its [*body*]{}, the convex hull of its vertices: $||\sigma||={\rm conv}(\{||x||:x\in\sigma\})$. We assume that the points $||x||$ for $x\in\sigma$ are affine independent, and so $||\sigma||$ is a geometric simplex. We also assume that disjoint simplices have disjoint bodies. The body of the complex $K$ is $||K||=\cup_{\sigma\in K}||\sigma||$, it is determined up to homeomorphism by $K$. Any point in $p\in||K||$ has a unique representation as a convex combination $p=\sum_{x\in
V(K)}\alpha_x||x||$ such that $\{x:\alpha_x>0\}\in K$.
A map $f:V(K)\to V(L)$ is called simplicial if it maps simplices to simplices, that is $\sigma\in K$ implies $f(\sigma)\in L$. In this case we define $||f||:||K||\to||L||$ by setting $||f||(||x||)=||f(x)||$ for vertices $x\in V(K)$ and taking an affine extension of this function to the bodies of each of the simplices in $K$. If $||K||$ and $||L||$ are $\2$-spaces (usually with an involution also given by simplicial maps), then we say that $f$ is a [*$\2$-map*]{} if $||f||$ is a $\2$-map. If $||K||$ is a $\2$-space we use $\ind(K)$ and $\coind(K)$ for $\ind(||K||)$ and $\coind(||K||)$, respectively.
Following the papers [@AFL; @Kriz; @MZ] we introduce the [*box complex*]{} $B_0(G)$ for any finite graph $G$. See [@MZ] for several similar complexes. We define $B_0(G)$ to be a simplicial complex on the vertices $V(G)\times\{1,2\}$. For subsets $S,T\subseteq V(G)$ we denote the set $S\times\{1\}\cup
T\times\{2\}$ by $S\uplus T$, following the convention of [@Mat; @MZ]. For $v\in V(G)$ we denote by $+v$ the vertex $(v,1)\in\{v\}\uplus\emptyset$ and $-v$ denotes the vertex $(v,2)\in\emptyset\uplus\{v\}$. We set $S\uplus T\in B_0(G)$ if $S\cap T=\emptyset$ and the complete bipartite graph with sides $S$ and $T$ is a subgraph of $G$. Note that $V(G)\uplus\emptyset$ and $\emptyset\uplus V(G)$ are simplices of $B_0(G)$.
The $\mathbb{Z}_2$-map $S\uplus T\mapsto T\uplus S$ acts simplicially on $B_0(G)$. It makes the body of the complex a free $\mathbb{Z}_2$-space.
We define the [*hom space*]{} $H(G)$ of $G$ to be the subspace consisting of those points $p\in||B_0(G)||$ that, when written as a convex combination $p=\sum_{x\in V(B_0(G))}\alpha_x||x||$ with $\{x:\alpha_x>0\}\in B_0(G)$ give $\sum_{x\in V(G)\uplus\emptyset}\alpha_x=1/2$.
Notice that $H(G)$ can also be obtained as the body of a [*cell complex*]{} $Hom(K_2,G)$, see [@BK], or a simplicial complex $B_{chain}(G)$, see [@MZ].
A useful connection between $B_0(G)$ and $H(G)$ follows from a combination of results of Csorba [@Cs] and Matoušek and Ziegler [@MZ].
\[csorba\] $||B_0(G)||\leftrightarrow\susp(H(G))$
[[**Proof.** ]{}]{}Csorba [@Cs] proves the $\2$-homotopy equivalence of $||B_0(G)||$ and the suspension of the body of yet another box complex $B(G)$ of $G$. As we mentioned, $\2$-homotopy equivalence implies $\2$-equivalence. Matoušek and Ziegler [@MZ] prove the $\2$-equivalence of $||B(G)||$ and $H(G)$. Finally for $\2$-spaces $S$ and $T$ if $S\to T$, then $\susp(S)\to\susp(T)$, therefore $||B(G)||\leftrightarrow H(G)$ implies $\susp(||B(G)||)\leftrightarrow\susp(H(G))$.
Note that Csorba [@Cs] proves, cf. also Živaljević [@Ziv], the $\2$-homotopy equivalence of $||B(G)||$ and $H(G)$, and therefore we could also claim $\2$-homotopy equivalence in Proposition \[csorba\].
Some earlier topological bounds {#subsect:bounds}
-------------------------------
A graph homomorphism is an edge preserving map from the vertex set of a graph $F$ to the vertex set of another graph $G$. If there is a homomorphism $f$ from $F$ to $G$, then it generates a simplicial map from $B_0(F)$ to $B_0(G)$ in the natural way. This map is a $\mathbb{Z}_2$-map and thus it shows $||B_0(F)||\to||B_0(G)||$. Here $||B_0(F)||\not\to||B_0(G)||$ can often be proved using the indexes or coindexes of these complexes and it implies the non-existence of a homomorphism from $F$ to $G$. A similar argument applies with the spaces $H(\cdot)$ in place of $||B_0(\cdot)||$.
Coloring a graph $G$ with $m$ colors can be considered as a graph homomorphism from $G$ to the complete graph $K_m$. The box complex $B_0(K_m)$ is the boundary complex of the $m$-dimensional [*cross-polytope*]{} (i.e., the convex hull of the basis vectors and their negatives in ${\mathbb R}^m$), thus $||B_0(K_m)||\cong S^{m-1}$ with a $\2$-homeomorphism and $\coind(B_0(G))\le\ind(B_0(G))\leq m-1$ is necessary for $G$ being $m$-colorable. Similarly, $\coind(H(G))\le{\rm ind}(H(G))\leq m-2$ is also necessary for $\chi(G)\leq m$ since $H(K_m)$ can be obtained from intersecting the boundary of the $m$-dimensional cross-polytope with the hyperplane $\sum x_i=0$, and therefore $H(K_m)\cong S^{m-2}$ with a $\2$-homeomorphism. These four lower bounds on $\chi(G)$ can be arranged in a single line of inequalities using Lemma \[suspension\] and Proposition \[csorba\]: $$\label{eq:chib1}
\chi(G)\ge\ind(H(G))+2\ge\ind(B_0(G))+1\ge\coind(B_0(G))+1\ge\coind(H(G))+2$$
In fact, many of the known proofs of Kneser’s conjecture can be interpreted as a proof of an appropriate lower bound on the (co)index of one of the above complexes. In particular, Bárány’s simple proof [@Bar] exhibits a map showing $S^{n-2k}\to H(KG(n,k))$ to conclude that ${\rm
coind}(H(KG(n,k)))\ge n-2k$ and thus $\chi(KG(n,k))\ge n-2k+2$. The even simpler proof of Greene [@Gre] exhibits a map showing $S^{n-2k+1}\to B_0(KG(n,k))$ to conclude that ${\rm coind}(B_0(KG(n,k)))\ge
n-2k+1$ and thus $\chi(KG(n,k))\ge n-2k+2$. Schrijver’s proof [@Schr] of $\chi(SG(n,k))\ge n-2k+2$ is a generalization of Bárány’s and it also can be interpreted as a proof of $S^{n-2k}\to H(SG(n,k))$. We remark that the same kind of technique is used with other complexes related to graphs, too. In particular, Lovász’s original proof [@LLKn] can also be considered as exhibiting a $\mathbb{Z}_2$-map from $S^{n-2k}$ to such a complex, different from the ones we consider here. For a detailed discussion of several such complexes and their usefulness in bounding the chromatic number we refer the reader to [@MZ].
The above discussion gives several possible “topological reasons” that can force a graph to be at least $t$-chromatic. Here we single out two such reasons. The statement of our results in Section \[sect:results\] becomes precise by applying the conventions given by the following definition.
\[defi:topres\] We say that a graph $G$ is [*topologically $t$-chromatic*]{} if $${\rm coind}(B_0(G))\ge t-1.$$ We say that a graph $G$ is [*strongly topologically $t$-chromatic*]{} if $${\rm coind}(H(G))\ge t-2.$$
By Equation (\[eq:chib1\]) if a graph is strongly topologically $t$-chromatic, then it is topologically $t$-chromatic, and if $G$ is topologically $t$-chromatic, then $\chi(G)\ge t$. In an upcoming paper [@up] we will show the existence of a graph for any $t\ge 4$ that is topologically $t$-chromatic but not strongly topologically $t$-chromatic. We will also show that the two notions have different consequences in terms of the local chromatic number for $t=4$.
The notion that a graph is (strongly) topologically $t$-chromatic is useful, as it applies to many widely studied classes of graphs. As we mentioned above, Bárány [@Bar] and Schrijver [@Schr] establish this for $t$-chromatic Kneser and Schrijver graphs. For the reader’s convenience we recall the proof here. See the analogous statement for generalized Mycielski graphs and (certain finite subgraphs of the) Borsuk graphs after we introduce those graphs.
\[b-s\][(Bárány; Schrijver)]{} The $t$-chromatic Kneser and Schrijver graphs are strongly topologically $t$-chromatic.
[[**Proof.** ]{}]{} We need to prove that $SG(n,k)$ is strongly topologically $(n-2k+2)$-chromatic, i.e., that ${\rm coind}(H(SG(n,k)))\ge n-2k$. The statement for Kneser graphs follows. For ${{\mbox{\boldmath$x$}}}\in S^{n-2k}$ let $H_{{\mbox{\boldmath\scriptsize$x$}}}$ denote the open hemisphere in $S^{n-2k}$ around ${\mbox{\boldmath$x$}}$. Consider an arrangement of the elements of $[n]$ on $S^{n-2k}$ so that each open hemisphere contains a stable $k$-subset, i.e., a vertex of $SG(n,k)$. It is not hard to check that identifying $i\in[n]$ with ${{\mbox{\boldmath$v$}}}_i/|{{\mbox{\boldmath$v$}}}_i|$ for ${{\mbox{\boldmath$v$}}}_i=(-1)^i(1,i,i^2,\dots,i^{n-2k})\in{\mathbb R}^{n-2k+1}$ provides such an arrangement. For each vertex $v$ of $SG(n,k)$ and ${{\mbox{\boldmath$x$}}}\in S^{n-2k}$ let $D_v({{\mbox{\boldmath$x$}}})$ denote the smallest distance of a point in $v$ from the set $S^{n-2k}\setminus H_{{\mbox{\boldmath\scriptsize$x$}}}$ and let $D({{\mbox{\boldmath$x$}}})=
\sum_{v\in V(SG(n,k))}D_v({{\mbox{\boldmath$x$}}})$. Note that $D_v({{\mbox{\boldmath$x$}}})>0$ if $v$ is contained in $H_{{\mbox{\boldmath\scriptsize$x$}}}$ and therefore $D({{\mbox{\boldmath$x$}}})>0$ for all ${\mbox{\boldmath$x$}}$. Let $f({{\mbox{\boldmath$x$}}}):={1\over2D({{\mbox{\boldmath\scriptsize$x$}}})}\sum_{v\in V(SG(n,k))}D_v({{\mbox{\boldmath$x$}}})||{+}v||+
{1\over2D(-{{\mbox{\boldmath\scriptsize$x$}}})}\sum_{v\in V(SG(n,k))}D_v(-{{\mbox{\boldmath$x$}}})||{-}v||$. This $f$ is a ${\mathbb Z}_2$-map $S^{n-2k}\to H(SG(n,k))$ proving the proposition.
Ky Fan’s result on covers of spheres and the Zig-Zag theorem {#subsect:kyfzag}
------------------------------------------------------------
The following result of Ky Fan [@kyfan] implies the Lyusternik-Schnirel’man version of the Borsuk-Ulam theorem. Here we state two equivalent versions of the result, all in terms of sets covering the sphere. See the original paper for another version generalizing another standard form of the Borsuk-Ulam theorem.
[**Ky Fan’s Theorem.**]{}
**
(i)
: Let $\cal A$ be a system of open (or a finite system of closed) subsets of $S^k$ covering the entire sphere. Assume a linear order $<$ is given on $\cal A$ and all sets $A\in\cal A$ satisfy $A\cap-A=\emptyset$. Then there are sets $A_1<A_2<\dots<A_{k+2}$ of $\cal A$ and a point ${{\mbox{\boldmath$x$}}}\in
S^k$ such that $(-1)^i {{\mbox{\boldmath$x$}}}\in A_i$ for all $i=1,\dots,k+2$.
(ii)
: Let $\cal A$ be a system of open (or a finite system of closed) subsets of $S^k$ such that $\cup_{A\in\cal A}(A\cup-A)=S^k$. Assume a linear order $<$ is given on $\cal A$ and all sets $A\in\cal A$ satisfy $A\cap-A=\emptyset$. Then there are sets $A_1<A_2<\dots<A_{k+1}$ of $\cal A$ and a point ${{\mbox{\boldmath$x$}}}\in S^k$ such that $(-1)^i {{\mbox{\boldmath$x$}}}\in A_i$ for all $i=1,\dots,k+1$.
The Borsuk-Ulam theorem is easily seen to be implied by version (i), that shows in particular, that $|{\cal A}|\ge k+2$. We remark that [@kyfan] contains the above statements only about closed sets. The statements on open sets can be deduced by a standard argument using the compactness of the sphere. We also remark that version (ii) is formulated a little differently in [@kyfan]. A place where one finds exactly the above formulation (for closed sets, but for any $\2$-space) is Bacon’s paper [@Bacon].
[**Zig-zag Theorem**]{} [*Let $G$ be a topologically $t$-chromatic finite graph and let $c$ be an arbitrary proper coloring of $G$ by an arbitrary number of colors. We assume the colors are linearly ordered. Then $G$ contains a complete bipartite subgraph $K_{\lceil{t\over 2}\rceil,\lfloor{t\over 2}\rfloor}$ such that $c$ assigns distinct colors to all $t$ vertices of this subgraph and these colors appear alternating on the two sides of the bipartite subgraph with respect to their order.*]{}
[[**Proof.** ]{}]{}We have $\coind(B_0(G))\ge t-1$, so there exists a $\2$-map $f:S^{t-1}\to
B_0(G)$. For any color $i$ we define a set $A_i\subset S^{t-1}$ letting ${{\mbox{\boldmath$x$}}}\in A_i$ if and only if for the minimal simplex $U_{{\mbox{\boldmath\scriptsize$x$}}}\uplus V_{{\mbox{\boldmath\scriptsize$x$}}}$ containing $f({{\mbox{\boldmath$x$}}})$ there exists a vertex $z\in U_{{\mbox{\boldmath\scriptsize$x$}}}$ with $c(z)=i$. These sets are open, but they do not necessarily cover the entire sphere $S^{t-1}$. Notice that $-A_i$ consists of the points ${{\mbox{\boldmath$x$}}}\in S^{t-1}$ with $-{{\mbox{\boldmath$x$}}}\in A_i$, which happens if and only if there exists a vertex $z\in
U_{-{\mbox{\boldmath\scriptsize$x$}}}$ with $c(z)=i$. Here $U_{-{\mbox{\boldmath\scriptsize$x$}}}=V_{{\mbox{\boldmath\scriptsize$x$}}}$. For every ${{\mbox{\boldmath$x$}}}\in
S^{t-1}$ either $U_{{\mbox{\boldmath\scriptsize$x$}}}$ or $V_{{\mbox{\boldmath\scriptsize$x$}}}$ is not empty, therefore we have $\cup_i(A_i\cup-A_i)=S^{t-1}$. Assume for a contradiction that for a color $i$ we have $A_i\cap-A_i\ne\emptyset$ and let ${\mbox{\boldmath$x$}}$ be a point in the intersection. We have a vertex $z\in U_{{\mbox{\boldmath\scriptsize$x$}}}$ and a vertex $z'\in V_{{\mbox{\boldmath\scriptsize$x$}}}$ with $c(z)=c(z')=i$. By the definition of $B_0(G)$ the vertices $z$ and $z'$ are connected in $G$. This contradicts the choice of $c$ as a proper coloring. The contradiction shows that $A_i\cap
-A_i=\emptyset$ for all colors $i$.
Applying version (ii) of Ky Fan’s theorem we get that for some colors $i_1<i_2<\dots<i_t$ and a point ${{\mbox{\boldmath$x$}}}\in S^{t-1}$ we have $(-1)^j{{\mbox{\boldmath$x$}}}\in
A_{i_j}$ for $j=1,2,\dots t$. This implies the existence of vertices $z_j\in
U_{(-1)^j{\mbox{\boldmath\scriptsize$x$}}}$ with $c(z_j)=i_j$. Now $U_{(-1)^j{\mbox{\boldmath\scriptsize$x$}}}=U_{{\mbox{\boldmath\scriptsize$x$}}}$ for even $j$ and $U_{(-1)^j{\mbox{\boldmath\scriptsize$x$}}}=V_{{\mbox{\boldmath\scriptsize$x$}}}$ for odd $j$. Therefore the complete bipartite graph with sides $\{z_j|\hbox{$j$ is even}\}$ and $\{z_j|\hbox{$j$
is odd}\}$ is a subgraph of $G$ with the required properties.
This result was previously established for Kneser graphs in [@kyfan2].
[*Remark 1.*]{} Since for any fixed coloring we are allowed to order the colors in an arbitrary manner, the Zig-zag Theorem implies the existence of several totally multicolored copies of $K_{\lceil{t\over 2}\rceil,\lfloor{t\over 2}\rfloor}$. For a uniform random order any fixed totally multicolored $K_{\lceil{t\over
2}\rceil,\lfloor{t\over 2}\rfloor}$ satisfies the zig-zag rule with probability $1/{t\choose\lfloor t/2\rfloor}$ if $t$ is odd and with probability $2/{t\choose t/2}$ if $t$ is even. Thus the Zig-zag Theorem implies the existence of many different totally multicolored subgraphs $K_{\lceil{t\over 2}\rceil,\lfloor{t\over 2}\rfloor}$ in $G$: ${t\choose\lfloor t/2\rfloor}$ copies for odd $t$ and ${t\choose t/2}/2$ copies for even $t$.
In the computation above we do not consider two subgraphs different if they are isomorphic with an isomorphism preserving the color of the vertices. With this convention, if the coloring uses only $t$ colors we get a totally multicolored $K_{\lceil{t\over 2}\rceil,\lfloor{t\over 2}\rfloor}$ subgraph with all possible colorings, and the number of these different subgraphs is exactly the lower bound stated. $\Diamond$
[**Proof of Theorems \[thm:lowb\] and \[thm:bip\].**]{}
Theorems \[thm:lowb\] and \[thm:bip\] are direct consequences of the Zig-zag theorem. Indeed, any vertex of the $\lfloor t/2\rfloor$ side of the multicolored complete bipartite graphs has at least $\lceil t/2\rceil$ differently colored neighbors on the other side. $\Box$
[*Remark 2.*]{} Theorem \[thm:lowb\] gives tight lower bounds for the local chromatic number of topologically $t$-chromatic graphs for odd $t$ as several examples of the next section will show. In the upcoming paper [@up] we will present examples that show that the situation is similar for even values of $t$. However, the graphs establishing this fact are [*not*]{} strongly topologically $t$-chromatic, whereas the graphs showing tightness of Theorem \[thm:lowb\] for odd $t$ are. This leaves open the question whether $\psi(G)\ge t/2+2$ holds for all strongly topologically $t$-chromatic graphs $G$ and even $t\ge4$. While we will prove this statement in [@up] for $t=4$ we do not know the answer for higher values of $t$. $\Diamond$
Upper bound {#sect:upb}
===========
In this section we present the combinatorial constructions that prove Theorems \[thm:upb\] and \[thm:gmycspec7\]. In both cases general observations on wide colorings (to be defined below) prove useful. The upper bound in either of Theorems \[thm:upb\] or \[thm:gmycspec7\] implies the existence of certain open covers of spheres. These topological consequences and the local chromatic number of Borsuk graphs are discussed in the last subsection of this section.
Wide colorings {#ss:wide}
--------------
We start here with a general method to alter a $t$-coloring and get a $(t+1)$-coloring showing that $\psi\le t/2+2$. It works if the original coloring was wide as defined below.
\[defi:wide\] A vertex coloring of a graph is called [*wide*]{} if the end vertices of all walks of length $5$ receive different colors.
Note that any wide coloring is proper, furthermore any pair of vertices of distance $3$ or $5$ receive distinct colors. Moreover, if a graph has a wide coloring it does not contain a cycle of length $3$ or $5$. For graphs that do not have cycles of length $3$, $5$, $7$, or $9$ any coloring is wide that assigns different colors to vertices of distance $1$, $3$ or $5$ apart. Another equivalent definition (considered in [@GyJS]) is that a proper coloring is wide if the neighborhood of any color class is an independent set and so is the second neighborhood.
\[lem:5ut\] If a graph $G$ has a wide coloring using $t$ colors, then $\psi(G)\le\lfloor
t/2\rfloor+2$.
[[**Proof.** ]{}]{}Let $c_0$ be the wide $t$-coloring of $G$. We alter this coloring by switching the color of the neighbors of the troublesome vertices to a new color. We define a vertex $x$ to be [*troublesome*]{} if $|c_0(N(x))|>t/2$. Assume the color $\beta$ is not used in the coloring $c_0$. For $x\in V(G)$ we let $$c(x)=\left\{\begin{array}{lll}\beta&&\hbox{if $x$ has a troublesome
neighbor}\\c_0(x)&&\hbox{otherwise.}\end{array}\right.$$
The color class $\beta$ in $c$ is the union of the neighborhoods of troublesome vertices. To see that this is an independent set consider any two vertices $z$ and $z'$ of color $\beta$. Let $y$ be a troublesome neighbor of $z$ and let $y'$ be a troublesome neighbor of $z'$. Both $c_0(N(y))$ and $c_0(N(y'))$ contain more than half of the $t$ colors in $c_0$, therefore these sets are not disjoint. We have a neighbor $x$ of $y$ and a neighbor $x'$ of $y'$ satisfying $c_0(x)=c_0(x')$. This shows that $z$ and $z'$ are not connected, as otherwise the walk $xyzz'y'x'$ of length $5$ would have two end vertices in the same color class.
All other color classes of $c$ are subsets of the corresponding color classes in $c_0$, and are therefore independent. Thus $c$ is a proper coloring.
Any troublesome vertex $x$ has now all its neighbors recolored, therefore $c(N(x))=\{\beta\}$. For the vertices of $G$ that are not troublesome one has $|c_0(N(x))|\le t/2$ and $c(N(x))\subseteq c_0(N(x))\cup\{\beta\}$, therefore $|c(N(x))|\le t/2+1$. Thus the coloring $c$ shows $\psi(G)\le t/2+2$ as claimed. $\Box$
We note that the coloring $c$ found in the proof uses $t+1$ colors and any vertex that sees the maximal number $\lfloor t/2\rfloor+1$ of the colors in its neighborhood must have a neighbor of color $\beta$. In particular, for odd $t$ one will always find two vertices of the same color in any $K_{(t+1)/2,(t+1)/2}$ subgraph.
Schrijver graphs {#subsect:schr}
----------------
In this subsection we prove Theorem \[thm:upb\] which shows that the local chromatic number of Schrijver graphs with certain parameters are as low as allowed by Theorem \[thm:lowb\]. We also prove Proposition \[prop:k2\] to show that for other Schrijver graphs the local chromatic number agrees with the chromatic number.
For the proof of Theorem \[thm:upb\] we will use the following simple lemma.
\[partav\] Let $u,v\subseteq [n]$ be two vertices of $SG(n,k)$. If there is a walk of length $2s$ between $u$ and $v$ in $SG(n,k)$ then $|v\setminus u|\leq s(t-2)$, where $t=n-2k+2=\chi(SG(n,k))$.
[[**Proof.** ]{}]{}Let $xyz$ be a length two walk in $SG(n,k)$. Since $y$ is disjoint from $x$, it contains all but at most $n-2k=t-2$ elements of $[n]\setminus x$. As $z$ is disjoint from $y$ it can contain at most $t-2$ elements not contained in $x$. This proves the statement for $s=1$.
Now let $x_0x_1\dots x_{2s}$ be a $2s$-length walk between $u=x_0$ and $v=x_{2s}$ and assume the statement is true for $s-1$. Since $|v\setminus u|\leq |v\setminus
x_{2s-2}|+|x_{2s-2}\setminus u|\leq (t-2)+(s-1)(t-2)$, the proof is completed by induction. $\Box$
We remark that Lemma \[partav\] remains true for $KG(n,k)$ with literally the same proof, but we will need it for $SG(n,k)$, this is why it is stated that way.
[**Theorem \[thm:upb\]**]{} (restated) [*If $t=n-2k+2>2$ is odd and $n\ge4t^2-7t$, then $$\psi(SG(n,k))=\left\lceil t\over 2\right\rceil+1.$$* ]{}
[[**Proof.** ]{}]{}We need to show that $\psi(SG(n,k))=(t+3)/2$. Note that the $t=3$ case is trivial as all $3$-chromatic graphs have local chromatic number $3$. The lower bound for the local chromatic number follows from Theorem \[thm:lowb\] and Proposition \[b-s\].
We define a wide coloring $c_0$ of $SG(n,k)$ using $t$ colors. From this Lemma \[lem:5ut\] gives the upper bound on $\psi(SG(n,k))$.
Let $[n]=\{1,\dots,n\}$ be partitioned into $t$ sets, each containing an odd number of consecutive elements of $[n]$. More formally, $[n]$ is partitioned into disjoint sets $A_1,\dots,A_t$, where each $A_i$ contains consecutive elements and $|A_i|=2p_i-1$. We need $p_i\ge2t-3$ for the proof, this is possible as long as $n\ge t(4t-7)$ as assumed.
Notice, that $\sum_{i=1}^t(p_i-1)=k-1$, and therefore any $k$-element subset $x$ of $[n]$ must contain more than half (i.e., at least $p_i$) of the elements in some $A_i$. We define our coloring $c_0$ by arbitrarily choosing such an index $i$ as the color $c_0(x)$. This is a proper coloring even for the graph $KG(n,k)$ since if two sets $x$ and $y$ both contain more than half of the elements of $A_i$, then they are not disjoint.
As a coloring of $KG(n,k)$ the coloring $c_0$ is not wide. We need to show that the coloring $c_0$ becomes wide if we restrict it to the subgraph $SG(n,k)$.
The main observation is the following: $A_i$ contains a single subset of cardinality $p_i$ that does not contain two consecutive elements. Let $C_i$ be this set consisting of the first, third, etc. elements of $A_i$. A vertex of $SG(n,k)$ has no two consecutive elements, thus a vertex $x$ of $SG(n,k)$ of color $i$ must contain $C_i$.
Consider a walk $x_0x_1\dots x_5$ of length $5$ in $SG(n,k)$ and let $i=c_0(x_0)$. Thus the set $x_0$ contains $C_i$. By Lemma \[partav\] $|x_4\setminus x_0|\leq 2(t-2)$. In particular, $x_4$ contains all but at most $2t-4$ elements of $C_i$. As $p_i=|C_i|\ge
2t-3$, this means $x_4\cap C_i\neq\emptyset.$ Thus the set $x_5$, which is disjoint from $x_4$, cannot contain all elements of $C_i$, showing $c_0(x_5)\neq i$. This proves that the coloring $c_0$ is wide, thus Lemma \[lem:5ut\] completes the proof of the theorem. $\Box$
Note that the smallest Schrijver graph for which the above proof gives $\psi(SG(n,k))<\chi(SG(n,k))$ is $G=SG(65,31)$ with $\chi(G)=5$ and $\psi(G)=4$. In Remark 4 below we show how the lower bound on $n$ can be lowered somewhat. After that we show that some lower bound is needed as $\psi(SG(n,2))=\chi(SG(n,2))$ for every $n$.
[*Remark 3.*]{} In [@EFHKRS] universal graphs $U(m,r)$ are defined for which it is shown that a graph $G$ can be colored with $m$ colors such that the neighborhood of every vertex contains fewer than $r$ colors if and only if a homomorphism from $G$ to $U(m,r)$ exists. The proof of Theorem \[thm:upb\] gives, for odd $t$, a $(t+1)$-coloring of $SG(n,k)$ (for appropriately large $n$ and $k$ that give chromatic number $t$) for which no neighborhood contains more than $(t+1)/2$ colors, thus establishing the existence of a homomorphism from $SG(n,k)$ to $U(t+1,(t+3)/2)$. This, in particular, proves that $\chi(U(t+1,(t+3)/2))\ge t$, which is a special case of Theorem 2.6 in [@EFHKRS]. It is not hard to see that this inequality is actually an equality. Further, by the composition of the appropriate maps, the existence of this homomorphism also proves that $U(t+1,(t+3)/2)$ is strongly topologically $t$-chromatic. $\Diamond$
[*Remark 4.*]{} For the price of letting the proof be a bit more complicated one can improve upon the bound given on $n$ in Theorem \[thm:upb\]. In particular, one can show that the same conclusion holds for odd $t$ and $n\ge
2t^2-4t+3$. More generally, we can show $\psi(SG(n,k))\le\chi(SG(n,k))-m=n-2k+2-m$ provided that $\chi(SG(n,k))\ge2m+3$ and $n\ge8m^2+16m+9$ or $\chi(SG(n,k))\ge4m+3$ and $n\ge20m+9$. The smallest Schrijver graph for which we can prove that the local chromatic number is smaller than the ordinary chromatic number is $SG(33,15)$ with $\chi=5$ but $\psi=4$. It has $1496$ vertices. (In general, one has $|V(SG(n,k))|={n\over k}{{n-k-1}\choose {k-1}}$, cf. Lemma 1 in [@Tal].) The smallest $n$ and $k$ for which we can prove $\psi(SG(n,k))<\chi(SG(n,k))$ is for the graph $SG(29,12)$ for which $\chi=7$ but $\psi\le6$.
We only sketch the proof. For a similar and more detailed proof see Theorem \[thm:gmycspec4\]. The idea is again to take a basic coloring $c_0$ of $SG(n,k)$ and obtain a new coloring $c$ by recoloring to a new color some neighbors of those vertices $v$ for which $|c_0(N(v))|$ is too large. The novelty is that now we do not recolor all such neighbors, just enough of them, and also the definition of the basic coloring $c_0$ is a bit different. Partition $[n]$ into $t=n-2k+2$ intervals $A_1,\dots,A_t$, each of odd length as in the proof of Theorem \[thm:upb\] and also define $C_i$ similarly to be the unique largest subset of $A_i$ not containing consecutive elements. For a vertex $x$ we define $c_0(x)$ to be the [*smallest*]{} $i$ for which $C_i\subseteq x$. Note that such an $i$ must exist. Now we define when to recolor a vertex to the new color $\beta$ if our goal is to prove $\psi(SG(n,k))\leq b:=t-m$, where $m>0$. We let $c(y)=\beta$ iff $y$ is the neighbor of a vertex $x$ having at least $b-2$ different colors [*smaller*]{} than $c_0(y)$ in its neighborhood. Otherwise, $c(y)=c_0(y)$. It is clear that $|c(N(x))|\leq b-1$ is satisfied, the only problem we face is that $c$ may not be a proper coloring. To avoid this problem we only need that the recolored vertices form an independent set. For each vertex $v$ define the index set $I(v):=\{j: v\cap C_j=\emptyset\}$. If $y$ and $y'$ are recolored vertices then they are neighbors of some $x$ and $x'$, respectively, where $I(x)$ contains $c_0(y)$ and at least $b-2$ indices smaller than $c_0(y)$ and $I(x')$ contains $c_0(y')$ and at least $b-2$ indices smaller than $c_0(y')$. Since $[n]-(x\cup y)=t-2$, there are at most $t-2$ elements in $\cup_{j\in I(x)}C_j$ not contained in $y$. The definition of $c_0$ also implies that at least one element of $C_j$ is missing from $y$ for every $j<c_0(y)$. Similarly, there are at most $t-2$ elements in $\cup_{j\in I(x')}C_j$ not contained in $y'$ and at least one element of $C_j$ is missing from $y'$ for every $j<c_0(y').$ These conditions lead to $y\cap y'\neq \emptyset$ if the sizes $|A_i|=2|C_i|-1$ are appropriately chosen. In particular, if $t\ge 2m+3$ and $|A_t|\ge 1,\, |A_{t-1}|\ge 2m+3,\, |A_{t-2}|\ge \dots\ge |A_{t-(2m+2)}|\ge
4m+5$, or $t\ge 4m+3$ and $|A_t|\ge 1,\, |A_{t-1}|\ge 3,\, |A_{t-2}|\ge \dots\ge |A_{t-(4m+2)}|\ge
5$, then the above argument leads to a proof of $\psi(SG(n,k))\leq t-m$. (It takes some further but simple argument why the last two intervals $A_i$ can be chosen smaller than the previous ones.) These two possible choices of the interval sizes give the two general bounds on $n$ we claimed sufficient for attaining $\psi(SG(n,k))\leq t-m$. The strengthening of Theorem \[thm:upb\] is obtained by the $m=(t-3)/2$ special case of the first bound. $\Diamond$
[**Proposition \[prop:k2\]**]{} (restated) [*$\psi(SG(n,2))=n-2=\chi(SG(n,2))$ for every $n\ge 4$.* ]{}
[[**Proof.** ]{}]{}In the $n=4$ case $SG(n,2)$ consists of a single edge and the statement of the proposition is trivial. Assume for a contradiction that $\psi(SG(n,2))\le n-3$ for some $n\ge5$ and let $c$ be a proper coloring of $SG(n,2)$ showing this with the minimal number of colors. As $\chi(SG(n,2))=n-2$ and any coloring of a graph $G$ with exactly $\chi(G)$ colors cannot show $\psi(G)<\chi(G)$ the coloring $c$ uses at least $n-1$ colors.
It is worth visualizing the vertices of $SG(n,2)$ as diagonals of an $n$-gon (see [@BjLo]). In other words, $SG(n,2)$ is the complement of the line graph of $D$, where $D$ is the complement of the cycle $C_n$. The color classes are independent sets in $SG(n,2)$, so they are either stars or triangles in $D$.
We say that a vertex $x$ [*sees*]{} the color classes of its neighbors. By our assumption every vertex sees at most $n-4$ color classes.
Assume a color class consists of a single vertex $x$. As $x$ sees at most $n-4$ of the at least $n-1$ color classes we can choose a different color for $x$. The resulting coloring attains the same local chromatic number with fewer colors. This contradicts the choice of $c$ and shows that no color class is a singleton.
A triangle color class is seen by all other edges of $D$. A star color class with center $i$ and at least three elements is seen by all vertices that, as edges of $D$, are not incident to $i$. For star color classes of two edges there can be one additional vertex not seeing the class. So every color class is seen by all but at most $n-2$ vertices. We double count the pairs of a vertex $x$ and a color class $C$ seen by $x$. On one hand every vertex sees at most $n-4$ classes. On the other hand all the color classes are seen by at least $\left({n\choose2}-n\right)-(n-2)$ vertices. We have $$(n-1)\left({n\choose2}-2n+2\right)\le\left({n\choose2}-n\right)(n-4),$$ and this contradicts our $n\ge5$ assumption. The contradiction proves the statement.
Generalized Mycielski graphs {#subsect:gmyc}
----------------------------
Another class of graphs for which the chromatic number is known only via the topological method is formed by generalized Mycielski graphs, see [@GyJS; @Mat; @Stieb]. They are interesting for us also for another reason: there is a big gap between their fractional and ordinary chromatic numbers (see [@LPU; @Tar]), therefore the local chromatic number can take its value from a large interval.
Recall that the Mycielskian $M(G)$ of a graph $G$ is the graph defined on $(\{0,1\}\times V(G))\cup \{z\}$ with edge set $E(M(G))=\{\{(0,v),(i,w)\}:
\{v,w\}\in E(G), i\in \{0,1\}\}\cup \{\{(1,v),z\}: v\in V(G)\}$. Mycielski [@Myc] used this construction to increase the chromatic number of a graph while keeping the clique number fixed: $\chi(M(G))=\chi(G)+1$ and $\omega(M(G))=\omega(G)$.
Following Tardif [@Tar], the same construction can also be described as the direct (also called categorical) product of $G$ with a path on three vertices having a loop at one end and then identifying all vertices that have the other end of the path as their first coordinate. Recall that the direct product of $F$ and $G$ is a graph on $V(F)\times V(G)$ with an edge between $(u,v)$ and $(u',v')$ if and only if $\{u,u'\}\in E(F)$ and $\{v,v'\}\in E(G)$. The generalized Mycielskian of $G$ (called a cone over $G$ by Tardif [@Tar]) $M_r(G)$ is then defined by taking the direct product of $P$ and $G$, where $P$ is a path on $r+1$ vertices having a loop at one end, and then identifying all the vertices in the product with the loopless end of the path as their first coordinate. With this notation $M(G)=M_2(G)$. These graphs were considered by Stiebitz [@Stieb], who proved that if $G$ is $k$-chromatic “for a topological reason” then $M_r(G)$ is $(k+1)$-chromatic for a similar reason. (Gyárfás, Jensen, and Stiebitz [@GyJS] also consider these graphs and quote Stiebitz’s argument a special case of which is also presented in [@Mat].) The topological reason of Stiebitz is in different terms than those we use in this paper but using results of [@BK] they imply strong topological $(t+d)$-chromaticity for graphs obtained by $d$ iterations of the generalized Mychielski construction starting, e.g, from $K_t$ or from a $t$-chromatic Schrijver graph. More precisely, Stiebitz proved that the body of the so-called neighborhood complex ${\cal N}(M_r(G))$ of $M_r(G)$, introduced in [@LLKn] by Lovász, is homotopy equivalent to the suspension of $||{\cal N}(G)||$. Since ${\rm susp}(S^n)\cong S^{n+1}$ this implies that whenever $||{\cal N}(G)||$ is homotopy equivalent to an $n$-dimensional sphere, then $||{\cal N}(M_r(G))||$ is homotopy equivalent to the $(n+1)$-dimensional sphere. This happens, for example, if $G$ is a complete graph, or an odd cycle. By a recent result of Björner and de Longueville [@BjLo] we also have a similar situation if $G$ is isomorphic to any Schrijver graph $SG(n,k)$. Notice that the latter include complete graphs and odd cycles.
It is known, that $||{\cal N}(F)||$ is homotopy equivalent to $H(F)$ for every graph $F$, see Proposition 4.2 in [@BK]. All this implies that ${\rm coind}(H(M_r(G)))={\rm coind}(H(G))+1$ whenever $H(G)$ is homotopy equivalent to a sphere, in particular, whenever $G$ is a complete graph or an odd cycle, or, more generally, a Schrijver graph. It is very likely that Stiebitz’s proof can be generalized to show that $H(M_r(G))\leftrightarrow\susp(H(G))$ and therefore ${\rm
coind}(H(M_r(G)))\ge{\rm coind}(H(G))+1$ holds always. Here we restrict attention to graphs $G$ with $H(G)$ homotopy equivalent to a sphere.
For an integer vector ${{\mbox{\boldmath$r$}}}=(r_1,\dots,r_d)$ with $r_i\ge1$ for all $i$ we let $M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(G)=M_{r_d}(M_{r_{d-1}}(\dots M_{r_1}(G)\ldots))$ denote the graph obtained by a $d$-fold application of the generalized Mycielski construction with respective parameters $r_1,\dots,r_d$.
\[prop:Stieb\] [(Stiebitz)]{} If $G$ is a graph for which $H(G)$ is homotopy equivalent to a sphere $S^h$ with $h=\chi(G)-2$ (in particular, $G$ is a complete graph or an odd cycle, or, more generally, a Schrijver graph) and ${{\mbox{\boldmath$r$}}}=(r_1,\dots,r_d)$ is arbitrary, then $M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(G)$ is strongly topologically $t$-chromatic for $t=\chi(M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(G))=\chi(G)+d$.
It is interesting to remark that $\chi(M_r(G))>\chi(G)$ does not hold in general if $r\ge 3$, e.g., for $\overline C_7$, the complement of the $7$-cycle, one has $\chi(M_3(\overline C_7))=\chi(\overline
C_7)=4$. Still, the result of Stiebitz implies that the sequence $\{\chi(M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(G))\}_{d=1}^{\infty}$ may avoid to increase only a finite number of times.
The fractional chromatic number of Mycielski graphs were determined by Larsen, Propp, and Ullman [@LPU], who proved that $\chi_f(M(G))=\chi_f(G)+{1\over {\chi_f(G)}}$ holds for every $G$. This already shows that there is a large gap between the chromatic and the fractional chromatic number of $M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(G)$ if $d$ is large enough and $r_i\ge 2$ for all $i$, since obviously, $\chi_f(M_r(F))\leq \chi_f(M(F))$ holds if $r\ge 2$. The previous result was generalized by Tardif [@Tar] who showed that $\chi_f(M_r(G))$ can also be expressed by $\chi_f(G)$ as $\chi_f(G)+{1\over{\sum_{i=0}^{r-1}(\chi_f(G)-1)^i}}$ whenever $G$ has at least one edge.
First we show that for the original Mycielski construction the local chromatic number behaves similarly to the chromatic number.
\[prop:myc2\] For any graph $G$ we have $$\psi(M(G))=\psi(G)+1.$$
[[**Proof.** ]{}]{}We proceed similarly as one does in the proof of $\chi(M(G))=\chi(G)+1$. Recall that $V(M(G))=\{0,1\}\times V(G)\cup\{z\}$.
For the upper bound consider a coloring $c'$ of $G$ establishing its local chromatic number and let $\alpha$ and $\beta$ be two colors not used by $c'$. We define $c((0,x))=c'(x)$, $c((1,x))=\alpha$ and $c(z)=\beta$. This proper coloring shows $\psi(M(G))\le\psi(G)+1$.
For the lower bound consider an arbitrary proper coloring $c$ of $M(G)$. We have to show that some vertex must see at least $\psi(G)$ different colors in its neighborhood.
We define the coloring $c'$ of $G$ as follows: $$c'(x)=\left\{\begin{array}{lll}c((0,x))&&\hbox{if }
c((0,x))\ne c(z)\\
c((1,x))&&\hbox{otherwise.}\end{array}\right.$$ It follows from the construction that $c'$ is a proper coloring of $G$. Note that $c'$ does not use the color $c(z)$.
By the definition of $\psi(G)$, there is some vertex $x$ of $G$ that has at least $\psi(G)-1$ different colors in its neighborhood $N_G(x)$. If $c'(y)=c(0,y)$ for all vertices $y\in N_G(x)$, then the vertex $(1,x)$ has all these colors in its neighborhood, and also the additional color $c(z)$. If however $c'(y)\ne c(0,y)$ for a neighbor $y$ of $x$, then the vertex $(0,x)$ sees all the colors $c'(N_G(x))$ in its neighborhood $N_{M(G)}(0,x)$, and also the additional color $c(0,y)=c(z)$. In both cases a vertex has $\psi(G)$ different colors in its neighborhood as claimed. $\Box$
We remark that $M_1(G)$ is simply the graph $G$ with a new vertex connected to every vertex of $G$, therefore the following trivially holds.
\[prop:myc1\] For any graph $G$ we have $$\psi(M_1(G))=\chi(G)+1.$$
$\Box$
For our first upper bound we apply Lemma \[lem:5ut\]. We use the following result of Gyárfás, Jensen, and Stiebitz [@GyJS]. The lemma below is an immediate generalization of the $l=2$ special case of Theorem 4.1 in [@GyJS]. We reproduce the simple proof from [@GyJS] for the sake of completeness.
\[gyarfasek\] [([@GyJS])]{} If $G$ has a wide coloring with $t$ colors and $r\ge7$, then $M_r(G)$ has a wide coloring with $t+1$ colors.
[[**Proof.** ]{}]{}As there is a homomorphism from $M_r(G)$ to $M_7(G)$ if $r>7$ it is enough to give the coloring for $r=7$. We fix a wide $t$-coloring $c_0$ of $G$ and use the additional color $\gamma$. The coloring of $M_7(G)$ is given as $$c((v,x))=\left\{\begin{array}{lll}\gamma&&\hbox{$v$ is the vertex at
distance $3$, $5$ or $7$ from the loop}\\
c_0(x)&&\hbox{otherwise.}\end{array}\right.$$ It is straightforward to check that $c$ is a wide coloring. $\Box$
We can apply the results of Stiebitz and Gyárfás et al. recursively to give tight or almost tight bounds for the local chromatic number of the graphs $M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(G)$ in many cases:
\[nagymyc\] If $G$ has a wide $t$-coloring and ${{\mbox{\boldmath$r$}}}=(r_1,\ldots,r_d)$ with $r_i\ge7$ for all $i$, then $\psi(M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(G))\le\frac{t+d}2+2$.
If $H(G)$ is homotopy equivalent to a sphere $S^h$, then $\psi(M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(G))\ge\frac{h+d}2+2$.
[[**Proof.** ]{}]{}For the first statement we apply Lemma \[gyarfasek\] recursively to show that $M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(G)$ has a wide $(t+d)$-coloring and then apply Lemma \[lem:5ut\].
For the second statement we apply the result of Stiebitz recursively to show that $H(M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(G))$ is homotopy equivalent to $S^{h+d}$. As noted in the preliminaries this implies $\coind(H(M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(G)))\ge h+d$. By Theorem \[thm:lowb\] the statement follows. $\Box$
[**Theorem \[thm:gmycspec7\]**]{} (restated) [*If ${{\mbox{\boldmath$r$}}}=(r_1,\ldots,r_d)$, $d$ is odd, and $r_i\ge 7$ for all $i$, then $$\psi(M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(K_2))=\left\lceil d\over2\right\rceil+2.$$*]{}
[**Proof.**]{} Notice that for ${{\mbox{\boldmath$r$}}}=(r_1,\ldots,r_d)$ with $d$ odd and $r_i\ge7$ for all $i$ the lower and upper bounds of Corollary \[nagymyc\] give the exact value for the local chromatic number $\psi(M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(K_2))=(d+5)/2$. This proves the theorem. $\Box$
Notice that a similar argument gives the exact value of $\psi(G)$ for the more complicated graph $G=M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(SG(n,k))$ whenever $n+d$ is odd, $r_i\ge
7$ for all $i$, and $n\ge 4t^2-7t$ for $t=n-2k+2$. This follows from Corollary \[nagymyc\] via the wide colorability of $SG(n,k)$ for $n\ge
4t^2-7t$ shown in the proof of Theorem \[thm:upb\] and Björner and de Longueville’s result [@BjLo] about the homotopy equivalence of $H(SG(n,k))$ to $S^{n-2k}$.
We summarize our knowledge on $\psi(M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(K_2))$ after proving the following theorem, which shows that almost the same upper bound as in Corollary \[nagymyc\] is implied from the relaxed condition $r_i\ge4$.
\[thm:gmycspec4\] For ${{\mbox{\boldmath$r$}}}=(r_1,\ldots,r_d)$ with $r_i\ge4$ for all $i$ one has $$\psi(M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(G))\leq \psi(G)+\left\lfloor d\over 2\right\rfloor+2.$$ Moreover, for $G\cong K_2$, the following slightly sharper bound holds: $$\psi(M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(K_2))\le\left\lceil d\over2\right\rceil+3.$$
[[**Proof.** ]{}]{}We denote the vertices of $Y:=M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(G)$ in accordance to the description of the generalized Mycielski construction via graph products. That is, a vertex of $Y$ is a sequence $a_1a_2\dots a_du$ of length $(d+1)$, where $\forall i: \ a_i\in \{0,1,\dots,r_i\}\cup \{*\}$, $u\in V(G)\cup \{*\}$ and if $a_i=r_i$ for some $i$ then necessarily $u=*$ and $a_j=*$ for every $j>i$, and this is the only way $*$ can appear in a sequence. To define adjacency we denote by $\hat P_{r_i+1}$ the path on $\{0,1,\dots,r_i\}$ where the edges are of the form $\{i-1,i\}, i\in
\{1,\dots, r_i\}$ and there is a loop at vertex $0$. Two vertices $a_1a_2\dots a_du$ and $a_1'a_2'\dots
a_d'u'$ are adjacent in $Y$ if and only if $$u=*\hbox{ or }u'=*\hbox{ or }\{u,u'\}\in E(G)\hbox{ and}$$ $$\forall i:\ \ a_i=*\hbox{ or }a_i'=*\hbox{ or }\{a_i, a_i'\}\in E(\hat
P_{r_i+1}).$$
Our strategy is similar to that used in Remark 4. Namely, we give an original coloring $c_0$ and identify the set of “troublesome” vertices for this coloring and recolor most of the neighbors of these vertices to a new color.
Let us fix a coloring $c_G$ of $G$ with at most $\psi(G)-1$ colors in the neighborhood of a vertex. Let the colors we use in this coloring be called $0,-1,-2$, etc. Now we define $c_0$ as follows. $$c_0(a_1\dots a_du)=\left\{\begin{array}{lll}c_G(u)&&
\hbox{if }\forall i:a_i\leq 2\\
i&&\hbox{if $a_i\ge 3$ is odd and $a_j\leq 2$ for all $j<i$}\\
0&&\hbox{if $\exists i:a_i\ge4$ is even and $a_j\leq 2$ for all $j<i$}
\end{array}\right.$$ It is clear that vertices having the same color form independent sets, i.e., $c_0$ is a proper coloring. Notice that if a vertex has neighbors of many different “positive” colors, then it must have many coordinates that are equal to $2$. Now we recolor most of the neighbors of these vertices.
Let $\beta$ be a color not used by $c_0$ and set $c(a_1\dots a_du)=\beta$ if $|\{i: a_i\hbox{ is odd}\}|> d/2$. (In fact, it would be enough to give color $\beta$ only to those of the above vertices, for which the first $\lfloor{d\over 2}\rfloor$ odd coordinates are equal to $1$. We recolor more vertices for the sake of simplicity.) Otherwise, let $c(a_1\dots a_du)=c_0(a_1\dots a_du)$.
First, we have to show that $c$ is proper. To this end we only have to show that no pair of vertices getting color $\beta$ can be adjacent. If two vertices, ${{\mbox{\boldmath$x$}}}=x_1\dots x_dv_x$ and ${{\mbox{\boldmath$y$}}}=y_1\dots y_dv_y$ are colored $\beta$ then both have more than $d/2$ odd coordinates (among their first $d$ coordinates). Thus there is some common coordinate $i$ for which $x_i$ and $y_i$ are both odd. This implies that they cannot be adjacent.
Now we show that for any vertex ${\mbox{\boldmath$a$}}$ we have $|c(N({{\mbox{\boldmath$a$}}}))\cap\{1,\ldots,d\}|\le d/2$. Indeed, if $|c_0(N({{\mbox{\boldmath$a$}}}))\cap\{1,\ldots,d\}|>d/2$ we have ${{\mbox{\boldmath$a$}}}=a_1\dots a_du$ with more than $d/2$ coordinates $a_i$ that are even and positive. Furthermore, the first $\lfloor d/2\rfloor$ of these coordinates should be $2$. Let $I$ be the set of indices of these first $\lfloor d/2\rfloor$ even and positive coordinates. We claim that $c(N({{\mbox{\boldmath$a$}}}))\cap\{1,\ldots,d\}\subseteq I$. This is so, since if a neighbor has an odd coordinate somewhere outside $I$, then it cannot have $*$ at the positions of $I$, therefore it has more than $d/2$ odd coordinates and it is recolored by $c$ to the color $\beta$.
It is also clear that no vertex can see more than $\psi(G)-1$ “negative” colors in its neighborhood in either coloring $c_0$ or $c$. Thus the neighborhood of any vertex can contain at most $\lfloor
d/2\rfloor+(\psi(G)-1)+2$ colors, where the last $2$ is added because of the possible appearance of colors $\beta$ and $0$ in the neighborhood. This proves $\psi(Y)\le d/2+\psi(G)+2$ proving the first statement in the theorem.
For $G\cong K_2$ the above gives $\psi(M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(K_2))\leq
\lfloor d/2\rfloor +4$ which implies the second statement for odd $d$. For even $d$ the bound of the second statement is $1$ less. We can gain $1$ as follows. When defining $c$ let us recolor to $\beta$ those vertices ${{\mbox{\boldmath$a$}}}=a_1\dots a_du$, too, for which the number of odd coordinates $a_i$ is exactly ${d\over 2}$ and $c_G(u)=-1$. The proof proceeds similarly as before but we gain $1$ by observing that those vertices who see $-1$ can see only ${d\over 2}-1$ “positive” colors. $\Box$
We collect the implications of Theorems \[thm:gmycspec7\], \[thm:gmycspec4\] and Propositions \[prop:myc2\] and \[prop:myc1\]. It would be interesting to estimate the value $\psi(M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(K_2))$ for the missing case ${{\mbox{\boldmath$r$}}}=(3,\ldots,3)$. We have $\lceil d/2\rceil+2\le\psi\le d+2$ in this case.
\[cor:mycgap\] For ${{\mbox{\boldmath$r$}}}=(r_1,\ldots,r_d)$ we have $$\psi(M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(K_2))=\left\{\begin{array}{lll}
(d+5)/2&&\hbox{if $d$ is odd and }\forall i:r_i\ge7\\
\lceil d/2\rceil+2\hbox{ or }\lceil d/2\rceil+3&&
\hbox{if }\forall i:r_i\ge4\\
d+2&&\hbox{if }r_d=1\hbox{ or }\forall i:r_i=2.
\end{array}\right.$$
$\Box$
[*Remark 5.*]{} The improvement for even $d$ given in the last paragraph of the proof of Theorem \[thm:gmycspec4\] can also be obtained in a different way we explain here. Instead of changing the rule for recoloring, we can enforce that a vertex can see only $\psi(G)-2$ negative colors. This can be achieved by setting the starting graph $G$ to be $M_4(K_2)\cong C_9$ instead of $K_2$ itself and coloring this $C_9$ with the pattern $-1,0,-1,-2,0,-2,-3,0,-3$ along the cycle. One can readily check that every vertex can see only one non-$0$ color in its neighborhood.
The same trick can be used also if the starting graph is not $K_2$ or $C_9$, but some large enough Schrijver graph of odd chromatic number. Coloring it as in the proof of Lemma \[lem:5ut\] (using the wide coloring as given in the proof of Theorem \[thm:upb\]), we arrive to the same phenomenon if we use the new color $\beta=0$. $\Diamond$
[*Remark 6.*]{} Gyárfás, Jensen, and Stiebitz [@GyJS] use generalized Mycielski graphs to show that another graph they denote by $G_k$ is $k$-chromatic. The way they prove it is that they exhibit a homomorphism from $M_{{\mbox{\boldmath\scriptsize$r$}}}^{(k-2)}(K_2)$ to $G_k$ for ${{\mbox{\boldmath$r$}}}=(4,\dots,4)$. The existence of this homomorphism implies that $G_k$ is strongly topologically $k$-chromatic, thus its local chromatic number is at least $k/2+1$. We do not know any non-trivial upper bound for $\psi(G_k)$. Also note that [@GyJS] gives universal graphs for the property of having a wide $t$-coloring. By Lemma \[lem:5ut\] this graph has $\psi\le t/2+2$. On the other hand, since any graph with a wide $t$-coloring admits a homomorphism to this graph, and we have seen the wide colorability of some strongly topologically $t$-chromatic graphs, it is strongly topologically $t$-chromatic, as well. This gives $\psi\ge t/2+1$. $\Diamond$
Borsuk graphs and the tightness of Ky Fan’s theorem {#subsect:ctopI}
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\[defi:Bogr\] The Borsuk graph $B(n,\alpha)$ of parameters $n$ and $0<\alpha<2$ is the infinite graph whose vertices are the points of the unit sphere in ${\mathbb
R}^n$ (i.e., $S^{n-1}$) and its edges connect the pairs of points with distance at least $\alpha$.
One easily sees that $\chi(B(n,\alpha))\ge n+1$, and, as Lovász [@LLgomb] remarks, this statement is equivalent to the Borsuk-Ulam theorem. For $\alpha\ge\sqrt{2+2/n}$ this lower bound is sharp, see [@LLgomb; @Mat] (cf. also the proof of Corollary \[cor:Borpsi\] below).
The local chromatic number of Borsuk graphs for large enough $\alpha$ can also be determined by our methods. First we want to argue that Theorem \[thm:lowb\] is applicable for this infinite graph. Lovász gives in [@LLgomb] a finite graph $G_P\subseteq B(n,\alpha)$ which has the property that its neighborhood complex ${\cal N}(G_P)$ is homotopy equivalent to $S^{n-1}$. Now we can continue the argument the same way as in the previous subsection: Proposition 4.2 in [@BK] states that ${\cal N}(F)$ is homotopy equivalent to $H(F)$ for every graph $F$, thus ${\rm
coind}(H(G_P))\ge n-1$, i.e., $G_P$ is (strongly) topologically $(n+1)$-chromatic. As $G_P\subseteq B(n,\alpha)$ we have $\lceil{{n+3}\over 2}\rceil\leq\psi(G_P)\le\psi(B(n,\alpha))$ by Theorem \[thm:lowb\].
The following lemma shows the special role of Borsuk graphs among strongly topologically t-chromatic graphs. It will also show that our earlier upper bounds on the local chromatic number have direct implications for Borsuk graphs.
\[gh\] A finite graph $G$ is strongly topologically $(n+1)$-chromatic if and only if for some $\alpha<2$ there is a graph homomorphism from $B(n,\alpha)$ to $G$.
[[**Proof.** ]{}]{}For the if part consider the finite graph $G_P\subseteq B(n,\alpha)$ given by Lovász [@LLgomb] satisfying $\coind(H(G_P))\ge n-1$. If there is a homomorphism from $B(n,\alpha)$ to $G$, it clearly gives a homomorphism also from $G_P$ to $G$ which further generates a $\2$-map from $H(G_P)$ to $H(G)$. This proves $\coind(H(G))\ge n-1$.
For the only if part, let $f:S^{n-1}\to H(G)$ be a $\2$-map. For a point ${{\mbox{\boldmath$x$}}}\in S^{n-1}$ write $f({{\mbox{\boldmath$x$}}})\in H(G)$ as the convex combination $f({{\mbox{\boldmath$x$}}})=\sum\alpha_v({{\mbox{\boldmath$x$}}})||{+}v||+\sum\beta_v({{\mbox{\boldmath$x$}}})||{-}v||$ of the vertices of $||B_0(G)||$. Here the summations are for the vertices $v$ of $G$, $\sum\alpha_v({{\mbox{\boldmath$x$}}})=\sum\beta_v({{\mbox{\boldmath$x$}}})=1/2$, and $\{v:\alpha_v({{\mbox{\boldmath$x$}}})>0\}\uplus\{v:\beta_v({{\mbox{\boldmath$x$}}})>0\}\in B_0(G)$. Note that $\alpha_v$ and $\beta_v$ are continuous as $f$ is continuous and $\beta_v({{\mbox{\boldmath$x$}}})=\alpha_v(-{{\mbox{\boldmath$x$}}})$ by the equivariance of $f$. Set $\varepsilon=1/(2|V(G)|)$. For ${{\mbox{\boldmath$x$}}}\in S^{n-1}$ select an arbitrary vertex $v=g({{\mbox{\boldmath$x$}}})$ of $G$ with $\alpha_v\ge\varepsilon$. We claim that $g$ is a graph homomorphism from $B(n,\alpha)$ to $G$ if $\alpha$ is close enough to $2$. By compactness it is enough to prove that if we have vertices $v$ and $w$ of $G$ and sequences ${{\mbox{\boldmath$x$}}}_i\to{{\mbox{\boldmath$x$}}}$ and ${{\mbox{\boldmath$y$}}}_i\to-{{\mbox{\boldmath$x$}}}$ of points in $S^{n-1}$ with $g({{\mbox{\boldmath$x$}}}_i)=v$ and $g({{\mbox{\boldmath$y$}}}_i)=w$ for all $i$, then $v$ and $w$ are connected in $G$. But since $\alpha_v$ is continuous we have $\alpha_v({{\mbox{\boldmath$x$}}})\ge\varepsilon$ and similarly $\beta_w({{\mbox{\boldmath$x$}}})=\alpha_w(-{{\mbox{\boldmath$x$}}})\ge\varepsilon$ and so $+v$ and $-w$ are contained in the smallest simplex of $B_0(G)$ containing $f({{\mbox{\boldmath$x$}}})$ proving that $v$ and $w$ are connected. $\Box$
By Lemma \[gh\] either of Theorems \[thm:upb\] or \[thm:gmycspec7\] implies that the above given lower bound on $\psi(B(n,\alpha))$ is tight whenever $\chi(B(n,\alpha))$ is odd, that is, $n$ is even, and $\alpha<2$ is close enough to $2$. The following corollary uses Lemma \[lem:5ut\] directly to have an explicit bound on $\alpha$.
\[cor:Borpsi\] If $n$ is even and $\alpha_n\le\alpha<2$, then $$\psi(B(n,\alpha))={n\over 2}+2,$$ where $\alpha_n=2\cos{\arccos(1/n)\over10}$.
Note that $\alpha_n\le\alpha_2=2\cos(\pi/30)<1.99$.
[[**Proof.** ]{}]{}The lower bound on $\psi(B(n,\alpha))$ follows from the discussion preceding Lemma \[gh\]. The upper bound follows from Lemma \[lem:5ut\] as long as we can give a wide $(n+1)$-coloring $c_0$ of the graph $B(n,\alpha)$.
To this end we use the standard $(n+1)$-coloring of $B(n,\alpha)$ (see, e.g., [@LLgomb; @Mat]). Consider a regular simplex $R$ inscribed into the unit sphere $S^{n-1}$ and color a point ${{\mbox{\boldmath$x$}}}\in S^{n-1}$ by the facet of $R$ intersected by the segment from the origin to ${\mbox{\boldmath$x$}}$. If this segment meets a lower dimensional face then we arbitrarily choose a facet containing this face.
We let $\varphi=2\arccos(\alpha/2)$. Clearly, ${\mbox{\boldmath$x$}}$ and ${\mbox{\boldmath$y$}}$ is connected if and only if the length of the shortest arc on $S^{n-1}$ connecting $-{\mbox{\boldmath$x$}}$ and ${\mbox{\boldmath$y$}}$ is at most $\varphi$. Therefore ${\mbox{\boldmath$x$}}$ and ${\mbox{\boldmath$y$}}$ are connected by a walk of length $5$ if and only if the length of this same minimal arc is at most $5\varphi$. For the coloring $c_0$ the length of the shortest arc between $-{\mbox{\boldmath$x$}}$ and ${\mbox{\boldmath$y$}}$ for two vertices ${\mbox{\boldmath$x$}}$ and ${\mbox{\boldmath$y$}}$ colored with the same color is at least $\arccos(1/n)$. Hence $c_0$ is wide. To make sure this holds for $\alpha=\alpha_n$ one has to color the $n+1$ vertices of the simplex $R$ with different colors. $\Box$
Our investigations of the local chromatic number led us to consider the following function $Q(h)$. The question of its values was independently asked by Micha Perles motivated by a related question of Matatyahu Rubin[^3].
\[q\] For a nonnegative integer parameter $h$ let $Q(h)$ denote the minimum $l$ for which $S^h$ can be covered by open sets in such a way that no point of the sphere is contained in more than $l$ of these sets and none of the covering sets contains an antipodal pair of points.
Ky Fan’s theorem implies $Q(h)\ge {h\over 2}+1$. Either of Theorems \[thm:upb\] or \[thm:gmycspec7\] implies the upper bound $Q(h)\leq {h\over 2}+2$. Using the concepts of Corollary \[cor:Borpsi\] and Lemma \[lem:5ut\] one can give an explicit covering of the sphere $S^{2l-3}$ by open subsets where no point is contained in more than $l$ of the sets and no set contains an antipodal pair of points. In fact, the covering we give satisfies a stronger requirement and proves that version (ii) of Ky Fan’s theorem is tight, while version (i) is almost tight.
\[kyftight\] There is a configuration ${\cal A}$ of $k+2$ open (closed) sets such that $\cup_{A\in\cal A}(A\cup-A)=S^k$, all sets $A\in\cal A$ satisfy $A\cap-A=\emptyset$, and no ${{\mbox{\boldmath$x$}}}\in S^k$ is contained in more than $\left\lceil k+1\over2\right\rceil$ of these sets.
Furthermore, for every ${\mbox{\boldmath$x$}}$ the number of sets in $\cal A$ containing either ${\mbox{\boldmath$x$}}$ or $-{\mbox{\boldmath$x$}}$ is at most $k+1$.
[[**Proof.** ]{}]{} First we construct closed sets. Consider the unit sphere $S^k$ in ${\mathbb R}^{k+1}$. Let $R$ be a regular simplex inscribed in the sphere. Let $B_1,\ldots,B_{k+2}$ be the subsets of the sphere obtained by the central projection of the facets of $R$. These closed sets cover $S^k$. Let $C_0$ be the set of points covered by at least $\left\lceil k+3\over2\right\rceil$ of the sets $B_i$. Notice that $C_0$ is the union of the central projections of the $\lfloor{{k-1}\over 2}\rfloor$-dimensional faces of $R$. For odd $k$ let $C=C_0$, while for even $k$ let $C=C_0\cup C_1$, where $C_1$ is the set of points in $B_1$ covered by exactly $k/2+1$ of the sets $B_i$. Thus $C_1$ is the central projection of the ${k\over 2}$-dimensional faces of a facet of $R$. Observe that $C\cap-C=\emptyset$. Take $0<\delta<\hbox{dist}(C,-C)/2$ and let $D$ be the open $\delta$-neighborhood of $C$ in $S^k$. For $1\le i\le k+2$ let $A_i=B_i\setminus D$. These closed sets cover $S^k\setminus D$ and none of them contain a pair of antipodal points. As $D\cap-D=\emptyset$ we have $\cup_{i=1}^{k+2}(A_i\cup-A_i)=S^k$. It is clear that every point of the sphere is covered by at most $\left\lceil
k+1\over2\right\rceil$ of the sets $A_i$ proving the first statement of the corollary.
For the second statement note that if each set $B_i$ contains at least one of a pair of antipodal points, then one of these points belongs to $C$ and is therefore not covered by any of the sets $A_i$. Note also, that for odd $k$ the second statement follows also from the first.
To construct open sets as required we can simply take the open $\varepsilon$-neighborhoods of $A_i$. For small enough $\varepsilon>0$ they maintain the properties required in the corollary. $\Box$
\[kyft2\] There is a configuration of $k+3$ open (closed) sets covering $S^k$ none of which contains a pair of antipodal points, such that no ${{\mbox{\boldmath$x$}}}\in S^k$ is contained in more than $\lceil{{k+3}\over
2}\rceil$ of these sets and for every ${{\mbox{\boldmath$x$}}}\in S^k$ the number of sets that contain one of ${{\mbox{\boldmath$x$}}}$ and $-{{\mbox{\boldmath$x$}}}$ is at most $k+2$.
[[**Proof.** ]{}]{}For closed sets consider the sets $A_i$ in the proof of Corollary \[kyftight\] together with the closure of $D$. For open sets consider the open $\varepsilon$-neighborhoods of these sets for suitably small $\varepsilon>0$. $\Box$
Note that covering with $k+3$ sets is optimal in Corollary \[kyft2\] if $k\ge 3$. By the Borsuk-Ulam Theorem (form (i)) fewer than $k+2$ open (or closed) sets not containing antipodal pairs of points is not enough to cover $S^k$. If we cover with $k+2$ sets (open or closed), then it gives rise to a proper coloring of $B(k+1,\alpha)$ for large enough $\alpha$ in a natural way. This coloring uses the optimal number $k+2$ of colors, therefore it has a vertex with $k+1$ different colors in its neighborhood. A compactness argument establishes from this that there is a point in $S^k$ covered by $k+1$ sets. A similar argument gives that $k+2$ in Corollary \[kyftight\] is also optimal if $k\ge 3$.
\[qbound\] $${h\over 2}+1\leq Q(h)\leq {h\over 2}+2.$$
[[**Proof.** ]{}]{}The lower bound is implied by Ky Fan’s theorem. The upper bound follows from Corollary \[kyft2\]. $\Box$
Notice that for odd $h$ Corollary \[qbound\] gives the exact value $Q(h)={h+3\over 2}$. For $h$ even we either have $Q(h)={h\over 2}+1$ or $Q(h)={h\over 2}+2$. It is trivial that $Q(0)=1$. In [@up] we will show $Q(2)=3$. This was independently proved by Imre Bárány b. For $h>2$ even it remains open whether the lower or the upper bound of Corollary \[qbound\] is exact.
Circular colorings {#sec:circ}
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In this section we show an application of the Zig-zag Theorem for the circular chromatic number of graphs. This will result in the partial solution of a conjecture by Johnson, Holroyd, and Stahl [@JHS] and in a partial answer to a question of Hajiabolhassan and Zhu [@HZ] concerning the circular chromatic number of Kneser graphs and Schrijver graphs, respectively. We also answer a question of Chang, Huang, and Zhu [@CHZ] concerning the circular chromatic number of iterated Mycielskians of complete graphs.
The circular chromatic number of a graph was introduced by Vince [@Vin] under the name star chromatic number as follows.
\[defi:circ\] For positive integers $p$ and $q$ a coloring $c:V(G)\to [p]$ of a graph $G$ is called a [*$(p,q)$-coloring*]{} if for all adjacent vertices $u$ and $v$ one has $q\leq |c(u)-c(v)|\leq p-q$. The [*circular chromatic number*]{} of $G$ is defined as $$\chi_c(G)=\inf\left\{{p\over q}:\hbox{\rm\ there is a $(p,q)$-coloring of }
G\right\}.$$
It is known that the above infimum is always attained for finite graphs. An alternative description of $\chi_c(G)$, explaining its name, is that it is the minimum length of the perimeter of a circle on which we can represent the vertices of $G$ by arcs of length $1$ in such a way that arcs belonging to adjacent vertices do not overlap. For a proof of this equivalence and for an extensive bibliography on the circular chromatic number we refer to Zhu’s survey article [@Zhu].
It is known that for every graph $G$ one has $\chi(G)-1<\chi_c(G)\leq \chi(G)$. Thus $\chi_c(G)$ determines the value of $\chi(G)$ while this is not true the other way round. Therefore the circular chromatic number can be considered as a refinement of the chromatic number.
Our main result on the circular chromatic number is Theorem \[thm:circ\]. Here we restate the theorem with the explicit meaning of being topologically $t$-chromatic.
[**Theorem \[thm:circ\]**]{} (restated) [*For a finite graph $G$ we have $\chi_c(G)\ge\coind(B_0(G))+1$ if $\coind(B_0(G))$ is odd.* ]{}
[[**Proof.** ]{}]{}Let $t=\coind(B_0(G))+1$ be an even number and let $c$ be a $(p,q)$-coloring of $G$. By the Zig-zag Theorem there is a $K_{{t\over 2},{t\over 2}}$ in $G$ which is completely multicolored by colors appearing in an alternating manner in its two sides. Let these colors be $c_1<c_2<\dots<c_t$. Since the vertex colored $c_i$ is adjacent to that colored $c_{i+1}$, we have $c_{i+1}\ge c_i+q$ and $c_t\ge c_1+(t-1)q$. Since $t$ is even, the vertices colored $c_1$ and $c_t$ are also adjacent, therefore we must have $c_t-c_1\leq p-q$. The last two inequalities give $p/q\ge t$ as needed. $\Box$
This result has been independently obtained by Meunier [@meunier] for Schrijver graphs.
Circular chromatic number of even chromatic Kneser and Schrijver graphs
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Johnson, Holroyd, and Stahl [@JHS] considered the circular chromatic number of Kneser graphs and formulated the following conjecture. (See also as Conjecture 7.1 and Question 8.27 in [@Zhu].)
[**Conjecture**]{} (Johnson, Holroyd, Stahl [@JHS]): For any $n\ge 2k$ $$\chi_c(KG(n,k))=\chi(KG(n,k)).$$
It is proven in [@JHS] that the above conjecture holds if $k=2$ or $n=2k+1$ or $n=2k+2$.
Lih and Liu [@LihLiu] investigated the circular chromatic number of Schrijver graphs and proved that $\chi_c(SG(n,2))=n-2=\chi(SG(n,2))$ whenever $n\neq 5$. (For $n=2k+1$ one always has $\chi_c(SG(2k+1,k))=2+{1\over k}$.) It was conjectured in [@LihLiu] and proved in [@HZ] that for every fixed $k$ there is a threshold $l(k)$ for which $n\ge
l(k)$ implies $\chi_c(SG(n,k))=\chi(SG(n,k))$. This clearly implies the analogous statement for Kneser graphs, for which the explicit threshold $l(k)=2k^2(k-1)$ is given in [@HZ]. At the end of their paper [@HZ] Hajiabolhassan and Zhu ask what is the minimum $l(k)$ for which $n\ge l(k)$ implies $\chi_c(SG(n,k))=\chi(SG(n,k))$. We show that no such threshold is needed if $n$ is even.
\[cor:JHS\] The Johnson-Holroyd-Stahl conjecture holds for every even $n$. Moreover, if $n$ is even, then the stronger equality $$\chi_c(SG(n,k))=\chi(SG(n,k))$$ also holds.
[[**Proof.** ]{}]{}As $t$-chromatic Kneser graphs and Schrijver graphs are topologically $t$-chromatic, Theorem \[thm:circ\] implies the statement of the corollary. $\Box$
As mentioned above this result has been obtained independently by Meunier [@meunier].
We show in Subsection \[oddsch\] that for odd $n$ the situation is different.
Circular chromatic number of Mycielski graphs and Borsuk graphs
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The circular chromatic number of Mycielski graphs was also studied extensively, cf. [@CHZ; @Fan; @HZMyc; @Zhu]. Chang, Huang, and Zhu [@CHZ] formulated the conjecture that $\chi_c(M^d(K_n))=\chi(M^d(K_n))=n+d$ whenever $n\ge d+2$. Here $M^d(G)$ denotes the $d$-fold iterated Mycielskian of graph $G$, i.e., using the notation of Subsection \[subsect:gmyc\] we have $M^d(G)=M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(G)$ with ${{\mbox{\boldmath$r$}}}=(2,\dots,2)$. The above conjecture was verified for the special cases $d=1,2$ in [@CHZ], where it was also shown that $\chi_c(M^d(G))\leq \chi(M^d(G))-1/2$ if $\chi(G)=d+1$. A simpler proof for the above special cases of the conjecture was given (for $d=2$ with the extra condition $n\ge 5$) in [@Fan]. Recently Hajiabolhassan and Zhu [@HZMyc] proved that $n\ge 2^d+2$ implies $\chi_c(M^d(K_n))=\chi(M^d(K_n))=n+d$. Our results show that $\chi_c(M^d(K_n))=\chi(M^d(K_n))=n+d$ always holds if $n+d$ is even. This also answers the question of Chang, Huang, and Zhu asking the value of $\chi_c(M^n(K_n))$ (Question 2 in [@CHZ]). The stated equality is given by the following immediate consequence of Theorem \[thm:circ\].
\[cor:Mycirc\] If $H(G)$ is homotopy equivalent to the sphere $S^h$, ${{\mbox{\boldmath$r$}}}$ is a vector of positive integers, and $h+d$ is even, then $\chi_c(M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(G))\ge d+h+2$.
In particular, $\chi_c(M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(K_n))=n+d$ whenever $n+d$ is even.
[[**Proof.** ]{}]{}The condition on $G$ implies ${\rm coind}(H(M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(G)))=h+d$ by Stiebitz’s result [@Stieb] (cf. the discussion and Proposition \[prop:Stieb\] in Subsection \[subsect:gmyc\]), which further implies ${\rm coind}(B_0(M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(G)))=h+d+1$. This gives the conclusion by Theorem \[thm:circ\].
The second statement follows by the homotopy equivalence of $H(K_n)$ with $S^{n-2}$ and the chromatic number of $M_{{\mbox{\boldmath\scriptsize$r$}}}^{(d)}(K_n)$ being $n+d$. $\Box$
The above mentioned conjecture of Chang, Huang, and Zhu for $n+d$ even is a special case with ${{\mbox{\boldmath$r$}}}=({2,2,\dots,2})$ and $n\ge d+2$. Since $n+n$ is always even, the answer $\chi_c(M^n(K_n))=2n$ to their question also follows.
Corollary \[cor:Mycirc\] also implies a recent result of Lam, Lin, Gu, and Song [@LLGS] who proved that for the generalized Mycielskian of odd order complete graphs $\chi_c(M_r(K_{2m-1}))=2m$.
Lam, Lin, Gu, and Song [@LLGS] also determined the circular chromatic number of the generalized Mycielskian of even order complete graphs. They proved $\chi_c(M_r(K_{2m}))=2m+1/(\lfloor(r-1)/m\rfloor+1)$. This result can be used to bound the circular chromatic number of the Borsuk graph $B(2s,\alpha)$ from above.
\[thm:Borcirc\] For the Borsuk graph $B(n,\alpha)$ we have
(i)
: $\chi_c(B(n,\alpha))=n+1$ if $n$ is odd and $\alpha\ge\sqrt{2+2/n}$;
(ii)
: $\chi_c(B(n,\alpha))\to n$ as $\alpha\to2$ if $n$ is even.
[[**Proof.** ]{}]{}The lower bound of part (i) immediately follows from Theorem \[thm:circ\] considering again the finite subgraph $G_P$ of $B(n,\alpha)$ defined in [@LLgomb] and already mentioned in the proof of Lemma \[gh\]. The matching upper bound is provided by $\chi(B(n,\alpha))=n+1$ for the range of $\alpha$ considered, see [@LLgomb].
For (ii) we have $\chi_c(B(n,\alpha))>\chi(B(n,\alpha))-1\ge n$. For an upper bound we use that $\chi_c(M_r(K_n))\to n$ if $r$ goes to infinity by the result of Lam, Lin, Gu, and Song [@LLGS] quoted above. By the result of Stiebitz [@Stieb] and Lemma \[gh\] we have a graph homomorphism from $B(n,\alpha)$ to $M_r(K_n)$ for any $r$ and large enough $\alpha$. As $(p,q)$-colorings can be defined in terms of graph homomorphisms (see [@BH]), we have $\chi_c(G)\le\chi_c(H)$ if there exists a graph homomorphism from $G$ to $H$. This finishes the proof of part (ii) of the theorem. $\Box$
[*Remark 7.*]{} By Theorem \[thm:Borcirc\] (ii) we have a sequence of $(p_i,q_i)$-colorings of the graphs $B(n,\alpha_i)$ where $n$ is even such that $\alpha_i\to2$ and $p_i/q_i\to n$. By a direct construction we can show that a single function $g:S^{n-1}\to C$ is enough. Here $C$ is a circle of unit perimeter. We need $$\label{eqb}
\inf\{\hbox{\rm dist}_C(g({{\mbox{\boldmath$x$}}}),g({{\mbox{\boldmath$y$}}})):\{{{\mbox{\boldmath$x$}}},{{\mbox{\boldmath$y$}}}\}\in E(B(n,\alpha))\}\to 1/n\hbox{ as
$\alpha<2$ goes to }2.$$ The distance ${\rm dist}_C(\cdot,\cdot)$ is measured along the cycle $C$. Clearly, if $p/q>n$ and we split $C$ into $p$ arcs $a_1,\ldots,a_p$ of equal length and color the point ${{\mbox{\boldmath$x$}}}$ with $i$ if $g({{\mbox{\boldmath$x$}}})\in a_i$, then this is a $(p,q)$-coloring of $B(n,\alpha)$ for $\alpha$ close enough to $2$.
For $n=2$ any $\2$-map $g:S^1\to C$ satisfies Equation (\[eqb\]). Let $n>2$. The map $g$ to be constructed must not be continuous by the Borsuk-Ulam theorem. Let us choose a set $H$ of $n-1$ equidistant points in $C$ and for $b\in C$ let $T(b)$ denote the unique set of $n/2$ equidistant points in $C$ containing $b$.
We consider $S^{n-1}$ as the [*join*]{} of the sphere $S^{n-3}$ and the circle $S^1$. All points in $S^{n-1}$ are now either in $S^{n-3}$, or in $S^1$, or in the interval connecting a point in $S^{n-3}$ to a point in $S^1$. We define $g$ on $S^{n-3}$ such that it takes values only from $H$ and it is a proper coloring of $B(n-2,1.9)$. We define $g$ on $S^1$ such that if ${\mbox{\boldmath$y$}}$ goes a full circle around $S^1$ with uniform velocity, then its image $g({{\mbox{\boldmath$y$}}})$ covers an arc of length $2/n$ of $C$ and it also moves with uniform velocity. Notice that although $g$ is not continuous on $S^1$, the set $T(g({{\mbox{\boldmath$y$}}}))$ depends on ${{\mbox{\boldmath$y$}}}\in S^1$ in a continuous manner. Also note that for a point ${{\mbox{\boldmath$x$}}}\in S^1$ the images $g({{\mbox{\boldmath$x$}}})$ and $g(-{{\mbox{\boldmath$x$}}})$ are $1/n$ apart on $C$ and $T(g({{\mbox{\boldmath$x$}}}))\cup T(g(-{{\mbox{\boldmath$x$}}}))$ is a set of $n$ equidistant points.
Let ${{\mbox{\boldmath$x$}}}\in S^{n-3}$ and ${{\mbox{\boldmath$y$}}}\in S^1$. Assume that a point ${\mbox{\boldmath$z$}}$ moves with uniform velocity from ${\mbox{\boldmath$x$}}$ to ${\mbox{\boldmath$y$}}$ along the interval connecting them. We define $g$ on this interval such that $g({{\mbox{\boldmath$z$}}})$ moves with uniform velocity along $C$ covering an arc of length at most $1/n$ from $g({{\mbox{\boldmath$x$}}})$ to a point in $T(g({{\mbox{\boldmath$y$}}}))$. The choice of the point in $T(g({{\mbox{\boldmath$y$}}}))$ is uniquely determined unless $g({{\mbox{\boldmath$x$}}})\in T(g(-{{\mbox{\boldmath$x$}}}))$. In the latter case we make an arbitrary choice of the two possible points for the destination of the image $g({{\mbox{\boldmath$z$}}})$.
It is not hard to prove that the function $g$ defined above satisfies Equation (\[eqb\]). $\Diamond$
Circular chromatic number of odd chromatic Schrijver graphs {#oddsch}
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In this subsection we show that the parity condition on $\chi(SG(n,k))$ in Corollary \[cor:JHS\] is relevant, for odd chromatic Schrijver graphs the circular chromatic number can be arbitrarily close to its lower bound.
\[thm:oddsch\] For every $\varepsilon>0$ and every odd $t\ge3$ if $n\ge t^3/\varepsilon$ and $t=n-2k+2$, then $$1-\varepsilon<\chi(SG(n,k))-\chi_c(SG(n,k))<1.$$
The second inequality is well-known and holds for any graph. We included it only for completeness. To prove the first inequality we need some preparation. We remark that the bound on $n$ in the theorem is not best possible. Our method proves $\chi(SG(n,k))-\chi_c(SG(n,k))\ge1-1/i$ if $i$ is a positive integer and $n\ge6(i-1){t\choose3}+t$.
First we extend our notion of wide coloring.
For a positive integer $s$ we call a vertex coloring of a graph $s$-wide if the two end vertices of any walk of length $2s-1$ receive different colors.
Our original wide colorings are $3$-wide, while $1$-wide simply means proper. Gyárfás, Jensen, and Stiebitz [@GyJS] investigated $s$-wide colorings (in different terms) and mention (referring to a referee in the $s>2$ case) the existence of homomorphism universal graphs for $s$-wide colorability with $t$ colors. We give a somewhat different family of such universal graphs. In the $s=2$ case the color-criticality of the given universal graph is proven in [@GyJS] implying its minimality among graphs admitting $2$-wide $t$-colorings. Later in Subsection \[critical\] we generalize this result showing that the members of our family are color-critical for every $s$. Thus they must be minimal and therefore isomorphic to a retract of the corresponding graphs given in [@GyJS].
\[Wst\] Let $H_s$ be the path on the vertices $0,1,2,\dots,s$ ($i$ and $i-1$ connected for $1\le i\le s$) with a loop at $s$. We define $W(s,t)$ to be the graph with $$V(W(s,t))=\{(x_1\dots x_t): \forall i\ x_i\in\{0,1,\dots,s\},\exists!i\
x_i=0,\ \exists j\ x_j=1\},$$ $$E(W(s,t))=\{\{x_1\dots x_t,y_1\dots y_t\}: \forall i\ \{x_i,y_i\}\in
E(H_s)\}.$$
Note that $W(s,t)$ is an induced subgraph of the direct power $H_s^t$.
\[Wuni\] A graph $G$ admits an $s$-wide coloring with $t$ colors if and only if there is a homomorphism from $G$ to $W(s,t)$.
[[**Proof.** ]{}]{}For the if part color vertex ${{\mbox{\boldmath$x$}}}=x_1\dots x_t$ of $W(s,t)$ with $c({{\mbox{\boldmath$x$}}})=i$ if $x_i=0$. Any walk between two vertices colored $i$ either has even length or contains two vertices ${{\mbox{\boldmath$y$}}}$ and ${{\mbox{\boldmath$z$}}}$ with $y_i=z_i=s$. These ${{\mbox{\boldmath$y$}}}$ and ${{\mbox{\boldmath$z$}}}$ are both at least at distance $s$ apart from both ends of the walk, thus our coloring of $W(s,t)$ with $t$ colors is $s$-wide. Any graph admitting a homomorphism $\varphi$ to $W(s,t)$ is $s$-widely colored with $t$ colors by $c_G(v):=c(\varphi(v))$.
For the only if part assume $c$ is an $s$-wide $t$-coloring of $G$ with colors $1,\dots,t$. Let $\varphi(v)$ be an arbitrary vertex of $W(s,t)$ if $v$ is an isolated vertex of $G$. For a non-isolated vertex $v$ of $G$ let $\varphi(v)={{\mbox{\boldmath$x$}}}=x_1\dots x_t$ with $x_i=\min(s,d_i(v))$, where $d_i(v)$ is the distance of color class $i$ from $v$. It is clear that $x_i=0$ for $i=c(v)$ and for no other $i$, while $x_i=1$ for the colors of the neighbors of $v$ in $G$. Thus the image of $\varphi$ is indeed in $V(W(s,t))$. It takes an easy checking that $\varphi$ is a homomorphism. $\Box$
The following lemma is a straightforward extension of the argument given in the proof of Theorem \[thm:upb\].
\[SGswide\] If $n\ge (2s-2)t^2-(4s-5)t$ then $SG(n,k)$ admits an $s$-wide $t$-coloring.
[[**Proof.** ]{}]{} We use the notation introduced in the proof of Theorem \[thm:upb\].
Let $n\ge t(2(s-1)(t-2)+1)$ as in the statement and let $c_0$ be the coloring defined in the mentioned proof. The lower bound on $n$ now allows to assume that $|C_i|\ge(s-1)(t-2)+1$. We show that $c_0$ is $s$-wide.
Consider a walk $x_0x_1\dots x_{2s-1}$ of length $(2s-1)$ in $SG(n,k)$ and let $i=c_0(x_0)$. Then $C_i\subseteq x_0$. By Lemma \[partav\] $|x_0\setminus x_{2s-2}|\leq(s-1)(t-2)<|C_i|$. Thus $x_{2s-2}$ is not disjoint from $C_i$. As $x_{2s-1}$ is disjoint from $x_{2s-2}$, it does not contain $C_i$ and thus its color is not $i$. $\Box$
\[WM\] $W(s,t)$ admits a homomorphism to $M_s(K_{t-1})$.
[[**Proof.** ]{}]{}Recall our notation for the (iterated) generalized Mycielskians from Subsection \[subsect:gmyc\].
We define the following mapping from $V(W(s,t))$ to $V(M_s(K_{t-1}))$.
$$\varphi(x_1\dots x_t):=\left\{\begin{array}{lll}(s-x_t,i)&&\hbox{if }
x_t\neq x_i=0\\(s,*)&&\hbox{if }x_t=0.\end{array}\right.$$
One can easily check that $\varphi$ is indeed a homomorphism. $\Box$
[**Proof of Theorem \[thm:oddsch\].**]{} By Lemma \[SGswide\], if $n\ge(2s-2)t^2-(4s-5)t$, then $SG(n,k)$ has an $s$-wide $t$-coloring, thus by Proposition \[Wuni\] it admits a homomorphism to $W(s,t)$. Composing this with the homomorphism given by Lemma \[WM\] we conclude that $SG(n,k)$ admits a homomorphism to $M_s(K_{t-1})$, implying $\chi_c(SG(n,k))\leq\chi_c(M_s(K_{t-1}))$.
We continue by using Lam, Lin, Gu, and Song’s result [@LLGS], who proved, as already quoted in the previous subsection, that $\chi_c(M_s(K_{t-1}))=t-1+{1\over\left\lfloor2s-2\over t-1\right\rfloor+1}$ if $t$ is odd. Thus, for odd $t$ and $i>0$ integer we choose $s=(t-1)(i-1)/2+1$ and $\chi(SG(n,k))-\chi_c(SG(n,k))=t-\chi_c(SG(n,k))\ge1-1/i$ follows from the $n\ge6(i-1){t\choose3}+t$ bound.
To get the form of the statement claimed in the theorem we choose $i=\lfloor1/\varepsilon\rfloor+1$. $\Box$
Further remarks
===============
Color-criticality of $W(s,t)$ {#critical}
-----------------------------
In this subsection we prove the edge color-criticality of the graphs $W(s,t)$ introduced in the previous section. This generalizes Theorem 2.3 in [@GyJS], see Remark 8 after the proof.
For every integer $s\ge 1$ and $t\ge 2$ the graph $W(s,t)$ has chromatic number $t$, but deleting any of its edges the resulting graph is $(t-1)$-chromatic.
[[**Proof.** ]{}]{}$\chi(W(s,t))\ge t$ follows from the fact that some $t$-chromatic Schrijver graphs admit a homomorphism to $W(s,t)$ which is implied by Lemma \[SGswide\] and Proposition \[Wuni\]. The coloring giving vertex ${{\mbox{\boldmath$x$}}}=x_1\dots x_t$ of $W(s,t)$ color $i$ iff $x_i=0$ is proper proving $\chi(W(s,t))\leq t$.
We prove edge-criticality by induction on $t$. For $t=2$ the statement is trivial as $W(s,t)$ is isomorphic to $K_2$. Assume that $t\ge 3$ and edge-criticality holds for $t-1$. Let $\{x_1\dots
x_t,y_1\dots y_t\}$ be an edge of $W(s,t)$ and $W'$ be the graph remaining after removal of this edge. We need to give a proper $(t-1)$-coloring $c$ of $W'$.
Let $i$ and $j$ be the coordinates for which $x_i=y_j=0$. We have $x_j=y_i=1$, in particular, $i\neq j$. Let $r$ be a coordinate different from both $i$ and $j$. We may assume without loss of generality that $r=1$, and also that $y_1\ge x_1$. Coordinates $i$ and $j$ make sure that $x_2x_3\dots x_t$ and $y_2y_3\dots y_t$ are vertices of $W(s,t-1)$, and in fact, they are connected by an edge $e$.
A proper $(t-2)$-coloring of the graph $W(s,t-1)\setminus e$ exists by the induction hypothesis. Let $c_0$ be such a coloring. Let $\alpha$ be a color of $c_0$ and $\beta$ a color that does not appear in $c_0$. We define the coloring $c$ of $W'$ as follows: $$c(z_1z_2\dots z_t)=\left\{\begin{array}{lll}
\alpha&&\hbox{if }z_1<x_1,\ x_1-z_1\hbox{ is even}\\
\beta&&\hbox{if }z_1<x_1,\ x_1-z_1\hbox{ is odd}\\
\alpha&&\hbox{if }z_1=x_1=1,\ z_i\ne1\hbox{ for }i>1\\
\beta&&\hbox{if }z_1>x_1,\ z_i=x_i\hbox{ for }i>1\\
c_0(z_2z_3\dots z_t)&&\hbox{otherwise.}
\end{array}\right.$$
It takes a straightforward case analysis to check that $c$ is a proper $(t-1)$-coloring of $W'$. $\Box$
[*Remark 8.*]{} Gyárfás, Jensen, and Stiebitz [@GyJS] proved the $s=2$ version of the previous theorem using a homomorphism from their universal graph with parameter $t$ to a generalized Mycielskian of the same type of graph with parameter $t-1$. In fact, our proof is a direct generalization of theirs using very similar ideas. Behind the coloring we gave is the recognition of a homomorphism from $W(s,t)$ to $M_{3s-2}(W(s,t-1))$. $\Diamond$
Hadwiger’s conjecture and the Zig-zag theorem
---------------------------------------------
Hadwiger’s conjecture, one of the most famous open problems in graph theory, states that if a graph $G$ contains no $K_{r+1}$ minor, then $\chi(G)\leq r$. For detailed information on the history and status of this conjecture we refer to Toft’s survey [@Toft]. We only mention that even $\chi(G)=O(r)$ is not known to be implied by the hypothesis for general $r$.
As a fractional and linear approximation version, Reed and Seymour [@RS] proved that if $G$ has no $K_{r+1}$ minor then $\chi_f(G)\leq 2r$. This means that graphs with $\chi_f(G)$ and $\chi(G)$ appropriately close and not containing a $K_{r+1}$ minor satisfy $\chi(G)=O(r)$.
We know that the main examples of graphs in [@SchU] for $\chi_f(G)<<\chi(G)$ (Kneser graphs, Mycielski graphs) satisfy the hypothesis of the Zig-zag theorem, therefore their $t$-chromatic versions must contain $K_{\lceil{t\over 2}\rceil,\lfloor{t\over 2}\rfloor}$ subgraphs. (We mention that for strongly topologically $t$-chromatic graphs this consequence, in fact, the containment of $K_{a,b}$ for every $a,b$ satisfying $a+b=t$, was proven by Csorba, Lange, Schurr, and Wassmer [@CsLSW].) However, a $K_{\lceil{t\over 2}\rceil,\lfloor{t\over 2}\rfloor}$ subgraph contains a $K_{\lfloor{t\over 2}\rfloor +1}$ minor (just take a matching of size $\lfloor{{t-2}\over 2}\rfloor$ plus one point from each side of the bipartite graph) proving the following statement which shows that the same kind of approximation is valid for these graphs, too.
\[Hadw\] If a topologically $t$-chromatic graph contains no $K_{r+1}$ minor, then $t<2r.$
$\Box$
[**Acknowledgments:**]{} We thank Imre Bárány, Péter Csorba, Gábor Elek, László Fehér, László Lovász, Jiří Matoušek, and Gábor Moussong for many fruitful conversations that helped us to better understand the topological concepts used in this paper.
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[^1]: Research partially supported by the Hungarian Foundation for Scientific Research Grant (OTKA) Nos. T037846 and T046376.
[^2]: Research partially supported by the Hungarian Foundation for Scientific Research Grant (OTKA) Nos. T037846 and T046234.
[^3]: We thank Imre Bárány [@b] and Gil Kalai [@GK] for this information.
|
---
abstract: 'In this paper, we are concerned with the valuation of Catastrophic Mortality Bonds and, in particular, we examine the case of the Swiss Re Mortality Bond 2003 as a primary example of this class of assets. This bond was the first Catastrophic Mortality Bond to be launched in the market and encapsulates the behaviour of a well-defined mortality index to generate payoffs for bondholders. Pricing this type of bonds is a challenging task and no closed form solution exists in the literature. In our approach, we adapt the payoff of such a bond in terms of the payoff of an Asian put option and present a new approach to derive model-independent bounds exploiting comonotonic theory as illustrated in [@prime1] for the pricing of Asian options. We carry out Monte Carlo simulations to estimate the bond price and illustrate the strength of the bounds.'
author:
- Raj Kumari Bahl and Sotirios Sabanis
title: 'Model-Independent Price Bounds for Catastrophic Mortality Bonds'
---
Introduction
============
In the present day world, many financial institutions face the risk of unexpected fluctuations in human mortality and clearly, this risk has two aspects. On one side, life insurers paying death benefits will suffer an economic loss if actual rates of mortality are in excess of those expected, due to catastrophic events such as a severe outbreak of an epidemic or a major man-made or natural disaster. This side of the risk is known in the literature by the name of *mortality risk*. On the other hand, pension plan sponsors, as well as insurance companies providing retirement annuities, are subject to *longevity risk*, that is, the risk that people outlive their expected lifetimes. For these institutions, the longer the life-span of people, the greater the period of time over which retirement income must be paid and, hence, the larger the financial liability.
An unanticipated change in mortality rates will affect all policies in force. Therefore, as opposed to the random variations between lifetimes of individuals, it cannot be diversified away by increasing the size of the portfolio. Reinsurance is one possible solution to the problem, but its capacity is usually limited. Alternatively, the risk may be naturally hedged or reduced through balancing products. For example, an insurance company may sell life insurance to the same customers who are buying life annuities. The resulting combination would then reduce the company’s exposure to future changes in mortality, consequently permitting a reduction of capital reserves held in respect of mortality or longevity risk. This idea of compensating longevity risk by mortality risk is often referred to as *natural hedging*. However, this strategy, as [@Cox] pointed out, may be cost prohibitive and may not be practical in some circumstances.
As a result, a natural remedy to tackle these risks has emerged in the form of what is known as mortality securitization which manifests itself in the form of *mortality-linked securities* abbreviated in the literature as *MLSs*. These securities provide a tool in the hands of insurers to transfer their mortality-sensitive exposures to a vested number of investors in the capital market, offering them a reasonable risk premium in return. Mortality-linked securities differ from their longevity counterparts in the sense that while the former have their cash flows linked to a mortality index, the latter are based upon survivor index. For a more detailed review of the two type of bonds, one can refer to [@Melnick]. In fact mortality-linked securities are also known as *Extreme Mortality Bonds* or *EMBs* or *Catastrophe (CAT) Mortality Bonds* or *CATM bonds* since they are triggered by a catastrophic evolution of death rates of one or more populations. These bonds are extremely lucrative to the investors because of their potential of providing diversification to the portfolio. The generous return on these bonds generally does not bear any correlation with the return on other investments, such as fixed income or equities. From the point of view of the reinsurer these instruments act as ‘Alternative Risk Transfer’ (ART) mechanisms.
The pioneering MLS was the Swiss Re mortality bond (Vita I) issued in 2003 which is the prime focus of this paper. This was followed up by the EIB/BNP longevity bond issued in 2004 ([@Blake]; [@Lane]). For the former, the principal of the bond would have been reduced if there had been a catastrophic mortality event during the life of the bond, therefore allowing Swiss Re to reduce some of its exposure to extreme mortality risk. On the contrary, the latter was a 25-year longevity bond, which was intended for UK pension funds with exposures to longevity risk. This bond took the form of an annuity bond with annual coupon payments tied to the realized survival rates for some English and Welsh males. However, it did not get the same reception as the Swiss Re bond. Swiss Re followed up the success of VITA I by launching five more series of VITA bonds with the latest one being VITA VI which will cover extreme mortality events in Australia, Canada and the UK over a 5 year term from January 2016. Apart from this Swiss Re also experimented with a multi-peril bond called “Mythen Re" which synthesized catastrophe and mortality risks, obtaining 200 million US dollars in protection for North Atlantic hurricane and UK extreme mortality risk. Many other reinsurance giants such as Scottish Re and Munich Re have also issued a score of other mortality bonds. We refer readers to [@Blake2008], [@Coughlan], [@Zhou] and [@Chen3] for further details. In fact it is interesting to note that Swiss Re has also launched an innovative ‘Longevity Trend Bond’ called the Swiss Re Kortis bond in December 2010. Interested readers can refer to [@Chen3] and [@Hunt]. A more up to date list of developments connected to mortality and longevity securities and markets can be found in [@Tan] and [@Liu]. As an aftereffect of these innovative securities, a number of valuation approaches on MLS’s have germinated. [@Huang] classify the approaches into the following four heads:
- *Risk-adjusted process or no-arbitrage pricing:* Under this approach, the first step is to estimate the distribution of future mortality rates in the real-world probability measure. Then the real-world distribution is transformed to its risk-neutral counterpart, on the basis of the actual prices of mortality-linked securities observed in the market. Finally, the price of a mortality-linked security can be calculated by discounting, at the risk-free interest rate, its expected payoff under the identified risk-neutral probability measure. An important point underlying this approach is that it takes into account the actual prices as given. The need of market prices makes the implementation of this approach difficult. One way to effectively use the no arbitrage approach is to use a stochastic mortality model, which is, at the very beginning, defined in the real-world measure and fitted to past data. The model is then calibrated to market prices, yielding a risk-neutral mortality process from which security prices are calculated. For instance, [@Cairns2006] calibrate a two-factor mortality model to the price of the BNP/EIB longevity bond.
<!-- -->
- *The Wang transform:* It is the approach given by [@Wang1], [@Wang2] which consists of employing a distortion operator that transforms the underlying distribution into a risk-adjusted distribution and the MLS price is the expected value under the risk-adjusted probability discounted by risk-free rate. The Wang transform was first employed for mortality-linked securities by [@Lin2], and subsequently by other researchers including [@Dowd] and [@Denuit2007]. Based on the positive dependence characteristic of the mortality in catastrophe areas, [@Shang09] develop a pricing model for catastrophe mortality bonds with comonotonicity and a jump-difusion process. Pointing out there is no unique risk-neutral probability in this incomplete market settings, they use the Wang transform method to price the bond. Unless a very simple mortality model is assumed, parameters in the distortion operator are not unique if we are not given sufficient market price data. For example, when [@Chen2009] used their extended Lee-Carter model with transitory jump effects to price a mortality bond, they were required to estimate three parameters in the Wang transform. To solve for these three parameters, Chen and Cox assumed that they were equal, but such an assumption is not easy to justify. In fact [@Pelsser] has questioned the Wang transform by stating that it is not a universal financial measure for financial and insurance pricing. For more details one can refer to [@Goovaerts1] and [@Lauschagne].
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- *Instantaneous Sharpe Ratio:* [@Milevsky] propose that the expected return on the MLS equals the risk free rate plus the Sharp ratio times its standard deviation.
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- *The utility-based valuation:* The utility based method defines an investor’s utility function and maximizes an agent’s expected utility subject to wealth constraints to obtain the MLS equilibrium. For an elaborate discussion one can review [@Tsai], [@Cox10], [@Hainaut] and [@Dahl].
Apart from the aforesaid methods [@Beelders] and [@Chen10] use the extreme value theory to measure mortality risk of the 2003 Swiss Re Bond. For an interesting summary of other methods to price MLS’s one can refer to [@Shang], [@Zhou13], [@Tan] and [@Liu].
The methods available in literature for the pricing of MLS’s offer only a limited application due to restrictions such as availability of price information or specific utility functions. The difficulty in pricing MLS’s stems from the fact that the MLS market is incomplete as the underlying mortality rates are usually untradeable in financial markets. As a result, the usual no-arbitrage pricing method can only provide a price range or a price bound, instead of a single value.
Surprisingly, mortality linked securities, apart from their present day form seem to have a long history. In the 17th and 18th centuries, so-called *‘tontines’*, which were named after the Neapolitan banker Lorenzo Tonti, had been offered by several governments ([@Weir]). Within these schemes, investors made a one-time payment, and annual dividends were distributed among the survivors. Hence, while still relying on the investor’s survival, his payoffs were connected to the mortality experience among the pool of subscribers. These issues were particularly successful in France, but due to high interest payments, they soon became precarious for the crown’s financial situation (see [@Jennings]). However, this was not only the case with tontines; life annuities, which presented another large share of the royal debt, were also offered at highly favourable conditions from the investors’ perspective. This carelessness was exploited by the Genevan entrepreneur Jacob Bouthillier Beaumont in the scheme attributed to him (cf. [@Jennings]). Here, annuities were subscribed on the lives of a group of Genevan girls for the account of Genevan investors. Thus, their payoffs were directly linked to the survival of the Genevan “madmoiselles", and due to the “generous" assumptions of the French authorities, the schemes were initially highly profitable for the Genevans, the real victim being the French taxpayer. These speculations came to an abrupt end with the French Revolution in 1789, for which the budgetary crises caused by the careless borrowing was, undoubtedly, one major reason. Until the beginning of this century, there has not been another public issue of a mortality linked security, however, there are indications of recent private transactions resembling the tontine scheme (see [@Dowd]). For a more detailed overview of the history of mortality contingent securities the reader is referred to [@Bauer] and [@Luis].
Today, all around the world, investment banks and other financial service providers are working on the idea of trading longevity risk, and the first mortality trading desks have been installed solidifying that “betting on the time of death is set”.[^1]
This paper is concerned with finding price bounds for the Swiss Re mortality catastrophe bond by expressing its payoff in the form of an Asian put option and using the methodology adopted by [@prime1] to find a price range for Asian options.
The rest of this paper is organised as follows: the next section describes the structure of the Swiss Re Bond and expresses its payoff in the form of an Asian put option. Section 3 shows derivations of the lower bound for the aforesaid bond using comonotonicity. In Section 4, we use the same to derive upper bounds for the Swiss Re Bond. In Section 5, we illustrate the computation of bounds by choosing specific models for mortality index. Section 6 portrays numerical results for the derived theory and compares the results with Monte Carlo price of the bond price. Appropriate figures that highlight comparisons among the bounds have also been furnished. The concluding section presents conclusions and avenues for further research.
Design of the Swiss Re Bond
===========================
As pointed out in the introduction, the financial capacity of the life insurance industry to pay catastrophic death losses from natural or man-made disasters is limited. To expand its capacity to pay catastrophic mortality losses, Swiss Re procured about 400 million in coverage from institutional investors in lieu of its first pure mortality security. The reinsurance giant issued a three year bond in December 2003 with maturity on January 1, 2007. To carry out the transaction, Swiss Re set up a special purpose vehicle (SPV) called Vita Capital Ltd. This enabled the corresponding cash flows to be kept off Swiss Re’s balance sheet. The principal is subject to mortality risk which is defined in terms of an index $q_{t_{i}}$ in year $t_{i}$. This mortality index was constructed as a weighted average of mortality rates (deaths per 100,000) over age, sex (male 65% and female 35%) and nationality (US 70%, UK 15%, France 7.5%, Italy 5% and Switzerland 2.5%) and is given below. $$\label{2.0}
q_{t_{i}} = \sum_{j}C_{j}\sum_{k}A_{k}\left(G^{m}q_{k,j,t_{i}}^{m}+G^{f}q_{k,j,t_{i}}^{f}\right)$$ where $q_{k,j,t_{i}}^{m}$ and $q_{k,j,t_{i}}^{f}$ are the respective mortality rates (deaths per 100,000) for males and females in the age group $k$ for country $j$, $C_{j}$ is the weight attached to country $j$, $A_{k}$ is the weight attributed to age group $k$ (same for males and females) and $G^{m}$ and $G^{f}$ are the gender weights applied to males and females respectively.
The Swiss Re bond was a principal-at-risk bond. If the index $q_{t_{i}}$ ($t_{i}$ = 2004, 2005 or 2006 for $i=1,2,3$ respectively) exceeds $K_{1}$ of the actual 2002 level, $q_{0}$, then the investors will have a reduced principal payment. The following equation describes the principal loss percentage, in year $t_{i}$: $$\label{2.1}
L_{i}=\begin{cases}
0 & \text{if } q_{t_{i}}\leq K_{1}q_{0}\\
\frac{\left(q_{t_{i}}-K_{1}q_{0}\right)}{\left(K_{2}-K_{1}\right)q_{0}} & \text{if } K_{1}q_{0}<q_{t_{i}}\leq K_{2}q_{0}\\1 & \text{if }q_{t_{i}}> K_{2}q_{0}\end{cases}$$ In particular, for the case of Swiss Re Bond, $K_{1}=1.3$ and $K_{2}=1.5$. In lieu of having their principal at risk, investors received quarterly coupons equal to the three-month U.S. LIBOR plus 135 basis points. There were 12 coupons in all with a coupon value of $$\label{2.1a}
CO_{j}=\begin{cases}
\left(\frac{SP+LI_{j}}{4}\right).C & \text{if } j=\frac{1}{4},\frac{2}{4},...,\frac{11}{4},\\
\left(\frac{SP+LI_{j}}{4}.C+X_{T}\right) & \text{if } j=3,\end{cases}$$ where $SP$ is the spread value which is 1.35%, $LI_{j}$ are the LIBOR rates, $C=\$400$ million, $T=t_{3}$ and $X_{T}$ is a random variable representing the proportion of the principal returned to the bondholders on the maturity date such that $$\label{2.2}
X_{T}=C\left(1-\sum_{i=1}^{3}L_{i}\right)^{+},$$ where $\sum_{i=1}^{3}L_{i}$ is the aggregate loss ratio at $t_{3}$. However, there was no catastrophe during the term of the bond. The discounted cash flow (DC) of payments is given by $$\label{2.2a}
DC\left(r\right)=\sum_{i=1}^{12}\frac{CO_\frac{i}{4}}{\left(1+\frac{r}{4}\right)^{i}}$$ where $r$ is the nominal annual interest rate.
Further define $$Y_{T}=-{\displaystyle \int_{0}^{T}}\rho\left(t\right)dt$$ where $\rho(t)$ is the US LIBOR at time $t$. As a result, the risk-neutral value at time 0 of the random principal returned at the termination of the bond is $$P=\mbox{E\ensuremath{_{Q}}\ensuremath{\left[e^{-Y_{T}}X_{T}\right]}}$$ where $Q$ is the risk-neutral measure. However, under the assumption of independence of $Y_{T}$ and $X_{T}$, this reduces to $$P=\mbox{E\ensuremath{_{Q}}\ensuremath{\left[e^{-Y_{T}}\right]}}\mbox{E\ensuremath{_{Q}}\ensuremath{\left[X_{T}\right]}}$$ However, for all practical purposes, the literature assumes $$\label{2.3}
P=e^{-rT}\mbox{E\ensuremath{_{Q}}\ensuremath{\left[X_{T}\right]}}$$ where $\mbox{E\ensuremath{_{Q}}\ensuremath{\left[X_{T}\right]}}$ is the expected value under the risk-neutral measure $Q$ and $r$ is the risk-free rate of interest. In subsequent writing, we drop $Q$ from the above expression.
The Principal Payoff of Swiss Re Bond as that of an Asian-type Put Option
-------------------------------------------------------------------------
In fact, we can write $X_{T}$ given in in a more compact form similar to the payoff of the Asian put option as shown below: $$\label{2.4}
X_{T}=D\left({q_{0}-\displaystyle \sum_{i=1}^{3}}5\left(q_{t_{i}}-1.3q_{0}\right)^{+}\right)^{+}$$ with $$\label{2.5}
D=\frac{C}{q_{0}}$$ and the strike price equal to $q_{0}$. For the sake of simplicity, we use $q_{i}$ in place of $q_{t_{i}}$ and define $$\label{2.6}
S_{{i}}=5\left(q_{i}-1.3q_{0}\right)^{+}$$ and $$\label{2.7}
S=\displaystyle \sum_{i=1}^{3}S_{i}$$ Using - in and plugging the result into , we have: $$\label{2.8}
P=De^{-rT}\mbox{E\ensuremath{\left[\left(q_{0}-S\right)^{+}\right]}}$$ Our interest lies in the calculation of reasonable bounds for $P$. We invoke Jensen’s inequality for computing the lower bounds and present our findings in the subsequent sections. We exploit this inequality twice and note that in order to maintain uniformity of having a convex function at each step, it is beneficial to consider the call counterpart of the payoff of Swiss Re Bond rather than . We nomenclate this payoff as $P_{1}$, i.e., we have $$\label{2.9}
P_{1}=De^{-rT}\mbox{E\ensuremath{\left[\left(S-q_{0}\right)^{+}\right]}}$$ We then exploit the put-call parity for Asian options to achieve the bounds for the payoff in question.
Put-Call Parity for the Swiss Re Bond
-------------------------------------
We now derive the put-call parity relationship for the Swiss Re Bond. For any real number $a$, we have: $$\label{2.10}
\left(a\right)^{+}-\left(-a\right)^{+}=a$$ So we obtain $$e^{-rT}\left(\sum_{i=1}^{3}S_{i}-q_{0}\right)^{+}-e^{-rT}\left(q_{0}-\sum_{i=1}^{3}S_{i}\right)^{+}=e^{-rT}\left(\sum_{i=1}^{3}S_{i}-q_{0}\right)$$ On taking expectations on both sides, we obtain $$e^{-rT}\textbf{E}\left[\left(\sum_{i=1}^{3}S_{i}-q_{0}\right)^{+}\right]-e^{-rT}\textbf{E}\left[\left(q_{0}-\sum_{i=1}^{3}S_{i}\right)^{+}\right]=e^{-rT}\textbf{E}\left[\sum_{i=1}^{3}S_{i}-q_{0}\right]$$ Finally, on multiplying by $D$ and expanding the definition of $S_{i}$, we have $$P_{1}-P=De^{-rT}\textbf{E}\left[\sum_{i=1}^{3}5\left(q_{i}-1.3q_{0}\right)^{+}-q_{0}\right]$$ $$\label{2.11}
\Rightarrow P_{1}-P=De^{-rT}\left[5\sum_{i=1}^{3}e^{rt_{i}}C\left(1.3q_{0},t_{i}\right)-q_{0}\right],$$ where $C\left(K,t_{i}\right)$ denotes the price of a European call on the mortality index with strike $K$, maturity $t_{i}$ and current mortality value $q_{0}$. This option would be in-the-money if the mortality index is more than $1.3q_{0}$ which is the trigger level of Swiss Re bond. Clearly, such instruments are not available for trading in the market at present. But a complete life market is in the making and we feel such securities will soon be introduced (c.f. [@Blake3] and [@Blake2008]). The pay-off structures, i.e. the design of the issued securities and the mortality contingent payments should be developed to appear attractive to investors and the re-insurer. Although, the Swiss Re bond was fully subscribed and press reports highlight that investors were quite satisfied with it (e.g. *Euroweek*, 19 December 2003), the market for mortality linked securities still needs innovations such as vanilla options on mortality index to provide flexible hedging solutions. Investors of the Swiss Re bond included a large number of pension funds as they could view this bond as a powerful hedging instrument. The underlying mortality risk associated with the bond is correlated with the mortality risk of the active members of a pension plan. If a catastrophe occurs, the reduction in the principal would be offset by reduction in pension liability of these pension funds. Moreover, the bond offers a considerably higher return than similarly rated floating rate securities (c.f. [@Blake]). In a manner similar to [@Bauer], we feel the success of the life market hinges upon flexibility. As a result, such option-type structures enable re-insurer to keep most of the capital while at the same time being hedged against catastrophic mortality situation. [@Cox3] present an interesting note on the trigger level of $1.3q_{0}$ in context of 2004 tsunami in Asia and Africa. A mortality option of the above type would become extremely useful in such a case. [@Tsai] and [@Cheng] decompose the terminal payoff of the Swiss Re bond into two call options.
Equation gives the required put-call parity relation between the Swiss Re mortality bound and its call counterpart. Define $$\label{2.13}
G=De^{-rT}\left[5\sum_{i=1}^{3}e^{rt_{i}}C\left(1.3q_{0},t_{i}\right)-q_{0}\right]$$ Clearly, if we bound $P_{1}$ by bounds $l_{1}$ and $u_{1}$, then the corresponding bounds for the Swiss Re mortality bond are as follows $$\label{2.14}
\left(l_{1}-G\right)^{+} \leq P \leq \left(u_{1}-G\right)^{+}$$
Lower Bounds for the Swiss Re Bond
==================================
We now proceed to work out appropriate lower bounds for the terminal value of the principal paid in the Swiss Re Bond. For this we first calculate bounds for the following Asian-type call option $$\label{4.1.1}
P_{1}=De^{-rT}\textbf{E}\ensuremath{\left[\left({\displaystyle \sum_{i=1}^{n}}S_{i}-q_{0}\right)^{+}\right]}$$ with $T=t_{n}$ and $n=3$. The interval $\left[0,T\right]$ consists of the monitoring times $t_{1}, t_{2},...,t_{n-1}$. The undercurrent of the theory presented in this section is the paper by [@prime1]. In an attempt to estimate the value of the Asian call option, the authors derive four lower bounds namely trivial, $LB_{1}$, $LB_{t}^{\left(1\right)}$ and $LB_{t}^{\left(2\right)}$, which are sharper in increasing order in sense of their proximity to the actual value of the Asian call. The underlying assumption they make in deriving these bounds is that European call prices with arbitrary strikes and maturities are available in the market. Although, as our previous discussion indicates, such securities with the underlying as the mortality index have not appeared on the horizon as yet, but would be indispensable for the development of a complete life market. The first step towards designing of such securities is the need for a benchmark longevity index. The formation of Life and Longevity Markets Association (LLMA) in 2010 was an important milestone in this direction. The LLMA promotes the development of a liquid trading market in longevity and mortality-related risk, of the type that exists for Insurance Linked Securities (ILS) and other large trend risks like interest rates and inflation. There have been a few mortality indices created by various parties but we still lack a benchmark. [@Menioux] throws light on various longevity indices.
Invoking Jensen’s inequality, we have $$\begin{aligned}
\label{4.1.3}
\textbf{E}\ensuremath{\left[\left({\displaystyle \sum_{i=1}^{n}}S_{i}-q_{0}\right)^{+}\right]} & \geq & \textbf{E}\ensuremath{\left[\left(5{\displaystyle \sum_{i=1}^{n}}\left(\textbf{E}\left(q_{i}|\Lambda\right)-1.3q_{0}\right)^{+}-q_{0}\right)^{+}\right]}.\end{aligned}$$ We now define $$\label{4.1.4}
Z_{i}=5\left(\textbf{E}\left(q_{i}|\Lambda\right)-1.3q_{0}\right)^{+}; i=1,2,...,n$$ As a result in , we have obtained $$\label{4.1.5}
\textbf{E}\ensuremath{\left[\left({\displaystyle \sum_{i=1}^{n}}S_{i}-q_{0}\right)^{+}\right]} \geq \textbf{E}\ensuremath{\left[\left({\displaystyle \sum_{i=1}^{n}}Z_{i}-q_{0}\right)^{+}\right]}$$
On investigating the relationship between $\textbf{E}\ensuremath{\left[{\displaystyle \sum_{i=1}^{n}}S_{i}\right]}$ and $\textbf{E}\ensuremath{\left[{\displaystyle \sum_{i=1}^{n}}Z_{i}\right]}$, we find that $$\begin{aligned}
\label{4.1.6}
\textbf{E}\ensuremath{\left[{\displaystyle \sum_{i=1}^{n}}S_{i}\right]}
& \geq & \textbf{E}\ensuremath{\left[{\displaystyle \sum_{i=1}^{n}}Z_{i}\right]}.\end{aligned}$$ On lines of , define $$\label{4.1.8}
Z=\displaystyle \sum_{i=1}^{n}Z_{i}$$ so that we can rewrite as $$\label{4.1.10}
\mbox{E\ensuremath{\left[\left(S-q_{0}\right)^{+}\right]}}\geq \mbox{E\ensuremath{\left[\left(Z-q_{0}\right)^{+}\right]}}$$ In fact, the two sides of the inequality in are essentially the stop-loss premiums of $S$ and $Z$. Thus, we have obtained $$\label{4.1.11}
S \geq_{sl} Z$$ or $$S\geq_{\mbox{sl }}{\displaystyle \sum_{i=1}^{n}}\left(\textbf{E}\left(q_{i}|\Lambda\right)-1.3q_{0}\right)^{+}$$ Now, suitably tailoring the inequality to suit our need of the Asian-type call option by multiplying by the discount factor at time $T$, we obtain $$\label{4.1.12}
P_{1}\geq De^{-rT}\textbf{E}\ensuremath{\left[\left({\displaystyle \sum_{i=1}^{n}}5\left(\textbf{E}\left(q_{i}|\Lambda\right)-1.3q_{0}\right)^{+}-q_{0}\right)^{+}\right].}$$ To exploit the theory of comonotonicity see for example in [@1], we now have to show that the lower bound for $S$, i.e. $S^{l}$ can be formulated as the sum of stop-loss premiums. This task becomes trivial if we can choose the conditioning variable $\Lambda$ in such a way that $\textbf{E}\left(q_{i}|\Lambda\right)$ is either increasing or decreasing for every $i$, so that the vector: $\textbf{q}{}^{\textbf{l}}=\left(\textbf{E}\ensuremath{\left(q_{1}|\Lambda\right)},\ldots,\textbf{E}\ensuremath{\left(q_{n}|\Lambda\right)}\right)$ is comonotonic. This automatically implies that the vector: $\textbf{Z}{}^{\textbf{l}}=\left(Z_{1},\ldots,Z_{n}\right)$ is comonotonic. As a result we have
$$\begin{aligned}
\label{4.1.15}
\textbf{E}\ensuremath{\left[\left(S-q_{0}\right)^{+}\right]} & \geq & {\displaystyle \sum_{i=1}^{n}}\textbf{E}\ensuremath{\left[\left(Z_{i}-F_{Z_{i}}^{-1}\left(F_{Z}\left(q_{0}\right)\right)\right)^{+}\right]},\end{aligned}$$
where $F_{X}^{-1}$ is the generalized inverse defined in the usual way: $$\label{4.1.15a}
F_{X}^{-1}\left(p\right)=\inf \{x \in\mathbb{R}|F_{X}\left(x\right)\geq p\},\;\;p \in \left[0,1\right]$$
Further, by the definition of cdf, we have $$\label{4.1.16}
F_{Z}\left(q_{0}\right)=\textbf{P}\left[Z\leq q_{0}\right]=\textbf{P}\left[{\displaystyle \sum_{j=1}^{n}Z_{j}}\leq q_{0}\right]=\textbf{P}\left[{\displaystyle \sum_{j=1}^{n}5\left(\textbf{E}\left(q_{j}|\Lambda\right)-1.3q_{0}\right)^{+}}\leq q_{0}\right].$$ Thus, we have been able to obtain a stop-loss lower bound for $S=\sum_{i=1}^{n}S_{i}$ by conditioning on an arbitrary random variable $\Lambda$, i.e., $$\label{4.1.17}
P_{1}\geq De^{-rT}{\displaystyle \sum_{i=1}^{n}}\textbf{E}\ensuremath{\left[\left(5\left(\textbf{E}\left(q_{i}|\Lambda\right)-1.3q_{0}\right)^{+}-F_{Z_{i}}^{-1}\left(F_{Z}\left(q_{0}\right)\right)\right)^{+}\right]}.$$
The Trivial Lower Bound
-----------------------
In case, if the random variable $\Lambda$ is independent of the mortality evolution $\left\{ q_{t}\right\} _{t\geq0}$, the bound in simply reduces to:$$\label{4.2.1}
P_{1}\geq De^{-rT}\textbf{E}\ensuremath{\left[\left({\displaystyle \sum_{i=1}^{n}}5\left(\textbf{E}\left(q_{i}\right)-1.3q_{0}\right)^{+}-q_{0}\right)^{+}\right]}$$ or even more precisely as the outer expectation is redundant $$\label{4.2.2}
P_{1}\geq De^{-rT}\ensuremath{\left({\displaystyle \sum_{i=1}^{n}}5\left(\textbf{E}\left(q_{i}\right)-1.3q_{0}\right)^{+}-q_{0}\right)^{+}.}$$ Under the assumption of the existence of an Equivalent Martingale Measure (EMM), Q, the discounted mortality process is a martingale, so that $$\label{4.2.4}
\textbf{E}\left[q_{t}\right]=q_{0}e^{rt}.$$ If we substitute this in equation , we obtain a very rough lower bound for the Asian-type call option $$\label{4.2.5}
P_{1}\geq Ce^{-rT}\left({\displaystyle \sum_{i=1}^{n}5\left(e^{rt_{i}}-1.3\right)^{+}}-1\right)^{+}=:\mbox{ lb}_{0}.$$ In the light of put-call parity derived in section 2, the trivial lower bound for the Swiss Re mortality bond is given as $$\label{4.2.6}
P\geq \left(\mbox{ lb}_{0}-G\right)^{+}=:\mbox{ SWLB}_{0}.$$ where G is defined in .
The Lower Bound $\mbox{SWLB}_{1}$
---------------------------------
To improve upon the trivial lower bound, we choose $\Lambda=q_{1}$ in . Using the martingale argument for the discounted mortality process $$\textbf{E}\left[q_{i}|q_{1}\right]=\textbf{E}\left[e^{rt_{i}}e^{-rt_{i}}q_{i}|q_{1}\right]=e^{r\left(t_{i}-t_{1}\right)}q_{1}.$$ Then the random vector $\left(q_{1},e^{r\left(t_{2}-t_{1}\right)}q_{1},\ldots,e^{r\left(t_{n}-t_{1}\right)}q_{1}\right)$ is comonotone. Equation then reduces to $$\label{4.3.2}
P_{1}\geq De^{-rT}{\displaystyle \sum_{i=1}^{n}}\textbf{E}\ensuremath{\left[\left(5\left(e^{r\left(t_{i}-t_{1}\right)}q_{1}-1.3q_{0}\right)^{+}-F_{Z_{i}}^{-1}\left(F_{Z}\left(q_{0}\right)\right)\right)^{+}\right]},$$ where by the definition of cdf, we have $$F_{Z}\left(q_{0}\right)=\textbf{P}\left[Z\leq q_{0}\right]=\textbf{P}\left[{\displaystyle \sum_{j=1}^{n}}5\left(e^{r\left(t_{j}-t_{1}\right)}q_{1}-1.3q_{0}\right)^{+}\leq q_{0}\right]$$ $$\Rightarrow F_{Z}\left(q_{0}\right)=\textbf{P}\left[{\displaystyle \sum_{j=1}^{n}}5\left(e^{r\left(t_{j}-t_{1}\right)}\frac{q_{1}}{q_{0}}-1.3\right)^{+}\leq 1\right].$$
Now, as the left hand side of the inequality within the probability is an increasing function in $q_{1}/q_{0}$, we have that $Z\leq q_{0}$ if and only if $q_{1}\leq xq_{0}$, where we substitute $x$ for $q_{1}/q_{0}$ in the above probability and obtain its value by solving $$\label{4.3.2.1}
{\displaystyle \sum_{i=1}^{n}}\left(e^{r\left(t_{i}-t_{1}\right)}x-1.3\right)^{+}=0.2$$
As a result, we have $$\label{4.3.2.2}
F_{Z}\left(q_{0}\right)=F_{q_{1}}\left(xq_{0}\right)=F_{Z_{i}}\left(5q_{0}\left(e^{r\left(t_{i}-t_{1}\right)}x-1.3\right)^{+}\right)\;\;\forall i$$
Plugging into , the lower bound reduces to [ $$\begin{aligned}
\label{4.3.3}
P_{1} & \geq & 5De^{-rT}{\displaystyle \sum_{i=1}^{n}}\textbf{E}\ensuremath{\left[\left(\left(e^{r\left(t_{i}-t_{1}\right)}q_{1}-1.3q_{0}\right)^{+}-q_{0}\left(e^{r\left(t_{i}-t_{1}\right)}x-1.3\right)^{+}\right)^{+}\right]}
\nonumber \\
& = & 5De^{-rT}{\displaystyle \sum_{i=1}^{n}}e^{r\left(t_{i}-t_{1}\right)}\textbf{E}\ensuremath{\left[\left(\left(q_{1}-\frac{1.3q_{0}}{e^{r\left(t_{i}-t_{1}\right)}}\right)^{+}-q_{0}\left(x-\frac{1.3}{e^{r\left(t_{i}-t_{1}\right)}}\right)^{+}\right)^{+}\right]}
\nonumber \\
& = & 5De^{-rT}{\displaystyle \sum_{i=1}^{n}}e^{r\left(t_{i}-t_{1}\right)}\textbf{E}\ensuremath{\left[\left(q_{1}-q_{0}\left(\frac{1.3}{e^{r\left(t_{i}-t_{1}\right)}}+\left(x-\frac{1.3}{e^{r\left(t_{i}-t_{1}\right)}}\right)^{+}\right)\right)^{+}\right]}
\nonumber \\
& = &5D{\displaystyle \sum_{i=1}^{n}}e^{-r\left(T-t_{i}\right)}C\left(q_{0}.\max\left(x,\,\frac{1.3}{e^{r\left(t_{i}-t_{1}\right)}}\right),\; t_{1}\right)=:\,\mbox{lb}_{1}.\end{aligned}$$ ]{} where $C\left(K,t_{1}\right)$ denotes the price of a European call on the mortality index with strike K, maturity $t_{1}$ and current mortality index $q_{0}$. The function $\mbox{lb}_{1}$ provides a lower bound for the Asian-type call option in terms of European calls at each of the times such that these contracts have maturity $t_{1}$ and strike $q_{0}.\max\left(x,\,\frac{1.3}{e^{r\left(t_{i}-t_{1}\right)}}\right)$ at the $i$th time point. This bound holds for any arbitrage-free market model and is a significant improvement over the trivial bound given in . Invoking the put-call parity derived in section 2, the corresponding lower bound for the Swiss Re mortality bond is given as $$\label{4.3.8}
P\geq \left(\mbox{ lb}_{1}-G\right)^{+}=:\mbox{ SWLB}_{1}.$$ where G is defined in .
A Model-independent Lower Bound
-------------------------------
As the next step, we suggest that the bound $\mbox{ SWLB}_{1}$ can be improved by imposing the following additional assumption $$\label{4.5.1}
{\displaystyle \sum_{i=1}^{n}q_{i}\geq_{sl}\left(\sum_{i=1}^{j-1}q_{0}^{\left(1-t_{i}/t\right)}q_{t}^{t_{i}/t}+{\displaystyle \sum_{i=j}^{n}}e^{r\left(t_{i}-t\right)}q_{t}\right)}$$ for $0\leq t\leq T$ and $j=\min\left\{ i\,:\, t_{i}\geq t\right\}$. Clearly, [ $$\begin{aligned}
\label{4.5.5}
{\displaystyle \sum_{i=1}^{n}}5\left(\textbf{E}\left(q_{i}|q_{t}\right)-1.3q_{0}\right)^{+} & = & \sum_{i=1}^{j-1}5\left(\textbf{E}\left(q_{i}|q_{t}\right)-1.3q_{0}\right)^{+}+{\displaystyle \sum_{i=j}^{n}}5\left(\textbf{E}\left(q_{i}|q_{t}\right)-1.3q_{0}\right)^{+}
\nonumber\\ & = & \sum_{i=1}^{j-1}5q_{0}\left(\left(\frac{q_{t}}{q_{0}}\right)^{t_{i}/t}-1.3\right)^{+}+{\displaystyle \sum_{i=j}^{n}}5q_{0}\left(\frac{q_{t}}{q_{0}}e^{r\left(t_{i}-t\right)}-1.3\right)^{+}
\nonumber\\
& =: & S^{l_{2}}.\end{aligned}$$ ]{}
Evidently, $S^{l_{2}}$ is the same as $Z$ in with $\Lambda$ being replaced by $q_{t}$ and thus from , we have $$\label{4.5.6}
S \geq_{sl} S^{l_{2}}$$
As before, let $j=\min\left\{ i\,:\, t_{i}\geq t\right\}$. Consider the components of $S^{l_{2}}$ in equation and define $\textbf{Y}=\left(Y_{1},\ldots,Y_{n}\right)$, where $$Y_{i}=\begin{cases}
5q_{0}\left(\left(\frac{q_{t}}{q_{0}}\right)^{t_{i}/t}-1.3\right)^{+} & i<j\\
5q_{0}\left(\left(\frac{q_{t}}{q_{0}}\right)e^{r\left(t_{i}-t\right)}-1.3\right)^{+} & i\geq j
\end{cases}$$ $i=1,2,...,n$. Clearly, **Y** is comonotonic since its components are strictly increasing functions of a single variable $q_{t}$. So, the stop-loss transform of $S^{l_{2}}$ can be written as the sum of stop-loss transform of its components (see for example in [@1]), i.e., $$\label{4.5.7}
\textbf{E}\ensuremath{\left[\left(S^{l_{2}}-q_{0}\right)^{+}\right]={\displaystyle \sum_{i=1}^{n}}\textbf{E}\ensuremath{\left[\left(Y_{i}-F_{Y_{i}}^{-1}\left(F_{S^{l_{2}}}\left(q_{0}\right)\right)\right)^{+}\right]}}$$ where $F_{S^{l_{2}}}\left(q_{0}\right)$ is the distribution function of $S^{l_{2}}$ evaluated at $q_{0}$ such that for an arbitrary $t$, we have: $$\begin{aligned}
F_{S^{l_{2}}}\left(q_{0}\right) & = & \textbf{P}\left[S^{l_{2}}\leq q_{0}\right]
\nonumber\\
& = & \textbf{P}\left[\sum_{i=1}^{j-1}5q_{0}\left(\left(\frac{q_{t}}{q_{0}}\right)^{t_{i}/t}-1.3\right)^{+}+{\displaystyle \sum_{i=j}^{n}}5q_{0}\left(\left(\frac{q_{t}}{q_{0}}\right)e^{r\left(t_{i}-t\right)}-1.3\right)^{+}\leq q_{0}\right]
\nonumber\\
& = & \textbf{P}\left[\sum_{i=1}^{j-1}\left(\left(\frac{q_{t}}{q_{0}}\right)^{t_{i}/t}-1.3\right)^{+}+{\displaystyle \sum_{i=j}^{n}}\left(\left(\frac{q_{t}}{q_{0}}\right)e^{r\left(t_{i}-t\right)}-1.3\right)^{+}\leq 0.2\right].
\nonumber\end{aligned}$$
Clearly, $S^{l_{2}}\leq q_{0}$ if and only if $q_{t}\leq xq_{0}$, where we substitute $x$ for $q_{t}/q_{0}$ in the above expression and obtain its value by solving: $$\label{4.5.9}
\sum_{i=1}^{j-1}\left(x^{t_{i}/t}-1.3\right)^{+}+{\displaystyle \sum_{i=j}^{n}}\left(xe^{r\left(t_{i}-t\right)}-1.3\right)^{+}= 0.2.$$ As a result, we have: $$F_{S^{l_{2}}}\left(q_{0}\right)=F_{q_{t}}\left(xq_{0}\right)=\begin{cases}
F_{Y_{i}}\left(5q_{0}\left(x^{t_{i}/t}-1.3\right)^{+}\right)\: & i<j\\
F_{Y_{i}}\left(5q_{0}\left(xe^{r\left(t_{i}-t\right)}-1.3\right)^{+}\right)\: & i\geq j
\end{cases}$$ Using this result in equation and recalling the definition of the Asian-type call option given in along with the stop-loss order relationship between $S$ and $S^{l_{2}}$ as given by equation , we obtain [ $$\begin{aligned}
\label{4.5.10}
P_{1} & \geq & De^{-rT}\left({\displaystyle \sum_{i=1}^{n}}\textbf{E}\ensuremath{\left[\left(Y_{i}-F_{Y_{i}}^{-1}\left(F_{S^{l_{2}}}\left(q_{0}\right)\right)\right)^{+}\right]}\right)
\nonumber\\
& = & Ce^{-rT}\Bigg(\sum_{i=1}^{j-1}\textbf{E}\left[\left(5\left(\left(\frac{q_{t}}{q_{0}}\right)^{t_{i}/t}-1.3\right)^{+}-5\left(x^{t_{i}/t}-1.3\right)^{+}\right)^{+}\right] \nonumber\\
& {} {} & {} \;\;\;\;\;\;\;\;+{\displaystyle \sum_{i=j}^{n}\textbf{E}}\left[\left(5\left(\left(\frac{q_{t}}{q_{0}}\right)e^{r\left(t_{i}-t\right)}-1.3\right)^{+}-5\left(xe^{r\left(t_{i}-t\right)}-1.3\right)^{+}\right)^{+}\right]\Bigg) \nonumber\\
& = & 5Ce^{-rT}\Bigg(\sum_{i=1}^{j-1}\frac{1}{q_{0}^{t_{i}/t}}\textbf{E}\left[\left(q_{t}^{t_{i}/t}-q_{0}^{t_{i}/t}\left(1.3+\left(x^{t_{i}/t}-1.3\right)^{+}\right)\right)^{+}\right] \nonumber\\
& {} {} & {} \;\;\;\;\;\;\;\;+{\displaystyle \sum_{i=j}^{n}\frac{e^{r\left(t_{i}-t\right)}}{q_{0}}\textbf{E}}\left[\left(q_{t}-q_{0}\left(\frac{1.3}{e^{r\left(t_{i}-t\right)}}+\left(x-\frac{1.3}{e^{r\left(t_{i}-t\right)}}\right)^{+}\right)\right)^{+}\right]\Bigg) \nonumber\\
& = & 5De^{-rT}\Bigg(\sum_{i=1}^{j-1}q_{0}^{1-t_{i}/t}\textbf{E}\left[\left(q_{t}^{t_{i}/t}-q_{0}^{t_{i}/t}.\max\left(x^{t_{i}/t},\,1.3\right)\right)^{+}\right] \nonumber\\
& {} {} & {} \;\;\;\;\;\;\;\;+{\displaystyle \sum_{i=j}^{n}}e^{rt_{i}}C\left(q_{0}.\max\left(x,\,\frac{1.3}{e^{r\left(t_{i}-t\right)}}\right),\; t\right)\Bigg) \nonumber\\
& =: & \,\mbox{lb}_{t}^{\left(2\right)}\end{aligned}$$ ]{} In fact, $\mbox{lb}_{t}^{\left(2\right)}$ is a lower bound for all $t$ and so it can be maximized with respect to $t$ to yield the optimal lower bound as given below: $$\label{4.5.19}
P_{1}\geq\max_{0\leq t\leq T}\mbox{lb}_{t}^{\left(2\right)}.$$
On choosing $t=t_{1}$ implies $j=1$ and so equation reduces to and we obtain $$\mbox{lb}_{1}^{\left(2\right)}=\mbox{lb}_{1}.$$ As a result we have $$\max_{0\leq t\leq T}\mbox{lb}_{t}^{\left(2\right)} \geq \mbox{lb}_{1}.$$ Clearly, once again, as in the previous sections, we have $$\label{4.5.20}
P\geq \left(\mbox{lb}_{t}^{\left(2\right)}-G\right)^{+}=:\mbox{SWLB}_{t}^{\left(2\right)}.$$ where G is defined in . We now move on to the derivation of an upper bound for the price of Swiss Re bond in the next section.
Upper Bounds for the Swiss Re Bond
==================================
We now derive a couple of upper bounds for the Swiss Re bond.
A First Upper Bound
-------------------
This section will focus on finding an upper bound for the bond in question by using comonotonicity theory. Define the comonotonic counterpart of $\textbf{q}{}=\left(q_{1},...,q_{n}\right)$ as $\textbf{q}{}^{\textbf{u}}=\left(F_{S_{1}}^{-1}\left(U\right),...,F_{S_{n}}^{-1}\left(U\right)\right)$ where $U \sim U\left(0,1\right)$. Further define $$\label{4.1.1a}
S^{c}={\displaystyle \sum_{i=1}^{n}}F_{S_{i}}^{-1}\left(U\right)={\displaystyle \sum_{i=1}^{n}}S_{i}^{c}.$$ Clearly, $$\label{4.1.2a}
S \leq_{cx} S^{c}$$ where $cx$ denotes convex ordering (see for example in [@1]). In other words, $$\label{4.1.3a}
\textbf{E}\ensuremath{\left[\left({\displaystyle \sum_{i=1}^{n}}S_{i}-q_{0}\right)^{+}\right]} \leq \textbf{E}\ensuremath{\left[\left({\displaystyle \sum_{i=1}^{n}}S_{i}^{c}-q_{0}\right)^{+}\right]}= {\displaystyle \sum_{i=1}^{n}}\textbf{E}\ensuremath{\left[\left(S_{i}-F_{S_{i}}^{-1}\left(F_{S^{c}}\left(q_{0}\right)\right)\right)^{+}\right]}.$$ As a result, an upper bound for the call counterpart of the Swiss Re bond is given as [ $$\begin{aligned}
\label{4.1.4a}
P_{1} & \leq & De^{-rT}{\displaystyle \sum_{i=1}^{n}}\textbf{E}\ensuremath{\left[\ensuremath{\left(S_{i}-F_{S_{i}}^{-1}\left(F_{S^{c}}\left(q_{0}\right)\right)\right)^{+}}\right]}\nonumber\\
& = & 5De^{-rT}{\displaystyle \sum_{i=1}^{n}}\textbf{E}\ensuremath{\left[\left(q_{i}-\left(1.3q_{0}+\frac{F_{S_{i}}^{-1}\left(F_{S^{c}}\left(q_{0}\right)\right)}{5}\right)\right)^{+}\right]}\nonumber\\
& = & 5De^{-rT}{\displaystyle \sum_{i=1}^{n}e^{rt_{i}}C\left(1.3q_{0}+\frac{F_{S_{i}}^{-1}\left(F_{S^{c}}\left(q_{0}\right)\right)}{5},t_{i}\right)}.\end{aligned}$$ ]{}As a result we can write the upper bound given above as $$\label{4.1.5a}
P_{1} \leq 5De^{-rT}{\displaystyle \sum_{i=1}^{n}e^{rt_{i}}C\left(1.3q_{0}+\frac{F_{S_{i}}^{-1}\left(x\right)}{5},t_{i}\right)}$$ where $x\in\left(0,1\right)$ is the solution of the equation $$\label{4.1.6a}
{\displaystyle \sum_{i=1}^{n}F_{S_{i}}^{-1}\left(x\right)=q_{0}}.$$ We now seek to express the inverse distribution function of $S_{i}$ in terms of that of $q_{i}$. Let $$\label{4.1.7a}
y_{i} = F_{S_{i}}^{-1}\left(x\right);\;y_{i} \geq 0$$ $$\begin{aligned}
\label{4.1.8a}
\Rightarrow x & = & F_{S_{i}}\left(y_{i}\right) \nonumber\\
& = & P\left[5\left(q_{i}-1.3q_{0}\right)^{+} \leq y_{i}\right] \nonumber\\
& = & 1-P\left[5\left(q_{i}-1.3q_{0}\right)^{+} > y_{i}\right] \nonumber\\
& = & 1-P\left[q_{i}>1.3q_{0}+\frac{y_{i}}{5}\right] \nonumber\\
& = & F_{q_{i}}\left(1.3q_{0}+\frac{y_{i}}{5}\right).\end{aligned}$$ $$\label{4.1.9a}
\Rightarrow y_{i} = 5\left(F_{q_{i}}^{-1}\left(x\right)-1.3q_{0}\right).$$
From equations , and , we conclude that the upper bound is given as $$\label{4.1.10a}
P_{1}\leq 5De^{-rT}{\displaystyle \sum_{i=1}^{n}e^{rt_{i}}C\left(F_{q_{i}}^{-1}\left(x\right),t_{i}\right)}=: \mbox{ub}_{1}.$$ where using equations and , we see that $x$ solves the following equation $$\label{4.1.11a}
{\displaystyle \sum_{i=1}^{n}F_{q_{i}}^{-1}\left(x\right)=\frac{q_{0}}{5}\left(1+6.5n\right)}.$$
As in the case of lower bounds, invoking the put-call parity of section 2, we have for the Swiss Re bond $$\label{5.10}
P\leq \left(\mbox{ub}_{1}-G\right)^{+}=:\mbox{SWUB}_{1}.$$ where G is defined in .
An Improved Upper Bound by conditioning
---------------------------------------
We now seek to obtain a sharper upper bound for the Swiss Re bond. This is possible if we assume that some additional information is available concerning the stochastic nature of $\left(q_{1},q_{2},...,q_{n}\right)$. That is, if we can find a random variable $\Lambda$, with a known distribution, such that the individual conditional distributions of $q_{i}$ given the event $\Lambda=\lambda$ are known for all $i$ and all possible values of $\lambda$.
Define $$\label{4.22}
S^{u}={\displaystyle \sum_{i=1}^{n}}F_{S_{i}|\Lambda}^{-1}\left(U\right)={\displaystyle \sum_{i=1}^{n}}S_{i}^{u}.$$ where $U \sim U\left(0,1\right)$. Then we have $$\label{4.21}
S \leq_{cx} S^{u} \leq_{cx} S^{c}$$ where we Now let $\textbf{q}{}^{\textbf{u}}=\left(S_{1}^{u},...,S_{n}^{u}\right)$. Since $\left(F_{S_{1}|\Lambda=\lambda}^{-1},...,F_{S_{n}|\Lambda=\lambda}^{-1}\right)$ is comonotonic, we have, $$\label{4.23}
F_{S^{u}|\Lambda=\lambda}^{-1}\left(p\right)={\displaystyle \sum_{i=1}^{n}}F_{S_{i}|\Lambda=\lambda}^{-1}\left(p\right),\;p \in \left(0,1\right).$$ It follows that, in this case $$\label{4.24}
{\displaystyle \sum_{i=1}^{n}}F_{S_{i}|\Lambda=\lambda}^{-1}\left(F_{S^{u}|\Lambda=\lambda}\left(q_{0}\right)\right)=q_{0}.$$ and so we have $$\label{4.25}
f\left(\lambda\right)=\textbf{E}\ensuremath{\left[\left({\displaystyle \sum_{i=1}^{n}}S_{i}^{u}-q_{0}\right)^{+}\middle|\Lambda=\lambda\right]} = {\displaystyle \sum_{i=1}^{n}}\textbf{E}\ensuremath{\left[\left(S_{i}-F_{S_{i}|\Lambda=\lambda}^{-1}\left(F_{S^{u}|\Lambda=\lambda}\left(q_{0}\right)\right)\right)^{+}\middle|\Lambda=\lambda\right]}.$$ By applying the tower property and using the convex order relationship given by , we obtain an upper bound for the call counterpart of the Swiss Re bond, i.e., [ $$\begin{aligned}
\label{4.26}
P_{1} & \leq & De^{-rT}\textbf{E}\ensuremath{\left[\left(S^{u}-q_{0}\right)^{+}\right]}\nonumber \\
& = & De^{-rT}\textbf{E}\ensuremath{\left[f\left(\lambda\right)\right]}\nonumber \\
& = & De^{-rT}{\displaystyle \sum_{i=1}^{n}}{\displaystyle \int_{-\infty}^{\infty}}\textbf{E}\ensuremath{\left[\ensuremath{\left(S_{i}-F_{S_{i}|\Lambda=\lambda}^{-1}\left(F_{S^{u}|\Lambda=\lambda}\left(q_{0}\right)\right)\right)^{+}\middle|\Lambda=\lambda}\right]}dF_{\Lambda}\left(\lambda\right)\nonumber\\
& = & 5De^{-rT}{\displaystyle \sum_{i=1}^{n}}{\displaystyle \int_{-\infty}^{\infty}}\textbf{E}\ensuremath{\left[\left(q_{i}-\left(1.3q_{0}+\frac{F_{S_{i}|\Lambda=\lambda}^{-1}\left(F_{S^{u}|\Lambda=\lambda}\left(q_{0}\right)\right)}{5}\right)\right)^{+}\middle|\Lambda=\lambda\right]}dF_{\Lambda}\left(\lambda\right).\end{aligned}$$ ]{}Given the event $\Lambda=\lambda$, let $x$ be the solution to the following equation. $$\label{4.272}
{\displaystyle \sum_{i=1}^{n}}F_{S_{i}|\Lambda=\lambda}^{-1}\left(x\right)=q_{0}.$$ Further, we see from equation , that $x=F_{S^{u}|\Lambda=\lambda}\left(q_{0}\right)$. It therefore follows, as a result of equation 93 of [@1] that an upper bound for the call counterpart of the Swiss Re bond is given as $$\label{4.28}
P_{1} \leq 5De^{-rT}{\displaystyle \sum_{i=1}^{n}}{\displaystyle \int_{-\infty}^{\infty}}\textbf{E}\ensuremath{\left[\left(q_{i}-\left(1.3q_{0}+\frac{F_{S_{i}|\Lambda=\lambda}^{-1}\left(x\right)}{5}\right)\right)^{+}\middle|\Lambda=\lambda\right]}dF_{\Lambda}\left(\lambda\right).$$ where $x$ is obtained by solving . Moreover, it is straightforward to write $$\label{4.28a}
F_{S_{i}|\Lambda=\lambda}^{-1}\left(x\right) = 5\left(F_{q_{i}|\Lambda=\lambda}^{-1}\left(x\right)-1.3q_{0}\right).$$ As a result, the upper bound can be rewritten as $$\label{4.28b}
P_{1} \leq 5De^{-rT}{\displaystyle \sum_{i=1}^{n}}{\displaystyle \int_{-\infty}^{\infty}}\textbf{E}\ensuremath{\left[\left(q_{i}-F_{q_{i}|\Lambda=\lambda}^{-1}\left(x\right)\right)^{+}\middle|\Lambda=\lambda\right]}dF_{\Lambda}\left(\lambda\right)=:\mbox{ub}_{t}^{\left(1\right)}$$ where $x \in \left(0,1\right)$ can be obtained by solving the equation $$\label{4.28c}
{\displaystyle \sum_{i=1}^{n}}F_{q_{i}|\Lambda=\lambda}^{-1}\left(x\right)=\frac{q_{0}}{5}\left(1+6.5n\right).$$
Since this is is an upper bound for all $t$, it follows that we can find the optimal upper bound by minimising equation over $t\in \left[0,T\right]$. As before, invoking the put-call parity of section 2, we have for the Swiss Re bond $$\label{4.29}
P \leq \left(\mbox{ub}_{t}^{\left(1\right)}-G\right)^{+}=:\mbox{SWUB}_{t}^{\left(1\right)}.$$ where G is defined in . As remarked earlier, this bound improves upon the unconditional bound given by .
Examples
========
We now derive lower and upper bounds by choosing specific models for the mortality index.
Black-Scholes Model
-------------------
Let us consider the case where the mortality evolution process $\left\{ q_{t}\right\}_{t\geq0}$ follows the Black-Scholes model (c.f. [@Black]) which we write as $q_{t}=e^{U_{t}}$, where $\left\{ U_{t}\right\} _{t\geq0}$ is defined as: $$\label{4.6.1}
U_{t}=\log_{e}\left(q_{0}\right)+\left(r-\frac{\sigma^{2}}{2}\right)t+\sigma W_{t}^{*}$$ where $\left\{ W_{t}^{*}\right\} _{t\geq0}$ denotes a standard Brownian motion so that $W_{t}^{*}\sim N\left(0,t\right)$. As a result $$\label{4.6.2}
U_{t}\sim N\left(\log_{e}q_{0}+\left(r-\frac{\sigma^{2}}{2}\right)t,\,\sigma^{2}t\right)$$ We now derive lower and upper bounds for this model on the lines of $\mbox{SWLB}_{t}^{\left(2\right)}$ and $\mbox{SWUB}_{t}^{\left(1\right)}$ respectively.
### The Lower Bound $\mbox{SWLB}_{t}^{\left(BS\right)}$
We know that if $\left(X,\, Y\right)\sim\mbox{BVN}\left(\mu_{X},\mu_{Y},\sigma_{X}^{2},\sigma_{Y}^{2},\rho\right)$ where $BVN$ stands for bivariate normal distribution, the conditional distribution of the lognormal random variable $e^{X}$, given the event $e^{Y}=y$ is given as $$\label{4.6.3}
F_{e^{X}|e^{Y}=y}\left(x\right)=\Phi\left(\frac{\log_{e}x-\left(\mu_{X}+\rho\frac{\sigma_{X}}{\sigma_{Y}}\left(\log_{e}y-\mu_{Y}\right)\right)}{\sigma_{X}\sqrt{1-\rho^{2}}}\right).$$ where $\Phi$ denotes the c.d.f. of standard normal distribution. Given the time points $t_{i}$, $t$ for each $i$, let $\rho$ be the correlation between $U_{t_{i}}$ and $U_{t}$. Then, from , it is evident that: $\left(U_{t_{i}},U_{t}\right)\sim\mbox{BVN}\left(\mu_{U_{t_{i}}},\mu_{U_{t}},\sigma_{U_{t_{i}}}^{2},\sigma_{U_{t}}^{2},\rho\right)$, where the same equation specifies $\mu_{U_{t_{i}}},\mu_{U_{t}},\sigma_{U_{t_{i}}}^{2}$ and $\sigma_{U_{t}}^{2}$. Also as $q_{t}=e^{U_{t}}$, we have from equation that the distribution function of $q_{i}$ conditional on the event $q_{t}=s_{t}$ is given as $$F_{q_{i}|q_{t}=s_{t}}\left(x\right)=\Phi\left(a\left(x\right)\right)$$ where $a\left(x\right)$ is given by $$\label{4.6.4}
a\left(x\right)=\frac{\log_{e}x-\left(\log\left(q_{0}\left(\frac{s_{t}}{q_{0}}\right)^{\rho\sqrt{\frac{t_{i}}{t}}}\right)+\left(r-\frac{\sigma^{2}}{2}\right)\left(t_{i}-\rho\sqrt{t_{i}t}\right)\right)}{\sigma\sqrt{t_{i}\left(1-\rho^{2}\right)}}.$$ As the differentiation of c.d.f. yields the p.d.f., therefore the conditional density function of $q_{i}$ given $q_{t}=s_{t}$ satisfies the following equation: $$\label{4.6.5}
f_{q_{i}|q_{t}=s_{t}}\left(x\right)=\frac{1}{x\sigma\sqrt{t_{i}\left(1-\rho^{2}\right)}}\phi\left(a\left(x\right)\right),$$ where $\phi$ denotes the p.d.f. of standard normal distribution. We consider the following proposition before unraveling the improved lower bound.
If we assume that the mortality evolution process $\left\{ q_{t}\right\} _{t\geq0}$ be defined as $q_{t}=e^{U_{t}}$ where $U_{t}$ is given in equation , the conditional expectation of $q_{i}$ given $q_{t}$ is given by the expression $$\label{4.6.6}
\textbf{E}\left(q_{i}|q_{t}\right)=\begin{cases}
q_{0}\left(\frac{q_{t}}{q_{0}}\right)^{\frac{t_{i}}{t}}e^{\frac{\sigma^{2}t_{i}}{2t}\left(t-t_{i}\right)}\;\;\; & t_{i}<t,\\
q_{t}e^{r\left(t_{i}-t\right)} & t_{i}\geq t.
\end{cases}$$ We utilize this expression to obtain a lower bound for Asian call option under the Black-Scholes setting. Define: $S^{l_{3}}=\sum_{i=1}^{n}Y_{i}$, where exploiting , under the Black-Scholes case, $Y_{i}$, $i=1,2,...,n$ are given by $$Y_{i}=\begin{cases}
5q_{0}\left(\left(\frac{q_{t}}{q_{0}}\right)^{t_{i}/t}e^{\frac{\sigma^{2}t_{i}}{2t}\left(t-t_{i}\right)}-1.3\right)^{+}\;\; & i<j\\
5q_{0}\left(\left(\frac{q_{t}}{q_{0}}\right)e^{r\left(t_{i}-t\right)}-1.3\right)^{+} & i\geq j
\end{cases}$$ Evidently, $\textbf{Y}=\left(Y_{1},\ldots,Y_{n}\right)$ is comonotonic and so we have $$\label{4.6.8}
\textbf{E}\ensuremath{\left[\left(S^{l_{3}}-q_{0}\right)^{+}\right]={\displaystyle \sum_{i=1}^{n}}\textbf{E}\ensuremath{\left[\left(Y_{i}-F_{Y_{i}}^{-1}\left(F_{S^{l_{3}}}\left(q_{0}\right)\right)\right)^{+}\right]}},$$ where $F_{S^{l_{3}}}\left(q_{0}\right)$ is the distribution function of $S^{l_{3}}$ evaluated at $q_{0}$. For an arbitrary t, we have $$\begin{aligned}
\label{4.6.9}
F_{S^{l_{3}}}\left(q_{0}\right) & = & \textbf{P}\left[S^{l_{3}}\leq q_{0}\right]
\nonumber\\
& = & \textbf{P}\left[\sum_{i=1}^{j-1}5q_{0}\left(\left(\frac{q_{t}}{q_{0}}\right)^{t_{i}/t}e^{\frac{\sigma^{2}t_{i}}{2t}\left(t-t_{i}\right)}-1.3\right)^{+}+{\displaystyle \sum_{i=j}^{n}}5q_{0}\left(\left(\frac{q_{t}}{q_{0}}\right)e^{r\left(t_{i}-t\right)}-1.3\right)^{+}\leq q_{0}\right]
\nonumber\\
& = & \textbf{P}\left[\sum_{i=1}^{j-1}\left(\left(\frac{q_{t}}{q_{0}}\right)^{t_{i}/t}e^{\frac{\sigma^{2}t_{i}}{2t}\left(t-t_{i}\right)}-1.3\right)^{+}+{\displaystyle \sum_{i=j}^{n}}\left(\left(\frac{q_{t}}{q_{0}}\right)e^{r\left(t_{i}-t\right)}-1.3\right)^{+}\leq 0.2\right].\end{aligned}$$ As in the previous section, we substitute $x$ for $q_{t}/q_{0}$ and solve for $x$, using the equation: $$\label{4.6.10}
\sum_{i=1}^{j-1}\left(x^{t_{i}/t}e^{\frac{\sigma^{2}t_{i}}{2t}\left(t-t_{i}\right)}-1.3\right)^{+}+{\displaystyle \sum_{i=j}^{n}}\left(xe^{r\left(t_{i}-t\right)}-1.3\right)^{+}=0.2.$$ This is indeed straight forward, noting that the left hand side of this equation is strictly increasing in $x$. This yields: $$F_{S^{l_{3}}}\left(q_{0}\right)=F_{q_{t}}\left(xq_{0}\right)=\begin{cases}
F_{Y_{i}}\left(5q_{0}\left(x^{t_{i}/t}e^{\frac{\sigma^{2}t_{i}}{2t}\left(t-t_{i}\right)}-1.3\right)^{+}\right)\: & i<j,\\
F_{Y_{i}}\left(5q_{0}\left(xe^{r\left(t_{i}-t\right)}-1.3\right)^{+}\right)\: & i\geq j.
\end{cases}$$ Substituting this in equation , recalling the stop-loss order relationship between $S$ and $S^{l_{2}}$ as given by equation , applying it for $S^{l_{3}}$, splitting the terms and multiplying by the averaged discount factor as done in the last section, we obtain [ $$\begin{aligned}
\label{4.6.11}
P_{1} & \geq & De^{-rT}\left({\displaystyle \sum_{i=1}^{n}}\textbf{E}\ensuremath{\left[\left(Y_{i}-F_{Y_{i}}^{-1}\left(F_{S^{l_{3}}}\left(q_{0}\right)\right)\right)^{+}\right]}\right)
\nonumber\\
& = & Ce^{-rT}\Bigg(\sum_{i=1}^{j-1}\textbf{E}\left[\left(5\left(\left(\frac{q_{t}}{q_{0}}\right)^{t_{i}/t}e^{\frac{\sigma^{2}t_{i}}{2t}\left(t-t_{i}\right)}-1.3\right)^{+}-5\left(x^{t_{i}/t}e^{\frac{\sigma^{2}t_{i}}{2t}\left(t-t_{i}\right)}-1.3\right)^{+}\right)^{+}\right] \nonumber\\
& {} {} & {} \;\;\;\;\;\;\;\;+{\displaystyle \sum_{i=j}^{n}\textbf{E}}\left[\left(5\left(\left(\frac{q_{t}}{q_{0}}\right)e^{r\left(t_{i}-t\right)}-1.3\right)^{+}-5\left(xe^{r\left(t_{i}-t\right)}-1.3\right)^{+}\right)^{+}\right]\Bigg) \nonumber\\
& = & 5Ce^{-rT}\Bigg(\sum_{i=1}^{j-1}\frac{1}{q_{0}^{t_{i}/t}}\textbf{E}\left[\left(q_{t}^{t_{i}/t}e^{\frac{\sigma^{2}t_{i}}{2t}\left(t-t_{i}\right)}-q_{0}^{t_{i}/t}\left(1.3+\left(x^{t_{i}/t}e^{\frac{\sigma^{2}t_{i}}{2t}\left(t-t_{i}\right)}-1.3\right)^{+}\right)\right)^{+}\right] \nonumber\\
& {} {} & {} \;\;\;\;\;\;\;\;+{\displaystyle \sum_{i=j}^{n}\frac{e^{r\left(t_{i}-t\right)}}{q_{0}}\textbf{E}}\left[\left(q_{t}-q_{0}\left(\frac{1.3}{e^{r\left(t_{i}-t\right)}}+\left(x-\frac{1.3}{e^{r\left(t_{i}-t\right)}}\right)^{+}\right)\right)^{+}\right]\Bigg) \nonumber\\
& = & 5De^{-rT}\Bigg(\sum_{i=1}^{j-1}q_{0}^{1-t_{i}/t}\textbf{E}\left[\left(q_{t}^{t_{i}/t}e^{\frac{\sigma^{2}t_{i}}{2t}\left(t-t_{i}\right)}-q_{0}^{t_{i}/t}.\max\left(x^{t_{i}/t}e^{\frac{\sigma^{2}t_{i}}{2t}\left(t-t_{i}\right)},\,1.3\right)\right)^{+}\right] \nonumber\\
& {} {} & {} \;\;\;\;\;\;\;\;+{\displaystyle \sum_{i=j}^{n}}e^{rt_{i}}C\left(q_{0}.\max\left(x,\,\frac{1.3}{e^{r\left(t_{i}-t\right)}}\right),\, t\right)\Bigg)\end{aligned}$$ ]{} We denote the term within the first summation as $E_{1}$ and its value is given below. $$\label{4.6.12}
\textbf{E}_{1}=5q_{0}\left(e^{rt_{i}}\Phi\left(d_{1ai}\right)-\max\left(x^{t_{i}/t}e^{\frac{\sigma^{2}t_{i}}{2t}\left(t-t_{i}\right)},\,1.3\right).\Phi\left(d_{2ai}\right)\right)$$ where $d_{2ai}$ and $d_{1ai}$ are given respectively as $$\label{4.6.13}
d_{2ai}=\frac{-\log_{e}\left(\frac{da_{i}}{q_{0}}\right)+\left(r-\frac{\sigma^{2}}{2}\right)t}{\sigma\sqrt{t}}$$ $$\label{4.6.14}
d_{1ai}=d_{2ai}+\sigma\frac{t_{i}}{\sqrt{t}}$$ and $da_{i}$ is given as $$\label{4.6.15}
da_{i}=q_{0}.\left(\max\left(x^{t_{i}/t},\,\frac{1.3}{e^{\frac{\sigma^{2}t_{i}}{2t}\left(t-t_{i}\right)}}\right)\right)^{t/t_{i}}$$ Inserting in , we achieve the lower bound $\mbox{lb}_{t}^{\left(BS\right)}$ as follows $$\begin{aligned}
\label{4.6.16}
P_{1} & \geq & 5De^{-rT}\Bigg(\sum_{i=1}^{j-1}q_{0}\left(e^{rt_{i}}\Phi\left(d_{1ai}\right)-\max\left(x^{t_{i}/t}e^{\frac{\sigma^{2}t_{i}}{2t}\left(t-t_{i}\right)},\,1.3\right).\Phi\left(d_{2ai}\right)\right) \nonumber\\
& & {} \;\;\;\;\;\;\;\;\;+{\displaystyle \sum_{i=j}^{n}}e^{rt_{i}}C\left(q_{0}.\max\left(x,\,\frac{1.3}{e^{r\left(t_{i}-t\right)}}\right),\, t\right)\Bigg) \nonumber\\
& =: & \,\mbox{lb}_{t}^{\left(BS\right)}.\end{aligned}$$ The bound $\mbox{lb}_{t}^{\left(BS\right)}$ can undergo treatment similar to $\mbox{lb}_{t}^{\left(2\right)}$ in sense of maximization with respect to $t$ yielding $$\label{4.6.17}
P_{1}\geq\max_{0\leq t\leq T}\mbox{lb}_{t}^{\left(BS\right)}.$$ An interesting comment in the passing is that as we calculate $\textbf{E}\left[q_{i}|q_{t}\right]$ explicitly, rather than finding a lower bound for it, clearly $\mbox{lb}_{t}^{\left(BS\right)}$ improves on $\mbox{lb}_{t}^{\left(2\right)}$ in the case where $\left\{ q_{t}\right\}$ follows the Black-Scholes model. Again, as before, exploiting the put-call parity, $$\label{4.6.18}
P\geq \left(\mbox{lb}_{t}^{\left(BS\right)}-G\right)^{+}=:\mbox{SWLB}_{t}^{\left(BS\right)}.$$ where G is defined in .
### The Upper Bound $\mbox{SWUB}_{t}^{\left(BS\right)}$
In section 4.2, we have shown that the upper bound $\mbox{SWUB}_{1}$ can be improved by assuming that there exists a random variable $\Lambda$ such that $\text{Cov}\left(X_{i}, \Lambda\right) \neq 0\;\forall i$. Suppose this assumption is true here and the mortality index $\left\{ q_{t}\right\}_{t\geq0}$ depends on an underlying standard Brownian motion $\{ W_{t}\}_{t \in \left[0,T\right]}$. Then, from equation , we see that an upper bound for the call counterpart of the Swiss Re bond is given as $$\label{5.2.1}
P_{1} \leq 5De^{-rT}{\displaystyle \sum_{i=1}^{n}}{\displaystyle \int_{-\infty}^{\infty}}\textbf{E}\ensuremath{\left[\ensuremath{\left(q_{i}-F_{q_{i}|W_t=w}^{-1}\left(x\right)\right)^{+}\middle|W_t=w}\right]}d\Phi\left(\frac{w}{\sqrt{t}}\right)$$ where using , we see that $x$ is obtained by solving the following equation $$\label{5.2.2}
{\displaystyle \sum_{i=1}^{n}}F_{q_{i}|W_t=w}^{-1}\left(x\right)=\frac{q_{0}}{5}\left(1+6.5n\right).$$
An explicit formula for the conditional inverse distribution function of $q_{i}$ given the event $W_t=w$, is provided by the following result.
\[prop2\] Under the assumptions of the Black-Scholes model, conditional on the event $W_t=w$, the conditional distribution function of $q_{i}$ is given by $$\label{5.2.3}
F_{q_{i}|W_t=w}^{-1}=\begin{cases}
q_{0}e^{\left(r-\frac{\sigma^{2}}{2}\right)t_{i}+\sigma\frac{t_{i}}{t}w+\sigma\sqrt{\frac{t_{i}}{t}\left(t-t_{i}\right)}\Phi^{-1}\left(x\right)}\;\;\; & i<j,\\
q_{0}e^{\left(r-\frac{\sigma^{2}}{2}\right)t_{i}+\sigma w+\sigma\sqrt{\left(t_{i}-t\right)}\Phi^{-1}\left(x\right)} & i\geq j.
\end{cases}$$ where $j = min\{i: t_{i} \geq t\}$.
Let us set $X = \sigma W_{t_{i}}$, $Y=W_{t}$ and $y=e^{w}$ in . Then we obtain the following expression for the conditional distribution function of $e^{\sigma W_{t_{i}}}$ given the event $W_{t}=w$. $$\label{5.2.4}
F_{e^{\sigma W_{t_{i}}}|W_{t}=w}\left(s\right)=\Phi\left(\frac{\log_{e}s-\rho\sigma\sqrt{\frac{t_{i}}{t}}w}{\sigma\sqrt{t_{i}\left(1-\rho^{2}\right)}}\right).$$ It then follows that $F_{e^{\sigma W_{t_{i}}}|W_{t}=w}\left(s\right)=x$ if and only if $$s=F^{-1}_{e^{\sigma W_{t_{i}}}|W_{t}=w}\left(x\right)=e^{\rho\sigma\sqrt{\frac{t_{i}}{t}}w+\sigma\sqrt{t_{i}\left(1-\rho^{2}\right)}\Phi^{-1}\left(x\right)}$$ We can then obtain equation by noting that $\rho=\sqrt{\left(t_{i}\wedge t\right)\left(t_{i}\vee t\right)}$ and the following expression for the inverse conditional distribution function of $q_{i}$ given $W_{t}=w$. $$F_{q_{i}|W_t=w}^{-1}=q_{0}e^{\left(r-\frac{\sigma^{2}}{2}\right)t_{i}}F^{-1}_{e^{\sigma W_{t_{i}}}|W_{t}=w}$$ This completes the proof.
It is of note that $F_{q_{i}|W_t=w}^{-1}$ is continuous when $t=t_{i}$ (that is if, for some $i$, we have $i=j$). From equation , we then wish to solve the following for $x$. $$\label{5.2.5}
\sum_{i=1}^{j-1}e^{\left(r-\frac{\sigma^{2}}{2}\right)t_{i}+\sigma\frac{t_{i}}{t}w+\sigma\sqrt{\frac{t_{i}}{t}\left(t-t_{i}\right)}\Phi^{-1}\left(x\right)}+{\displaystyle \sum_{i=j}^{n}}e^{\left(r-\frac{\sigma^{2}}{2}\right)t_{i}+\sigma w+\sigma\sqrt{\left(t_{i}-t\right)}\Phi^{-1}\left(x\right)}=0.2+1.3n.$$ As a result, using equation, the improved upper bound for the call counterpart of the Swiss Re bond in the Black-Scholes case is given by the following set of equations $$\begin{aligned}
\label{5.2.5a}
P_{1} & \leq & 5Ce^{-rT}{\displaystyle \int_{-\infty}^{\infty}}\Bigg(\sum_{i=1}^{n}e^{\left(r-\frac{\sigma^{2}\left(t_{i}\wedge t\right)^{2}}{2t_{i}t}\right)t_{i}+\sigma\frac{t_{i}\wedge t}{t}w}\Phi\left(c_{1}^{\left(i\right)}\right)-\left(0.2+1.3n\right)\left(1-x\right)\Bigg)d\Phi\left(\frac{w}{\sqrt{t}}\right) \nonumber\\
& =: & \,\mbox{ub}_{t}^{\left(BS\right)},\end{aligned}$$
$$\label{5.2.6}
c_{1}^{\left(i\right)}=\begin{cases}
\sigma\sqrt{\frac{t_{i}}{t}\left(t-t_{i}\right)}-\Phi^{-1}\left(x\right)\;\;\; & i<j,\\
\sigma\sqrt{\left(t_{i}-t\right)}-\Phi^{-1}\left(x\right) & i\geq j.
\end{cases}$$
where $x \in \left(0,1\right)$ solves equation . The optimal upper bound in this case is then given by minimising equation over $t\in \left[0,T\right]$. As before, invoking the put-call parity of section 2, we have for the Swiss Re bond $$\label{5.2.7}
P \leq \left(\mbox{ub}_{t}^{1}-G\right)^{+}=:\mbox{SWUB}_{t}^{\left(BS\right)}$$ where G is defined in .
Log Gamma Distribution
----------------------
The log Gamma distribution is a particular type of transformed Gamma distribution. The mortality index ‘$q$’ is said to follow log Gamma distribution if $$\label{4.0.3}
\frac{\log_{e}q-\mu}{\sigma}=x\sim Gamma\left(p,a\right),$$ where $\mu, \sigma, p$ and $a$ are parameters ($>0$) and $log$ is the natural logarithm. Useful references for reading about transformed gamma distribution are [@Johnson2], [@Vitiello] and [@Cheng].
### The Lower Bound $\mbox{SWLB}_{t}^{\left(LG\right)}$
For the log-gamma distribution we obtain the following compact expression for ${lb}_{t}^{\left(2\right)}$ and then subtract $G$ from it to obtain $\mbox{SWLB}_{t}^{\left(LG\right)}$. [ $$\begin{aligned}
\label{4.0.10}
\mbox{lb}_{t}^{\left(2\right)}& = & 5Ce^{-rT}\Bigg(\sum_{i=1}^{j-1}q_{0}^{-t_{i}/t}\left(\frac{e^{\frac{t_{i}}{t}\mu}}{\left(\sigma^{"}\right)^{p}}\left[1-G\left(d_{2}^{'},\;p,\sigma^{"}\right)\right]-K_{1}\left[1-G\left(d_{2}^{'},\;p\right)\right]\right) \nonumber\\
& {} {} & {} \;\;\;\;\;\;\;\;+{\displaystyle \sum_{i=j}^{n}}\frac{e^{r\left(t_{i}-t\right)}}{q_{0}}\left(q_{0}e^{rt}\left[1-G\left(d_{1},\;p\right)\right]-K_{2}\left[1-G\left(d_{2},\;p\right)\right]\right)\Bigg)\end{aligned}$$ ]{}
where we have $$\sigma^{"}=1-\sigma^{'}\frac{t_{i}}{t},\; \sigma^{'}=1-\left(q_{0}e^{rt-\mu}\right)^{1/p},\; d_{2}^{'}=\frac{lnd_{1}^{'}-\mu}{\sigma},\; d_{1}^{'}=q_{0}\left(1.3+\left(x^{t_{i}/t}-1.3\right)^{+}\right)^{t/t_{i}},$$ $$K_{1}=\left(d_{1}^{'}\right)^{t_{i}/t},\;K_{2}=q_{0}\left(\frac{1.3}{e^{r\left(t_{i}-t\right)}}+\left(x-\frac{1.3}{e^{r\left(t_{i}-t\right)}}\right)^{+}\right),\;d_{1}=\frac{lnK_{2}-\mu}{q_{0}e^{rt-\mu}-1},\;d_{2}=d_{1}+lnK_{2}-\mu$$
$$G\left(x,p\right)={\displaystyle \int_{0}^{x}\frac{1}{\Gamma\left(p\right)}x^{p-1}e^{-x}dx},$$ and $$G\left(x,p,\sigma^{"}\right)={\displaystyle \int_{0}^{x}\frac{\left(\sigma^{"}\right)^{p}}{\Gamma\left(p\right)}x^{p-1}e^{-\left(\sigma^{"}x\right)}dx}.$$
Numerical Results
=================
The stage is now set to investigate the applications of the theory derived in the previous sections. We have successfully obtained a number of lower bounds and an upper bound for the Swiss Re bond in sections 3 and 4. In section 5 we have furnished a couple of examples. We now test these vis-a-vis the well-known Monte Carlo estimate for the Swiss Re bond. We assume that $C=1$ in all the examples. We first carry out this working under the well known [@Black] model in finance and then for a couple of transformed distributions. The nomenclature for the bounds has already been specified in sections 3 and 4. In tables 1 and 2, we assume that the mortality evolution process $\left\{ q_{t}\right\} _{t\geq0}$ obeys the Black-Scholes model, specified by the following stochastic differential equation (SDE) $$dq_{t}=rq_{t}dt+\sigma q_{t}dW_{t}.$$ In order to simulate a path, we will consider the value of the mortality index in the three years that form the term of the bond, i.e., $n=3$. In fact we consider the time points as $t_{1}=1,...,t_{n}=T=3$. We invoke the following equation to generate the mortality evolution: $$\label{4.0.1}
q_{t_{j}}=q_{t_{j-1}}\exp\left[\left(r-\frac{1}{2}\sigma^{2}\right)\delta t+\sigma\sqrt{\delta t}Z_{j}\right]\;\;\; Z_{j}\sim N\left(0,1\right),\;\;\; j=1,2,\ldots,n$$ We highlight below the parameter choices in accordance with [@Lin]. The value of the interest rate is varied in table 1 while table 2 experiments with the variation in the base value of the mortality index while assuming a zero interest rate. Parameter choices for tables 1 and 2 with $t$ specified in terms of years are: $$q_{0}=0.008453,\; T=3,\; t_{0}=0,\; n=3, \; \sigma=0.0388.$$
Table 2 is followed by figures 1-3. While figures 1 and 2 depict comparisons between the bounds, figure 3 portrays the price bounds for the Swiss Re bond generated by the Black-Scholes model. We will let MC denote the Monte Carlo estimate for the Swiss Re bond.
Table 1 reflects that the relative difference ($=\frac{|bound-MC|}{MC}$) between any bound and the benchmark Monte Carlo estimate increases with an increase in the interest rate for a fixed value of the base mortality index $q_{0}$. This observation is echoed by figure 1. On the other hand, figure 2 depicts the difference between the Monte Carlo estimate of the Swiss Re bond and the derived bounds. The bound $\mbox{SWLB}_{t}^{\left(BS\right)}$ fares much better than $\mbox{SWLB}_{1}$. The absolute difference between the estimated price and the bounds increase as the value of the base mortality index is increased and then there is a switch and this gap begins to diminish. This observation is supported by the fact that an increase in the starting value of mortality increases the possibility of a catastrophe which leads to the washing out of the principal or in other words the option goes out of money.
We now consider an additional example. Assume that the mortality rate ‘$q$’ obeys the four-parameter transformed Normal ($S_{u}$) Distribution (for details see [@Johnson] and [@Johnson2]) which is defined as follows
$$\label{4.0.2}
sinh^{-1}\left(\frac{q-\alpha}{\beta}\right)=x\sim N\left(\mu,\sigma^{2}\right),$$
where $\alpha, \beta, \mu$ and $\sigma$ are parameters ($\beta, \sigma > 0$) and $sinh^{-1}$ is the inverse hyperbolic sine function.
For table 3, we vary the interest rate as in table 1 and use the parameter set employed by [@Tsai]. The aforesaid authors use the mortality catastrophe model of [@Lin] to generate the data and then utilize the quantile-based estimation of [@Slifker] to estimate the parameters of the $S_{u}$-fit. The initial mortality rate and time points are same as for tables 1 and 2. The following arrays present the values of the parameters for the three years 2004, 2005 and 2006 that were covered by the Swiss Re bond. $$\alpha=[0.008399, 0.008169, 0.007905],\;\beta=[0.000298, 0.000613, 0.000904],$$ $$\mu=[0.70780, 0.58728, 0.58743]\;\text{and}\;\sigma=[0.67281, 0.50654, 0.42218].$$
The value of $\mbox{SWLB}_{t}^{\left(2\right)}$ in table 3 has been calculated by using ‘Numerical Integration’ in MATLAB since the first term in can not be calculated mathematically. Table 3 adds weight to the claim that the bounds are extremely tight for a large class of models assuming a variety of distributions for the mortality index. Finally in tables 4 and 5, we experiment with log gamma distribution by varying the interest rate in table 4 and the base mortality rate in the the latter. The parameters are chosen as in [@Cheng] who employ an approach similar to [@Tsai] outlined above with $q_{0}=.0088$ but use maximum likelihood estimation to obtain the parameters of the fitted log gamma distribution. As before, the following arrays present the year wise parameters $$p=[61.6326, 64.2902, 71.8574],\;a=[0.0103, 0.0098, 0.0080],$$ $$\mu=[-5.2452, -5.4600, -5.7238]\;\text{and}\;\sigma=[7.4\times10^{-5}, 9.5\times10^{-5}, 9.4\times10^{-5}].$$ Tables 4 and 5 clearly shows that even for non-normal universe, the bounds are extremely precise. Figures 4-6 are drawn on the lines of figures 1-3 and strongly support our observation.
\[h\]
$\;$r$\;$ $\mbox{SWLB}_{0}\;\;\;\;$ $\mbox{SWLB}_{1}\;\;\;\;$ $\;\;\mbox{SWLB}_{t}^{\left(BS\right)}\;\;$ $\;MC\;\;\;\;\;\;\;$ $\;\;\mbox{SWUB}_{t}^{\left(BS\right)}\;\;$ $\mbox{SWUB}_{1}\;\;\;$
----------- --------------------------- --------------------------- --------------------------------------------- ---------------------- --------------------------------------------- ------------------------- -- -- --
0.035 0.899130889131 0.899130889153 0.899131577419 0.899130939229 0.899131588500 0.899131637780
0.030 0.913324024542 0.913324024546 0.913324256506 0.913324120543 0.913324317265 0.913324320930
0.025 0.927447505802 0.927447505803 0.927447580428 0.927447582074 0.927447605312 0.927447619324
0.020 0.941626342686 0.941626342687 0.941626365600 0.941626356704 0.941626369727 0.941626384749
0.015 0.955935721003 0.955935721003 0.955935727716 0.955935715489 0.955935732230 0.955935736078
0.010 0.970419124546 0.970419124546 0.970419126422 0.970419112046 0.970419126802 0.970419129772
0.005 0.985101139986 0.985101139986 0.985101140486 0.985101142704 0.985101140840 0.985101141738
0.000 0.999995778016 0.999995778016 0.999995778143 0.999995730679 0.999995778175 0.999995778584
\[h\]
$\mbox{q}_{0}\;$ $\mbox{SWLB}_{0}\;\;\;\;$ $\mbox{SWLB}_{1}\;\;\;\;$ $\;\;\mbox{SWLB}_{t}^{\left(BS\right)}\;\;$ $\;MC\;\;\;\;\;\;$ $\;\;\mbox{SWUB}_{t}^{\left(BS\right)}\;\;$ $\mbox{SWUB}_{1}\;\;\;$
------------------ --------------------------- --------------------------- --------------------------------------------- -------------------- --------------------------------------------- ------------------------- -- --
0.007 1.000000000000 1.000000000000 1.000000000000 1.000000000000 1.000000000000 1.000000000000
0.008 0.999999915252 0.999999915252 0.999999915252 0.999999935586 0.999999915253 0.999999915253
0.008453 0.999995778016 0.999995778016 0.999995778143 0.999995730679 0.999995778175 0.999995778584
0.009 0.999821987943 0.999821987950 0.999822025863 0.999816103329 0.999822374801 0.999822875816
0.010 0.978292691035 0.978310383929 0.978503560221 0.978738658828 0.978292691184 0.986262918347
0.011 0.572750782004 0.610962124258 0.610962123857 0.652440509315 0.572755594265 0.877336305502
0.012 0.000000000000 0.040209774144 0.040209770810 0.094615386164 0.000000000000 0.395672911251
0.013 0.000000000000 0.000000000000 0.000000000000 0.001662471990 0.000000000000 0.083466184427
0.014 0.000000000000 0.000000000000 0.000000000000 0.000003376858 0.000000000000 0.008942985848
\[h\]
$\;$r$\;$ $\mbox{SWLB}_{0}\;\;\;\;$ $\mbox{SWLB}_{1}\;\;\;\;$ $\mbox{SWLB}_{t}^{\left(2\right)}\;\;\;\;\;$ $\;MC\;\;\;\;\;\;\;$ $\mbox{SWUB}_{1}\;\;\;$
----------- --------------------------- --------------------------- ---------------------------------------------- ---------------------- ------------------------- -- -- -- --
0.035 0.883255461690 0.884321427702 0.885548150429 0.884689900254 0.886806565750
0.030 0.903403981323 0.904010021303 0.904693957669 0.904223406591 0.905481788285
0.025 0.921607066867 0.921935518851 0.922291170235 0.922030679118 0.922759498340
0.020 0.938407830149 0.938576980454 0.938747560828 0.938598989786 0.939010425492
0.015 0.954287129641 0.954369722665 0.954444088119 0.954415686473 0.954582647473
0.010 0.969639544072 0.969677756802 0.969706604343 0.969683647402 0.969774875755
0.005 0.984762743262 0.984779521693 0.984789115795 0.984784143646 0.984820459036
0.000 0.999861354235 0.999868375732 0.999870879263 0.999871208429 0.999884274666
\[h\]
$\;$r$\;$ $\mbox{SWLB}_{0}\;\;\;\;$ $\mbox{SWLB}_{1}\;\;\;\;$ $\mbox{SWLB}_{t}^{\left(LG\right)}\;\;\;\;$ $\;MC\;\;\;\;\;\;\;$ $\mbox{SWUB}_{1}\;\;\;\;$
----------- --------------------------- --------------------------- --------------------------------------------- ---------------------- --------------------------- -- -- --
0.035 0.848032774815 0.848424044790 0.855969730838 0.854167495147 0.866104360048
0.030 0.873577023530 0.873813448730 0.879110918003 0.878026709161 0.887240130128
0.025 0.897102805167 0.897242672829 0.900881660116 0.900486935408 0.907283088297
0.020 0.918896959517 0.918977921696 0.921421185493 0.921030195924 0.926366403383
0.015 0.939240965474 0.939286791779 0.940888331577 0.941092453291 0.944633306794
0.010 0.958403723326 0.958429070674 0.959452704643 0.959485386732 0.962230654370
0.005 0.976635430514 0.976649121750 0.977286229664 0.977322136745 0.979302971605
0.000 0.994162849651 0.994170066411 0.994555652671 0.994698510161 0.995987334250
\[h\]
$\mbox{q}_{0}\;$ $\mbox{SWLB}_{0}\;\;\;\;$ $\mbox{SWLB}_{1}\;\;\;\;$ $\mbox{SWLB}_{t}^{\left(LG\right)}\;\;\;\;\;$ $\;MC\;\;\;\;\;\;\;\;$ $\mbox{SWUB}_{1}\;\;\;\;$
------------------ --------------------------- --------------------------- ----------------------------------------------- ------------------------ --------------------------- -- -- --
0.008 0.999766066714 0.999766066846 0.999772840362 0.999793281502 0.999779562417
0.0088 0.994162849651 0.994170066411 0.994555652671 0.994686720835 0.995987334250
0.009 0.989104987071 0.989146149900 0.989952105693 0.990012775483 0.993383346708
0.01 0.876692543049 0.888049181230 0.896376305638 0.891609413788 0.958189590379
0.011 0.410971060715 0.596089667857 0.596089667857 0.568675584083 0.837207974723
0.012 0.000000000000 0.271045973760 0.271045973760 0.207081909248 0.613838720959
0.013 0.000000000000 0.082740708460 0.082740708460 0.045779872978 0.381822437531
0.014 0.000000000000 0.012702023135 0.012702023135 0.006694089214 0.212229375395
0.015 0.000000000000 0.000000000000 0.000000000000 0.000883157236 0.110420349200
0.016 0.000000000000 0.000000000000 0.000000000000 0.000084710726 0.055539272591
0.017 0.000000000000 0.000000000000 0.000000000000 0.000004497045 0.027576845294
0.018 0.000000000000 0.000000000000 0.000000000000 0.000000019842 0.013697961783
Conclusions
===========
Mortality forecasts are extremely significant in the management of life insurers and private pension plans. Securitization and construction of mortality bonds has become an important part of capital market solutions. Prior to the launch of the Swiss Re bond in 2003, life insurance securitization was not designed to handle mortality risk.
This article investigates the designing of price bounds for the Swiss Re mortality bond 2003. As stated in [@Deng], an incomplete mortality market that has no arbitrage guarantees the existence of at least one risk-neutral measure termed the equivalent martingale measure $Q$ that can be used for calculating fair prices of mortality securities. We rely on this fact and devise model-independent bounds for the mortality security in question. To the best of our knowledge, there is only one earlier publication by [@Huang] in direction of price bounds for the Swiss Re bond. However, these authors propose gain-loss bonds that suffer from model risk. Our results assume the trading of vanilla options written on the mortality index, as in that case one can use the market price of these options to create bounds which are truly model independent. A worthy observation is that the stimulant for the present work is the theory of comonotonicity. One can therefore easily extend this approach for computing tight bounds for other mortality and longevity linked securities.
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---
abstract: 'We study the effect of a time-dependent driving field with a large amplitude on a system composed of two coupled qubits (two-level systems). Using the rotating wave approximation (RWA) makes it possible to find simple conditions for resonant excitation of the four-level system. We find that the resonance conditions include the coupling strength between the qubits. Numerical simulations confirm the qualitative conclusions following from the RWA. To reveal the peculiarities of resonant transitions caused by the quasi-level motion and crossing in a periodic driving field, we use Floquet states, which determine the precise intermediate states of the system. Calculating the quasi-energy states of the multi-level system makes it possible to find the transition probabilities and build interference patterns for the transition probabilities. The interference patterns demonstrate the possibility of obtaining various pieces of information about the qubits, since the positions of transition-probability maxima depend on various system parameters, including the coupling strength between the qubits.'
author:
- 'A. M. Satanin'
- 'M. V. Denisenko'
- Sahel Ashhab
- Franco Nori
title: Amplitude spectroscopy of two coupled qubits
---
Introduction
============
Recently much attention has been focused on the spectroscopy of Josephson junction-superconducting circuits with a weak link which can be considered as “macroscopic atoms” with sizes of the order of tens or hundreds of micrometers [@Nori1; @Nori3]. Single Josephson-junction qubits are characterized by relatively long relaxation times (tens of microseconds) which allows to consider them as one of the most promising elements for the realization of quantum information processors [@Zagoskin]. Spectroscopic investigations of artificial “Josephson atoms” are carried out at sufficiently low temperatures in the micro-wave and millimeter-wave regions, since the spectral lines of Josephson junctions are located in that spectral region. However, practical measurements are not simple because stable tunable micro-wave sources are not easy to produce in this range. Measurement difficulties are connected with the frequency dependence of the dispersion and damping, as well as with strict requirements to impedance control tolerances which limit the application of broadband spectroscopy.
In this regard several groups have used amplitude spectroscopy [@Oliver; @Berns1; @Sillanpa; @Berns2; @Ruder] which obtains information by means of the response function “sweep” over the signal amplitude and some control parameter (an applied magnetic flux or a bias). This method may be applied to systems with crossing energy levels between which the transitions can be realized by changing external parameters. The frequency of such a driving field can be orders of magnitude lower than the distance between levels. This means that the system evolves adiabatically, except for the immediate vicinity of quasi-crossing levels, where Landau-Zener quantum coherent transitions and Stückelberg interference can be observed [@Landau; @Zener; @Stuckelberg; @Majorana] (see Ref. for an overview). The main advantage of this type of spectroscopy is that the system is investigated in wide ranges of the amplitude change. Thus, in alternating fields, multiphoton processes and Landau-Zener transitions, also observed earlier in Ref. , take place.
For a driven two-level system, drastic effects on the tunneling rate arise from quasi-energy crossing and anticrossing [@Autler; @Shirley]. At certain amplitudes of the driving field, dynamical localization and trapping of the system into a non-linear resonance can take place [@Scully; @Grossmann; @Agarwal; @Ian; @Gawryluk; @Zhang; @Tuorila; @Wubs; @Childress]. As the parameters are changed when the level-approaching and level-crossing take place, the effects of band-to-band tunneling (Landau-Zener transitions) can occur. In the language of wave functions, an interference of different zone states, predicted by Stückelberg [@Stuckelberg] is possible. As applied to qubits these effects have lately been discussed in numerous works [@Ashhab; @Son; @Sun; @Sun2; @Plotz; @Wang; @Hausinger; @Xu; @Du; @Ferron; @Denisenko; @Hijii; @Ditzhuijzen; @Wang2; @Chotorlishvili; @Gasparinetti].
Coupled qubits have also been created (see, e.g., Ref. ). In these works the basic parameters of qubits and coupling constants have been measured and also some relaxation characteristics of coupled qubits have been studied. Rabi-spectroscopy of two coupled qubits both experimentally and theoretically have been investigated in publications [@Majer; @Steffen; @Plantenberg; @Groot; @Shevchenko3]. Recently different schemes of controlled coupling between two or more qubits have been proposed [@You; @Ashhab2; @Ashhab3; @Izmalkov2]. However, at present there are no studies of the way the coupled qubits behave in strong fields. Meanwhile, the extension of the amplitude spectroscopy method makes it possible to give much information about coupled multi-qubit clusters.
The goal of this work is to describe quantum-mechanical phenomena in a system of coupled qubits from the point of view of quasi-energy states at different parameters of multi-level systems. It is possible to control the magnetic fluxes (biases) which penetrate the circuits and we will take these bias parameters to be dependent on time [@Berkley; @Izmalkov; @Majer; @Steffen; @van; @der; @Ploeg; @Plantenberg; @Izmalkov2; @Groot; @Shevchenko3]. Although generally, for spectroscopic investigations, the response dependence on the frequency is studied, we will focus our attention here on the response dependence on the signal amplitude and the control parameters. Our approach differs from the one used in Ref. , where the density matrix equation was used to determine the steady state populations of coupled qubits. We assume here that the qubits are relevant for quantum information processing only in the case when they have negligible dissipation. In this case, to analyze the dynamics of the system it is most natural to proceed directly from the Schrödinger equation, which allows us to understand the dynamics and to identify features of the evolution of systems in strong alternating fields.
First, we shall investigate the nonlinear time dynamics of the coupled qubits by using the RWA In this approximation the system exhibits generalized Rabi resonances where the role of the coupling parameter may be investigated. Secondly, for high-field amplitude excitations we shall apply the quasi-energy representation to understand the influence of the driving field on the transition probabilities and the population of the energy levels. Using the RWA and numerical calculations of quasi-energy levels as a function of the driving field we will be able to demonstrate that the effect of the quasi-energy avoided crossing leads to drastically increased transition probabilities between the qubits steady states. Finally, we shall develop a numerical method for calculating the transition probabilities in the quasi-energy representation and build interference patterns for the transition probabilities. The quasi-energy basis allows us to analyze the influence of phase fluctuations on the observable effects that have not previously been studied in previous works. As we demonstrate in the following, the peaks of the transition probabilities between the directly coupled states shift if the inter-qubit coupling changes, the indirectly coupled states the peak positions are not affected by the inter qubit coupling. This effect can be observed in experiments using the technique of amplitude spectroscopy. It will be demonstrated that Landau-Zener-Stückelberg interferometry or amplitude spectroscopy may be considered as a tool to obtain the coupling parameter by seeing the shift of the peak of the resonances (the population maxima).
EQUATION OF MOTION OF COUPLED QUBITS
====================================
The main features of coupled qubits system behavior can be understood in the framework of the Hamiltonian: $$H=-\frac{1}{2}\left(
\begin{smallmatrix}
\epsilon_{1}+\epsilon_{2}+J & \Delta_{2} & \Delta_{1} & 0\\
\Delta_{2} & \epsilon_{1}-\epsilon_{2}-J & 0 & \Delta_{1}\\
\Delta_{1} & 0 & -\epsilon_{1}+\epsilon_{2}-J & \Delta_{2}\\
0 & \Delta_{1} & \Delta_{2} & -(\epsilon_{1}+\epsilon_{2})+J
\end{smallmatrix}
\right),\label{1}%$$ where $\epsilon_{i}$ is the control parameter of qubit $i$ ($i=1,\,2$), $\Delta_{i}$ is the corresponding tunneling matrix element, and the parameter $J$ quantifies the strength of the interaction between the qubits. The form of the Hamiltonian differs from [@Majer; @Storcz; @Temchenko] only by a simple redefinition of parameters.
Near the half-flux quantum point, each flux qubit experiences a double-well potential and the tunneling energy through the potential barrier separating the wells becomes $\Delta_{i}$. The wells correspond to currents of magnitude $I_{i}$ circulating in opposite directions along the loop, and the above Hamiltonian is actually written in this circulating current basis. Following Ref. , in a constant field the control parameters $\epsilon_{i}$ can be expressed in terms of the bias $f_{i}=\Phi^{\rm{ext}}_{i}/\Phi_{0}$ ($\Phi^{\rm{ext}}_{i}$ is the flux threading the qubit loop (magnetic flux), penetrating circuit $i$, $\Phi_{0}$ is the flux quantum) by the relation $$\epsilon_{i}=\epsilon^{0}_{i}\left(f_{i}-\frac{1}{2}\right),\label{2}%$$ where $\epsilon^{0}_{i}=2|I_{i}|\Phi_{0}$. The parameters $\epsilon_{i}$ and $\Delta_{i}$ determine the spectrum of the uncoupled qubits ($J=0$): $
E_{i}=\pm\frac{1}{2}\sqrt{\epsilon^{2}_{i}+\Delta^{2}_{i}}.
$ The ferromagnetic/antiferromagnetic interaction between the qubits is characterized by the coupling strength $J=\pm|J|$. With the help of an additional superconducting circuit it is possible to realize ferromagnetic as well as antiferromagnetic interactions between the qubits [@Izmalkov2]. For a planar circuit the antiferromagnetic interaction is determined by the expression $\frac{|J|}{2}=M_{12}I_{1}I_{2}$, where $M_{12}$ is the mutual inductance.
The state of the system can be represented by four amplitudes $C_{\alpha}(t)$, $\alpha=1,... ,4$, so that $|\Psi\rangle=\sum C_{\alpha}(t)|\alpha\rangle$, where $|\alpha\rangle$ is the basis of the time-independent Hamiltonian Eq. (\[1\]). The spectrum $E_{\alpha}$ and eigenvectors $|\alpha\rangle$ of the Hamiltonian ([\[1\]]{}) are not difficult to find.
To study the time-dependent evolution of the coupled qubits we use the eigenstates of the Hamiltonian Eq. (\[1\]) as the basis, since expanding in this basis is a well controlled procedure. Let us now consider the case when the control parameters $\epsilon_{1,2}$ are time-dependent. For the case of coupled qubits, we introduce a driving field of the form $$\epsilon_{1}(t)=\epsilon_{10}+A_{1}\cos(\omega_{1}t+\theta_1),\:\:\:\:\: \epsilon_{2}(t)=\epsilon_{20}+A_{2}\cos(\omega_{2}t+\theta_2).\label{4}%$$ For simplicity, we will only discuss the case when driving fields of only one frequency $\omega=\omega_{1}=\omega_{2}$ are applied to the system and the two fields have the same phase shift $\theta=\theta_{1}=\theta_{2}$. In this paper, we also assume that the system is subject to a sequence of synchronized pulses of alternating fields whose duration is much longer than the period of the field. At the same time, we take into account the fluctuations in the arrival times of pulses and their durations against a fixed period of the field [@Shirley].
We will solve the time-dependent Schrödinger equation to determine the resonant conditions of the qubits, $$i\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle=H(t)|\Psi(t)\rangle.\label{5}%$$ We perform the canonical transformation: $$|\Psi(t)\rangle=U(t)|\overline{\Psi}(t)\rangle\, \label{6}%$$ where the unitary matrix $
U(t)=\exp{[iS(t)/2\hbar]},
$ with $$S(t)=\phi_{1}(t)\left(\begin{smallmatrix}
I & 0\\
0 & -I
\end{smallmatrix} \right)+
\phi_{2}(t)\left(\begin{smallmatrix}
\sigma_{z} & 0 \\
0 & \sigma_{z}
\end{smallmatrix}\right)\\+Jt\left(\begin{smallmatrix}
\sigma_{z} & 0\\
0 & -\sigma_{z}
\end{smallmatrix}\right),\label{8}%$$ and phases $\phi_{1,2}(t)=\epsilon_{(1, 2)0}t+\frac{A_{1,2}}{\hbar\omega}\sin{\omega t}$. The transformed Hamiltonian $\overline{H}$ has the following form
$$\begin{gathered}
\overline{H}(t)=-\frac{\Delta_{1}}{2}\sum^{\infty}_{n=
-\infty}{J_{n}\left(\frac{A_{1}}{\hbar\omega}\right)}\times
\left(
\begin{smallmatrix}
0 & 0 & e^{-i((\epsilon_{10}+J)/\hbar+n\omega)t} & 0\\
0 & 0 & 0 & e^{-i((\epsilon_{10}-J)/\hbar+n\omega)t}\\
e^{i((\epsilon_{10}+J)/\hbar+n\omega)t} & 0 & 0 & 0 \\
0 & e^{i((\epsilon_{10}-J)/\hbar+n\omega)t} & 0 & 0
\end{smallmatrix}
\right)
\\-\frac{\Delta_{2}}{2}\sum^{\infty}_{n=
-\infty}{J_{n}\left(\frac{A_{2}}{\hbar\omega}\right)}\times\left(
\begin{smallmatrix}
0 & e^{-i((\epsilon_{20}+J)/\hbar+n\omega)t}& 0 & 0\\
e^{i((\epsilon_{20}+J)/\hbar+n\omega)t}& 0 & 0 & 0 \\
0 & 0 & 0 & e^{-i((\epsilon_{20}-J)/\hbar+n\omega)t}\\
0 & 0 & e^{i((\epsilon_{20}-J)/\hbar+n\omega)t} & 0
\end{smallmatrix}
\right),\label{9}%\end{gathered}$$
where the following relation for Bessel functions was used $$\exp{\left(i\frac{A}{\hbar\omega}\sin{\omega t}\right)}=
\sum_{n}{J_{n}\left(\frac{A}{\hbar\omega}\right)\exp{\left(in\omega t\right)}}.$$ From Eq. ([\[9\]]{}) it follows that the resonance conditions are given by $\epsilon_{10}\pm J+n\hbar\omega\approx0$, $\epsilon_{20}\pm J+n\hbar\omega\approx0$, and à *population trapping* is controlled by the two conditions $J_{n}(\frac{A_{1}}{\hbar\omega})=0$ and $J_{n}(\frac{A_{2}}{\hbar\omega})=0$. It is evident that in this case the resonance conditions depend on the coupling constant. In the RWA in the Hamiltonian Eq. ([\[9\]]{}) fast oscillating components can be neglected with the exception of those for which the resonance conditions are satisfied. Then the Hamiltonian describing the slow dynamics will have the simple matrix form which we can find, in general, the four quasi-energies.
It should be noted that the obtained results are valid in the framework of the RWA [@Scully] and cannot describe the system dynamics at an arbitrary amplitude time-dependent field. To leave the framework of the RWA limitations we will apply the numerical solution of the Schrödinger equation in the next section. Recent studies beyond the RWA can be found in Refs. \[\].
QUASI-ENERGIES AND TRANSITION AMPLITUDES IN A STRONG DRIVING FIELD
==================================================================
To obtain results for high-field amplitudes a quasi-energy representation is used. This representation gives the precise intermediate system state in a periodically-driven field with an arbitrary amplitude and allows to detect the peculiarities of resonant transitions caused by the motion and crossing of quasi-levels when the field changes.
Quasi-energies of multi-level systems
-------------------------------------
Let us consider the Hamiltonian of a multi-level system and let us take it to be periodic with period $T = 2\pi/\omega$ $$H(t)=H(t+T). \label{10}$$ According to Floquet’s theorem, the general solution of the Schrödinger equation can be decomposed into the complete set of functions $$|\Psi_{k}(t)\rangle=|\Phi_{k}(t)\rangle e^{-iQ_{k}t/\hbar},\quad
|\Phi_{k}(t+T)\rangle=|\Phi_{k}(t)\rangle, \label{11}%$$ where the functions $|\Phi_{k}(t)\rangle$ are the solutions of the equation $$\left(H(t)-i\hbar\frac{\partial}{\partial t}\right)|\Phi_{k}(t)\rangle=Q_{k}|\Phi_{k}(t)\rangle,\label{12}$$ and the real parameter $Q_{k}$ is called the quasi-energy [@Shirley; @quasienergy]($k$ is the quantum number determining the quasi-energy).
The quasi-energies $Q_{k}$ and eigenfunctions $|\Phi_{k}(0)\rangle$ at the initial moment of time (which may be chosen arbitrarily [@Shirley]) are found by the solution $$F(T)|\Phi_{k}(0)\rangle=e^{-iQ_{k}T/\hbar}|\Phi_{k}(0)\rangle, \label{13}$$ where $F(T)=\hat{P}\exp(-i\int^{T}_{0}{H(t)dt}/\hbar)$, $\hat{P}$ is the chronological ordering operator. The value of the functions $|\Phi_{k}(t)\rangle$ at any moment of time are obtained from the equation (\[12\]). Since quasi-energies are not uniquely defined $Q^{'}_{k}=Q_{k}+n\hbar\omega$, we will depict them in the first “Brillouin” zone ($0<Q_{k}<\hbar\omega$).
Expanding the periodic functions $|\Phi_{k}(t)\rangle$ in Fourier series [@Autler; @Shirley; @quasienergy] can be used to find the quasi-energies. The coefficients of the Fourier series in turn satisfy an infinite-dimensional system of linear equations which is approximately solved by a finite-dimensional approximation. In this work the form of the functions $Q_{k}$ is found numerically. First, we do not need to work with large-size sub-matrices; secondly, this approach allows us to obtain a controllable approximate solution.
An arbitrary wave function may be expanded in the complete Floquet basis $$|\Psi(t)\rangle=\sum_{k}{c_{k}}|\Phi_{k}(t)\rangle e^{-iQ_{k}t/\hbar}, \label{14}$$ where the coefficients $c_{k}$ are defined by the initial wave function: $c_{k}=\langle\Phi_{k}(0)|\Psi(0)\rangle$. So the Floquet time-evolutional operator can be found from Eq. (\[14\]): $$F(t,0)=\sum{|\Phi_{k}(t)\rangle e^{-iQ_{k}t/\hbar}\langle\Phi_{k}(0)|}.\label{15}%$$
Let us take the system to be initially in the state $|\alpha\rangle$, which is a steady state of the time-independent Hamiltonian Eq. (\[1\]). Let us also suppose that the electromagnetic pulse has an unknown phase. The transition probability into the excited state $|\beta\rangle$ of the Hamiltonian Eq. (\[1\]), averaged over the relative phase, is described by the following expression: $$P_{\alpha\rightarrow\beta}(t)=\sum_{k,l}
e^{-i(Q_{k}-Q_{l})t/\hbar}
\sum_{n}M_{k}^{(n)}(t)M_{l}^{*(n)}(t),\label{16}%$$ where $$M_{k}^{(n)}(t)=\frac{1}{T}\int^{T}_{0}{e^{-in\omega \tau}
\langle\beta|\Phi_{k}(\tau+t)\rangle\langle\Phi_{k}(\tau)|\alpha\rangle
d\tau}.\label{17}%$$ Notice that the sum with respect to $n$ appears in Eq. (\[16\]) because the Fourier expansion of the Floquet states has been used in the intermediate manipulations.
The expression Eq. (\[15\]) manifests that in a strong field the system evolution occurs through the intermediate quasi-energy states of qubits. It may be shown that the transition probability Eq. (\[16\]) in general contains strongly oscillating-in-time terms which may be be canceled when the time interval $t$ is long enough. The exception is the contribution of the states with almost equal quasi-energies. After averaging the expansion for the probability Eq. (\[16\]) by the time interval $t$ we find $$\overline{P}_{\alpha\rightarrow\beta} = \sum_{k}\sum_{n, n^{'}}\left |\langle\beta|\Phi^{(n-n^{'})}_{k}\rangle\langle\Phi^{(n)}_{k}|\alpha\rangle\right |^{2} ,\label{18}$$ where the Fourier components are defined by the relation $$|\Phi^{(n)}_{k}\rangle=\frac{1}{T}\int^{T}_{0}\!\!{e^{in\omega t}\;|\Phi_{k}(t)\rangle dt}.\label{19}%$$ The transition probabilities for different harmonics can be calculated according to Eq. (\[18\]). To do that we solve numerically Eq. (\[12\]) and take the Fourier components according to Eq. (\[19\]).
Numerical results for coupled qubits in a strong driving field
--------------------------------------------------------------
We now present numerical results of the coupled qubits response in a strong driving field. We will use the language of quasi-energies crossing which depend on the system parameters. It is well known that when the amplitude of the driving field and control parameter change, the quasi-energies of different symmetry classes may cross but if they are of the same symmetry class they form an anticrossing. As a result the transition amplitudes may change drastically for such parameters [@Autler; @Shirley; @quasienergy; @Grifoni]. Special attention will be paid to the dependence of the level populations on the interaction parameter. As was recently shown [@van; @der; @Ploeg], the interaction parameter can be varied over a wide range by using an intermediate coupler which, for instance, may be an additional Josephson loop placed between the two main qubit loops. So, we are going to investigate here the behavior of the level populations as a function of the coupling parameter of the qubits.
First we shall depict a 3D plot of the qubit energy dependence on the control parameter and the coupling parameter. Figure \[fig1\](a) shows the energy surfaces for the time-independent Hamiltonian Eq. (\[1\]) (when $A=0$). Figure \[fig1\](b) shows the transformation of the dispersion surfaces to quasi-energy surfaces when the time-dependent field is applied to the qubits. In order to understand what quasi-energies cross, we have depicted some of the characteristic cross sections of the quasi-energy surface in Fig. \[fig1\](b).
. (a) Energies $E_\alpha$ of the Hamiltonian (\[1\]), and (b) the quasi-energies $Q_k$ as functions of the control parameters $\epsilon_0 = \epsilon_{20} = 2 \epsilon_{10}$ and the coupling parameter $J$. We used the qubit parameters: $\Delta_{2}/h = 1.5 \Delta_{1}/h = 0.45$ GHz, $\omega/2\pi = 1$ GHz, and $A_{2}/h = 2A_{1}/h = 7$ GHz.](Satanin_Fig_1.eps){width="8.5cm" height="12cm"}
{width="12cm" height="7cm"}
{width="12cm" height="7cm"}
{width="17cm" height="10cm"}
The dependencies of the quasi-energies and transition probabilities on the control parameter, at a given amplitude of the alternating field, were investigated. In Fig. \[fig2\](a, b) the quasi-energies are shown as functions of the control parameter $ \epsilon_0=\epsilon_{20}=\lambda\epsilon_{10}$ for $J=0$ (a) and $J/h=-0.1$ GHz (b), respectively. In this case, a set of quasi-energy level crossings which produce additional peaks for transition probabilities between the eigenstates of the Hamiltonian Eq. (\[1\]) is observed, in Fig. \[fig2\](c, d). Several examples of the quasi-energy level crossings in Fig. \[fig2\] and their coincidence with the resonance peaks are shown by the gray vertical dotted lines.
The quasi-energy dependence, in Fig. \[fig2\](b), on the control parameter can be easily understood in the framework of perturbation theory. We shall explain the meaning of the quasi-energy levels formation, which is shown in Fig. \[fig2\](a). Let us mentally draw a set of lines parallel to the vertical axis at distances $n\hbar\omega$ from each other and then move the fragments of dispersion curves from each line to the first Brillouin zone ($0<Q_{k}<\hbar\omega$). It is shown below that the obtained picture will approximately correspond to the pictures shown in Fig. \[fig2\]. As can be seen from Fig. \[fig2\](a) the dependence of quasi-energies on the parameter $\epsilon_{0}$ is very simple at $\epsilon_{0}\gg\Delta$: the quasi-energies behave in accordance with the almost linear laws of dispersion of the uncoupled qubits (defined by $\hbar\omega$ module). The above explanation also provides a key to understanding the meaning of Fig. \[fig1\] (b). Notice that when $\epsilon_{0}\sim\Delta$, the curvature of the qubits dispersion plays an important role in the formation of the resonance peaks \[see Fig. \[fig2\](d)\].
Figure \[fig3\](a) shows the dependence of the four quasi-levels of two non-interacting qubits in an alternating field. In the RWA, the dependence of the quasi-energies on the driving amplitude may be found approximately from the average Hamiltonian defined by Eq. (\[9\]). The inclusion of the interaction leads to an effective repulsion of quasi-energy levels \[Fig. \[fig3\](b)\]. At the same time the populations have peaks when the quasi-levels approach each other \[Fig. \[fig3\](c)\]. Also this effect occurs for interacting qubits.
. The transition probabilities: $\overline{P}_{1\rightarrow2}$ (a), $\overline{P}_{1\rightarrow3}$ (b), and $\overline{P}_{1\rightarrow4}$ (c), as a function of the control parameter $\epsilon_{0}=\epsilon_{20} = 2\epsilon_{10}$ for different coupling constants $J$: black dotted lines $J=0$, dashed blue $J/h =-0.3$ GHz, and continuous red $J/h =-0.8$ GHz. Here we have set: $\Delta_{2}/h = 1.5 \Delta_{1}/h = 0.45$ GHz, $\omega/2\pi = 1$ GHz, and $A_{2}/h = 2A_{1}/h = 7$ GHz.](Satanin_Fig_5.eps){width="7cm" height="11cm"}
. The transition probabilities: $\overline{P}_{1\rightarrow2}$ (a), $\overline{P}_{2\rightarrow4}$ (b), and $\overline{P}_{1\rightarrow4}$ (c) as functions of the control parameter $\epsilon_0 = \epsilon_{20} = 2 \epsilon_{10}$ and the coupling parameter $J$. The qubit parameters are the same as in Fig. \[fig5\] and the field amplitudes used here are $A_{2}/h= 2A_{1}/h = 7$ GHz. The color bar is the same as in Fig. \[fig5\].](Satanin_Fig_6.eps){width="7cm" height="13cm"}
As can be seen from Fig. \[fig3\](a, b), the quasi-energies exhibit a nontrivial dependence on the field amplitude for the two coupling parameters $J=0$ \[see Fig. \[fig3\](a)\] and $J = -0.1$ GHz \[see Fig. \[fig3\](b)\]. In this case, the appearance of additional quasi-energy crossings and the formation of new peaks for the transition probabilities might be possible here \[see Fig. \[fig3\](c, d)\]. The circles show additional quasi-energy levels crossing and their coincidence with resonance peaks (the gray dashed lines in Fig. \[fig3\]).
The dependencies of the transition probabilities between the states of two qubits built at one time according to the alternating field amplitude and the control parameter are more informative and obvious. The interference patterns in Fig. \[fig4\] for the interacting qubits are qualitatively understandable on the basis of the results given in section II. The positions of the “bright spots” on the probability diagrams, at definite values of $\epsilon_{0}$ and the field amplitudes, coincide with the positions of the given transitions on the dependence of the quasi-energies on the amplitude, as shown in Fig. \[fig3\]. We see that the system possesses a distinct behavior depending on the coupling parameter $J$, which causes a shift of the peaks depending on $J$ along the bias direction. Figure \[fig5\] clarifies the radical change that the interaction between the qubits makes on the level populations. First, from the RWA analysis follows that a shift of the resonance peaks as a function of the coupling constant should be observed. These shifts can be seen in Fig. \[fig5\](a) and (b) (in the transitions $\overline{P}_{1\rightarrow2}$ (a) and $\overline{P}_{1\rightarrow3}$ (b) the shifts with increasing the coupling constant of qubits are shown by the arrows). Secondly, for the transitions $\overline{P}_{1\rightarrow4}$ the resonance peaks do not move when the coupling parameter $J$ is changed.
We note that in order to calculate the level populations for the coupled qubits in Fig. \[fig5\], a definite relationship between the control parameters: $\epsilon_{0}=\epsilon_{20}(t)=\lambda \epsilon_{10}(t)$ (where $\lambda$ is a parameter that determines the slope lines in the plane $\epsilon_{20}$ and $\epsilon_{10}$) has been assumed. The analysis in the framework of the RWA (Section II) has shown that the locations of the resonance peaks are given by the following conditions: $$\begin{aligned}
\epsilon_{20}+J+n\hbar\omega\approx0,\quad ( 1\rightarrow2),\label{20} \\
\epsilon_{10}-J+n^{'}\hbar\omega\approx0, \quad (2\rightarrow4),\nonumber\end{aligned}$$ and $$\begin{aligned}
\epsilon_{10}+J+m\hbar\omega\approx0, \quad (1\rightarrow3),\label{21} \\
\epsilon_{20}-J+m^{'}\hbar\omega\approx0, \quad ( 3\rightarrow4).\nonumber\end{aligned}$$ We can see in Figs. \[fig2\], \[fig3\] and Fig. \[fig5\] for the populations as well as for the interference patterns in Fig. \[fig4\](b, d), that the resonance peaks undergo a shift by a distance $| J |$ for the transitions $1 \rightarrow 2$ \[see Fig. \[fig4\](b) and Fig. \[fig5\](a)\] and $3\rightarrow 4$. At the same time, for the transitions $1 \rightarrow 3$ \[see Fig. \[fig4\](d) and Fig. \[fig5\](b)\] and $2 \rightarrow 4$, the peaks are shifted by a distance $| J | / \lambda$. Also shown in the figures is the fact that due to the chosen relations between the parameters (for example, when $\lambda=2$) and the relevant conditions ($\epsilon_{0}+J+n\hbar\omega\approx0$ and $2\epsilon_{0}+J+m\hbar\omega\approx0$) the “bright” resonances of a quantum-coherent tunneling in the transition $1 \rightarrow 3$ \[Fig. \[fig4\](d)\] are seen twice as often than for the transition $1\rightarrow 2$ \[Fig. \[fig4\](b)\]. Depending on the sign of the coupling constant $J$ (ferromagnetic or antiferromagnetic coupling), there is a shift of the resonance peaks to the right or to the left.
Also note that the transitions to a higher excited level are due to virtual transitions that are possible when both of the paired resonance conditions Eq. ([\[20\]]{}) and/or Eq. ([\[21\]]{}) can be fulfilled with the participation of second and third intermediate levels, respectively. A characteristic feature of this transition is the absence of peaks at integer values of the control parameter of the qubits, and the lack of resonance shifts when the coupling constant $J$ is changed. The positions of the resonance peaks (for $1 \rightarrow 4$) for fixed $J$ are determined by $ \epsilon_{0}=\frac{s\hbar\omega}{\lambda+1}$, where $s\equiv n+n^{'}=m+m^{'}$, and do not depend on the coupling constant.
Thus, Figs. \[fig5\](a, b) demonstrate the shift of peaks when increasing the parameter $J$ and that agrees qualitatively with the results of the analysis on the basis of the RWA (see section II). These conclusions manifest the fact that the experimental study of the response of a system of coupled qubits will make it possible to obtain some additional information and in particular determine the qubit coupling parameter.
In Fig. \[fig6\](a, b) we show the dependence of the population for the transitions $1 \rightarrow2$ and $2\rightarrow 4$ depending on the control parameter and the interaction parameters of qubits. For the selected slope parameter, $\lambda = 2$ is a clearly visible position of the resonance peaks, defined by Eq. (\[20\]). Resonant lines defined by Eq. (\[21\]) look quite similar. In contrast, the resonance peaks in Fig. \[fig6\](c) for the transition $1\rightarrow 4$ are determined by the intermediate states, so according to Eqs. (\[20\]) and Eq. (\[21\]) these will be located at the intersection of the lines.
CONCLUSIONS
===========
In this work we have presented results on the behavior of two interacting qubits in a strong driving field. The principal difference of our approach from the works devoted to the laser spectroscopic investigations of multi-level atomic systems is that we study the excitation probability dependencies on the applied field amplitude and the control parameter at a fixed frequency of the applied field.
For a better understanding of the effects of driving fields on a multi-level system we use the RWA, which allows to find simple conditions of the system resonant excitation. We have shown that these conditions differ from those that occur in the case of a single qubit. The most important result here is that these conditions of the resonant excitation include the interaction qubit constant. The realized numerical simulation confirms the qualitative conclusions as follows from the RWA.
Our results show that the change of the field amplitude and the control parameter have a strong effect on the system dynamics. At the same time, the quasi-energy basis proves to be the most adequate for describing states in periodic time-dependent fields. The quasi-energy representation gives the precise intermediate states of a system in a driving field with an arbitrary amplitude and allows to detect the peculiarities of resonant transitions caused by quasi-level motion and crossing as a function of changing parameters. This numerical method of calculating quasi-energy states of multi-level systems made it possible to find the transition probabilities in a quasi-energy representation and build interference patterns for the transition probabilities. The interference patterns obtained are very sensitive to the coupling strength of the qubits, suggesting a method to extract the value of the coupling parameter. The other parameters of the qubits, in particular the tunneling rates, also significantly affect the interference pattern so they can also be obtained in experiments.
The RWA as well as the numerical calculations of quasi-energy levels of the qubits in the strong driving field has shown that the effect of avoided crossings leads to drastically increased transition probabilities between the qubits steady states. Surprisingly, the peaks of the transition probabilities between the directly coupled states shift with changing the inter-qubit coupling $J$, but for indirectly coupled states the peak positions are not affected by $J$. This effect should be observed in experiments using the technique of amplitude spectroscopy.
The theory developed in this work should allow to extend the technique of amplitude spectroscopy used earlier for a single qubit [@Oliver; @Berns1; @Sillanpa; @Berns2; @Ruder] to more complicated systems. Clearly, amplitude spectroscopy can be used for studying the spectra of artificial quantum objects: quantum wells, quantum dots, quantum wires etc. in which the distances between energy levels are significantly smaller than in atomic systems. We thank Sergey Shevchenko for useful comments on the manuscript. SA and FN acknowledge partial support from the LPS, NSA, ARO, NSF grant No. 0726909, Grant-in-Aid for Scientific Research (S), MEXT Kakenhi on Quantum Cybernetics, and the JSPS-FIRST program. This work was funded in part by the Federal Program of the Russian Ministry of Education and Science through the contract No. 07.514.11.4012, and the RFBR Grants No. 11-02-97058-a and 12-07-00546-a. M.V.D. was financially supported by the “Dinastia” fund.
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|
---
abstract: |
We report on improvements made over the past two decades to our adaptive treecode N-body method (HOT). A mathematical and computational approach to the cosmological N-body problem is described, with performance and scalability measured up to 256k ($2^{18}$) processors. We present error analysis and scientific application results from a series of more than ten 69 billion ($4096^3$) particle cosmological simulations, accounting for $4 \times
10^{20}$ floating point operations. These results include the first simulations using the new constraints on the standard model of cosmology from the Planck satellite. Our simulations set a new standard for accuracy and scientific throughput, while meeting or exceeding the computational efficiency of the latest generation of hybrid TreePM N-body methods.
author:
- |
Michael S. Warren\
\
\
bibliography:
- 'refs.bib'
- '../zotero.bib'
title: |
2HOT: An Improved Parallel Hashed Oct-Tree\
N-Body Algorithm for Cosmological Simulation\
[*Best Paper Finalist, SC ’13.*]{}\
---
Introduction
============
We first reported on our parallel N-body algorithm (HOT) 20 years ago [@warren93] (hereafter WS93). Over the same timescale, cosmology has been transformed from a qualitative to a quantitative science. Constrained by a diverse suite of observations [@smoot92; @spergel03; @tegmark04; @riess04; @planckcollaboration13], the parameters describing the large-scale Universe are now known to near 1% precision. In this paper, we describe an improved version of our code (2HOT), and present a suite of simulations which probe the finest details of our current understanding of cosmology.
Computer simulations enable discovery. In the words of the Astronomy and Astrophysics Decadal Survey, “Through computer modeling, we understand the deep implications of our very detailed observational data and formulate new theories to stimulate further observations” [@nationalresearchcouncil10]. The only way to accurately model the evolution of dark matter in the Universe is through the use of advanced algorithms on massively parallel computers (see [@kuhlen12] for a recent review). The origin of cosmic structure and the global evolution of the Universe can be probed by selecting a set of cosmological parameters, modeling the growth of structure, and then comparing the model to the observations (Figure \[fig:Planck\]).
\
\[fig:Planck\]
Computer simulations are playing an increasingly important role in the modern scientific method, yet the exponential pace of growth in the size of calculations does not necessarily translate into better tests of our scientific models or increased understanding of our Universe. Anywhere the relatively slow growth in the capacity of human attention intersects with the exponential explosion of information, new tensions are created. The timespan between the completion of a large simulation and the publication of scientific results based upon it is now often a year or more, and is growing longer instead of shorter. In the application described here, the sheer complexity of managing the volume of information in many layers of data and code has required additional software tools to be developed. We have written substantially more lines of software for data analysis, generating initial conditions, testing and task management than are present in the 2HOT code base. The scale of simulations requires most of these ancillary tools to be parallel as well.
High-performance computing (HPC) allows us to probe more questions with increased resolution and reduced statistical uncertainty, leading to new scientific discoveries. However, reducing the statistical errors more often than not uncovers systematic errors previously masked by statistical variance. Addressing these details takes us out of realm of HPC into applied mathematics, software engineering and data analysis. However, without progress on all fronts, the over-arching scientific questions can not be answered. A corollary of this point is that making a code faster is often a poor investment when the aim is to answer a particular scientific question. More important than speed is the code’s applicability to the problem, correctness, and even less tangible properties such as robustness and maintainability. For those reasons, we focus here on the wide variety of changes made to 2HOT over the past two decades which have enabled us to produce the state-of-the-art scientific results presented in Section \[sec:science\].
One of our first scientific N-body simulations of dark matter [@warren91a] used 1.1 million particles and was performed on the 64-node Caltech/JPL Mark III hypercube in 1990. The simulation was completed in 60 hours, sustaining 160 Mflop/s with a parallel efficiency of 85%. In 2012 we used 2HOT on 262 thousand processors with over one trillion ($10^{12}$) particles, sustaining in excess of 1.6 Petaflops with a parallel efficiency of 90% [@warren12]. Since our first parallel treecode simulations, the message-passing programming model, time to solution and parallel efficiency are nearly the same, but the problem size has increased by a factor of a million, and performance a factor of 10 million.
Since WS93, HOT was been extended and optimized to be applicable to more general problems such as incompressible fluid flow with the vortex particle method [@ploumhans02] and astrophysical gas dynamics with smoothed particle hydrodynamics [@fryer02a; @fryer06; @ellinger12]. The code also won the Gordon Bell performance prize and price/performance prize in 1997 [@warren97a] and 1998 [@warren98]. It was an early driver of Linux-based cluster architectures [@warren97a; @warren97b; @warren03] and helped call attention to power issues [@warren02a; @feng03]. Perhaps surprisingly (given that WS93 was presented at the same conference as the draft MPI 1.0 standard), the fundamental HPC abstractions in the code have changed little over two decades, while more significant changes have been required in its mathematical and cosmological underpinnings.
Mathematical Approach
=====================
Equations of Motion
-------------------
The mathematical equations governing the evolution of structure in an expanding Universe are generally solved using comoving coordinates, $
\vec{x} = \vec{r}/a(t) $. $\vec{r}$ is the “proper” coordinate, while the scale factor $a(t)$ is defined via the Friedmann equation \[eq:scale\] (H/H\_0)\^2 = \_R/a\^4 + \_M/a\^3 + \_k/a\^2 + \_[DE]{} in terms of the Hubble parameter $H \equiv \dot{a}/a$ and the densities of the various components of the Universe; radiation in the form of photons and ultra-relativistic particles ($\Omega_R$), mass in the form of cold dark matter and baryons ($\Omega_M$), spatial curvature ($\Omega_k$) and dark energy or a cosmological constant ($\Omega_{DE}$). The particle dynamics are defined in terms of the motion relative to the background model, the scale factor and the acceleration due to gravity [@peebles80], \[eq:eom\] [d dt]{} + 2 [ a]{} = - [1 a\^3]{} \_[i j]{} [G m\_j \_[ij]{} \^3]{} Cosmological evolution codes most often account for cold dark matter, baryons and dark energy. The Boltzmann solvers which calculate the power spectrum of density perturbations use all of the components, including photons and massless and massive neutrinos. For precise computations, it is now necessary to include these other species. Using the parameters of the Planck 2013 cosmological model, the age of the Universe is 3.7 million years older if photons and radiation from massless neutrinos are not treated correctly. The linear growth factor from redshift 99 (an expansion of 100) changes by almost 5% (from 82.8 to 79.0) under the same circumstances. 2HOT integrates directly with the computation of the background quantities and growth function provided by CLASS [@lesgourgues11], either in tabular form or by linking directly with the CLASS library, and thereby supports any cosmology which can be defined in CLASS. 2HOT additionally maintains the ability to calculate the scale factor and linear growth factor analytically (when radiation or non-trivial dark energy is not included) in order to be able to directly compare with codes which do not yet support them.
Multipole Methods
-----------------
Using $N$ particles to represent the Universe, treecodes and fast multipole methods reduce the $N^2$ scaling of the right-hand side of equation to $O(N)$ or $O(N \log N)$—a significant savings for current cosmological simulations which use $N$ in the range of $10^{10}$ to $10^{12}$.
### Background Subtraction {#sec:bs}
Large cosmological simulations present a unique set of challenges for multipole methods. The Universe is nearly uniform at large scales. This means the resultant acceleration on a particle from distant regions is a sum of large terms which mostly cancel. We can precisely quantify this effect by looking at the variance of density in spheres of radius $r$, which is an integral of the power spectrum convolved with a top-hat window, \_0\^(dk/k) \_k\^2 W(kr)\^2 For a sphere of radius 100 Mpc/h, the variance is 0.068 of the mean value for the standard model. This value scales with the growth of cosmic structure over time, so at the beginning of a simulation it will be a factor of 50-100 lower. At early times when we calculate the acceleration from a 100 Mpc cell in one direction, 99% of that value will cancel with a cell in the opposite direction, leaving a small remainder (the “peculiar” acceleration). This implies that the error tolerance needed for these large cells is 100 times stricter than for the short-range interactions. For larger volumes or earlier starting times, even more accuracy is required. This suggests that eliminating the background contribution from the partial acceleration terms would be beneficial.
The mathematical equations describing the evolving Universe subtract the uniform background, accounting for it in the evolution of the scale factor $a(t)$. Fourier-based codes do this automatically, since the DC component has no dynamical effect. For treecodes, the proper approach is less obvious. Essentially, we wish to convert the always-positive mass distribution into density perturbations $\delta
\rho / \rho$. These density contrasts can be positive or negative, making the gravitational problem analogous to an electrostatics problem, with positive and negative charges.
Since we wish to retain the particle-based representation of the density, the background subtraction can be obtained by adding the multipole expansion of a cube of uniform negative density to each interaction. Since the multipole expansion of a cube is fairly simple due to symmetries, this can be done with a few operations if the multipole expansions are with respect to the cell centers (rather than the center of mass). This in turn adds a few operations to the interaction routines, since dipole moments are now present. At scales near the inter-particle separation, this approach breaks down, since any empty cells which would be ignored in a direct summation must be accounted for, as well as requiring high-order expansions for neighboring cells with only a few particles, which would normally be calculated with cheaper monopole interactions. Rather than modify each interaction for the near field, we define a larger cube which approximately surrounds the local region of empty and single particle cells and calculate the background acceleration within the surrounding cell (Figure \[fig:bs\]). This acceleration term can be done with a multipole and local expansion, or our current approach of using the analytic expression for the force inside a uniform cube [@waldvogel76; @seidov00].
A subtle point is that in the far-field we only want to subtract the uniform background expansion up to the same order as the multipole expansion of the matter to minimize the error. If a cube of particles is expanded to order $p=4$, the $p=6$ and higher multipoles from the background are not included, so they should not be subtracted. Using background subtraction increases the cost of each interaction somewhat, but results in a huge improvement in overall efficiency, since many fewer interactions need to be computed. At early times we have measured an improvement of a factor of five. The multipole acceptance criterion (MAC) based on an absolute error also becomes much better behaved, leading to improved error behavior as well.
### Multipole Error Bounds
A critical ingredient of any optimized multipole method is the mathematical machinery to bound or estimate the error in the interactions. The methods we previously developed [@salmon94; @warren95a] allow us to dynamically decide between using different orders of expansion or refinement, automatically choosing the most efficient method to achieve a given accuracy.
The expressions we derived in [@warren95a] support methods which use both multipole and local expansions (cell-cell interactions) and those which use only multipole expansions (cell-body interactions with $\Delta=0$). The scaling of these methods with $N$ depends on precisely how the error is constrained while increasing $N$, but generally methods which support cell-cell interactions scale as $O(N)$ and those that do not scale as $O(N \log N)$. Our experience has been that using $O(N)$-type algorithms for cosmological simulation exposes some undesirable behaviors. In particular, the behavior of the errors near the outer regions of local expansions are highly correlated. To suppress the accumulation of these errors, the accuracy of the local expansion must be increased, or their spatial scale reduced to the point where the benefit of the $O(N)$ method is questionable, at least at the modest accuracies of current cosmological simulations. For this reason, we have focused on the implementation and optimization of an $O(N \log N)$ method.
Consider a configuration of sources as in Figure \[fig:mac\]. The sources are contained within a “source” cell, $\sV$ of radius $b_{max}$, while the field is evaluated at separation $\vD$ from $\vx_0$, the center of “sink” cell $\sW$.
In terms of an arbitrary Green’s function, $G$, the field is: () &=& \_dG(-) () \[eq:green\]
Expanding G around $\vR_0 = \vx_0 - \vy_0$ in a Taylor series leads to the Cartesian multipole expansion:
() = \_[n=0]{}\^p [(-1)\^n n!]{} (\_0)\
(\_0 + ) + \_[(p)]{}()
where $\Phi_{(p)}$ is the error term, and the moment tensor is defined relative to a center, $\vz$ as: M\^[(n)]{}() = d (- )\^[(n)]{} () We have used a notational shorthand in which $\vv^{(n)}$ indicates the n-fold outer product of the vector $\vv$ with itself, while $\odot$ indicates a tensor inner-product and $\pnG$ indicates the rank-$n$ tensor whose components are the partial derivatives of G in the Cartesian directions. We can further expand the result by writing $\Mn(\vy_0 + \vD)$ as a sum over powers of the components of $\vD$, and then recollecting terms (see Eqns 12-14 in [@warren95a]).
While the mathematical notation above is compact, translating this representation to an optimized interaction routine is non-trivial. The expression for the force with $p=8$ in three dimensions begins with $3^8 = 6561$ terms. We resort to metaprogramming, translating the intermediate representation of the computer algebra system [@wolfram99] directly into `C` code. This approach is capable of producing the necessary interaction routines through $p=8$ without human intervention. A better approach would combine a compiler with knowledge of the computing architecture into the symbolic algebra system, allowing very high-level optimizations using mathematical equivalences that are lost once the formulae are expressed in a general programming language. To our knowledge, no such system currently exists.
We have also investigated support for pseudo-particle [@kawai01] and kernel-independent [@ying04] approaches which abstract the multipole interactions to more easily computed equations. For instance, the pseudo-particle method allows one to represent the far field of many particles as a set of pseudo-particle monopole interactions. We have found that such approaches are not as efficient as a well-coded multipole interaction routine in the case of gravitational or Coulombic interactions, at least up to order $p=8$.
Time Integration
----------------
The original version of HOT integrated the equations of motion using the leapfrog techniques described in [@efstathiou85], with a logarithmic timestep at early times. This approach has proven inadequate for high-accuracy simulations. Fortunately, the theory for symplectic time integration in a comoving background was developed by [@quinn97], which we have fully adopted. The advantages of this integrator are discussed in detail in [@springel05]. We calculate the necessary integrals for the “drift” and “kick” operators in arbitrary cosmologies with code added to the background calculations in CLASS [@lesgourgues11]. We additionally restrict the changes of the timestep to exact factors of two, rather than allowing incremental changes at early times. Any change of timestep breaks the symplectic property of the integrator, but making occasional larger adjustments rather than continuous small adjustment (as is done in GADGET2 [@springel05]) appears to provide slightly better convergence properties. We have also modified 2HOT to save “checkpoint” files which maintain the leapfrog offset between position and velocity. This allows the code to maintain 2nd-order accuracy in the time integration when restarting from a saved file. Otherwise, the initial (first order) step in the leapfrog scheme can lead to detectable errors after restarting at early times.
Boundary Conditions
-------------------
Periodic boundary conditions have been applied to multipole methods in a variety of ways, but most often are variants of the Ewald method [@hernquist91]. For 2HOT, we have adopted the approach described in [@challacombe97], which is based on the central result of Nijboer & De Wette (1957) [@nijboer57]. Effectively the same method in a Cartesian basis was first used in a cosmological simulation by Metchnik [@metchnik2009fast]. This method sums the infinite series of each relevant combination of powers of the co-ordinates, which can be taken outside the sum of periodic replicas (since the multipole expansion of each replica is identical). These pre-computed coefficients are then used in a local expansion about the center of the volume. We use $p=8$ and $ws=2$, which accounts for the boundary effects to near single-precision floating point accuracy (one part in $10^{-7}$). The computational expense of this approach is about 1% of the total force calculation for the local expansion, and 5-10% for the 124 boundary cubes, depending on the overall accuracy tolerance.
Force Smoothing
---------------
The standard practice in cosmological N-body simulations is to smooth the forces at small scales, usually with a Plummer or spline [@springel05] kernel. We have implemented these smoothing kernels in 2HOT, as well as the additional kernels described by Dehnen [@dehnen01]. Dehnen concludes that the optimal softening method uses a compensating kernel, with forces that are higher than the Newtonian force at the outer edge of the smoothing kernel, which compensates for the lower forces in the interior and serves to reduce the bias in the force calculation. Our tests confirm these conclusions, and we use Dehnen’s $K1$ compensating kernel for our simulations, except for the tests comparing directly to other codes.
Computational Approach
======================
Domain Decomposition
--------------------
The *space-filling curve domain decomposition* approach we proposed in WS93 has been widely adopted in both application codes (e.g. [@griebel99; @springel05; @jetley08; @wu12]) and more general libraries [@parashar96; @macneice00]. Our claim that such orderings are also beneficial for improving memory hierarchy performance has also been validated [@springel05; @mellor-crummey99]. We show an example of a 3-d decomposition of 3072 processor domains in Figure \[fig:morton\].
The mapping of spatial co-ordinates to integer keys described in WS93 converts the domain decomposition problem into a generalized parallel sort. The method we use is similar to the sample sort described in [@solomonik10], with the on-node portion done with an American flag radix sort [@mcllroy93]. After using the samples to determine the edges of the processor domains, in the initial HOT implementation the data was moved using a loop over all pairs of processors needing to exchange data. We converted the data exchange to use `MPI_Alltoall()` for improved scalability. This exposed problems in the implementation of Alltoall on large machines for both OpenMPI and the Cray system MPI. The first “scalability surprise” was related to the way buffers were managed internally in OpenMPI, with the number of communication buffers scaling as the number of processes squared. This did not allow our code to run on more than 256 24-core nodes using OpenMPI. We had to rewrite the implementation of Alltoall using a hierarchical approach, with only one process per node relaying messages to other nodes. The second was a “performance surprise” as defined by [@thakur10], where replacing the Cray system implementation of `MPI_Alltoall()` with a trivial implementation using a loop over all pairs of processes exchanging data led to a huge performance improvement when using more than 32k processors. Note that after the initial decomposition, the Alltoall communication pattern is very sparse, since particles will only move to a small number of neighboring domains during a timestep. This also allows significant optimization of the sample sort, since the samples can be well-placed with respect to the splits in the previous decomposition.
Tree Construction and Traversal
-------------------------------
The parallel tree construction in WS93 used a global concatenation of a set of “branch” nodes from each processor to construct the tree at levels coarser than the individual processor domains. While this is an adequate solution up to a few thousand processors, at the level of tens of thousands of domains and larger, it leads to unacceptable overhead. Most of the nodes communicated and stored will never be used directly, since the local traversal will only probe that deeply in the tree near its own spatial domain. Instead of a global concatenation, we proceed with a pairwise hierarchical aggregation loop over $i$ up to $log_2\thinspace N_{proc}$ by exchanging branch nodes between nearest neighbors in the 1-d space-filling curve, incrementally updating the tree with those nodes, then doing the same with the $2^i$-th neighbor. This provides a minimal set of shared upper-level nodes for each processor domain, and has demonstrated its scalability to 256k processors.
In [@warren95a] we describe a tree traversal abstraction which enables a variety of interactions to be expressed between “source” and “sink” nodes in tree data structures. This abstraction has since been termed *dual-tree traversal* [@gray01; @yokota12]. The dual-tree traversal is a key component of our method to increase the instruction-level parallelism in the code to better enable new CPU and GPU architectures (see Section \[sec:ilp\]).
During the tree traversal we use the same request/reply protocol described in WS93 using the global key labels assigned during the tree construction phase. Additional bits to label the source processor have been added to the hcells to support machines with up to $2^{18}$ processors. Our initial approach to hiding latency in the tree traversal was recast in the form of an active message abstraction. We believe that such event-driven handlers are more robust and less error-prone to implement correctly [@ousterhout96]. We currently use our own implementation of active messages within MPI, which we call “Asynchronous Batched Messages” (ABM). ABM is a key component of our ability to overlap communication and computation and hide message latency. MPI has supported one-sided communications primitives for many years, but their performance is often worse than regular point-to-point communication. It is likely that synchronization and locking overheads and complexity are to blame [@balaji11]. Newer implementations of active messages [@willcock10] are an attractive alternative, which we plan to implement as time allows.
Improving instruction-level parallelism {#sec:ilp}
---------------------------------------
In WS93 we used the fact that particles which are spatially near each other tend to have very similar cell interaction lists. By updating the particles in an order which takes advantage of their spatial proximity, we improved the performance of the memory hierarchy. Going beyond this optimization with dual-tree traversal, we can bundle a set of $m$ source cells which have interactions in common with a set of $n$ sink particles (contained within a sink cell), and perform the full $m \times n$ interactions on this block. This further improves cache behavior on CPU architectures, and enables a simple way for GPU co-processors to provide reasonable speedup, even in the face of limited peripheral bus bandwidth. We can further perform data reorganization on the source cells (such as swizzling from an array-of-structures to a structure-of-arrays for SIMD processors) to improve performance, and have this cost shared among the $n$ sinks. In an $m \times n$ interaction scheme, the interaction vector for a single sink is computed in several stages, which requires writing the intermediate results back to memory multiple times, in contrast to the WS93 method which required only one write per sink. For current architectures, the write bandwidth available is easily sufficient to support the $m \times n$ blocking.
Taking advantage of instruction-level parallelism is essential. In the past, obtaining good CPU performance for gravitational kernels often required hand-tuned assembly code. Implementing the complex high-order multipole interactions using assembly code would be extremely difficult. Fortunately, the gcc compiler comes to the rescue with vector intrinsics [@stallman89]. We use gcc’s `vector_size` attribute, which directs the compiler to use `SSE` or `AVX` vector instructions for the labeled variables. By providing the interaction functions with the appropriately aligned and interleaved data, gcc is able to obtain near optimal SIMD performance from C code.
We have also implemented our gravitational interaction functions with both CUDA and OpenCL kernels on NVIDIA GPUs, obtaining single-precision performance of over 2 Tflops on a K20x (Table \[tab:ukernel\]). We have implemented these kernels within 2HOT and demonstrated a 3x speedup over using the CPU alone. The ultimate performance of our code on hybrid GPU architectures depends on the ability of the to perform a highly irregular tree-traversal quickly enough to provide the necessary flow of floating-point intensive gravitational interactions. A parallel scan and sort based on our space-filling curve key assignment is one example of a successful approach [@bedorf12].
We have generally achieved near 40% of peak (single-precision) CPU performance on the supercomputers we have ported our code to over the past 20 years. We are working toward demonstrating the performance of 2HOT on Titan, using 18,688 NVIDIA K20x GPUs. With 25% of peak performance, we would obtain near 20 Tflops on that machine.
Managing the Simulation Pipeline
--------------------------------
In order to better integrate the various codes involved, and to simplify the management of the multiple configuration files per simulation, we have developed a Python [@vanrossum95] metaprogramming environment to translate a high-level description of a simulation into the specific text configuration files and shell scripts required to execute the entire simulation pipeline. Without this environment, it would be extremely difficult to guarantee consistency among the various components, or to reproduce earlier simulations after new features have been added to the individual software agents. It also allows us to programatically generate the configuration of thousands of simulations at once, that would previously have to be configured manually.
### Task Management
Modern simulation pipelines present a complex task for queueing systems. Given the flexibility of 2HOT, which can run on an arbitrary number of processors, or be interrupted with enough notice to write a checkpoint, we would like to control our tasks using higher-level concepts. We wish to specify the general constraints on a simulation task and have the system perform it in an efficient manner with as little human attention as possible. For example, “Please run our simulation that will require 1 million core-hours using as many jobs in sequence as necessary on at least 10,000 cores at a time, but use up to 2x as many cores if the wait for them to become available does not increase the overall wallclock time, and allow our job to be pre-empted by higher-priority jobs by sending a signal at least 600 seconds in advance.” Optimal scheduling of such requests from hundreds of users on a machine with hundreds of thousands of processors is NP-hard, but there seems to be ample room for improvement over the current systems, even without an “optimal” solution.
Data analysis often requires many smaller tasks, which queueing systems and MPI libraries have limited support for as well. We have developed an additional Python tool called `stask`. It allows us to maintain a queue inside a larger PBS or Moab allocation which can perform multiple smaller simulations or data analysis tasks. It has also proven useful to manage tens of thousands of independent tasks for `MapReduce` style jobs on HPC hardware. For instance, we have used this approach to generate 6-dimensional grids of cosmological power spectra, as well as perform Markov-Chain Monte Carlo analyses.
### Checkpoints and I/O
2HOT reads and writes single files using collective MPI/IO routines. We use our own self-describing file format (SDF), which consists of ASCII metadata describing raw binary particle data structures. I/O requirements are driven primarily by the frequency of checkpoints, which is in turn set by the probability of failure during a run. For the production simulations described here, we experience a hardware failure which ends the job about every million CPU hours (80 wallclock hours on 12288 CPUs). Writing a 69 billion particle file takes about 6 minutes, so checkpointing every 4 hours with an expected failure every 80 hours costs 2 hours in I/O and saves 4-8 hours of re-computation from the last permanently saved snapshot. At LANL, we typically obtain 5-10 Gbytes/sec on a Panasas filesystem. We have demonstrated the ability to read and write in excess of 20 Gbytes/sec across 160 Lustre OSTs on the filesystem at ORNL. By modifying our internal I/O abstraction to use MPI/IO across 4 separate files to bypass the Lustre OST limits, we have obtained I/O rates of 45 Gbytes/sec across 512 OSTs. These rates are sufficient to support simulations at the $10^{12}$ particle scale at ORNL, assuming the failure rate is not excessive.
### Version Control of Source Code and Data
To assure strict reproducibility of the code and scripts used for any simulation and to better manage development distributed among multiple supercomputer centers, we use the `git` version control system [@torvalds05] for all of the codes in the simulation pipeline, as well as our Python configuration system. We additionally automatically propagate the `git` tags into the metadata included in the headers of the data which is produced from the tagged software.
### Generating Initial Conditions
We use the Boltzmann code CLASS [@lesgourgues11; @blas11] to calculate the power spectrum of density fluctuations for a particular cosmological model. A particular realization of this power spectrum is constructed using a version of 2LPTIC [@crocce06] we have modified to support more than $2^{31}$ particles and use the FFTW3 library.
### Data Analysis
One of the most important analysis tasks is generating halo catalogs from the particle data by identifying and labeling groups of particles. We use `vfind` [@pfitzner98] implemented with the HOT library to perform both friend-of-friends (FOF) and isodensity halo finding. More recently, we have adopted the ROCKSTAR halo finder [@behroozi13], contributing some scalability enhancements to that software, as well as interfacing it with SDF. Our plans for future data analysis involve developing interfaces to the widely-adopted `yt` Project [@turk11], as well as contributing the parallel domain decomposition and tree traversal technology described here to `yt`.
Many of the mathematical routines we developed over the years as needed for our evolution or analysis codes have been replaced with superior implementations. The GSL [@galassi07] and FFTW [@frigo98] libraries have been particularly useful.
Scalability and Performance
===========================
In Table \[tab:nbody-historical\] we show the performance of our N-body code on a sample of the major supercomputer architectures of the past two decades. It is perhaps interesting to note that now a single core has more memory and floating-point performance than the fastest computer in the world in 1992 (the Intel Delta, on which we won our first Gordon Bell prize [@warren92b]). We show a typical breakdown among different phases of our code in Table \[tab:phases\], and single processor performance in Table \[tab:ukernel\].
Year Site Machine Procs Tflop/s
------ --------- ------------------- -------- --------- -- --
2012 OLCF Cray XT5 (Jaguar) 262144 1790
2012 LANL Appro (Mustang) 24576 163
2011 LANL SGI XE1300 4096 41.7
2006 LANL Linux Networx 448 1.88
2003 LANL HP/Compaq (QB) 3600 2.79
2002 NERSC IBM SP-3(375/W) 256 0.058
1996 Sandia Intel (ASCI Red) 6800 0.465
1995 JPL Cray T3D 256 0.008
1995 LANL TMC CM-5 512 0.014
1993 Caltech Intel Delta 512 0.010
: Performance of HOT on a variety of parallel supercomputers spanning 20 years of time and five decades of performance.[]{data-label="tab:nbody-historical"}
[*computation stage*]{} [*time*]{} (sec)
------------------------------------- ------------------
Domain Decomposition 12
Tree Build 24
Tree Traversal 212
Data Communication During Traversal 26
Force Evaluation 350
Load Imbalance 80
Total (56.8 Tflops) 704
: Breakdown of computation stages in a single timestep from a recent $4096^3$ particle simulation using 2HOT on 12288 processors of Mustang at LANL. The force evaluation consisted of 1.05e15 hexadecapole interactions, 1.46e15 quadrupole interactions and 4.68e14 monopole interactions, for a total of 582,000 floating point operations per particle. Reducing the accuracy parameter to a value consistent with other methods would reduce the operation count by more than a factor of three.[]{data-label="tab:phases"}
We present strong scaling results measured on Jaguar in Figure \[fig:scaling\]. These benchmarks represent a single timestep, but are representative of all aspects of a production simulation, including domain decomposition, tree construction, tree traversal, force calculation and time integration, but do not include I/O (our development allocation was not sufficient to perform this set of benchmarks if they had included I/O). Also, note that these results were using the code prior to the implementation of background subtraction, so the error tolerance was set to a value resulting in about 4 times as many interactions as the current version of the code would require for this system.
Processor Gflop/s
------------------------------------ ---------
2530-MHz Intel P4 (icc) 1.17
2530-MHz Intel P4 (SSE) 6.51
2600-MHz AMD Opteron 8435 13.88
2660-MHz Intel Xeon E5430 16.34
2100-MHz AMD Opteron 6172 (Hopper) 14.25
PowerXCell 8i (single SPE) 16.36
2200-MHz AMD Opteron 6274 (Jaguar) 16.97
2600-MHz Intel Xeon E5-2670 (AVX) 28.41
1300-MHz NVIDIA M2090 GPU (16 SMs) 1097.00
732-MHz NVIDIA K20X GPU (15 SMs) 2243.00
: Single core/GPU performance in Gflop/s obtained with our gravitational micro-kernel benchmark for the monopole interaction. All numbers are for single-precision calculations, calculated using 28 flops per interaction.[]{data-label="tab:ukernel"}
![Scaling on Jaguar measured in June 2012.[]{data-label="fig:scaling"}](strong_scaling){width="3.25in"}
Error Analysis
==============
Verifying the correctness of a large simulation is a complex and difficult process. Analogous to the “distance ladder” in astronomy, where no single technique can measure the distances at all scales encountered in cosmology, we must use a variety of methods to check the results of our calculations. As an example, using the straightforward Ewald summation method to calculate the force on a single particle in a $4096^3$ simulation requires over $10^{14}$ floating point operations (potentially using 128-bit quadruple precision), so it is impractical to use for more than a very small sample of particles. However, it can be used to verify a faster method, and the faster method can be used to check the accuracy of the forces in a much larger system. Eventually, we reach the stage where we can use 2HOT itself to check lower-accuracy results by adjusting the accuracy parameter within the code (as long as we are willing to pay the extra cost in computer time for higher accuracy).
Additionally, writing simple tests to verify the behavior of individual functions is essential. We have used Cython [@behnel11] to wrap the functions in 2HOT, allowing them to be tested from within a more flexible and efficient Python environment. In Figure \[fig:p8err\] we show one such example, showing the expected behavior of various orders of multipole interactions vs distance.
We also can compare the results of 2HOT with other codes, and investigate the convergence properties of various parameters. One must always keep in mind that convergence testing is necessary, but not sufficient, to prove correctness. In a complex system there may be hidden parameters that are not controlled for, or variables that interact in an unexpected way, reducing the value of such tests. Having two methods agree also does not prove that they are correct, only that they are consistent.
In Figure \[fig:pspech\_10\_cps2\] we show the sensitivity of the power spectrum to adjustments in various code parameters, as well as comparing with the widely used GADGET2 [@springel05] code. The power spectrum is a sensitive diagnostic of errors at all spatial scales, and can detect deficiencies in both the time integration and force accuracy. We can conclude from these graphs that 2HOT with the settings used for our scientific results (an error tolerance of $10^{-5}$) produces power spectra accurate to 1 part in 1000 at intermediate and large scales, with parameters such as the smoothing length and starting redshift dominating over the force errors at small scales. 2HOT also systematically differs from GADGET2 at scales corresponding to the switch between tree and particle-mesh, an effect also observed when comparing GADGET2 with perturbation theory results at high redshift [@taruya12].
Scientific Results {#sec:science}
==================
The number of objects in the Universe of a given mass is a fundamental statistic called the mass function. The mass function is sensitive to cosmological parameters such as the matter density, $\Omega_m$, the initial power spectrum of density fluctuations, and the dark energy equation of state. Especially for very massive clusters (above $10^{15}$ solar masses \[$M_\odot/h$\]) the mass function is a sensitive probe of cosmology. For these reasons, the mass function is a major target of current observational programs [@planckcollaboration13a]. Precisely modeling the mass function at these scales is an enormous challenge for numerical simulations, since both statistical and systematic errors conspire to prevent the emergence of an accurate theoretical model (see [@reed12] and references therein). The dynamic range in mass and convergence tests necessary to model systematic errors require multiple simulations at different resolutions, since even a $10^{12}$ particle simulation does not have sufficient statistical power by itself.
![A plot of the mass function from four recent $4096^3$ particle simulations computed with 2HOT. The scale of the computational volume changes by a factor of two between each simulation (so the particle mass changes by factors of 8). We plot our data divided by the fit of Tinker08 [@tinker08] on a linear $y$-axis. The figure shows the simulations are internally consistent but deviate from the Tinker08 fit at large scales. Open symbols are used for halos with 100-1000 particles, showing consistency at the 1% level down to 200 particles per halo.[]{data-label="fig:ds2013mf"}](ds2013mfsc){width="1.05\columnwidth"}
Our HOT code was an instrumental part of the first calculations to constrain the mass function at the 10% level [@warren06] with a series of sixteen $1024^3$ simulations performed in 2005, accounting for about $4 \times 10^{18}$ floating point operations. These results were further refined to a 5% level of accuracy with the addition of simulations from other codes, and the use of a more observationally relevant spherical overdensity (SO) mass definition [@tinker08]. With our suite of simulations (twelve $4096^3$ simulations, with an aggregate volume of thousands of cubic Gpc, using roughly 20 million core-hours and accounting for $4 \times 10^{20}$ floating point operations), we are able to probe effects at the 1% level in the SO mass function above $10^{15} M_\odot/h$ for the first time.
Some highlights of our scientific results for the mass function of dark matter halos (Figure \[fig:ds2013mf\]) are:
- We provide the first mass function calculated from a suite of simulations using the new standard Planck 2013 cosmology (with a $4096^3$ particle simulation and six $2048^3$ simulations completed and shared with our collaborators within 30 days of the publication of the Planck 2013 results). Changes in the parameters from the previous WMAP7 model are large enough that extrapolations from the other cosmologies [@angulo10; @angulo12] are likely subject to systematic errors which are large compared to the statistical precision of our results.
- We find the Tinker08 [@tinker08] result underestimates the mass function at scales of $10^{15} M_\odot/h$ by about 5% when compared with the older WMAP1 cosmological model it was calibrated against.
- For the Planck 2013 cosmology, the Tinker08 mass function is 10-15% low at large scales, due to the added systematic effect of non-universality in the underlying theoretical model.
- We identify a systematic error stemming from the improper growth of modes near the Nyquist frequency, due to the discrete representation of the continuous Fourier modes in the ideal input power spectrum with a fixed number of particles. This is a resolution dependent effect which is most apparent when using particle masses larger than $10^{11} M_\odot$ (corresponding to using less than 1 particle per cubic Mpc/h). Uncertainty in the appropriate correction and consequences of this effect appear to be the dominant source of systematic error in our results, where statistical uncertainties prevent us from ruling out a 1% underestimate of the mass function at scales of $2 \times 10^{15}
M_\odot/h$ and larger. If uncontrolled, this discretization error confounds convergence tests which attempt to isolate the effects of the starting redshift of the simulation [@luki07; @reed12], since the error becomes larger at higher starting redshifts.
- We are in direct conflict with recent results [@watson12] (see their Figure 13) which find the SO mass function to be lower than the the Tinker08 result at high masses. Potential explanations would be insufficient force accuracy of the CUBEP$^3$M code [@harnois-deraps12] (c.f. their Figure 7 showing force errors of order 50% at a separation of a few mesh cells), with a secondary contribution from initial conditions that did not use 2LPT [@crocce06] corrections (more recent simulations in [@watson13] appear consistent with our results up to $2
\times 10^{15} M_\odot/h$).
Conclusion
==========
Using the background subtraction technique described in Section \[sec:bs\] improved the efficiency of our treecode algorithm for cosmological simulations by about a factor of three when using a relatively strict tolerance ($10^{-5}$), resulting in a total absolute force error of about 0.1% of the typical force. We have evidence that accuracy at this level is required for high-precision scientific results, and we have used that tolerance for the results presented here. That accuracy requires about 600,000 floating point operations per particle (coming mostly from $\sim$2000 hexadecapole interactions). Relaxing the error parameter by a factor of 10 (reducing the total absolute error by a factor of three) reduces the operation count per particle to 200,000.
We can compare our computational efficiency with the 2012 Gordon Bell Prize winning TreePM N-body application [@ishiyama12] which used 140,000 floating point operations per particle. The $\theta$ parameter for the Barnes-Hut algorithm in that work was not specified, so it is difficult to estimate the effective force accuracy in their simulation. Modulo being able to precisely compare codes at the same accuracy, this work demonstrates that a pure treecode can be competitive with TreePM codes in large periodic cosmological volumes. The advantage of pure treecodes grows significantly as applications move to higher resolutions in smaller volumes, use simulations with multiple hierarchical resolutions, and require non-periodic boundary conditions.
Our experience with HOT over the past twenty years perhaps provides a reasonable baseline to extrapolate for the next ten years. The Intel Delta machine provided 512 single processor nodes running at 40 MHz and no instruction-level parallelism (concurrency of 512). The benchmark we ran on Jaguar had 16,384 16-core nodes running at 2.2GHz and 4-wide single-precision multiply-add SSE instructions (concurrency of 2.1 million). The performance difference for HOT of 180,000 between these machines is nicely explained from a factor of 55 in clock rate, a factor of 4096 in concurrency, and the loss of about 20% in efficiency. (Most of the efficiency loss is simply the fact that the gravitational inner loop can not balance multiplies and adds, so FMA instructions can not be fully utilized).
Looking to the future, if we guess clock rates go down a factor of two for better power utilization, and we lose up to a factor of two in efficiency, we would need an additional factor of 2000 in concurrency to reach an exaflop. A factor of 64 is gained going to 256-wide vector operations, leaving us with 32x as many cores. A machine with 8 million cores is daunting, but measured logarithmically the jump from $\log_2(512) = 9$ on the Delta to $\log_2(262144) = 18$ on Jaguar is twice as large as the jump from Jaguar to an exaflop machine with $\log_2(N_{cores})$ of 23. Assuming the hardware designers make sufficient progress on power and fault-tolerance challenges, the basic architecture of 2HOT should continue to serve at the exascale level.
Acknowledgments
===============
We gratefully acknowledge John Salmon for his many contributions to the initial version of HOT, and helpful comments on a draft version of this manuscript. We thank Mark Galassi for his memory management improvements to 2HOT and Ben Bergen for assistance with the OpenCL implementation. We thank the Institutional Computing Program at LANL for providing the computing resources used for our production simulations. This research used resources of the Oak Ridge Leadership Computing Facility at Oak Ridge National Laboratory, which is supported by the Office of Science of the Department of Energy under Contract DE-AC05-00OR22725. This research also used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. This research was performed under the auspices of the National Nuclear Security Administration of the U. S. Department of Energy under Contract DE-AC52-06NA25396.
|
---
abstract: 'We have studied a sample of Large Magellanic Cloud red giant binaries that lie on sequence E in the period–luminosity plane. We show that their combined light and velocity curves unambiguously demonstrate that they are binaries showing ellipsoidal variability. By comparing the phased light and velocity curves of both sequence D and E variables, we show that the sequence D variation – the Long Secondary Period – is not caused by ellipsoidal variability. We also demonstrate several further differences between stars on sequences D and E. These include differences in velocity amplitude, in the distribution of eccentricity, and in the correlations of velocity amplitude with luminosity and period. We also show that the sequence E stars, unlike stars on sequence D, do not show any evidence of a mid-infrared excess that would indicate circumstellar dust.'
author:
- |
C. P. Nicholls$^{1}$[^1], P. R. Wood$^{1}$and M.-R. L. Cioni$^{2}$\
$^{1}$Research School of Astronomy and Astrophysics, Australian National University, Cotter Road, Weston Creek ACT 2611, Australia\
$^{2}$Centre for Astrophysics Research, University of Hertfordshire, College Lane, Hatfield, AL10 9AB, UK
bibliography:
- 'bibliographynew.bib'
date: 'Accepted 2010 February 18. Received 2010 February 18; in original form 2009 December 10.'
nocite:
- '[@wood99mn]'
- '[@ogle04]'
- '[@ogleellipsoidal]'
- '[@wood99mn]'
- '[@wood99mn]'
- '[@ogleellipsoidal]'
- '[@pasquinimn]'
- '[@ogleellipsoidal]'
- '[@ogleellipsoidal]'
- '[@blum06mn; @sagemn]'
- '[@ogleellipsoidal]'
title: Ellipsoidal Variability and the Difference between Sequence D and E Red Giants
---
\[firstpage\]
stars: AGB and post-AGB – binaries: close – stars: oscillations
Introduction
============
Long Period Variables (LPVs) are known to fall on different sequences in the period–luminosity plane [Wood et al. 1999; Soszyński et al. 2004a; @ita04; @fraser05; @oglep-l; @fraser08]. One of these sequences, known as sequence E, is thought to consist of red giants in close binary systems showing ellipsoidal variability, although this has not been unambiguously demonstrated. A small number of the binaries appear to be eclipsing and others appear to have unexpectedly eccentric orbits, based on their light curve shape (Soszyński et al. 2004b).
A star in a close binary will have its Roche Lobe distorted by the tidal influence of its orbiting companion. When the star begins to fill its Roche Lobe it takes on an elongated, or ellipsoidal shape, becoming an ellipsoidal variable. The velocity variations of such stars are dominated by the orbital motion, but light variability is caused mainly by the change in apparent surface area as the star orbits around its companion. Because of this, the light curve of an ellipsoidal variable shows two maxima and minima per orbit – two cycles for every one cycle of the velocity curve. This is an easy way to unambiguously identify ellipsoidal variables and we use this test here.
Four of the other sequences of LPVs – A, B, C$'$ and C – are known to harbour radially pulsating variables (Wood et al. 1999). A fifth sequence, sequence D, contains stars which show Long Secondary Periods. The origin of Long Secondary Periods (LSPs) remains something of a mystery, though several attempts have been made to discover it [Wood et al. 1999; @hinkle02; @olivierwood03; @sequenceDstars; @seqDpaper]. Due to their overlap in the period–luminosity diagram, some authors [e.g. Soszyński et al. 2004b, @soszynski07] have suggested that stars on sequences D and E may be fundamentally the same – binaries showing ellipsoidal variability.
Here we study a sample of sequence E binaries. In particular we present new radial velocity data derived from VLT spectra, which we use alongside MACHO light curves. We show that the variations of stars on sequences D and E are caused by different mechanisms. Preliminary results for three stars are given in [@betsy].
Observations and Data Reduction
===============================
The observations and data reduction for this sample are the same as for our recently published sequence D sample [@seqDpaper]. The spectra were taken using the FLAMES/ GIRAFFE spectrograph (Pasquini et al. 2002) on the European Southern Observatory’s Very Large Telescope (VLT), on 21 nights from 2003 November to 2006 March. Radial velocities were calculated via cross-correlation with the <span style="font-variant:small-caps;">iraf</span> task *fxcor*. The reader is referred to [@seqDpaper] for details. Table \[vtable\] shows the radial velocities calculated for part of our sample for a few dates. The full table, with radial velocities of our whole sample for all dates, is available online.
----------- ---------- ---------- ---------- ----------
2954.8508 0.00 247.1836 270.6885 265.7394
3005.8604 0.00 242.5239 260.0903 258.5819
3067.6028 0.00 237.8895 225.7872 239.5893
3091.5571 0.00 238.4238 251.1337 265.9271
3280.8611 272.6101 255.7782 284.0382 244.0299
----------- ---------- ---------- ---------- ----------
: Radial Velocities of Sequence E Stars. Stars are identified by their MACHO numbers.[]{data-label="vtable"}
Results
=======
Plots of phased light and velocity variations for all the stars in our sequence E sample are shown in Figs. \[phased1\] and \[phased2\]. A clear doubling of the phased light curve with respect to the phased velocity curve can be seen for all stars in our sample. This behaviour unambiguously demonstrates that these stars are ellipsoidal variables. Many of the stars show lightcurves with equal maxima and unequal minima, which suggests that one end of the ellipsoid is hotter and brighter than the other. From Figs. \[phased1\] and \[phased2\] we see that the deepest minimum occurs at mean velocity during decreasing radial velocity, i.e. when the red giant is behind its companion. Therefore the inner end of the red giant ellipsoid – the end closest to the companion – is the cooler end, and the side furthest from the companion is hotter. This is most likely due to gravity darkening towards the companion.
{width="90.00000%"}
{width="90.00000%"}
A histogram of the velocity amplitudes of our sample is shown in Fig. \[vhist\]. The majority of values lie between 15 and $55\ \rm{km\,s^{-1}}$, and the mean velocity amplitude is $43.3\ \rm{km\,s^{-1}}$.
![A histogram of the velocity amplitude for our sample.[]{data-label="vhist"}](fig3){width="50.00000%"}
We have made a binary fit to the velocity curves of our sequence E red giants [see section 2.5 of @seqDpaper]. We calculated the mass function, $$f(m) = \frac{K^3 P}{2 \pi G} = \frac{m^3 \sin ^3 i}{(m+M)^2} ,$$ using the observed values for period ($P$) and velocity semiamplitude ($K$) (Table \[orbital\]). Using the calculated mass function of our binary fit and assuming a total system mass of $2\ \rm{M_{\odot}}$, we calculated an estimated companion mass for each of the stars in our sample. As expected, for most systems the companion is less massive than the red giant, but for two stars this resulted in companions with significantly higher mass than the red giant. This is unlikely since the more massive companion should evolve to the red giant stage first, and even if the current red giant was originally the less massive star, its companion would most likely now be a white dwarf of lower mass than the red giant. In these two cases we chose the total mass to be such that the red giant and its companion had equal mass. This results in higher masses for these stars (total system masses of $\sim 4.5 \rm{M_{\odot}}$.) The mass estimates are shown in Table \[orbital\], alongside the other parameters of the binary fit to the velocity. Here $\gamma$ is the system velocity, $e$ is the eccentricity, $\omega$ is the angle of periastron, $T$ is the date of periastron, $a \sin i$ is the semimajor axis of the red giant’s orbit, $f(m)$ is the mass function, $M$ is the mass of the red giant, and $m$ the mass of its companion. We assume $\sin i = 1$ when calculating the masses. The errors for each element are shown on the line below each star.
------------- ------------ ------------ ------------ ------------- ------------ -------- ------------ -------------- ------ ------
77.7429.189 281.40 25.99 0.05 275.38 3332.5 110.34 56.58 0.2003 1.07 0.93
$\pm$ 0.57 $\pm$ 0.80 $\pm$ 0.03 $\pm$ 28.52 $\pm$ 8.8 $\pm$ 1.75 $\pm$ 0.0186
77.7548.68 247.20 8.54 0.07 170.67 3502.6 442.83 74.55 0.0285 1.52 0.48
$\pm$ 0.19 $\pm$ 0.24 $\pm$ 0.03 $\pm$ 25.63 $\pm$ 32.9 $\pm$ 2.22 $\pm$ 0.0024
77.7672.98 250.71 32.77 0.06 38.99 2989.4 156.13 100.92 0.5679 2.27 2.27
$\pm$ 0.75 $\pm$ 1.10 $\pm$ 0.03 $\pm$ 29.11 $\pm$ 12.3 $\pm$ 3.42 $\pm$ 0.0575
77.7673.79 252.37 24.01 0.02 236.49 3064.4 131.35 62.30 0.1887 1.09 0.91
$\pm$ 0.41 $\pm$ 0.59 $\pm$ 0.02 $\pm$ 61.77 $\pm$ 22.6 $\pm$ 1.55 $\pm$ 0.0140
77.7789.152 255.04 14.03 0.05 198.55 3024.2 157.73 43.67 0.0451 1.44 0.56
$\pm$ 0.74 $\pm$ 0.97 $\pm$ 0.08 $\pm$ 90.81 $\pm$ 40.5 $\pm$ 3.05 $\pm$ 0.0094
77.7790.72 275.00 15.35 0.19 262.20 3326.7 328.79 97.85 0.1167 1.22 0.78
$\pm$ 0.83 $\pm$ 0.92 $\pm$ 0.06 $\pm$ 13.11 $\pm$ 10.2 $\pm$ 6.02 $\pm$ 0.0214
77.7791.115 350.88 21.47 0.03 275.04 3032.0 139.18 59.02 0.1429 1.17 0.83
$\pm$ 0.30 $\pm$ 0.35 $\pm$ 0.02 $\pm$ 40.51 $\pm$ 16.3 $\pm$ 0.97 $\pm$ 0.0070
77.7910.41 238.06 20.41 0.08 10.03 3075.8 191.46 76.94 0.1673 1.13 0.87
$\pm$ 0.55 $\pm$ 0.66 $\pm$ 0.04 $\pm$ 21.82 $\pm$ 11.8 $\pm$ 2.53 $\pm$ 0.0163
77.7910.77 255.34 44.35 0.08 279.65 2987.6 65.72 57.38 0.5890 2.36 2.36
$\pm$ 1.67 $\pm$ 2.83 $\pm$ 0.04 $\pm$ 38.55 $\pm$ 7.1 $\pm$ 3.67 $\pm$ 0.1131
77.7912.111 231.16 14.76 0.06 223.73 3075.1 128.89 37.50 0.0428 1.44 0.56
$\pm$ 0.45 $\pm$ 0.62 $\pm$ 0.04 $\pm$ 41.73 $\pm$ 14.7 $\pm$ 1.57 $\pm$ 0.0054
77.7914.74 305.31 16.33 0.02 272.80 3271.8 282.24 91.04 0.1276 1.20 0.80
$\pm$ 0.11 $\pm$ 0.14 $\pm$ 0.01 $\pm$ 24.26 $\pm$ 18.9 $\pm$ 0.79 $\pm$ 0.0033
------------- ------------ ------------ ------------ ------------- ------------ -------- ------------ -------------- ------ ------
The eccentricity of a binary system is expected to decrease over time due to circularising tidal forces [@zahn], provided no mechanism is in place that will stabilise or increase the orbital eccentricity. Thus binaries with one or more red giant components are expected to describe relatively circular orbits. We confirmed this theory by calculating the circularisation time for our stars, using the formula given in [@soker00]. The median circularisation time of our sample is only $\sim$ 3500 y. For comparison, the time a $1\ \rm{M_{\odot}}$, $Z=0.008$ red giant takes to double its radius when $R\sim 30\ \rm{R_{\odot}}$ is $\sim 2 \times 10^7$ y [using the evolutionary tracks of @girardi]. The value of $30\ \rm{R_{\odot}}$ was chosen as a typical Roche Lobe radius for these stars (see the velocity amplitude calculations given later in this section). A doubling of the radius is an estimate of the evolution time over which the tides have had time to act. Ten of the eleven stars in our sample have $e < 0.1$, and the mean eccentricity of the sample is 0.07. Therefore the majority of our sample of red giant binaries have fairly circular orbits, as expected. However some sequence E stars do show confoundingly high eccentricities (see Soszyński et al. 2004b), and this is a problem that must be solved in the future.
The relation between velocity amplitude and period for our sample and for the sequence D sample of [@seqDpaper] is plotted in Fig. \[vamp-p\]. In a binary system, for given masses, a longer period means a wider orbit, and thus a smaller amplitude in velocity variation. Fig. \[vamp-p\] shows velocity amplitude decreases with increasing period for the sequence E stars, as expected. However the sequence D stars show the same velocity amplitudes for all periods and do not follow the expected binary relation. A vivid demonstration of the disparity between these two samples is found in the region where their periods overlap. At periods between 200 and 450 days, the difference in velocity amplitude between the sequence D and sequence E samples is as much as 30 $\rm{km\,s^{-1}}$.
![Full velocity amplitude plotted against orbital period for our sequence E sample (blue points) and the sequence D sample of [@seqDpaper] (red crosses).[]{data-label="vamp-p"}](fig4){width="50.00000%"}
Ellipsoidal variability is only visible once a star has substantially filled its Roche Lobe. Stars in wider orbits (with lower velocity amplitudes) will fill their Roche Lobes when they are further up the giant branch, and thus will be more luminous than stars in closer orbits. For a given luminosity (and radius) there is a range of velocity amplitudes the star may have, the maximum of which is dictated by the size of the closest possible orbit the companion can occupy. Therefore, for our ellipsoidal variables we expect that the maximum velocity amplitude should decrease with increasing luminosity, but that stars may occupy velocity amplitudes below the maximum for a given luminosity. This is shown in Fig. \[k-vamp\] which gives a plot of $K$ magnitude against velocity amplitude for our sequence E stars. In order to define the upper limit of velocity amplitude for a given luminosity, we calculated the minimum orbital separation for a theoretical sample of binaries with equal mass components (mass ratios of $q = 1$) in which the red giant is filling its Roche Lobe. Using the approximation given in [@eggleton], we calculated the Roche Lobe radius $r_{L}$ to be $0.37$, in units of orbital separation. Therefore the minimum orbital separation for a Roche-Lobe filling binary with $q = 1$ is $a=\frac{r_{L}}{0.37}=2.7r_{L}$. We substitute for $r_{L}$ the radius expected for a given luminosity in these stars, calculated from a fit made to the radius–$K$ magnitude data for our sample. Finally we calculated the maximum velocity amplitude assuming a circular orbit and components of mass $1\ \rm{M_{\odot}}$. This velocity amplitude upper limit is shown by the solid blue line in Fig. \[k-vamp\].
Most stars lie where expected in Fig. \[k-vamp\] but two stars lie above the upper limit line. These are the two systems with higher mass components, mentioned earlier. As both these systems have components of around $2.3\ \rm{M_{\odot}}$, we also calculated the maximum velocity amplitude for stars of this mass, and this is shown by the green dashed line in Fig. \[k-vamp\]. One star lies above this limit. It appears somewhat atypical, as it has a significantly higher effective temperature than the rest of the sample ($\sim5200 K$ compared to the median $4200 K$) and hence a smaller radius. It therefore does not overflow its Roche Lobe as Fig. \[k-vamp\] suggests.
Table \[properties\] gives the radius, luminosity, effective temperature and orbital separation data for our sample. For the majority of stars, these properties were calculated using the 2MASS $J$ and $K$ magnitudes. However for the star 77.7910.77, which lies outside the green dashed limit in Fig. \[k-vamp\], the properties were calculated from OGLE $V-I$ values, which are more reliable for warmer stars. Due to this star’s warmer temperature, we suspect it could be a helium core burning clump star or a blue loop star.
------------- ------- --------- --------- --------
77.7429.189 48.92 593.16 4071.43 121.83
77.7548.68 92.03 1917.72 3980.39 307.67
77.7672.98 56.13 869.34 4181.83 201.84
77.7673.79 42.24 591.59 4378.57 136.84
77.7789.152 52.85 841.10 4274.26 154.60
77.7790.72 92.71 1736.69 3868.59 252.27
77.7791.115 49.66 805.48 4362.15 142.23
77.7910.41 72.37 1790.14 4412.02 175.91
77.7910.77 38.58 983.43 5202.34 114.76
77.7912.111 47.41 604.76 4155.77 135.12
77.7914.74 70.70 1079.15 3933.14 227.86
------------- ------- --------- --------- --------
: Sequence E Star Properties[]{data-label="properties"}
![$K$ magnitude (from 2MASS) plotted against velocity amplitude for our sequence E stars. The point size shows relative light amplitude in MACHO red. The solid blue curve gives the upper velocity limit for equal-mass components of $1\ \rm{M_{\odot}}$, and the dashed green curve gives the upper limit for equal-mass components of $2.3\ \rm{M_{\odot}}$. See text for details.[]{data-label="k-vamp"}](fig5){width="50.00000%"}
The mid-infrared colour of sequence E stars
-------------------------------------------
It has recently been shown [@dust] that variable red giants belonging to sequence D have a mid-infrared excess when compared to similar red giants without the Long Secondary Period that characterises sequence D stars. This mid-infrared excess is assumed to arise from circumstellar dust associated the presence of the LSP. Since the sequence E stars are close binary systems, some of which have quite eccentric orbits (Soszyński et al. 2004b), there is a possibility that these systems could have substantial circumstellar disks of dust and gas.
In order to investigate this possibility, we searched for a mid-infrared excess in sequence E stars in the LMC using mid-infrared data from the Spitzer Space Telescope SAGE survey (Blum et al. 2006; Meixner et al. 2006). The sequence E stars were obtained from the catalogue given by [@fraser08]. None of these objects were detected at 24$\mu$m in the SAGE survey but 262 were detected at 8$\mu$m. The light curves of these 262 stars were examined and it was found that only 184 could be considered as definite sequence E stars, i.e. ellipsoidal or eclipsing binary systems. The remainder were variables whose light curve characteristics showed that they belonged to sequence D or the pulsation sequences 1 to 4 [using the notation in @fraser08] or they were stars of indeterminate light curve type.
In order to see if the sequence E stars show a mid-infrared excess, we obtained a comparator sample of stars that were similar to the sequence E stars, but without binary companions. Such stars are the normal, non-varying field red giants in the LMC. Since the sequence E stars that were detected at 8$\mu$m have $12.5 <\ K\ < 14$ and $0.6 <\ J-K\ < 1.4$, we selected comparison sources from the SAGE survey with these characteristics along with the requirement of an 8$\mu$m detection. An examination of this sample of $\sim$40000 objects showed that sources fainter than $K\ = 13.5$ were near the 8$\mu$m faint detection limit and that this significantly biased the population of detected stars in favour of objects with a higher $K$-\[8\] colour. We therefore omitted such stars and we compared sequence E stars and non-varying field red giants only in the interval $12.5 <\ K\ < 13.5$. Finally, all variable stars in the catalogue of @fraser08 were removed from the list of comparison sources. This left 144 sequence E stars and $\sim$33000 non-varying field red giants. The $K$-\[8\] colour distributions of these two samples were then examined to search for an 8$\mu$m mid-infrared excess. A two sample K–S test gives a probability of up to 0.89 that the sequence E stars and non-varying field red giants come from the same underlying distribution in $K$-\[8\] colour. In other words, there is no evidence that sequence E variability generally leads to a significant amount of circumstellar dust and a mid-infrared excess.
Discussion and Conclusions
==========================
With new velocity curves, we now have strong evidence that the sequence E variation is indeed caused by ellipsoidal variability. This is shown by the doubling of the light curve with respect to the velocity curve in these stars, as in Figs. \[phased1\] and \[phased2\].
However, as we showed in [@seqDpaper], the phased light curves of sequence D variables – the stars with Long Secondary Periods – do not show this doubling phenomenon (see fig. 3 of that paper). Although it has been suggested that the sequence D and E variations may have a common origin due to their proximity in the period–luminosity plane (Soszyński et al. 2004b), and the possible existence of ellipsoidal shapes in their residual light curves [@soszynski07], no sequence D star has so far been found whose light executes two cycles during one velocity cycle. This seems to rule out ellipsoidal variation in sequence D stars.
A further difference between the stars on sequences D and E is demonstrated by their respective velocity amplitudes. The sequence E stars of the current sample show full velocity amplitudes of at least $15\ \rm{km\,s^{-1}}$ – some much larger – as can be seen in Fig. \[vhist\]. However as we showed in [@seqDpaper], stars with LSPs have much smaller velocity amplitudes, typically around $3.5\ \rm{km\,s^{-1}}$. This marked difference in the size of the velocity variations between sequence D and E – which can be seen in Fig. \[vamp-p\] – supports our assertion that they are caused by different mechanisms.
The sequence E stars, known binaries, show an expected spread in companion mass estimates. However this distribution seems very different for the sequence D stars. If treated as binaries, sequence D stars have companions that are small and absurdly similar ($\sim 0.09\ \rm{M_{\odot}}$). Additionally, the angles of periastron of sequence D stars are heavily biased towards large values, whereas we would expect a uniform distribution for binaries. Unfortunately we cannot compare the distribution of angle of periastron for the current sequence E sample: as most of these stars have almost circular orbits, angle of periastron is poorly defined (see the large errors in Table \[orbital\]).
The distribution of eccentricity is very different for sequence E and D stars, when the latter are treated as binaries. Most of the red giant binaries on sequence E have low-eccentricity orbits, as expected for their evolutionary state. However [@seqDpaper] showed that the sequence D stars have a significantly higher eccentricity, if we assume they are binaries (the mean eccentricity of that sample is 0.3). A two-sample K–S test gives a probability of less than $2.1 \times 10^{-6}$ that the sequence D and E eccentricity distributions come from the same underlying distribution.
Further evidence that sequence D stars are not ellipsoidal variables is given by the relation between luminosity and velocity amplitude in those stars. In Fig. \[k-vamp\], we showed that the sequence E stars generally populate an area delineated by a maximum velocity amplitude–luminosity relation, as expected for ellipsoidal variables. However, [@seqDpaper] showed that the sequence D stars do not show any correlation between these properties.
Finally, we have searched for a mid-infrared excess in sequence E stars since such an excess was found among the sequence D stars. No mid-infrared excess was found, demonstrating another difference from the sequence D stars.
In summary, we have demonstrated several significant differences between stars on sequences D and E. Most particularly, we have shown that Long Secondary Periods – the sequence D variation – are not caused by ellipsoidal variability, unlike the variation of the sequence E stars. The cause of Long Secondary Periods remains, at this time, a mystery.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful for the multiple allocations of VLT service time over several years for this extended series of observations (program identifiers 072.D-0387, 074.D-0098, 075.D-0090 and 076.D-0162).
This paper utilizes public domain data obtained by the MACHO Project, jointly funded by the US Department of Energy through the University of California, Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48, by the National Science Foundation through the Center for Particle Astrophysics of the University of California under cooperative agreement AST-8809616, and by the Mount Stromlo and Siding Spring Observatory, part of the Australian National University.
This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.
\[lastpage\]
[^1]: E-mail: nicholls@mso.anu.edu.au (CPN); wood@mso.anu.edu.au (PRW); m.cioni@herts.ac.uk (M-RLC)
|
---
abstract: 'We extend to arbitrary commutative base rings a recent result of Demeneghi that every ideal of an ample groupoid algebra over a field is an intersection of kernels of induced representations from isotropy groups, with a much shorter proof, by using the author’s Disintegration Theorem for groupoid representations. We also prove that every primitive ideal is the kernel of an induced representation from an isotropy group; however, we are unable to show, in general, that it is the kernel of an irreducible induced representation. If each isotropy group is finite (e.g., if the groupoid is principal) and if the base ring is Artinian (e.g., a field), then we can show that every primitive ideal is the kernel of an irreducible representation induced from isotropy.'
address: |
Department of Mathematics\
City College of New York\
Convent Avenue at 138th Street\
New York, New York 10031\
USA
author:
- Benjamin Steinberg
title: 'Ideals of étale groupoid algebras and Exel’s Effros-Hahn conjecture'
---
Introduction
============
The original Effros-Hahn conjecture [@EffHahn; @EffHahnBull] suggested that every primitive ideal of a crossed product of an amenable locally compact group with a commutative $C^*$-algebra should be induced from a primitive ideal of an isotropy group. The result was proved by Sauvageot [@Sauvage] for discrete groups and a more general result than the original conjecture was proved by Gootman and Rosenberg in [@Effhahnpf]. Crossed products of the above form are special cases of groupoid $C^*$-algebras and analogues of the Effros-Hahn conjecture in the groupoid setting were achieved by Renault [@RenaultEH] and Ionescu and Williams [@IonescuWilliams].
In [@mygroupoidalgebra], the author initiated the study of convolution algebras of ample groupoids over commutative rings with unit; see also [@operatorguys1]. R. Exel conjectured at the PARS meeting in Gramado, 2014 (and perhaps earlier) that an analogue of the Effros-Hahn conjecture should hold in this context. The author had developed in [@mygroupoidalgebra] a theory of induction from isotropy groups in this setting and had proven that inducing an irreducible representation from an isotropy group results in an irreducible representation of the groupoid algebra.
In [@ExelDoku], Dokuchaev and Exel showed that if a discrete group $G$ acts partially on a locally compact and totally disconnected space $X$, then every ideal of the partial crossed product $C_c(X,\Bbbk)\rtimes G$, where $C_c(X,\Bbbk)$ is the ring of locally constant, compactly supported functions from $X$ to the field $\Bbbk$, is an intersection of ideals induced from isotropy. Note that such partial crossed products are ample groupoid convolution algebras. Since in a $C^*$-algebra, every closed ideal is an intersection of primitive ideals, this result can be viewed as an analogue of Effros-Hahn for partial crossed products.
Demeneghi [@demeneghi] extended the result of Dokuchaev and Exel to arbitrary ample groupoid algebras over a field. Namely, he showed that each ideal is an intersection of kernels of induced representations from isotropy subgroups. His proof is rather indirect. First he develops a theory of induced representations for crossed products of the form $C_c(X,\Bbbk)\rtimes S$ where $S$ is an inverse semigroup acting on a locally compact and totally disconnected space $X$. Then he proves the result for such crossed products. Finally, he proves that groupoid convolution algebras are such crossed products using the full strength of his theory (and the converse is essentially true as well) and he shows that crossed product induction corresponds to groupoid induction under the isomorphism. His paper is around 50 pages in all.
In this paper, we prove that over an arbitrary base commutative ring each ideal of an ample groupoid convolution algebra is an intersection of kernels of induced representations from isotropy groups. Moreover, our proof is direct — circumventing entirely the crossed product machinery — and short. It relies on the author’s Disintegration Theorem [@groupoidbundles], which shows that modules for ample groupoid convolution algebras come from sheaves on the groupoid. This machinery is not very cumbersome to develop and is quite useful for analyzing irreducible representations, as was done in [@groupoidprimitive]. In future work, it will be shown that the Disintegration Theorem can be used to establish the isomorphism between inverse semigroup crossed products and groupoid algebras directly, without using induced representations.
We also obtain some new progress on Exel’s original conjecture on the structure of primitive ideals for groupoid algebras. Namely, we show that every primitive ideal is the kernel of a single representation induced from an isotropy group (rather than an infinite intersection of such kernels). We are, unfortunately, not able to show in general that it is the kernel of an irreducible representation induced from an isotropy group. We are, however, able to prove Exel’s version of the Effros-Hahn conjecture on primitive ideals if the base ring $R$ is Artinian and each isotropy group is either finite, or locally finite abelian with orders of elements invertible in $R/J(R)$ where $J(R)$ is the Jacobson radical of $R$ (e.g., if $R$ has characteristic zero).
Preliminaries
=============
This section summarizes definitions and results from [@mygroupoidalgebra] and [@groupoidbundles] that we use throughout. There are no new results in this section.
Groupoids
---------
Following Bourbaki, compactness will include the Hausdorff axiom throughout this paper. However, we do not require locally compact spaces to be Hausdorff. A topological groupoid $\mathscr G=(\mathscr G{^{(0)}},\mathscr G{^{(1)}})$ is *étale* if its domain map ${\mathop{\boldsymbol d}\nolimits}$ (or, equivalently, its range map ${\mathop{\boldsymbol r}\nolimits}$) is a local homeomorphism. In this case, identifying objects with identity arrows, we have that $\mathscr G{^{(0)}}$ is an open subspace of $\mathscr G{^{(1)}}$ and the multiplication map is a local homeomorphism. See, for example, [@Paterson; @resendeetale; @Exel].
Following [@Paterson], an étale groupoid is called *ample* if its unit space $\mathscr G{^{(0)}}$ is locally compact Hausdorff with a basis of compact open subsets. We shall say that an ample groupoid $\mathscr G$ is Hausdorff if $\mathscr G{^{(1)}}$ is Hausdorff.
A *local bisection* of an étale groupoid $\mathscr G$ is an open subset $U\subseteq \mathscr G{^{(1)}}$ such that both ${\mathop{\boldsymbol d}\nolimits}|_U$ and ${\mathop{\boldsymbol r}\nolimits}|_U$ are homeomorphisms with their images. The local bisections form a basis for the topology on $\mathscr G{^{(1)}}$ [@Exel]. The set ${\Gamma}(\mathscr G)$ of local bisections is an inverse monoid (cf. [@Lawson]) under the binary operation $$UV = \{\gamma\eta\mid \gamma\in U,\eta\in V,\ {\mathop{\boldsymbol d}\nolimits}(\gamma)={\mathop{\boldsymbol r}\nolimits}(\eta)\}.$$ The semigroup inverse is given by $U{^{-1}}= \{\gamma{^{-1}}\mid \gamma\in U\}$. The set ${\Gamma}_c(\mathscr G)$ of compact local bisections is an inverse subsemigroup of ${\Gamma}(\mathscr G)$ [@Paterson]. Note that $\mathscr G$ is ample if and only if ${\Gamma}_c(\mathscr G)$ is a basis for the topology on $\mathscr G{^{(1)}}$ [@Exel; @Paterson].
If $u\in \mathscr G{^{(0)}}$, then the *orbit* $\mathcal O_u$ of $u$ consists of all $v\in \mathscr G{^{(0)}}$ such that there is an arrow $\gamma$ with ${\mathop{\boldsymbol d}\nolimits}(\gamma)=u$ and ${\mathop{\boldsymbol r}\nolimits}(\gamma)=v$. The orbits form a partition of $\mathscr G{^{(0)}}$. A subset $X\subseteq \mathscr G{^{(0)}}$ is *invariant* if it is a union of orbits.
If $u\in \mathscr G{^{(0)}}$, the *isotropy group* of $\mathscr G$ at $u$ is $$G_u=\{\gamma\in \mathscr G{^{(1)}}\mid {\mathop{\boldsymbol d}\nolimits}(\gamma)=u={\mathop{\boldsymbol r}\nolimits}(\gamma)\}.$$ Isotropy groups of elements in the same orbit are isomorphic.
Ample groupoid algebras
-----------------------
Fix a commutative ring with unit $R$. The author [@mygroupoidalgebra] associated an $R$-algebra $R\mathscr G$ to each ample groupoid $\mathscr G$ as follows. We define $R\mathscr G$ to be the $R$-span in $R^{\mathscr G{^{(1)}}}$ of the characteristic functions $1_U$ of compact open subsets $U$ of $\mathscr G{^{(1)}}$. It is shown in [@mygroupoidalgebra Proposition 4.3] that $R\mathscr G$ is spanned by the elements $1_U$ with $U\in {\Gamma}_c(\mathscr G)$. If $\mathscr G{^{(1)}}$ is Hausdorff, then $R\mathscr G$ consists of the locally constant $R$-valued functions on $\mathscr G{^{(1)}}$ with compact support. Convolution is defined on $R\mathscr G$ by $$f\ast g(\gamma)=\sum_{{\mathop{\boldsymbol d}\nolimits}(\eta)={\mathop{\boldsymbol d}\nolimits}(\gamma)}f(\gamma \eta{^{-1}})g(\eta)=\sum_{\alpha\beta=\gamma}f(\alpha)g(\beta).$$ The finiteness of the sums is proved in [@mygroupoidalgebra]. The fact that the convolution belongs to $R\mathscr G$ comes from the computation $1_U\ast 1_V=1_{UV}$ for $U,V\in {\Gamma}_c(\mathscr G)$ [@mygroupoidalgebra].
The algebra $R\mathscr G$ is unital if and only if $\mathscr G{^{(0)}}$ is compact, but it always has local units (i.e., is a directed union of unital subrings) [@mygroupoidalgebra; @groupoidbundles]. A module $M$ over a ring $S$ with local units is termed *unitary* if $SM=M$. A module $M$ over a unital ring is unitary if and only if $1m=m$ for all $m\in M$. The category of unitary $S$-modules is denoted ${S\text{-}\mathrm{mod}}$. Notice that every simple module is unitary; $M$ is simple if $SM\neq 0$ and $M$ has no proper, non-zero submodules.
Induced modules
---------------
We recall from [@mygroupoidalgebra] the induction functor $${\mathop{\mathrm{Ind}}\nolimits}_u\colon {RG_u\text{-}\mathrm{mod}}{\longrightarrow}{R\mathscr G\text{-}\mathrm{mod}}.$$ For $u\in \mathcal G{^{(0)}}$, let $\mathscr Gu = {\mathop{\boldsymbol d}\nolimits}{^{-1}}(u)$ denote the set of all arrows starting at $u$. Then $G_u$ acts freely on the right of $\mathscr Gu$ by multiplication. Hence $R\mathscr Gu$ is a free right $RG_u$-module. A basis can be obtained by choosing, for each $v\in \mathcal O_u$, an arrow $\gamma_v\colon u{\longrightarrow}v$. We normally choose $\gamma_u=u$. There is a left $R\mathscr G$-module structure on $R\mathscr Gu$ given by $$\label{eq:action}
f\alpha = \sum_{{\mathop{\boldsymbol d}\nolimits}(\gamma)={\mathop{\boldsymbol r}\nolimits}(\alpha)}f(\gamma)\gamma\alpha$$ for $\alpha\in \mathscr Gu$ and $f\in R\mathscr G$. The $R\mathscr G$-action commutes with the $RG_u$-action by associativity and so $R\mathscr Gu$ is an $R\mathscr G$-$RG_u$-bimodule, unitary under both actions (as is easily checked).
The functor ${\mathop{\mathrm{Ind}}\nolimits}_u$ is defined by $${\mathop{\mathrm{Ind}}\nolimits}_u(M) = R\mathscr Gu\otimes_{RG_u}M.$$ This functor is exact by freeness of $R\mathscr Gu$ as a right $RG_u$-module and there is, in fact, an $R$-module direct sum decomposition $${\mathop{\mathrm{Ind}}\nolimits}_u(M) =\bigoplus_{v\in \mathcal O_u} \gamma_v\otimes M.$$ The action of $f\in R\mathscr G$ in these coordinates is given by $$\label{eq:action.ind}
f(\gamma_v\otimes m) =\sum_{w\in \mathcal O_u}\gamma_w\otimes \sum_{\gamma\colon v{\longrightarrow}w}f(\gamma)(\gamma_w{^{-1}}\gamma\gamma_v)m,$$ as is easily checked. An immediate corollary of is the following (see also [@demeneghi]).
\[p:annihilator.induced\] Let $u\in \mathscr G{^{(0)}}$ and $M$ an $RG_u$-module. Fix $\gamma_u\colon u{\longrightarrow}v$ for all $v\in \mathcal O_u$. Then the equality $${\mathop{\mathrm{Ann}}}({\mathop{\mathrm{Ind}}\nolimits}_u(M)) = \left\{f\in R\mathscr G\mid \forall v,w\in \mathcal O_u, \sum_{\gamma\colon v{\longrightarrow}w} f(\gamma)(\gamma_w{^{-1}}\gamma\gamma_v)\in {\mathop{\mathrm{Ann}}}(M)\right\}$$ holds.
A crucial result is that induction preserves simplicity.
\[t:induced.simple\] Let $M$ be a simple $RG_u$-module with $u\in \mathscr G{^{(0)}}$. Then ${\mathop{\mathrm{Ind}}\nolimits}_u(M)$ is a simple $R\mathscr G$-module. Moreover, the functor ${\mathop{\mathrm{Ind}}\nolimits}_u$ reflects isomorphism and ${\mathop{\mathrm{Ind}}\nolimits}_u(M)\cong {\mathop{\mathrm{Ind}}\nolimits}_v(N)$ implies $\mathcal O_u=\mathcal O_v$.
The Effros-Hahn conjecture for ample groupoids, first stated to the best of our knowledge by Exel at the PARS2014 conference in Gramado (see also [@ExelDoku]), says that each primitive ideal of $R\mathscr G$ is of the form ${\mathop{\mathrm{Ann}}}({\mathop{\mathrm{Ind}}\nolimits}_u(M))$ for some $u\in \mathscr G{^{(0)}}$ and simple $RG_u$-module $M$.
Modules over ample groupoid algebras
------------------------------------
The key ingredient to our approach is the author’s analogue [@groupoidbundles] of Renault’s Disintegration Theorem [@renaultdisintegration] for modules over groupoid algebras. Here $R$ will be a fixed commutative ring with unit and $\mathscr G$ an ample groupoid (not necessarily Hausdorff).
A *$\mathscr G$-sheaf* $\mathcal E=(E,p)$ consists of a space $E$, a local homeomorphism $p\colon E{\longrightarrow}\mathscr G{^{(0)}}$ and an action map $\mathscr G{^{(1)}}\times_{\mathscr G{^{(0)}}} E{\longrightarrow}E$ (where the fiber product is with respect to ${\mathop{\boldsymbol d}\nolimits}$ and $p$), denoted $(\gamma,e)\mapsto \gamma e$, satisfying the following axioms:
- $p(e)e=e$ for all $e\in E$;
- $p(\gamma e)={\mathop{\boldsymbol r}\nolimits}(\gamma)$ whenever $p(e)={\mathop{\boldsymbol d}\nolimits}(\gamma)$;
- $\gamma(\eta e)=(\gamma\eta)e$ whenever $p(e)={\mathop{\boldsymbol d}\nolimits}(\eta)$ and ${\mathop{\boldsymbol d}\nolimits}(\gamma)={\mathop{\boldsymbol r}\nolimits}(\eta)$.
A *$\mathscr G$-sheaf of $R$-modules* is a $\mathscr G$-sheaf $\mathcal E=(E,p)$ together with an $R$-module structure on each stalk $E_u=p{^{-1}}(u)$ such that:
- the zero section, $u\mapsto 0_u$ (the zero of $E_u$), is continuous;
- addition $E\times_{\mathscr G{^{(0)}}} E{\longrightarrow}E$ is continuous;
- scalar multiplication $R\times E{\longrightarrow}E$ is continuous;
- for each $\gamma\in \mathscr G{^{(1)}}$, the map $E_{{\mathop{\boldsymbol d}\nolimits}(\gamma)}{\longrightarrow}E_{{\mathop{\boldsymbol r}\nolimits}(\gamma)}$ given by $e\mapsto \gamma e$ is $R$-linear;
where $R$ has the discrete topology in the third item. Note that the first three conditions are equivalent to $(E,p)$ being a sheaf of $R$-modules over $\mathscr G{^{(0)}}$. Crucial to this paper is that $E_u$ is an $RG_u$-module for each $u\in \mathscr G{^{(0)}}$.
Note that the zero subspace $$\mathbf 0=\{0_u\mid u\in G{^{(0)}}\}$$ is an open subspace of $E$, being the image of a section of a local homeomorphism.
The *support* of $\mathcal E$ is $${\mathop{\mathrm{supp}}}(\mathcal E) = \{u\in \mathscr G{^{(0)}}\mid E_u\neq \{0_u\}\}.$$ Note that ${\mathop{\mathrm{supp}}}(\mathcal E)$ is an invariant subset of $\mathscr G_0$ but it need not be closed.
A *(global) section* of $\mathcal E$ is a continuous mapping $s\colon \mathscr G{^{(0)}}{\longrightarrow}E$ such that $p\circ s=1_{\mathscr G{^{(0)}}}$. Note that if $s\colon \mathscr G{^{(0)}}{\longrightarrow}E$ is a section, then its *support* $${\mathop{\mathrm{supp}}}(s) = s{^{-1}}(E\setminus \mathbf 0)$$ is closed. We denote by $\Gamma_c(\mathcal E)$ the set of (global) sections with compact support. Note that $\Gamma_c(\mathcal E)$ is an $R$-module with respect to pointwise operations and it be comes a unitary left $R\mathscr G$-module under the operation $$\label{eq:operation}
(fs)(u) =\sum_{{\mathop{\boldsymbol r}\nolimits}(\gamma)=u} f(\gamma)\gamma s({\mathop{\boldsymbol d}\nolimits}(\gamma)) = \sum_{v\in \mathcal O_u}\sum_{\gamma\colon v{\longrightarrow}u}f(\gamma)\gamma s(v).$$ See [@groupoidbundles] for details (where right actions and right modules are used).
If $e\in E_u$, then there is always a global section $s$ with compact support such that $s(u)=e$. Indeed, we can choose a neighborhood $U$ of $e$ such that $p|_U\colon U{\longrightarrow}p(U)$ is a homeomorphism with $p(U)$ open. Then we can find a compact open neighborhood $V$ of $u$ with $V\subseteq U$ and define $s$ to be the restriction of $(p|_U){^{-1}}$ on $V$ and $0$, elsewhere. Then $s$ is continuous, $s(u)=e$ and the support of $s$ is closed and contained in $V$, whence compact.
Conversely, if $M$ is a unitary left $R\mathscr G$-module, we can define a $\mathscr G$-sheaf of $R$-modules $\mathrm{Sh}(M)= (E,p)$ where $E_u =\varinjlim_{u\in U}1_UM$ (with the direct limit over all compact open neighborhoods $U$ of $u$ in $\mathscr G{^{(0)}}$) and if $\gamma\colon u{\longrightarrow}v$ and $[m]_u$ is the class of $m$ at $u$, then $\gamma[m]_u = [Um]_v$ where $U$ is any compact local bisection containing $\gamma$. Here $E=\coprod_{u\in \mathscr G{^{(0)}}} E_u$ has the germ topology. See [@groupoidbundles] for details.
\[t:disint\] The functor $\mathcal E\mapsto \Gamma_c(\mathcal E)$ is an equivalence between the category $\mathcal B\mathscr G_R$ of $\mathscr G$-sheaves of $R$-modules and ${R\mathscr G\text{-}\mathrm{mod}}$ with quasi-inverse $M\mapsto \mathrm{Sh}(M)$.
\[r:find.ideals\] As a consequence of Theorem \[t:disint\], $M\cong \Gamma_c(\mathrm{Sh}(M))$ for any unitary module $M$ and so we have an equality of annihilator ideals $${\mathop{\mathrm{Ann}}}(M)={\mathop{\mathrm{Ann}}}(\Gamma_c(\mathrm{Sh}(M))).$$ In particular, we have $I={\mathop{\mathrm{Ann}}}(\Gamma_c(\mathrm{Sh}(R\mathscr G/I)))$. Thus we can describe the ideal structure of $R\mathscr G$ in terms of the annihilators of modules of the form $\Gamma_c(\mathcal E)$.
The ideal structure of ample groupoid algebras
==============================================
In this section, we show how the Disintegration Theorem (Theorem \[t:disint\]) provides information about the ideal structure of $R\mathscr G$.
General ideals
--------------
Our main goal in this subsection is to prove the following theorem, generalizing a result of Demeneghi [@demeneghi] that was originally proved over fields. Demeneghi’s proof is indirect, via crossed products, and therefore quite long. Our proof is direct, using the Disintegration Theorem, and shorter, even including the 11 pages of [@groupoidbundles].
\[t:annih\] Let $\mathscr G$ be an ample groupoid and $R$ a ring. Let $\mathcal E=(E,p)$ be a $\mathscr G$-sheaf of $R$-modules. Then the equality $${\mathop{\mathrm{Ann}}}(\Gamma_c(\mathcal E))=\bigcap_{u\in \mathscr G{^{(0)}}} {\mathop{\mathrm{Ann}}}({\mathop{\mathrm{Ind}}\nolimits}_u(E_u))$$ holds. Consequently, every ideal $I\lhd R\mathscr G$ is an intersection of annihilators of induced modules.
The final statement follows from the first by the equivalence of categories in Theorem \[t:disint\] (cf. Remark \[r:find.ideals\]). So we prove the first statement. Let $I={\mathop{\mathrm{Ann}}}(\Gamma_c(\mathcal E))$ and put $J_u = {\mathop{\mathrm{Ann}}}({\mathop{\mathrm{Ind}}\nolimits}_u(E_u))$ for $u\in \mathscr G{^{(0)}}$. Set $J=\bigcap_{u\in \mathscr G{^{(0)}}}J_u$. Then we want to prove that $I=J$.
Fix $u\in \mathscr G{^{(0)}}$ and, for each $v\in \mathcal O_u$, choose $\gamma_v\colon u{\longrightarrow}v$. To show $I\subseteq J_u$, it suffices, by Proposition \[p:annihilator.induced\], to show that if $f\in I$, then, for each $v,w\in \mathcal O_u$, we have that $$\sum_{\gamma\colon v{\longrightarrow}w}f(\gamma)(\gamma_w{^{-1}}\gamma\gamma_v)\in {\mathop{\mathrm{Ann}}}(E_u).$$
So let $e\in E_u$ and let $s\in \Gamma_c(\mathcal E)$ with $s(u) =e$. Since $\mathscr G$ is ample, the set ${\mathop{\boldsymbol r}\nolimits}{^{-1}}(w)\cap {\mathop{\mathrm{supp}}}(f)$ is finite. Since $\mathscr G{^{(0)}}$ is Hausdorff, we can find $U\subseteq \mathscr G{^{(0)}}$ compact open with $v\in U$ and $U\cap {\mathop{\boldsymbol d}\nolimits}({\mathop{\boldsymbol r}\nolimits}{^{-1}}(w)\cap {\mathop{\mathrm{supp}}}(f))\subseteq \{v\}$.
Let $U_v$ be a compact local bisection containing $\gamma_v$ and $U_w$ a compact local bisection containing $\gamma_w$. Replacing $U_v$ by $UU_v$, we may assume that $${\mathop{\boldsymbol r}\nolimits}(U_v)\cap {\mathop{\boldsymbol d}\nolimits}({\mathop{\boldsymbol r}\nolimits}{^{-1}}(w)\cap {\mathop{\mathrm{supp}}}(f))\subseteq \{v\}.$$ By construction, the only elements in the support of $1_{U_w{^{-1}}}\ast f\ast 1_{U_v}$ with range $u$ are of the form $\gamma_w{^{-1}}\gamma\gamma_v$ with $\gamma\colon v{\longrightarrow}w$ and $f(\gamma)\neq 0$. As $1_{U_w{^{-1}}}\ast f\ast 1_{U_v}\in I$, we have by that $$\begin{aligned}
0&=(1_{U_w{^{-1}}}\ast f\ast 1_{U_v}s)(u) \\ &= \sum_{{\mathop{\boldsymbol r}\nolimits}(\alpha)=u}(1_{U_w{^{-1}}}\ast f\ast 1_{U_v})(\alpha)\alpha s({\mathop{\boldsymbol d}\nolimits}(\alpha)) \\ &=\sum_{\gamma\colon v{\longrightarrow}w}f(\gamma)(\gamma_w{^{-1}}\gamma\gamma_v)s(u) \\ &=\sum_{\gamma\colon v{\longrightarrow}w}f(\gamma)(\gamma_w{^{-1}}\gamma\gamma_v)e\end{aligned}$$ as required. Thus $I\subseteq J_u$ for all $u\in \mathscr G{^{(0)}}$.
Suppose now that $f\in J$ and let $s\in \Gamma_c(\mathcal E)$. Then $$\label{eq:mustcheck}
(fs)(v) = \sum_{u\in \mathcal O_v}\sum_{\gamma\colon u{\longrightarrow}v}f(\gamma)\gamma s(u).$$ Let us fix $u\in \mathcal O_v$ and let $\gamma_u=u$ and $\gamma_v\colon u{\longrightarrow}v$ be arbitrary. Then, since $f\in J\subseteq J_u$, we have by Proposition \[p:annihilator.induced\] that $$\sum_{\gamma\colon u{\longrightarrow}v}f(\gamma)(\gamma_v{^{-1}}\gamma) s(u)=0.$$ Multiplying on the left by $\gamma_v$ yields that, for each $u\in \mathcal O_v$, $$\sum_{\gamma\colon u{\longrightarrow}v}f(\gamma)\gamma s(u)=0$$ and so the right hand side of is $0$. Thus $f\in I$. This completes the proof.
More concretely, if $I\lhd R\mathscr G$ is an ideal, then following the construction of the proof we see that $$I = \bigcap_{u\in \mathscr G{^{(0)}}}{\mathop{\mathrm{Ann}}}\left({\mathop{\mathrm{Ind}}\nolimits}_u\left(\varinjlim_{u\in U}1_U(R\mathscr G/I)\right)\right)$$ where $\varinjlim_{u\in U} 1_U(R\mathscr G/I)$ has the $RG_u$-module structure $$\gamma[1_Uf+I] = [1_V1_Uf+I]$$ for $V$ any compact local bisection containing $\gamma$.
Primitive ideals
----------------
Recall that an ideal $I$ of a ring is *primitive* if it is the annihilator of a simple module. (Technically, we should talk about left primitive ideals since we are using left modules, but because groupoid algebras admit an involution, it doesn’t matter.) We prove that each primitive ideal is the annihilator of a single induced representation (rather than an intersection of such annihilators, as in Theorem \[t:annih\]). Unfortunately, we are not yet able to show, in general, that the induced representation is simple. Still, this is new progress toward Exel’s Effros-Hahn conjecture.
\[t:primitive.case\] Let $R$ be a commutative ring with unit and $\mathscr G$ an ample groupoid. Let $I\lhd R\mathscr G$ be a primitive ideal. Then $I={\mathop{\mathrm{Ann}}}({\mathop{\mathrm{Ind}}\nolimits}_u(M))$ for some $u\in \mathscr G{^{(0)}}$ and $RG_u$-module $M$.
By Theorem \[t:disint\] we may assume that our simple module with annihilator $I$ is of the form $\Gamma_c(\mathcal E)$ for some $\mathscr G$-sheaf $\mathcal E=(E,p)$ of $R$-modules. Let $u\in {\mathop{\mathrm{supp}}}(\mathcal E)$ and put $J_u = {\mathop{\mathrm{Ann}}}({\mathop{\mathrm{Ind}}\nolimits}_u(E_u))$. We claim that $I=J_u$. We know that $I\subseteq J_u$ by Theorem \[t:annih\]; we must prove the converse. Suppose that $J_u$ does not annihilate $\Gamma_c(\mathcal E)$. Then we can find a section $s$ with $J_us\neq \{0\}$. As $J_u$ is an ideal, $J_us$ is a submodule and so $J_us=\Gamma_c(\mathcal E)$. Let $0_u\neq e\in E_u$. Then there is a section $t\in \Gamma_c(\mathcal E)$ such that $t(u)=e$. Let $f\in J_u$ with $fs=t$. Then we have that $$\label{eq:big.helper}
e=t(u)=(fs)(u) = \sum_{v\in \mathcal O_u}\sum_{\gamma\colon v{\longrightarrow}u}f(\gamma)\gamma s(v).$$
Let us fix $v\in \mathcal O_u$ and fix $\gamma_v\colon u{\longrightarrow}v$; put $\gamma_u=u$. Then by the assumption $f\in J_u$ and Proposition \[p:annihilator.induced\], we have (since $\gamma_v{^{-1}}s(v)\in E_u$) that $$0=\sum_{\gamma\colon v{\longrightarrow}u} f(\gamma)(\gamma\gamma_v)(\gamma_v{^{-1}}s(v))=\sum_{\gamma\colon v{\longrightarrow}u}f(\gamma)\gamma s(v).$$ Thus the right hand side of is $0$, contradicting that $e\neq 0$. We conclude that $J_u\subseteq I$. This completes the proof.
We now show that the $RG_u$-module $M$ above can be chosen to be simple under some strong hypotheses on the base ring and isotropy groups. Let $J(S)$ denote the Jacobson radical of a ring $S$. A ring $S$ is called a *left max ring* if each non-zero left $S$-module has a maximal (proper) submodule. For example, any Artinian ring $S$ is a left max ring. Indeed, if $M\neq 0$, then $J(S)M\neq M$ by nilpotency of the Jacobson radical. But $M/J(S)M$ is then a non-zero $S/J(S)$-module and every non-zero module over a semisimple ring is a direct sum of simple modules and hence has a simple quotient. Thus $M$ has a maximal proper submodule. A result of Hamsher [@Hamsher] says that if $S$ is commutative, then $S$ is a left max ring if and only if $J(S)$ is $T$-nilpotent (e.g., if $J(S)$ is nilpotent) and $S/J(S)$ is von Neumann regular ring. We now establish the Effros-Hahn conjecture in the case that all isotropy group rings are left max rings.
\[t:left.max\] Let $R$ be a commutative ring and $\mathscr G$ an ample groupoid such that $RG_u$ is a left max ring for all $u\in \mathscr G{^{(0)}}$. Then the primitive ideals of $R\mathscr G$ are exactly the ideals of the form ${\mathop{\mathrm{Ann}}}({\mathop{\mathrm{Ind}}\nolimits}_u(M))$ where $M$ is a simple $RG_u$-module.
By Theorem \[t:induced.simple\] it suffices to show that any primitive ideal $I$ is of the form ${\mathop{\mathrm{Ann}}}({\mathop{\mathrm{Ind}}\nolimits}_u(M))$ where $M$ is a simple $RG_u$-module. By Theorem \[t:disint\] we may assume that our simple module with annihilator $I$ is of the form $\Gamma_c(\mathcal E)$ for some $\mathscr G$-sheaf $\mathcal E=(E,p)$ of $R$-modules. Let $u\in {\mathop{\mathrm{supp}}}(\mathcal E)$. We already know from the proof of Theorem \[t:primitive.case\] that $I = {\mathop{\mathrm{Ann}}}({\mathop{\mathrm{Ind}}\nolimits}_u(E_u))$. Let $N$ be a maximal submodule of $E_u$ (which exists by assumption on $RG_u$) and let $J={\mathop{\mathrm{Ann}}}({\mathop{\mathrm{Ind}}\nolimits}_u(E_u/N))$. Since $E_u/N$ is simple, it suffices to show that $J=I$. Clearly, $I\subseteq J$ (by Proposition \[p:annihilator.induced\]) since ${\mathop{\mathrm{Ann}}}(E_u)\subseteq {\mathop{\mathrm{Ann}}}(E_u/N)$. So it suffices to show that $J$ annihilates $\Gamma_c(\mathcal E)$.
Suppose that this is not the case. Then there exists $s\in \Gamma_c(\mathcal E)$ with $Js\neq 0$. Since $J$ is an ideal and $\Gamma_c(\mathcal E)$ is simple, we deduce $Js=\Gamma_c(\mathcal E)$. Let $e\in E_u\setminus N$ (using that $N$ is a proper submodule) and let $t\in \Gamma_c(\mathcal E)$ with $t(u)=e$. Then $t=fs$ with $f\in J$. Let us compute $$\label{eq:big.helper.2}
e=t(u)=(fs)(u) = \sum_{v\in \mathcal O_u}\sum_{\gamma\colon v{\longrightarrow}u}f(\gamma)\gamma s(v).$$
Let us fix $v\in \mathcal O_u$, fix $\gamma_v\colon u{\longrightarrow}v$ and set $\gamma_u=u$. Then by the assumption $f\in J$ and Proposition \[p:annihilator.induced\], we have (since $\gamma_v{^{-1}}s(v)\in E_u$) that $$\sum_{\gamma\colon v{\longrightarrow}u}f(\gamma)\gamma s(v)=\sum_{\gamma\colon v{\longrightarrow}u} f(\gamma)(\gamma\gamma_v)(\gamma_v{^{-1}}s(v))\in N.$$ We deduce from that $e\in N$, which is a contradiction. It follows that $J$ annihilates $\Gamma_c(\mathcal E)$ and so $I=J$.
We obtain as a consequence a special case of Exel’s Effros-Hahn conjecture, which will include the algebras of principal groupoids (or, more generally, groupoids with finite isotropy) over a field. Recall that a groupoid is called *principal* if all its isotropy groups are trivial.
It is well known that a group ring $RG$ is Artinian if and only if $R$ is Artinian and $G$ is finite [@PassmanBook]. A result of Villamayor [@villamayor] says that, for a group $G$ and commutative ring $R$, one has that $RG$ is von Neumann regular if and only if $R$ is von Neumann regular, $G$ is locally finite and the order of any element of $G$ is invertible in $R$. The reader should recall that any Artinian semisimple ring is von Neumann regular and also that any von Neumann regular ring has a zero Jacobson radical [@LamBook Cor. 4.24].
\[c:main\] Let $R$ be a commutative Artinian ring and $\mathscr G$ an ample groupoid such that each isotropy group of $\mathscr G$ is either finite or locally finite abelian with elements having order invertible in $R/J(R)$. Then the primitive ideals of $R\mathscr G$ are precisely the annihilators of modules induced from simple modules of isotropy group rings.
By Theorem \[t:left.max\] it suffices to show that $RG_u$ is a left max ring for each $u\in \mathscr G{^{(0)}}$. If $G_u$ is finite, then $RG_u$ is Artinian and hence a left max ring. Suppose that $G_u$ is locally finite abelian with each element of order invertible in $R/J(R)$. In particular, $RG_u$ is commutative and it suffices by Hamsher’s theorem to show that $J(RG_u)$ is nilpotent and $RG_u/J(RG_u)$ is von Neumann regular. First note that since $J(R)$ is nilpotent, we have that $J(R)RG_u$ is a nilpotent ideal of $RG_u$ and hence contained $J(RG_u)$. On the other hand, $RG_u/J(R)RG_u\cong (R/J(R))G_u$ is von Neumann regular by [@villamayor] and the hypotheses, and hence has a zero radical. Thus $J(RG_u)=J(R)RG_u$, and hence is nilpotent, and $RG_u/J(RG_u)$ is von Neumann regular. Therefore, $RG_u$ is a left max ring.
If $R$ is commutative Artinian of characteristic zero, then the order of any element of a locally finite group is invertible in $R/J(R)$ (which is a product of fields of characteristic zero) and so Corollary \[c:main\] applies if each isotropy group is either finite or locally finite abelian.
Acknowledgments {#acknowledgments .unnumbered}
---------------
The author would like to thank Enrique Pardo for pointing out Villamayor’s result [@villamayor] and that regular rings have zero Jacobson radical, leading to the current formulation of Corollary \[c:main\], which improves on a previous version.
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abstract: 'We investigate whether one can determine from the transcripts of U.S. Congressional floor debates whether the speeches represent support of or opposition to proposed legislation. To address this problem, we exploit the fact that these speeches occur as part of a discussion; this allows us to use sources of information regarding relationships between discourse segments, such as whether a given utterance indicates agreement with the opinion expressed by another. We find that the incorporation of such information yields substantial improvements over classifying speeches in isolation.'
author:
- |
Matt Thomas, Bo Pang, and Lillian Lee\
Department of Computer Science, Cornell University\
Ithaca, NY 14853-7501\
[ [mattthomas84@gmail.com]{}, [pabo@cs.cornell.edu]{}, [llee@cs.cornell.edu]{}]{}\
\
[**]{}
title: 'Get out the vote: Determining support or opposition from Congressional floor-debate transcripts'
---
Introduction {#sec:intro}
============
> [*One ought to recognize that the present political chaos is connected with the decay of language, and that one can probably bring about some improvement by starting at the verbal end.*]{} — Orwell, “Politics and the English language”
We have entered an era where very large amounts of politically oriented text are now available online. This includes both official documents, such as the full text of laws and the proceedings of legislative bodies, and unofficial documents, such as postings on weblogs (blogs) devoted to politics. In some sense, the availability of such data is simply a manifestation of a general trend of “everybody putting their records on the Internet”.[^1] The online accessibility of politically oriented texts in particular, however, is a phenomenon that some have gone so far as to say will have a potentially society-changing effect.
In the United States, for example, governmental bodies are providing and soliciting political documents via the Internet, with lofty goals in mind: [*electronic rulemaking*]{} (eRulemaking) initiatives involving the “electronic collection, distribution, synthesis, and analysis of public commentary in the regulatory rulemaking process”, may “\[alter\] the citizen-government relationship” [@Shulman+Schlosberg:02a]. Additionally, much media attention has been focused recently on the potential impact that Internet sites may have on politics[^2], or at least on political journalism[^3]. Regardless of whether one views such claims as clear-sighted prophecy or mere hype, it is obviously important to help people understand and analyze politically oriented text, given the importance of enabling informed participation in the political process.
Evaluative and persuasive documents, such as a politician’s speech regarding a bill or a blogger’s commentary on a legislative proposal, form a particularly interesting type of politically oriented text. People are much more likely to consult such evaluative statements than the actual text of a bill or law under discussion, given the dense nature of legislative language and the fact that (U.S.) bills often reach several hundred pages in length [@Smith+Robert+VanderWielen:05a]. Moreover, political opinions are explicitly solicited in the eRulemaking scenario.
In the analysis of evaluative language, it is fundamentally necessary to determine whether the author/speaker supports or disapproves of the topic of discussion. In this paper, we investigate the following specific instantiation of this problem: we seek to determine from the transcripts of U.S. Congressional floor debates whether each “speech” (continuous single-speaker segment of text) represents support for or opposition to a proposed piece of legislation. Note that from an experimental point of view, this is a very convenient problem to work with because we can automatically determine ground truth (and thus avoid the need for manual annotation) simply by consulting publicly available voting records.
####
Determining whether or not a speaker supports a proposal falls within the realm of [*sentiment analysis*]{}, an extremely active research area devoted to the computational treatment of subjective or opinion-oriented language (early work includes Wiebe and Rapaport , Hearst , Sack , and ; see Esuli for an active bibliography). In particular, since we treat each individual speech within a debate as a single “document”, we are considering a version of [*document-level sentiment-polarity classification*]{}, namely, automatically distinguishing between positive and negative documents [@Das+Chen:01a; @Pang+Lee+Vaithyanathan:02a; @Turney:02a; @Dave+Lawrence+Pennock:03a].
Most sentiment-polarity classifiers proposed in the recent literature categorize each document independently. A few others incorporate various measures of inter-document similarity between the texts to be labeled [@Agarwal+Bhattacharyya:05a; @Pang+Lee:05a; @Goldberg+Zhu:06a]. Many interesting opinion-oriented documents, however, can be linked through certain relationships that occur in the context of evaluative [*discussions*]{}. For example, we may find textual[^4] evidence of a high likelihood of [*agreement*]{} between two speakers, such as explicit assertions (“I second that!”) or quotation of messages in emails or postings (see but cf. ). Agreement evidence can be a powerful aid in our classification task: for example, we can easily categorize a complicated (or overly terse) document if we find within it indications of agreement with a clearly positive text.
Obviously, incorporating agreement information provides additional benefit only when the input documents are relatively difficult to classify individually. Intuition suggests that this is true of the data with which we experiment, for several reasons. First, U.S. congressional debates contain very rich language and cover an extremely wide variety of topics, ranging from flag burning to international policy to the federal budget. Debates are also subject to digressions, some fairly natural and others less so (e.g., “Why are we discussing this bill when the plight of my constituents regarding this other issue is being ignored?”)
Second, an important characteristic of persuasive language is that speakers may spend more time presenting evidence in support of their positions (or attacking the evidence presented by others) than directly stating their attitudes. An extreme example will illustrate the problems involved. Consider a speech that describes the U.S. flag as deeply inspirational, and thus contains only positive language. If the bill under discussion is a proposed flag-burning ban, then the speech is [*supportive*]{}; but if the bill under discussion is aimed at rescinding an existing flag-burning ban, the speech may represent [*opposition*]{} to the legislation. Given the current state of the art in sentiment analysis, it is doubtful that one could determine the (probably topic-specific) relationship between presented evidence and speaker opinion.
total train test development
----------------------------------------------------- ------- ------- ------ -------------
[speech segments]{} 3857 2740 860 257
[[[debate]{}]{}s]{} 53 38 10 5
average number of [speech segments]{}per [debate]{} 72.8 72.1 86.0 51.4
average number of speakers per [debate]{} 32.1 30.9 41.1 22.6
#### Qualitative summary of results
The above difficulties underscore the importance of enhancing standard classification techniques with new information sources that promise to improve accuracy, such as inter-document relationships between the documents to be labeled. In this paper, we demonstrate that the incorporation of agreement modeling can provide substantial improvements over the application of support vector machines (SVMs) in isolation, which represents the state of the art in the individual classification of documents. The enhanced accuracies are obtained via a fairly primitive automatically-acquired “agreement detector” and a conceptually simple method for integrating isolated-document and agreement-based information. We thus view our results as demonstrating the potentially large benefits of exploiting sentiment-related discourse-segment relationships in sentiment-analysis tasks.
Corpus {#sec:data}
======
This section outlines the main steps of the process by which we created our corpus (download site: www.cs.cornell.edu/home/llee/data/convote.html).
GovTrack () is an independent website run by Joshua Tauberer that collects publicly available data on the legislative and fund-raising activities of U.S. congresspeople. Due to its extensive cross-referencing and collating of information, it was nominated for a 2006 “Webby” award. A crucial characteristic of GovTrack from our point of view is that the information is provided in a very convenient format; for instance, the floor-debate transcripts are broken into separate HTML files according to the subject of the debate, so we can trivially derive long sequences of speeches guaranteed to cover the same topic.
We extracted from GovTrack all available transcripts of U.S. floor debates in the House of Representatives for the year 2005 (3268 pages of transcripts in total), together with voting records for all roll-call votes during that year. We concentrated on [[[debate]{}]{}s]{}regarding “controversial” bills (ones in which the losing side generated at least 20% of the speeches) because these debates should presumably exhibit more interesting discourse structure.
Each debate consists of a series of [*[speech segments]{}*]{}, where each segment is a sequence of uninterrupted utterances by a single speaker. Since [speech segments]{}represent natural discourse units, we treat them as the basic unit to be classified. Each [speech segment]{}was labeled by the vote ([“yea”]{}or [“nay”]{}) cast for the proposed bill by the person who uttered the [speech segment]{}.
We automatically discarded those [speech segments]{}belonging to a class of formulaic, generally one-sentence utterances focused on the yielding of time on the house floor (for example, “Madam Speaker, I am pleased to yield 5 minutes to the gentleman from Massachusetts”), as such [speech segments]{}are clearly off-topic. We also removed [speech segments]{} containing the term “amendment”, since we found during initial inspection that these speeches generally reflect a speaker’s opinion on an amendment, and this opinion may differ from the speaker’s opinion on the underlying bill under discussion.
We randomly split the data into training, test, and development (parameter-tuning) sets representing roughly 70%, 20%, and 10% of our data, respectively (see Table \[tab:data\]). The [speech segments]{}remained grouped by [debate]{}, with 38 [[[debate]{}]{}s]{}assigned to the training set, 10 to the test set, and 5 to the development set; we require that the [speech segments]{}from an individual [debate]{}all appear in the same set because our goal is to examine classification of [speech segments]{}in the context of the surrounding discussion.
Method {#sec:method}
======
The support/oppose classification problem can be approached through the use of standard classifiers such as support vector machines (SVMs), which consider each text unit in isolation. As discussed in Section \[sec:intro\], however, the conversational nature of our data implies the existence of various relationships that can be exploited to improve cumulative classification accuracy for [speech segments]{}belonging to the same [debate]{}. Our classification framework, directly inspired by , integrates both perspectives, optimizing its labeling of [speech segments]{}based on both individual [speech-segment]{}classification scores and preferences for groups of [speech segments]{}to receive the same label. In this section, we discuss the specific classification framework that we adopt and the set of mechanisms that we propose for modeling specific types of relationships.
Classification framework {#sec:method:graph}
------------------------
Let ${s}_1,{s}_2,\ldots, {s}_{{n}}$ be the sequence of [speech segments]{}within a given debate, and let ${{\cal Y}\xspace}$ and ${{\cal N}\xspace}$ stand for the “yea” and “nay” class, respectively. Assume we have a non-negative function ${\mbox{{\it ind}}\xspace}({s},{C})$ indicating the degree of preference that an individual-document classifier, such as an SVM, has for placing [speech-segment]{}${s}$ in class ${C}$. Also, assume that some pairs of [speech segments]{}have [*weighted links*]{} between them, where the non-negative [*strength*]{} (weight) ${{\mathit str}}({\ell})$ for a link ${\ell}$ indicates the degree to which it is preferable that the linked [speech segments]{}receive the same label. Then, any class assignment ${c}= {c}({s}_1),
{c}({s}_2),\ldots, {c}({s}_{{n}})$ can be assigned a [*cost*]{} $$\sum_{{s}}
{\mbox{{\it ind}}\xspace}({s},\overline{{c}}({s})) +
\sum_{{s},{s}':\: {c}({s}) \neq {c}({s}')}
\,
\sum_{\ell\; {\rm between}\, {{s}, {s}'} }{{\mathit str}}({\ell}),$$ where $\overline{{c}}({s})$ is the “opposite” class from ${{c}}({s})$. A [*minimum-cost*]{} assignment thus represents an optimum way to classify the [speech segments]{} so that each one tends not to be put into the class that the individual-document classifier disprefers, but at the same time, highly associated [speech segments]{}tend not to be put in different classes.
As has been previously observed and exploited in the NLP literature [@Pang+Lee:04a; @Agarwal+Bhattacharyya:05a; @Barzilay+Lapata:05a], the above optimization function, unlike many others that have been proposed for graph or set partitioning, can be solved [*exactly*]{} in an provably efficient manner via methods for finding minimum cuts in graphs. In our view, the contribution of our work is the examination of new types of relationships, not the method by which such relationships are incorporated into the classification decision.
Classifying [speech segments]{}in isolation
-------------------------------------------
In our experiments, we employed the well-known classifier [${\rm SVM}^{light}$]{} to obtain individual-document classification scores, treating ${{\cal Y}\xspace}$ as the positive class and using plain unigrams as features.[^5] Following standard practice in sentiment analysis [@Pang+Lee+Vaithyanathan:02a], the input to [${\rm SVM}^{light}$]{}consisted of normalized presence-of-feature (rather than frequency-of-feature) vectors. The ${\mbox{{\it ind}}\xspace}$ value for each [speech segment]{} ${s}$ was based on the signed distance ${{d}({s})\xspace}$ from the vector representing ${s}$ to the trained SVM decision plane: $${\mbox{{\it ind}}\xspace}({s},{{\cal Y}\xspace}) \stackrel{{\rm def}}{=}
\begin{cases}
1 & {{d}({s})\xspace}> 2{\sigma_{{s}}}; \cr
\left(1 + \frac{{{d}({s})\xspace}}{2{\sigma_{{s}}}}\right)/2 &
\vert {{d}({s})\xspace}\vert \leq 2{\sigma_{{s}}};
\cr
0 & {{d}({s})\xspace}< -2{\sigma_{{s}}}\end{cases}$$ where ${\sigma_{{s}}}$ is the standard deviation of ${d}({s})$ over all [speech segments]{}${s}$ in the debate in question, and ${\mbox{{\it ind}}\xspace}({s},{{\cal N}\xspace}) \stackrel{{\rm def}}{=} 1 - {\mbox{{\it ind}}\xspace}({s},{{\cal Y}\xspace})$.
We now turn to the more interesting problem of representing the preferences that [speech segments]{}may have for being assigned to the same class.
Relationships between [speech segments]{} {#sec:method:relationship}
-----------------------------------------
A wide range of relationships between text segments can be modeled as positive-strength links. Here we discuss two types of constraints that are considered in this work.
#### Same-speaker constraints:
In Congressional debates and in general social-discourse contexts, a single speaker may make a number of comments regarding a topic. It is reasonable to expect that in many settings, the participants in a discussion may be convinced to change their opinions midway through a debate. Hence, in the general case we wish to be able to express “soft” preferences for all of an author’s statements to receive the same label, where the strengths of such constraints could, for instance, vary according to the time elapsed between the statements. Weighted links are an appropriate means to express such variation.
However, if we assume that most speakers do not change their positions in the course of a discussion, we can conclude that all comments made by the same speaker must receive the same label. This assumption holds by fiat for the ground-truth labels in our dataset because these labels were derived from the single vote cast by the speaker on the bill being discussed.[^6] We can implement this assumption via links whose weights are essentially infinite. Although one can also implement this assumption via concatenation of same-speaker [speech segments]{}(see Section \[sec:eval:global\]), we view the fact that our graph-based framework incorporates both hard and soft constraints in a principled fashion as an advantage of our approach.
#### Different-speaker agreements
In House discourse, it is common for one speaker to make reference to another in the context of an agreement or disagreement over the topic of discussion. The systematic identification of instances of agreement can, as we have discussed, be a powerful tool for the development of intelligently selected weights for links between [speech segments]{}.
The problem of agreement identification can be decomposed into two sub-problems: identifying references and their targets, and deciding whether each reference represents an instance of agreement. In our case, the first task is straightforward because we focused solely on by-name references.[^7] Hence, we will now concentrate on the second, more interesting task.
We approach the problem of classifying references by representing each reference with a word-presence vector derived from a window of text surrounding the reference.[^8] In the training set, we classify each reference connecting two speakers with a positive or negative label depending on whether the two voted the same way on the bill under discussion[^9]. These labels are then used to train an SVM classifier, the output of which is subsequently used to create weights on [*agreement links*]{} in the test set as follows.
Let ${{d}({r})}$ denote the distance from the vector representing reference ${r}$ to the agreement-detector SVM’s decision plane, and let ${\sigma_{r}}$ be the standard deviation of ${d}({r})$ over all references in the [debate]{}in question. We then define the strength ${\mathit{agr}\xspace}$ of the [*agreement link*]{} corresponding to the reference as: $${\mathit{agr}\xspace}({r}) \stackrel{{\rm def}}{=} \begin{cases}
0 & {{d}({r})}< {\theta_{\mbox{agr}}\xspace}; \cr
{\alpha\xspace}\cdot {{d}({r})}/ 4{\sigma_{r}}& {\theta_{\mbox{agr}}\xspace}\le {{d}({r})}\le 4{\sigma_{r}}; \cr
{\alpha\xspace}& {{d}({r})}> 4{\sigma_{r}}.
\end{cases}$$ The free parameter ${\alpha\xspace}$ specifies the relative importance of the ${\mathit{agr}\xspace}$ scores. The threshold ${\theta_{\mbox{agr}}\xspace}$ controls the precision of the agreement links, in that values of ${\theta_{\mbox{agr}}\xspace}$ greater than zero mean that greater confidence is required before an agreement link can be added. [^10]
Evaluation {#sec:eval}
==========
This section presents experiments testing the utility of using [speech-segment]{}relationships, evaluating against a number of baselines. All reported results use values for the free parameter ${\alpha\xspace}$ derived via tuning on the development set. In the tables, [**boldface**]{} indicates the development- and test-set results for the [*development-set-optimal*]{} parameter settings, as one would make algorithmic choices based on development-set performance.
Preliminaries: Reference classification {#sec:eval:agr}
---------------------------------------
Recall that to gather inter-speaker agreement information, the strategy employed in this paper is to classify by-name references to other speakers as to whether they indicate agreement or not.
--------------------------------------------- ------------------- -------------------
[majority baseline]{} 81.51 80.26
[; [${\theta_{\mbox{agr}}\xspace}= 0$]{}]{} 84.25 81.07
[; [${\theta_{\mbox{agr}}\xspace}= 0$]{}]{} [[**86.99**]{}]{} [[**80.10**]{}]{}
--------------------------------------------- ------------------- -------------------
: \[tab:agr\] Agreement-classifier accuracy, in percent. “Amdmts”=“[speech segments]{}containing the word [‘amendment’]{}”. Recall that boldface indicates results for development-set-optimal settings.
To train our agreement classifier, we experimented with undoing the deletion of amendment-related [speech segments]{}in the training set. Note that such [speech segments]{}were [*never*]{} included in the development or test set, since, as discussed in Section \[sec:data\], their labels are probably noisy; however, including them in the [*training*]{} set allows the classifier to examine more instances even though some of them are labeled incorrectly. As Table \[tab:agr\] shows, using more, if noisy, data yields better agreement-classification results on the [development]{} set, and so we use that policy in all subsequent experiments. [^11]
An important observation is that precision may be more important than accuracy in deciding which agreement links to add: false positives with respect to agreement can cause [speech segments]{}to be incorrectly assigned the same label, whereas false negatives mean only that agreement-based information about other [speech segments]{}is not employed. As described above, we can raise agreement precision by increasing the threshold ${\theta_{\mbox{agr}}\xspace}$, which specifies the required confidence for the addition of an agreement link. Indeed, Table \[tab:agr-highprec\] shows that we can improve agreement precision by setting ${\theta_{\mbox{agr}}\xspace}$ to the (positive) mean agreement score ${\mu\xspace}$ assigned by the SVM agreement-classifier over all references in the given [[debate]{}]{}[^12]. However, this comes at the cost of greatly reducing agreement accuracy (development: 64.38%; test: 66.18%) due to lowered recall levels. Whether or not better [speech-segment]{}classification is ultimately achieved is discussed in the next sections.
-------------------------------------------------- ------------------- -------------------
[[Agreement classifier]{}]{}
[Devel. set]{} [Test set]{}
[${\theta_{\mbox{agr}}\xspace}= 0$]{} 86.23 82.55
[${\theta_{\mbox{agr}}\xspace}= {\mu\xspace}$]{} [[**89.41**]{}]{} [[**88.47**]{}]{}
-------------------------------------------------- ------------------- -------------------
: \[tab:agr-highprec\] Agreement-classifier precision.
[[Segment-based]{}[speech-segment]{}classification]{} {#sec:eval:local}
-----------------------------------------------------
#### Baselines
The first two data rows of Table \[tab:results-local\] depict baseline performance results. The [$\#(\mbox{``support''}) - \#(\mbox{``oppos''})$]{} baseline is meant to explore whether the [speech-segment]{}classification task can be reduced to simple lexical checks. Specifically, this method uses the signed difference between the number of words containing the stem “support” and the number of words containing the stem “oppos” (returning the majority class if the difference is 0). No better than 62.67% test-set accuracy is obtained by either baseline.
----------------------------------------------------------------------------------------- ------------------- -------------------
[majority baseline]{} 54.09 58.37
[$\#(\mbox{``support''}) - \#(\mbox{``oppos''})$]{} 59.14 62.67
[SVM]{}\[[[speech segment]{}]{}\] 70.04 66.05
[SVM + same-speaker links]{} 79.77 67.21
[SVM + same-speaker links]{}$\ldots$
[ ]{} [[+ ]{}[agreement links]{}, [${\theta_{\mbox{agr}}\xspace}= 0$]{}]{} [[**89.11**]{}]{} [[**70.81**]{}]{}
[ ]{} [[+ ]{}[agreement links]{}, [${\theta_{\mbox{agr}}\xspace}= {\mu\xspace}$]{}]{} 87.94 71.16
----------------------------------------------------------------------------------------- ------------------- -------------------
: \[tab:results-local\] [[Segment-based]{}[speech-segment]{}classification]{}accuracy, in percent.
--------------------------------------------------------------- ------------------- -------------------
[SVM]{}\[speaker\] 71.60 70.00
SVM + [agreement links]{}$\ldots$
[ ]{} with [${\theta_{\mbox{agr}}\xspace}= 0$]{} [[**88.72**]{}]{} [[**71.28**]{}]{}
[ ]{} with [${\theta_{\mbox{agr}}\xspace}= {\mu\xspace}$]{} 84.44 76.05
--------------------------------------------------------------- ------------------- -------------------
: \[tab:results-global\] [[Speaker-based]{}[speech-segment]{}classification]{}accuracy, in percent. Here, the initial SVM is run on the concatenation of all of a given speaker’s [speech segments]{}, but the results are computed over [speech segments]{}(not speakers), so that they can be compared to those in Table \[tab:results-local\].
#### Using relationship information
Applying an SVM to classify each [speech segment]{}in isolation leads to clear improvements over the two baseline methods, as demonstrated in Table \[tab:results-local\]. When we impose the constraint that all [speech segments]{}uttered by the same speaker receive the same label via “same-speaker links”, both test-set and development-set accuracy increase even more, in the latter case quite substantially so.
The last two lines of Table \[tab:results-local\] show that the best results are obtained by incorporating agreement information as well. The highest test-set result, 71.16%, is obtained by using a high-precision threshold to determine which agreement links to add. While the development-set results would induce us to utilize the standard threshold value of 0, which is sub-optimal on the test set, the [${\theta_{\mbox{agr}}\xspace}= 0$]{} agreement-link policy still achieves noticeable improvement over not using agreement links (test set: 70.81% vs. 67.21%).
[[Speaker-based]{}[speech-segment]{}classification]{} {#sec:eval:global}
-----------------------------------------------------
We use [speech segments]{}as the unit of classification because they represent natural discourse units. As a consequence, we are able to exploit relationships at the [speech-segment]{}level. However, it is interesting to consider whether we really need to consider relationships specifically between [speech segments]{}themselves, or whether it suffices to simply consider relationships between the [*speakers*]{} of the [speech segments]{}. In particular, as an alternative to using same-speaker links, we tried a [*[speaker-based]{}*]{} approach wherein the way we determine the initial individual-document classification score for each [speech segment]{}uttered by a person $p$ in a given debate is to run an SVM on the concatenation of [*all*]{} of $p$’s [speech segments]{}within that debate. (We also ensure that agreement-link information is propagated from [speech-segment]{} to speaker pairs.)
How does the use of same-speaker links compare to the concatenation of each speaker’s [speech segments]{}? Tables \[tab:results-local\] and \[tab:results-global\] show that, not surprisingly, the SVM individual-document classifier works better on the concatenated [speech segments]{}than on the [speech segments]{}in isolation. However, the effect on overall classification accuracy is less clear: the development set favors same-speaker links over concatenation, while the test set does not.
But we stress that the most important observation we can make from Table \[tab:results-global\] is that once again, the addition of agreement information leads to substantial improvements in accuracy.
“Hard” agreement constraints
----------------------------
Recall that in in our experiments, we created finite-weight agreement links, so that [speech segments]{}appearing in pairs flagged by our (imperfect) agreement detector can potentially receive different labels. We also experimented with [*forcing*]{} such [speech segments]{}to receive the same label, either through infinite-weight agreement links or through a [speech-segment]{}concatenation strategy similar to that described in the previous subsection. Both strategies resulted in clear degradation in performance on both the development and test sets, a finding that validates our encoding of agreement information as “soft” preferences.
On the development/test set split {#sec:eval:devtest}
---------------------------------
We have seen several cases in which the method that performs best on the development set does not yield the best test-set performance. However, we felt that it would be illegitimate to change the train/development/test sets in a post hoc fashion, that is, after seeing the experimental results.
Moreover, and crucially, it is very clear that using agreement information, encoded as preferences within our graph-based approach rather than as hard constraints, yields substantial improvements on both the development and test set; this, we believe, is our most important finding.
Related work
============
#### Politically-oriented text
Sentiment analysis has specifically been proposed as a key enabling technology in eRulemaking, allowing the automatic analysis of the opinions that people submit [@Shulman+al:05a; @Cardie+al:06a; @Kwon+Shulman+Hovy:06a]. There has also been work focused upon determining the political leaning (e.g., “liberal” vs. “conservative”) of a document or author, where most previously-proposed methods make no direct use of relationships between the documents to be classified (the “unlabeled” texts) [@Laver+Benoit+Garry:03a; @Efron:04a; @Mullen+Malouf:06a]. An exception is , who experimented with determining the political orientation of websites essentially by classifying the concatenation of all the documents found on that site.
Others have applied the NLP technologies of near-duplicate detection and topic-based text categorization to politically oriented text [@Yang+Callan:05a; @Purpura+Hillard:06a].
#### Detecting agreement
We used a simple method to learn to identify cross-speaker references indicating agreement. More sophisticated approaches have been proposed [@Hillard+Ostendorf+Shriberg:03a], including an extension that, in an interesting reversal of our problem, makes use of sentiment-polarity indicators within [speech segments]{}[@Galley+McKeown+Hirschberg+Shriberg:04a]. Also relevant is work on the general problems of dialog-act tagging [@Stolcke+al:00a], citation analysis [@Lehnert+Cardie+Riloff:90], and computational rhetorical analysis [@Marcu:00a; @Teufel+Moens:02a].
We currently do not have an efficient means to encode [*disagreement*]{} information as hard constraints; we plan to investigate incorporating such information in future work.
#### Relationships between the unlabeled items
consider sequential relations between different types of emails (e.g., between requests and satisfactions thereof) to classify messages, and thus also explicitly exploit the structure of conversations.
Previous sentiment-analysis work in different domains has considered inter-document similarity [@Agarwal+Bhattacharyya:05a; @Pang+Lee:05a; @Goldberg+Zhu:06a] or explicit inter-document references in the form of hyperlinks [@Agrawal+al:03a].
Notable early papers on graph-based semi-supervised learning include Blum and Chawla , Bansal, Blum, and Chawla , Kondor and Lafferty , and Joachims . Zhu maintains a survey of this area.
Recently, several alternative, often quite sophisticated approaches to [*collective classification*]{} have been proposed [@Neville+Jensen:00a; @Lafferty+McCallum+Pereira:01a; @Getoor+al:02a; @Taskar+al:02a; @Taskar+Guestrin+Koller:03a; @Taskar+Chatalbashev+Koller:04a; @McCallum+Wellner:04a]. It would be interesting to investigate the application of such methods to our problem. However, we also believe that our approach has important advantages, including conceptual simplicity and the fact that it is based on an underlying optimization problem that is provably and in practice easy to solve.
Conclusion and future work {#sec:conc}
==========================
In this study, we focused on very general types of cross-document classification preferences, utilizing constraints based only on speaker identity and on direct textual references between statements. We showed that the integration of even very limited information regarding inter-document relationships can significantly increase the accuracy of support/opposition classification.
The simple constraints modeled in our study, however, represent just a small portion of the rich network of relationships that connect statements and speakers across the political universe and in the wider realm of opinionated social discourse. One intriguing possibility is to take advantage of (readily identifiable) information regarding interpersonal relationships, making use of speaker/author affiliations, positions within a social hierarchy, and so on. Or, we could even attempt to model relationships between topics or concepts, in a kind of extension of collaborative filtering. For example, perhaps we could infer that two speakers sharing a common opinion on evolutionary biologist Richard Dawkins (a.k.a. “Darwin’s rottweiler”) will be likely to agree in a debate centered on Intelligent Design. While such functionality is well beyond the scope of our current study, we are optimistic that we can develop methods to exploit additional types of relationships in future work.
#### Acknowledgments {#acknowledgments .unnumbered}
We thank Claire Cardie, Jon Kleinberg, Michael Macy, Andrew Myers, and the six anonymous EMNLP referees for valuable discussions and comments. We also thank Reviewer 1 for generously providing additional [*post hoc*]{} feedback, and the EMNLP chairs Eric Gaussier and Dan Jurafsky for facilitating the process (as well as for allowing authors an extra proceedings page$\ldots$). This paper is based upon work supported in part by the National Science Foundation under grant no. IIS-0329064 and an Alfred P. Sloan Research Fellowship. Any opinions, findings, and conclusions or recommendations expressed are those of the authors and do not necessarily reflect the views or official policies, either expressed or implied, of any sponsoring institutions, the U.S. government, or any other entity.
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[^1]: It is worth pointing out that the United States’ Library of Congress was an extremely early adopter of Web technology: the THOMAS database (http://thomas.loc.gov) of congressional bills and related data was launched in January 1995, when Mosaic was not quite two years old and Altavista did not yet exist.
[^2]: E.g., “Internet injects sweeping change into U.S. politics”, Adam Nagourney, [*The New York Times*]{}, April 2, 2006.
[^3]: E.g., “The End of News?”, Michael Massing, [*The New York Review of Books*]{}, December 1, 2005.
[^4]: Because we are most interested in techniques applicable across domains, we restrict consideration to NLP aspects of the problem, ignoring external problem-specific information. For example, although most votes in our corpus were almost completely along party lines (and despite the fact that same-party information is easily incorporated via the methods we propose), we did not use party-affiliation data. Indeed, in other settings (e.g., a movie-discussion listserv) one may not be able to determine the participants’ political leanings, and such information may not lead to significantly improved results even if it were available.
[^5]: [${\rm SVM}^{light}$]{}is available at svmlight.joachims.org. Default parameters were used, although experimentation with different parameter settings is an important direction for future work [@Daelemans+Hoste:02a; @Munson+Cardie+Caruana:05a].
[^6]: We are attempting to determine whether a [speech segment]{}represents support or not. This differs from the problem of determining what the speaker’s actual opinion is, a problem that, as an anonymous reviewer put it, is complicated by “grandstanding, backroom deals, or, more innocently, plain change of mind (‘I voted for it before I voted against it’)”.
[^7]: One subtlety is that for the purposes of mining agreement cues (but [*not*]{} for evaluating overall support/oppose classification accuracy), we temporarily re-inserted into our dataset previously filtered [speech segments]{}containing the term “yield”, since the yielding of time on the House floor typically indicates agreement even though the yield statements contain little relevant text on their own.
[^8]: We found good development-set performance using the 30 tokens before, 20 tokens after, and the name itself.
[^9]: Since we are concerned with references that potentially represent relationships between [speech segments]{}, we ignore references for which the target of the reference did not speak in the [debate]{}in which the reference was made.
[^10]: Our implementation puts a link between just one arbitrary pair of [speech segments]{}among all those uttered by a given pair of apparently agreeing speakers. The “infinite-weight” same-speaker links propagate the agreement information to all other such pairs.
[^11]: Unfortunately, this policy leads to inferior [*test-set*]{} agreement classification. Section \[sec:eval:devtest\] contains further discussion.
[^12]: We elected not to explicitly tune the value of ${\theta_{\mbox{agr}}\xspace}$ in order to minimize the number of free parameters to deal with.
|
---
author:
- 'Alastair Fletcher, Vladimir Markovic'
title: Decomposing diffeomorphisms of the sphere
---
\[section\] \[theorem\][Result]{} \[theorem\][Fact]{} \[theorem\][Conjecture]{} \[theorem\][Definition]{} \[theorem\][Lemma]{} \[theorem\][Proposition]{} \[theorem\][Remark]{} \[theorem\][Corollary]{} \[theorem\][Facts]{} \[theorem\][Properties]{}
\[theorem\][Example]{}
Introduction
============
Background
----------
A bi-Lipschitz homeomorphism $f:X \rightarrow Y$ between metric spaces is a mapping $f$ such that $f$ and $f^{-1}$ satisfy a uniform Lipschitz condition, that is, there exists $L \geq 1$ such that $$\frac{ d_{X}(x,y) }{L} \leq d_{Y}(f(x),f(y)) \leq Ld_{X}(x,y)$$ for all $x,y \in X$. The smallest such constant $L$ is called the [*isometric distortion*]{} of $f$. In the metric space setting, a homeomorphism $f:X \rightarrow Y$ is called quasiconformal if there exists a constant $H \geq 1$ such that $$H_{f}(x) : = \limsup _{r \rightarrow 0} \frac
{\sup \{ d_{Y}(f(x),f(y)) : d_{X}(x,y) \leq r \}}
{\inf \{ d_{Y}(f(x),f(y)) : d_{X}(x,y) \geq r \}} \leq H$$ for all $x \in X$. The constant $H$ is called the [*conformal distortion*]{} of $f$. This definition coincides with the perhaps more familiar analytic definition of quasiconformal mappings in ${\mathbb{R}}^{n}$.
Let $S^{n}$ be the sphere of dimension $n$ and denote by $QC(S^{n})$ and $LIP(S^{n})$ the orientation preserving quasiconformal and bi-Lipschitz homeomorphisms, respectively, of $S^{n}$. An old central problem in this area is the following.
Let $f$ be in either $QC(S^{n})$ or $LIP(S^{n})$. Then $f$ can be written as a decomposition $f=f_{m} \circ \ldots \circ f_{1}$ where each $f_{k}$ has small conformal distortion or isometric distortion respectively.
The conjecture is known for the class $QC(S^{2})$ and is essentially a consequence of solving the Beltrami equation in the plane, see for example [@FM]. The quasisymmetric case $QC(S^{1})$ also follows from the dimension $2$ case.
It is well-known that every $L$-bi-Lipschitz homeomorphism between two intervals can be factored into bi-Lipschitz mappings with smaller isometric distortion $\alpha$. Such a factorisation can be written explicitly in the following way. Let $f:I \rightarrow I'$ be an $L$-bi-Lipschitz mapping. Then $f$ can be written as $f=f_{2} \circ f_{1}$, where $$f_{1}(x) = \int_{x_{0}}^{x} {\arrowvert}f'(t) {\arrowvert}^{\lambda}\: dt,$$ $x_{0} \in I$ is fixed, $\lambda = \log _{L} \alpha$, $f_{1}$ is $\alpha$-bi-Lipschitz and $f_{2} = f \circ f_{1}^{-1}$ is $L/\alpha$-bi-Lipschitz. It follows that to factorise an $L$-bi-Lipschitz mapping into $\alpha$-bi-Lipschitz mappings requires $N<\log_{\alpha}L + 1$ factors.
In dimension $2$, Freedman and He [@FH] studied the logarithmic spiral map $s_{k}(z) = z e^{ik \log {\arrowvert}z {\arrowvert}}$, which is an $L$-bi-Lipschitz mapping of the plane where ${\arrowvert}k {\arrowvert}= L - 1/L$. They showed that $s_{k}$ requires $N \geq {\arrowvert}k {\arrowvert}(\alpha^{2}-1)^{-1/2}$ factors to be represented as a composition of $\alpha$-bi-Lipschitz mappings. Gutlyanskii and Martio [@GM] studied a related class of mappings in dimension $2$, and generalized this to a class of volume preserving bi-Lipschitz automorphisms of the unit ball ${\mathbb{B}}^{3}$ in $3$ dimensions. Beyond these particular examples, however, very little is known about factorising bi-Lipschitz mappings in dimension $2$ and higher, and factorizing quasiconformal maps in dimension $3$ and higher.
A natural question to ask is whether diffeomorphisms of the sphere $S^{n}$ can be decomposed into diffeomorphisms that are $C^{1}$ close to the identity. The answer in general is negative as the exotic spheres of Milnor [@Milnor] provide an obstruction. In [@Milnor], it is shown that there exist topological $7$-spheres which are not diffeomorphic to the standard $7$-sphere $S^{7}$. In particular, one cannot in general find a $C^{1}$ path from the identity on $S^{6}$ to a given $C^{1}$ diffeomorphism.
There are two facts that might be obstructions to the factorisation theorem. One is the Milnor example. The second fact is that not all topological manifolds of dimension at least $5$ admit differentiable structures. On the other hand, a deep result of Sullivan [@Sullivan] states that they always admit a bi-Lipschitz structure. The recent results of Bonk, Heinonen and Wu [@Wu] which state that closed bi-Lipschitz manifolds where the transition maps have small enough distortion admit a $C^{1}$ structure, raises the question of whether a factorisation theorem in this case would contradict Sullivan’s theorem.
Main results
------------
Since some $C^{1}$ diffeomorphisms of $S^{n}$ cannot be decomposed into $C^{1}$ diffeomorphisms with derivative close to the identity, that suggests the question of trying to factor them into bi-Lipschitz mappings of small isometric distortion.
The main result of this paper states that one can find a path connecting the identity and any $C^{1}$ diffeomorphism of $S^{n}$ which is a composition of bi-Lipschitz paths, a notion that will be made more precise in §2.
\[mainthm\] Let $f:S^{n} \rightarrow S^{n}$ be a $C^{1}$ diffeomorphism. Then there exist bi-Lipschitz paths $A_{t},p^{1}_{t},p^{2}_{t}:S^{n} \rightarrow S^{n}$ for $t \in [0,1]$ such that $A_{0},p^{1}_{0}$ and $p^{2}_{0}$ are all the identity, and $A_{1} \circ p^{2}_{1} \circ p^{1}_{1} = f$.
It is not a priori true that a composition of bi-Lipschitz paths is another bi-Lipschitz path since issues arise at points of non-differentiability.
As a corollary to this theorem, we find that $C^{1}$ diffeomorphisms of the sphere $S^{n}$ can be decomposed into bi-Lipschitz mappings of arbitrarily small isometric distortion.
\[maincor\] Let $f:S^{n} \rightarrow S^{n}$ be a $C^{1}$ diffeomorphism. Given $\epsilon >0$, there exists $m \in {\mathbb{N}}$, depending on $f$, such that $f$ decomposes as $f=f_{m} \circ \ldots \circ f_{1}$, where $f_{k}$ is $(1+\epsilon)$-bi-Lipschitz with respect to the spherical metric $\chi$, and $\chi(f_{k}(x),x) < \epsilon$ for all $x \in S^{n}$ and for $k=1,\ldots,m$.
In §2, we will state several intermediate lemmas and prove Theorem \[mainthm\] and Corollary \[maincor\] assuming these lemmas hold. The proofs of the lemmas are postponed to §3.
Outline of proof
================
Some notation
-------------
We will first fix some notation. Let $S^{n} = {\mathbb{R}}^{n} \cup \{ \infty \}$ be the sphere of dimension $n$. Denote by $d$ the Euclidean metric on ${\mathbb{R}}^{n}$ and by $\chi$ the spherical metric on $S^{n}$, so that $$d(x,y) = {\arrowvert}x-y {\arrowvert},$$ for $x,y \in {\mathbb{R}}^{n}$ and $$\chi(x,y) = \frac{{\arrowvert}x-y {\arrowvert}}{\sqrt{1+{\arrowvert}x {\arrowvert}^{2}} \sqrt{1+{\arrowvert}y {\arrowvert}^{2}}}$$ for $x,y \in S^{n} \setminus \{ \infty \}$. If $y$ is the point at infinity, $$\chi(x, \infty) = \frac{1}{\sqrt{1+ {\arrowvert}x {\arrowvert}^{2}}}.$$ Let $B_{d}(x,r) = \{ y \in {\mathbb{R}}^{n}: d(x,y)\leq r\} $ and $B_{\chi}(x,r)=\{ y \in S^{n} : \chi (x,y)\leq r\}$ be the closed balls centred at $x$ of respectively Euclidean and spherical radius $r$. We say that a diffeomorphism $f$ is supported on a set $U \subset S^{n}$ if $f$ is the identity on the complement $S^{n}\setminus U$.
Diffeomorphisms supported on balls
----------------------------------
We first need to show that a $C^{1}$ diffeomorphism with a fixed point can be written as a composition of $C^{1}$ diffeomorphisms supported on spherical balls.
\[lemma1\] Let $f:S^{n} \rightarrow S^{n}$ be a $C^{1}$ diffeomorphism with at least one fixed point. Then there exist $x_{1},x_{2} \in S^{n}$ and $r_{1},r_{2}>0$ such that $f$ decomposes as $f=f^{2} \circ f^{1}$ where $f^{1},f^{2}$ are $C^{1}$ diffeomorphisms supported on spherical balls $B_{1}=B_{\chi}(x_{1},r_{1}),B_{2} = B_{\chi}(x_{2},r_{2})$ in $S^{n}$, and so that neither $B_{1}$ nor $B_{2}$ are $S^{n}$.
To prove the lemma, we will need to make use of the following result of Munkres [@Munkres Lemma 8.1] as formulated in [@Wilson].
\[thm1\] Let $h:{\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$ be an orientation preserving $C^{k}$ diffeomorphism for $1 \leq k \leq \infty$. Then there exists a $C^{k}$ diffeomorphism $\widetilde{h} :{\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$ which coincides with the identity near $0 \in {\mathbb{R}}^{n}$ and $h$ near infinity.
Suppose that $f:S^{n} \rightarrow S^{n}$ is a $C^{1}$ diffeomorphism with a fixed point in $S^{n}$. Identifying $S^{n}$ with $\overline{{\mathbb{R}}^{n}}$, without loss of generality we can assume $f$ fixes the point at infinity. Then by Theorem \[thm1\], there exists a $C^{1}$ diffeomorphism $\widetilde{f}$ and real numbers $r_{1},r_{2}>0$ such that $\widetilde{f} {\arrowvert}_{ B_{\chi}(0,r_{1}) }$ is the identity and $\widetilde{f} {\arrowvert}_{ B_{\chi}(\infty, r_{2})}$ is equal to $f$. We can then write $$f = \left ( f \circ \widetilde{f}^{-1} \right ) \circ \widetilde{f}$$ where $f^{2}:= f \circ \widetilde{f}^{-1}$ is supported on the ball $S^{n} \setminus B_{\chi}(\infty, r_{2}) $ and $f^{1}:=\widetilde{f}$ is supported on the ball $S^{n} \setminus B_{\chi}(0,r_{1})$.
Bi-Lipschitz paths
------------------
We shall postpone the proofs of the lemmas in this section until §3. Let us now define the notion of a bi-Lipschitz path.
\[bilippath\] Let $(X,d_{X})$ be a metric space. A path $h:[0,1] \rightarrow LIP(X)$ is called a *bi-Lipschitz path* if for every $\epsilon >0$, there exists $\delta >0$ such that if $s,t, \in [0,1]$ with ${\arrowvert}s-t {\arrowvert}< \delta$, the following two conditions hold:
1. for all $x \in X$, $d_{X}(h_{s} \circ h_{t}^{-1}(x),x) <\epsilon$;
2. we have that $h_{s} \circ h_{t}^{-1}$ is $(1+\epsilon)$-bi-Lipschitz with respect to $d_{X}$.
We need the following lemmas on bi-Lipschitz paths.
\[lemma10\] Let $h_{t}:[0,1] \rightarrow LIP({\mathbb{R}}^{n})$ be a bi-Lipschitz path with respect to $d$. Then $h_{t}:[0,1] \rightarrow LIP(S^{n})$ is a bi-Lipschitz path with respect to $\chi$.
\[lemma11\] Let $h_{t}:[0,1] \rightarrow LIP({\mathbb{R}}^{n})$ be a bi-Lipschitz path with respect to $d$ and let $g:S^{n} \rightarrow S^{n}$ be a Möbius transformation. Then the path $g \circ h_{t} \circ g^{-1}$ is bi-Lipschitz with respect to $\chi$ on $S^{n}$.
It can be shown that a bi-Lipschitz path $h_{t}:[0,1] \rightarrow LIP(M)$ on a closed manifold $M$ remains bi-Lipschitz after conjugation by a conformal map $g:M \rightarrow M$. The condition that $g$ is conformal cannot be weakened to $g$ being a diffeomorphism.
The following lemma is the main step in the proof of Theorem \[mainthm\].
\[thm2\] Let $f:{\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$ be a $C^{1}$ diffeomorphism supported in $B_{d}(0,1/3)$. Then there exists a path $h_{t}:[0,1] \rightarrow LIP({\mathbb{R}}^{n})$ which is bi-Lipschitz with respect to $d$, connecting the identity $h_{0}$ and $h_{1}=f$.
Proofs of the main results
--------------------------
Assuming the intermediate results above, the proof of Theorem \[mainthm\] proceeds as follows.
Let $f:S^{n} \rightarrow S^{n}$ be a $C^{1}$ diffeomorphism. There exists $A \in SO(n)$ such that $A \circ f$ has a fixed point in $S^{n}$. Note that if $n$ is even, then $f$ automatically has a fixed point and we can take $A$ to be the identity.
By Lemma \[lemma1\], we can write $A \circ f=f^{2}\circ f^{1}$ where $f^{i}$ is supported on the spherical ball $B_{i}$ for $i=1,2$. By standard spherical geometry, see e.g. [@V], for $i=1,2$, there exist Möbius transformations $g_{i}$ such that $g_{i}^{-1} \circ f^{i} \circ g_{i}$ is supported on $B_{d}(0,1/3)$.
Now, applying Lemma \[thm2\] to $g_{i}^{-1} \circ f^{i} \circ g_{i}$, we obtain two bi-Lipschitz paths $h_{t}^{i}$, for $i=1,2$, with respect to $d$ on ${\mathbb{R}}^{n}$. Consider the paths $$p_{t}^{i} = g_{i} \circ h_{t}^{i} \circ g_{i}^{-1}$$ for $i=1,2$, where $p_{0}^{i}$ is the identity and $p_{1}^{i} = f^{i}$.
It follows by Lemma \[lemma11\] that $p_{t}^{i}$ is bi-Lipschitz with respect to $\chi$ on $S^{n}$. Then $p_{t}^{2} \circ p_{t}^{1}$ is a composition of bi-Lipschitz paths, with respect to $\chi$, connecting the identity and $A \circ f$. Since $A^{-1} \in SO(n)$, there is a bi-Lipschitz path $A_{t}$ connecting the identity $A_{0}$ and $A_{1} = A^{-1}$. We conclude that $A_{t} \circ p_{t}^{2} \circ p_{t}^{1}$ is a composition of three bi-Lipschitz paths, which connects the identity and $f$. This completes the proof.
Let $\epsilon >0$. By Theorem \[mainthm\], $A_{t},p^{1}_{t}$ and $p^{2}_{t}$ are all bi-Lipschitz paths with respect to $\chi$ on $S^{n}$, $A_{0} \circ p_{0}^{2} \circ p_{0}^{1}$ is the identity and $A_{1} \circ p_{1}^{2} \circ p_{1}^{1}=f$.
Given a bi-Lipschitz path $h_{t}$, we can choose $0=t_{1}<t_{2}<\ldots < t_{j+1}=1$ such that $g_{k} = h_{k+1} \circ h_{k}^{-1}$ is $(1+\epsilon)$-bi-Lipschitz for $k=1,\ldots, j$ and $h_{1}= g_{j} \circ \ldots \circ g_{1}$. Applying this observation to the bi-Lipschitz paths $A_{t},p^{1}_{t}$ and $p^{2}_{t}$, there exists $j(1),j(2),j(3) \in{\mathbb{N}}$ such that $$\begin{aligned}
A_{1} &= A_{1,j(1)} \circ A_{1,j(1)-1} \circ \ldots \circ A_{1,1},\\
p_{1}^{1} &= p_{1,j(2)}^{1} \circ p_{1,j(2)-1}^{1} \circ \ldots \circ p_{1,1}^{1},\\
p_{1}^{2} &= p_{1,j(3)}^{2} \circ p_{1,j(3)-1}^{2} \circ \ldots \circ p_{1,1}^{2},\end{aligned}$$ and each map in these three decompositions is $(1+\epsilon)$-bi-Lipschitz with respect to $\chi$, and also only moves points in $S^{n}$ by at most spherical distance $\epsilon$. In view of $A_{1} \circ p_{1}^{2} \circ p_{1}^{1}=f$, this proves the theorem with $m=j(1)+j(2)+j(3)$.
Proofs of the Lemmas
====================
We will prove Lemma \[lemma10\] and Lemma \[lemma11\] first, before proving the main Lemma \[thm2\].
Proof of Lemma \[lemma10\]
--------------------------
Let $h_{t}:{\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$ be a bi-Lipschitz path with respect to $d$. Then each $h_{t}$ extends to a mapping $S^{n} \rightarrow S^{n}$ which fixes the point at infinity. Let $s,t \in [0,1]$ and consider the mapping $g= h_{s} \circ h_{t}^{-1}$. Since $h_{t}$ is a bi-Lipschitz path, choose $\delta>0$ small enough so that if ${\arrowvert}s-t {\arrowvert}< \delta$ then $d(g(x),x) < \epsilon$ for all $x \in {\mathbb{R}}^{n}$ and $g$ is $(1+\epsilon)$-bi-Lipschitz with respect to $d$.
Property (i) of Definition \[bilippath\] is satisfied for $\chi$ since $\chi(g(x),x) \leq d(g(x),x)$, for $x \in {\mathbb{R}}^{n}$, and $g$ fixes the point at infinity.
We now show that $h_{t}$ satisfies property (ii) of Definition \[bilippath\]. The fact that $h_{t}$ is a bi-Lipschitz path with respect to $d$ and the formula for the spherical distance give $$\begin{aligned}
\label{l2eq1}
\chi(g(x),g(y)) &= \frac{ {\arrowvert}g(x) - g(y) {\arrowvert}}{\sqrt{1+{\arrowvert}g(x) {\arrowvert}^{2}}\sqrt{1+ {\arrowvert}g(y) {\arrowvert}^{2}}}
\notag \\ &\leq \frac{ (1+\epsilon) {\arrowvert}x- y {\arrowvert}}{\sqrt{1+{\arrowvert}g(x) {\arrowvert}^{2}}\sqrt{1+ {\arrowvert}g(y) {\arrowvert}^{2}}}
\notag \\ &= (1+\epsilon)\chi(x,y) \left ( \frac{ 1+ {\arrowvert}x {\arrowvert}^{2}}{1 + {\arrowvert}g(x) {\arrowvert}^{2}} \right )^{1/2}
\left ( \frac{ 1+ {\arrowvert}y {\arrowvert}^{2}}{1 + {\arrowvert}g(y) {\arrowvert}^{2}} \right )^{1/2},\end{aligned}$$ for $x,y \in {\mathbb{R}}^{n}$. Since $d(g(x),x) < \epsilon$, it follows that $$\frac{1+{\arrowvert}x {\arrowvert}^{2}}{1+ ({\arrowvert}x {\arrowvert}+\epsilon ) ^{2}} \leq
\frac{1+{\arrowvert}x {\arrowvert}^{2}}{1+{\arrowvert}g(x) {\arrowvert}^{2}} \leq \frac{1+{\arrowvert}x {\arrowvert}^{2}}{1+ ({\arrowvert}x {\arrowvert}-\epsilon ) ^{2}}.$$ Therefore, $$\left ( 1 + \frac{\epsilon(\epsilon + 2 {\arrowvert}x {\arrowvert})}{1+{\arrowvert}x {\arrowvert}^{2}} \right)^{-1}
\leq \frac{ 1+ {\arrowvert}x {\arrowvert}^{2}}{1 + {\arrowvert}g(x) {\arrowvert}^{2}}
\leq \left ( 1 + \frac{\epsilon(\epsilon - 2 {\arrowvert}x {\arrowvert})}{1+{\arrowvert}x {\arrowvert}^{2}} \right)^{-1}$$ and so it follows that given $\epsilon>0$, we can choose $\epsilon'$ small enough so that $$\label{l2eq2}
\frac{1}{1+\epsilon'} \leq \frac{1+{\arrowvert}x {\arrowvert}^{2}}{1+ {\arrowvert}g(x) {\arrowvert}^{2}} \leq 1+\epsilon'$$ for all $x \in {\mathbb{R}}^{n}$. By (\[l2eq1\]) and (\[l2eq2\]), it follows that $$\label{l2eq4}
\chi(g(x),g(y)) \leq (1+\epsilon)(1+\epsilon')\chi(x,y),$$ for all $x,y \in {\mathbb{R}}^{n}$. We can conclude that given $\epsilon >0$, we can choose $\xi >0$ small enough so that $$\label{l2eq3}
\chi(g(x),g(y)) \leq (1+\xi) \chi(x,y)$$ for all $x,y \in {\mathbb{R}}^{n}$. The reverse inequality follows by applying (\[l2eq3\]) to $g^{-1}$. Therefore condition (ii) of Definition \[bilippath\] holds for $x,y \in {\mathbb{R}}^{n}$ with $\delta$, and $\xi$ playing the role of $\epsilon$.
Finally, if $x \in {\mathbb{R}}^{n}$ and $y= \infty$, then $$\chi(g(x),\infty) = \frac{1}{\sqrt{1+{\arrowvert}g(x) {\arrowvert}^{2}}} = \chi(x,\infty) \left (\frac{1+{\arrowvert}x {\arrowvert}^{2}}{1+{\arrowvert}g(x) {\arrowvert}^{2}}\right) ^{1/2}$$ and we then apply (\[l2eq2\]) as above. This completes the proof of Lemma \[lemma10\].
Proof of Lemma \[lemma11\]
--------------------------
Recall that $h_{t}$ is a bi-Lipschitz path with respect to $d$ on ${\mathbb{R}}^{n}$ and that $g:S^{n} \rightarrow S^{n}$ is a Möbius transformation. We can write $$g = C \circ B,$$ where $B:{\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$ is an affine map and $C$ is a spherical isometry. To see this, let $x\in S^n$ be the point such that $g(\infty)=x$. Then there exists a (non-unique) spherical isometry $C$ such that $C(\infty)=x$ and then the map $B=C^{-1} \circ g$ is affine.
We first show that $B \circ h_{t} \circ B^{-1}$ is a bi-Lipschitz path with respect to $d$ on ${\mathbb{R}}^{n}$. Since $B:{\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$ is an affine map, there is a real number $\alpha >0$ such that $$d(B(x),B(y)) = \alpha d(x,y),$$ for all $x,y \in {\mathbb{R}}^{n}$. Since $h_{t}$ is a bi-Lipschitz path with respect to $d$, write $f = h_{s} \circ h_{t}^{-1}$, with ${\arrowvert}s-t {\arrowvert}< \delta$ small enough so that $d(f(x),x) < \epsilon$ and $f$ is $(1+\epsilon)$-bi-Lipschitz with respect to $d$. Then $$\begin{aligned}
d(B(f(B^{-1}(x))),x) &= d( B(f(B^{-1}(x))),B(B^{-1}(x)) \\
&\leq \alpha d(f(B^{-1}(x)),B^{-1}(x)) \\
&< \alpha \epsilon,\end{aligned}$$ for all $x \in {\mathbb{R}}^{n}$. Therefore $B \circ h_{t} \circ B^{-1}$ satisfies condition (i) of Definition \[bilippath\] with $\delta$ and $\alpha \epsilon$. Next, $$\begin{aligned}
d(B(f(B^{-1}(x))),B(f(B^{-1}(y)))) &= \alpha d(f(B^{-1}(x)),f(B^{-1}(y))) \\
&\leq \alpha (1+\epsilon) d(B^{-1}(x),B^{-1}(y)) \\
&= (1+\epsilon) d(x,y)\end{aligned}$$ and so $B \circ h_{t} \circ B^{-1}$ satisfies condition (ii) of Definition \[bilippath\] with $\delta$ and $\epsilon$.
By Lemma \[lemma10\], $B \circ h_{t} \circ B^{-1}$ is also bi-Lipschitz with respect to $\chi$ on $S^{n}$. It remains to show that $C \circ B \circ h_{t} \circ B^{-1} \circ C^{-1}= g \circ h_{t} \circ g^{-1}$ is a bi-Lipschitz path with respect to $\chi$ on $S^{n}$.
Since $B \circ h_{t} \circ B^{-1}$ is a bi-Lipschitz path with respect to $\chi$, write $f = B \circ h_{s} \circ h_{t}^{-1} \circ B^{-1}$, with ${\arrowvert}s-t {\arrowvert}< \delta$ small enough so that $\chi(f(x),x) \leq \epsilon$ and $f$ is $(1+\epsilon)$-bi-Lipschitz with respect to $\chi$. Then $$\begin{aligned}
\chi (C(f(C^{-1}(x))),x) &= \chi ( C(f(C^{-1}(x))),C(C^{-1}(x)))\\
&= \chi (f(C^{-1}(x)),C^{-1}(x)) \\
&< \epsilon,\end{aligned}$$ for all $x \in S^{n}$. Therefore $C \circ B \circ h_{t} \circ B^{-1} \circ C^{-1}$ satisfies condition (i) of Definition \[bilippath\] with $\delta$ and $\epsilon$. Next, $$\begin{aligned}
\chi ( C(f(C^{-1}(x))), C(f(C^{-1}(y))) ) &= \chi (f(C^{-1}(x)), f(C^{-1}(y)) )\\
&\leq (1+\epsilon) \chi ( C^{-1}(x),C^{-1}(y))\\
&= (1+\epsilon)\chi (x,y),\end{aligned}$$ and so $C \circ B \circ h_{t} \circ B^{-1} \circ C^{-1}$ satisfies condition (ii) of Definition \[bilippath\] with $\delta$ and $\epsilon$. This completes the proof.
Proof of Lemma \[thm2\]
-----------------------
We first set some notation. If $g:{\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$ is differentiable at $x \in {\mathbb{R}}^{n}$, write $D_{x}g$ for the derivative of $g$ at $x$ and let $${\arrowvert}{\arrowvert}D_{x}g {\arrowvert}{\arrowvert}= \max_{y \in {\mathbb{R}}^{n} \setminus \{0 \} } \frac{ {\arrowvert}(D_{x}g)(y) {\arrowvert}}{{\arrowvert}y {\arrowvert}}$$ be the operator norm of the linear map $D_{x}g$. Note that we are regarding the derivative here as a mapping from ${\mathbb{R}}^{n}$ to ${\mathbb{R}}^{n}$ given by the matrix of partial derivatives $\partial g_{i}/ \partial x_{j}$, and not as a mapping between tangent spaces.
Recall that $f:{\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$ is a $C^{1}$ diffeomorphism supported on the ball $B_{0}:=B_{d}(0,1/3)$. Write $A_{t}:{\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$ for the translation $A_{t}(x_{1},x_{2},\ldots,x_{n}) = (x_{1}+t,x_{2},\ldots,x_{n})$ and define $B_{t} = A_{t}(B_{0})$. Write $e_{1}=(1,0,\ldots,0)$.
Define $g:{\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$ by $$g(x) = \left\{ \begin{array}{cl}
(A_{m} \circ f \circ A_{m}^{-1})(x) &\mbox{ if $x \in B_{m}$, \:\:\: $m \in {\mathbb{N}}$,} \\
x &\mbox{ otherwise.}
\end{array} \right.$$ Then $g$ is a propagated version of $f$, supported in $\cup _{m=1}^{\infty} B_{m}$. We can extend $g$ to a mapping on $S^{n}$ by defining $g$ to fix the point at infinity.
\[lemma3\] The map $g$ is $C^{1}$ on ${\mathbb{R}}^{n}$ and, further, satisfies the following properties:
1. $g$ is uniformly continuous on ${\mathbb{R}}^{n}$, that is, for all $\epsilon >0$, there exists $\delta >0$ such that for all $x,y \in {\mathbb{R}}^{n}$ satisfying ${\arrowvert}x-y {\arrowvert}< \delta$, we have ${\arrowvert}g(x)-g(y) {\arrowvert}<\epsilon$;
2. there exists $T>0$ such that $$\label{l3eq1}
{\arrowvert}{\arrowvert}D_{x}g{\arrowvert}{\arrowvert}\leq T$$ for all $x \in {\mathbb{R}}^{n}$;
3. there exists a function $\eta :[0,\infty]\rightarrow [0,\infty]$ for which $\eta(0)=0$, $\eta$ is continuous at $0$ and $$\label{l3eq2}
{\arrowvert}{\arrowvert}D_{x}g - D_{y}g {\arrowvert}{\arrowvert}\leq \eta ({\arrowvert}x-y{\arrowvert})$$ for all $x,y \in {\mathbb{R}}^{n}$. The function $\eta$ is the modulus of continuity of $Dg$.
Further, we may assume that $g^{-1}$ also satisfies these three conditions, by changing the constants and modulus of continuity if necessary.
First note that $f$ is $C^{1}$ by hypothesis, and satisfies the three claims of the lemma because it is supported in a compact subset of ${\mathbb{R}}^{n}$. Since $g$ is a propagated version of $f$, it satisfies the three claims of the lemma with the same constants as $f$. The last claim follows since $f^{-1}$ is also $C^{1}$, and $g^{-1}$ is a propagated version of $f^{-1}$.
For $t \in [0,1]$, let $$h_{t} = g^{-1} \circ A_{t}^{-1} \circ g \circ A_{t}.$$
By Lemma \[lemma3\] and [@V Lemma 1.54], which says that Euclidean translations in ${\mathbb{R}}^{n}$ are bi-Lipschitz with respect to $\chi$, $h_{t}$ is bi-Lipschitz with respect to both $d$ and $\chi$. The following lemma is elementary.
\[lemma9\] We have that $h_{0}$ is equal to the identity and $h_{1} =f$.
Observe that $h_{t}$ is a path that connects the identity and $f$ through bi-Lipschitz mappings, for $0 \leq t \leq 1$. We now want to show that this is a bi-Lipschitz path.
\[lemma6\] Given $\epsilon >0$, there exists $\delta>0$ such that if $s,t \in [0,1]$ satisfy ${\arrowvert}s-t {\arrowvert}<\delta$, then $$d( h_{s} \circ h_{t}^{-1}(x), x) \leq \epsilon,$$ for all $x \in {\mathbb{R}}^{n}$.
Writing $h_{s}\circ h_{t}^{-1}$ out in full gives $$\label{l6eq1}
h_{s} \circ h_{t}^{-1} = g^{-1} \circ A_{s}^{-1} \circ g \circ A_{s} \circ A_{t}^{-1} \circ g^{-1} \circ A_{t} \circ g.$$ Considering first the middle four functions in this expression, write $$\label{l6eq2}
P_{s,t}(x) = g \circ A_{s} \circ A_{t}^{-1} \circ g^{-1}(x).$$ Then the fact that $$d(g(x),g(y)) \leq \sup_{x} {\arrowvert}{\arrowvert}D_{x}g {\arrowvert}{\arrowvert}\cdot d(x,y),$$ and (\[l3eq1\]) gives $$\begin{aligned}
d( P_{s,t}(x), x ) &=
d( g(g^{-1}(x) + (s-t)e_{1}) , g(g^{-1}(x)) ) \\
&\leq T d( g^{-1}(x) + (s-t)e_{1} , g^{-1}(x) ) \\
&= T {\arrowvert}s-t {\arrowvert},\end{aligned}$$ for all $x \in {\mathbb{R}}^{n}$. Next, by using the the fact that translations are isometries of ${\mathbb{R}}^{n}$, the triangle inequality and the previous inequality applied to $x+te_{1}$, we obtain $$\begin{aligned}
\label{l3eq3}
\notag d( A_{s}^{-1} \circ P_{s,t} \circ A_{t}(x) , x) &= d( P_{s,t}(x+te_{1}) - se_{1},x) \\ \notag
&= d( P_{s,t}(x+te_{1}) , (x+te_{1}) + (s-t)e_{1} )\\ \notag
&\leq d( P_{s,t}(x+te_{1}) , (x+te_{1})) + d(x+te_{1},x+te_{1} + (s-t)e_{1}) \\
&\leq (T+1){\arrowvert}s-t {\arrowvert},\end{aligned}$$ for all $x \in {\mathbb{R}}^{n}$. Finally, we use (\[l3eq1\]) with $g^{-1}$ and (\[l3eq3\]) applied to $g(x)$ to obtain $$\begin{aligned}
d( h_{s} \circ h_{t}^{-1}(x),x ) &= d(g^{-1} \circ A_{s}^{-1} \circ P_{s,t} \circ A_{t} \circ g(x) , g^{-1}(g(x))) \\
&\leq T d (A_{s}^{-1} \circ P_{s,t} \circ A_{t} \circ g(x) , g(x) ) \\
&\leq T(T+1){\arrowvert}s-t {\arrowvert},\end{aligned}$$ for all $x \in {\mathbb{R}}^{n}$. We can therefore take $\delta = \epsilon / T(T+1)$.
\[lemma7\] Given $\epsilon >0$, there exists $\delta>0$ such that if $s,t \in [0,1]$ satisfy ${\arrowvert}s-t {\arrowvert}<\delta$, then $${\arrowvert}{\arrowvert}D_{x}(h_{s} \circ h_{t}^{-1}) - I {\arrowvert}{\arrowvert}< \epsilon$$ for all $x \in {\mathbb{R}}^{n}$, where $I$ is the identity mapping.
Recalling the strategy of the proof of the previous lemma, we will consider the middle six terms of (\[l6eq1\]) and work outwards. Recall the definition of $P_{s,t}$ from (\[l6eq2\]) and write $Q_{s,t}=A_{s}^{-1} \circ P_{s,t} \circ A_{t}$. Observe that $$D_{x}Q_{s,t} = D_{A_{t}(x)}P_{s,t}$$ and $$D_{x}P_{s,t} = D_{A_{s} \circ A_{t}^{-1} \circ g^{-1}(x)} g \circ D_{x}g^{-1}$$ since the derivative of $A_{t}$ is the identity. By this observation, the chain rule gives $$\label{l3eq4}
{\arrowvert}{\arrowvert}D_{x}(Q_{s,t}) - I {\arrowvert}{\arrowvert}= {\arrowvert}{\arrowvert}(D_{A_{s}\circ A_{t}^{-1}\circ g^{-1} \circ A_{t}(x)}g) \circ (D_{A_{t}(x)}g^{-1}) - I {\arrowvert}{\arrowvert}.$$ We can write the right hand side of (\[l3eq4\]) as $${\arrowvert}{\arrowvert}\left [ (D_{A_{s}\circ A_{t}^{-1}\circ g^{-1} \circ A_{t}(x)}g) - \left( (D_{A_{t}(x)}g^{-1}) \right )^{-1} \right ] \circ (D_{A_{t}(x)}g^{-1}) {\arrowvert}{\arrowvert}.$$ Using this, and applying the formula for the derivative of an inverse $(D_{A_{t}(x)}g^{-1})^{-1} = D_{g^{-1}(A_{t}(x))}g$ and (\[l3eq1\]) applied to $g^{-1}$, yields from (\[l3eq4\]) that $$\label{l3eq5}
{\arrowvert}{\arrowvert}D_{x}(Q_{s,t}) - I {\arrowvert}{\arrowvert}\leq
T {\arrowvert}{\arrowvert}(D_{A_{s}\circ A_{t}^{-1}\circ g^{-1} \circ A_{t}(x)}g) - (D_{g^{-1}\circ A_{t}(x)}g) {\arrowvert}{\arrowvert}.$$ We then apply (\[l3eq2\]) to the right hand side of (\[l3eq5\]) to give $$\begin{aligned}
\label{l3eq6}
{\arrowvert}{\arrowvert}D_{x}(Q_{s,t}) - I {\arrowvert}{\arrowvert}&\leq T \eta ( {\arrowvert}A_{s} \circ A_{t}^{-1} \circ g^{-1} \circ A_{t}(x) -g^{-1} \circ A_{t}(x) {\arrowvert})\\
\notag &= T \eta({\arrowvert}s-t{\arrowvert}),\end{aligned}$$ for all $x \in {\mathbb{R}}^{n}$.
Now, consider the derivative of $h_{s} \circ h_{t}^{-1} = g^{-1} \circ Q_{s,t} \circ g$. By the chain rule, we have $$\label{l3eq7}
{\arrowvert}{\arrowvert}D_{x}(g^{-1} \circ Q_{s,t} \circ g) - I {\arrowvert}{\arrowvert}= {\arrowvert}{\arrowvert}(D_{Q_{s,t}(g(x))}g^{-1}) \circ (D_{g(x)}Q_{s,t}) \circ (D_{x}g) - I {\arrowvert}{\arrowvert}.$$ We can write the right hand side of (\[l3eq7\]) as $${\arrowvert}{\arrowvert}(D_{Q_{s,t}(g(x))}g^{-1}) \circ \left [ D_{g(x)}Q_{s,t} - I \right ] \circ (D_{x}g)
+ (D_{Q_{s,t}(g(x))}g^{-1}) \circ (D_{x}g) - I {\arrowvert}{\arrowvert}.$$ Applying the triangle inequality and (\[l3eq1\]) for $g$ and $g^{-1}$ to this expression yields $$\label{l3eq8}
{\arrowvert}{\arrowvert}D_{x}(g^{-1} \circ Q_{s,t} \circ g) - I {\arrowvert}{\arrowvert}\leq T^{2} {\arrowvert}{\arrowvert}D_{g(x)}Q_{s,t} - I {\arrowvert}{\arrowvert}+ {\arrowvert}{\arrowvert}(D_{Q_{s,t}(g(x))}g^{-1}) \circ (D_{x}g) - I {\arrowvert}{\arrowvert}$$ We next apply (\[l3eq6\]) to the first term on the right hand side of (\[l3eq8\]), and re-write the second term to give $$\label{l3eq9}
{\arrowvert}{\arrowvert}D_{x}(g^{-1} \circ Q_{s,t} \circ g) - I {\arrowvert}{\arrowvert}\leq T^{3} \eta({\arrowvert}s-t {\arrowvert}) + {\arrowvert}{\arrowvert}\left [ D_{Q_{s,t}(g(x))}g^{-1} - (D_{x}g)^{-1} \right ] \circ (D_{x}g) {\arrowvert}{\arrowvert}$$ We use the formula $(D_{x}g)^{-1} = D_{g(x)}g^{-1}$ and (\[l3eq1\]) applied to $g$ on the second term on the right hand side of (\[l3eq9\]) to yield $${\arrowvert}{\arrowvert}D_{x}(g^{-1} \circ Q_{s,t} \circ g) - I {\arrowvert}{\arrowvert}\leq T^{3} \eta({\arrowvert}s-t {\arrowvert}) + T {\arrowvert}{\arrowvert}D_{Q_{s,t}(g(x))}g^{-1} - D_{g(x)}g^{-1} {\arrowvert}{\arrowvert}$$ Finally, (\[l3eq2\]) and (\[l3eq3\]) give $$\begin{aligned}
{\arrowvert}{\arrowvert}D_{x}(g^{-1} \circ Q_{s,t} \circ g) - I {\arrowvert}{\arrowvert}& \leq T^{3} \eta({\arrowvert}s-t {\arrowvert}) + T\eta( {\arrowvert}Q_{s,t}(g(x)) - g(x) {\arrowvert})\\
&\leq T^{3} \eta({\arrowvert}s-t {\arrowvert})+ T \eta ( (T+1) {\arrowvert}s-t {\arrowvert}).\end{aligned}$$ Since $\lim _{x \rightarrow 0} \eta(x)=0$, the lemma follows.
Lemmas \[lemma9\], \[lemma6\] and \[lemma7\] together show that $h_{t}$ is a bi-Lipschitz path with respect to $d$ connecting the identity and $f$. This completes the proof.
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abstract: |
This paper continues a study initiated in [@cf:GL], on the localization transition of a lattice free field on ${{\ensuremath{\mathbb Z}} }^d$ interacting with a quenched disordered substrate that acts on the interface when its height is close to zero. The substrate has the tendency to localize or repel the interface at different sites. A transition takes place when the average pinning potential $h$ goes past a threshold $h_c$: from a delocalized phase $h<h_c$, where the field is macroscopically repelled by the substrate to a localized one $h>h_c$ where the field sticks to the substrate. Our goal is to investigate the effect of the presence of disorder on this phase transition. We focus on the two dimensional case $(d=2)$ for which we had obtained so far only limited results. We prove that the value of $h_c({\beta})$ is the same as for the annealed model, for all values of ${\beta}$ and that in a neighborhood of $h_c$. Moreover we prove that, in contrast with the case $d\ge 3$ where the free energy has a quadratic behavior near the critical point, the phase transition is of infinite order $$\lim_{u\to 0+} \frac{ \log {\textsc{f}}({\beta},h_c({\beta})+u)}{(\log u)}= \infty.$$\
2010 *Mathematics Subject Classification: 60K35, 60K37, 82B27, 82B44*\
*Keywords: Lattice Gaussian Free Field, Disordered Pinning Model, Localization Transition, Critical Behavior, Disorder Relevance, Co-membrane Model*
address: 'IMPA - Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro / Brasil 22460-320'
author:
- Hubert Lacoin
title: 'Pinning and disorder relevance for the lattice Gaussian Free Field II: the two dimensional case'
---
Introduction
============
The aim of statistical mechanics is to obtain a qualitative understanding of natural phenomena of phase transitions by the study of simplified models, often built on a lattices. In general the Hamiltonian of a model of statistical mechanics is left invariant by the lattice symmetries: a prototypical example being the Ising model describing a ferromagnet.
However, one might argue that materials which are found in nature are usually not completely homogeneous and for this reason, physicists where led to considering systems in which the interaction terms, for example the potentials between nearest neighbor spins, are chosen by sampling a random field – which we call [*disorder*]{} – with good ergodic properties, often even a field of independent identically distributed random variables. An important question which arises is thus whether the results concerning the phase transition obtained for a model with homogeneous interactions referred to as *the pure system* (e.g. the Onsager solution of the two dimensional Ising Model) remain valid when a system where randomness of a very small amplitude is introduced.
In [@cf:Hcrit] A. B. Harris, gave a strikingly simple heuristical argument, based on renormalization theory consideration, to predict the effect of the introduction of a small amount of the system: in substance Harris’ criterion predict that if the phase transition of the pure system is sufficiently smooth, it will not be affected by small perturbation (disorder is then said to be *irrelevant*), while in the other cases the behavior of the system is affected by an arbitrary small addition of randomness (disorder is *relevant*). To be complete, let us mention also the existence of a boundary case for which the criterion yields no prediction (the *marginal disorder* case). The criterion however does not give a precise prediction concerning the nature of the phase transition when the disorder is relevant.
The mathematical verification of the Harris criterion is a very challenging task in general. In the first place, it can only be considered for the few special models of statistical mechanics for which we have a rigorous understanding of the critical properties of the pure system. In the last twenty years this question has been addressed, first by theoretical physicists (see e.g. [@cf:DHV] and references therein) and then by mathematicians [@cf:KZ; @cf:AZnew; @cf:Ken; @cf:QH2; @cf:DGLT; @cf:GLT; @cf:GLT2; @cf:GT_cmp; @cf:L; @cf:Trep] (see also [@cf:GB; @cf:G] for reviews), for a simple model of a 1-dimensional interface interacting with a substrate: for this model the interface is given by the graph of a random walk which takes random energy rewards when it touches a defect line. In this case, the pure system has the remarkable quality of being what physicists call [*exactly solvable*]{}, meaning that there exists an explicit expression for the free energy [@cf:Fisher].
This model under consideration in the present paper can be seen as a high-dimension generalization of the RW pinning model. The random walk is replaced by a random field ${{\ensuremath{\mathbb Z}} }^d\to {{\ensuremath{\mathbb R}} }$, and the random energies are collected when the graph of the field is close to the hyper-plane ${{\ensuremath{\mathbb Z}} }^d \times \{0\}$. While the pure model is not exactly solvable in that case, it has been studied in details and the nature of the phase transition is well known [@cf:BB; @cf:BDZ1; @cf:BV; @cf:CV; @cf:Vel].
On the other hand, the study of the disordered version of the model is much more recent [@cf:CM1; @cf:CM2; @cf:GL].
In [@cf:GL], we gave a close to complete description of the free energy diagram of the disordered model when $d\ge 3$:
- We identified the value of the disordered critical point, which is shown to coincide with that of the associated annealed model, regardless of the amplitude of disorder.
- We proved that for Gaussian disorder, the behavior of the free energy close to $h_c$ is quadratic, in contrasts with the annealed model for which the transitition is of first order.
- In case of general disorder, we proved that the quadratic upper-bound still holds, and found a polynomial lower bound with a different exponent.
Let us stress that the heuristic of our proof strongly suggests that the behavior of the free energy should be quadratic for a suitable large class of environments (those who satisfy a second moment assumption similar to ).
In the present paper, we choose to attack the case $d=2$, for which only limited results were obtained so far. We have seen in the proof of the main result [@cf:GL] that the critical behavior of the model is very much related to the extremal process of the field. The quadratic behavior of the free-energy in [@cf:GL Theorem 2.2] comes from the fact that high level sets of the Gaussian free field for $d\ge 3$ look like a uniformly random set with a fixed density (see [@cf:CCH]). In dimension $2$ however, the behavior of the extremal process is much more intricate, with a phenomenon of clustering in the level sets (see [@cf:BL; @cf:DZ2; @cf:Dav] or also [@cf:ABK] for a similar phenomenon for branching Brownian Motion). This yields results of a very different nature.
Model and results
=================
Given $\Lambda$ be a finite subset of ${{\ensuremath{\mathbb Z}} }^d$, we let $\partial {\Lambda}$ denote the internal boundary of $\Lambda$, $\mathring{{\Lambda}}$ the set of interior points of ${\Lambda}$, and $\partial^-{\Lambda}$ the set of point which are adjacent to the boundary, $$\label{boundary}\begin{split}
\partial {\Lambda}&:=\{x \in {\Lambda}: \, \exists y\notin {\Lambda}, \ x\sim y \},\\
\mathring{{\Lambda}}&:={\Lambda}\setminus \partial {\Lambda},\\
\partial^- {\Lambda}&:=\{x \in \mathring{{\Lambda}} : \, \exists y\in \partial {\Lambda}, \ x\sim y \}.
\end{split}$$ In general some of these sets could be empty, but throughout this work ${\Lambda}$ is going to be a large square. Given ${\widehat}\phi: {{\ensuremath{\mathbb Z}} }^d \to {{\ensuremath{\mathbb R}} }$, we define ${{\ensuremath{\mathbf P}} }^{{\widehat}\phi}_{{\Lambda}}$ to be the law of the lattice Gaussian free field $\phi=(\phi_x)_{x\in \Lambda}$ with boundary condition ${\widehat}\phi$ on $\partial {\Lambda}$. The field $\phi$ is a random function from ${\Lambda}$ to ${{\ensuremath{\mathbb R}} }$. It is satisfies $$\phi_x\,:=\, {\widehat}\phi_x \quad \text{ for every } x \in \partial {\Lambda},$$ and the distribution of $(\phi_x)_{x\in \mathring {\Lambda}}$ is given by $$\label{density}
{{\ensuremath{\mathbf P}} }^{{\widehat}\phi}_{\Lambda}({\mathrm{d}}\phi)=\frac{1}{\mathcal Z^{{\widehat}\phi}_{{\Lambda}}}
\exp\left(-\frac 1 2 {\sum_{\substack{(x,y)\in ({\Lambda})^2 \setminus (\partial {\Lambda})^2 \\ x\sim y}}}\frac{ (\phi_x-\phi_y)^2 }{2} \right)
\prod_{x\in \mathring{{\Lambda}}} {\mathrm{d}}\phi_x \, ,$$ where $\prod_{x\in \mathring{{\Lambda}}} {\mathrm{d}}\phi_x$ denotes the Lebesgue measure on ${{\ensuremath{\mathbb R}} }^{\mathring{{\Lambda}}}$ and $$\label{eq:defcalz}
\mathcal Z^{{\widehat}\phi}_{{\Lambda}}:= \int_{{{\ensuremath{\mathbb R}} }^{\mathring{{\Lambda}}}}
\exp\left(-
\frac 1 2 {\sum_{\substack{(x,y)\in ({\Lambda})^2 \setminus (\partial {\Lambda})^2 \\ x\sim y}}}
\frac{ (\phi_x-\phi_y)^2 }{2} \right) \prod_{x\in\mathring{{\Lambda}}} {\mathrm{d}}\phi_x \, ,$$ (one of the two $(1/2)$ factors is present to compensate that the edges are counted twice in the sum, the other one being the one usually present for Gaussian densities). In what follows we consider the case $$\Lambda=\Lambda_N:=\{0,\dots,N\}^d,$$ for some $N\in {{\ensuremath{\mathbb N}} }$. Note that we have $$\mathring{{\Lambda}}_N:= \{1,\dots, N-1\}^d.$$ We also introduce the notation ${\widetilde}{\Lambda}_N:= \{1,\dots, N\}^d$, and we simply write ${{\ensuremath{\mathbf P}} }^{{\widehat}\phi}_N$ for ${{\ensuremath{\mathbf P}} }^{{\widehat}\phi}_{{\Lambda}_N}$. We drop ${\widehat}\phi$ from our notation in the case where we consider zero boundary condition ${\widehat}\phi \equiv 0$.
We let ${\omega}=\{{\omega}_x\}_{x \in {{\ensuremath{\mathbb Z}} }^d}$ be the realization of a family of IID square integrable centered random variables (of law ${{\ensuremath{\mathbb P}} }$). We assume that they have finite exponential moments, or more precisely, that there exist constants ${\beta}_0, {\overline{{\beta}}}\in (0, \infty]$ such that $$\label{eq:assume-gl}
{\lambda}({\beta})\, :=\, \log {{\ensuremath{\mathbb E}} }[e^{{\beta}{\omega}_x}]\, < \, \infty\ \text{ for every } {\beta}\in (-{\beta}_0\, 2{\overline{{\beta}}}]\, .$$ For $x\in {\Lambda}_N$ set $\delta_x:= {\mathbf{1}}_{[-1,1]}(\phi_x)$. For ${\beta}>0$ and $h\in {{\ensuremath{\mathbb R}} }$, we define a modified measure ${{\ensuremath{\mathbf P}} }_{N, h}^{{\beta},{\omega},{\widehat}\phi}$ via the density $$\label{eq:modmeas}
\frac{{\mathrm{d}}{{\ensuremath{\mathbf P}} }^{{\beta},{\omega}, {\widehat}\phi}_{N,h}}{{\mathrm{d}}{{\ensuremath{\mathbf P}} }^{{\widehat}\phi}_N}(\phi)=\frac{1}{Z^{{\beta},{\omega},{\widehat}\phi}_{N,h}}\exp\left( \sum_{x\in {\widetilde}{\Lambda}_N}
({\beta}{\omega}_x-{\lambda}({\beta})+h)\delta_x\right)\, ,$$ where $$\label{eq:modZ}
Z^{{\beta},{\omega},{\widehat}\phi}_{N,h}:={{\ensuremath{\mathbf E}} }_N\left[ \exp\left( \sum_{x\in {\widetilde}{\Lambda}_N} ({\beta}{\omega}_x-{\lambda}({\beta})+h)\delta_x\right)\right].$$ Note that in the definition of ${{\ensuremath{\mathbf P}} }^{{\beta},{\omega},{\widehat}\phi}_{N,h}$, the sum $\left(\sum_{x\in {\widetilde}{\Lambda}_N}\right)$ can be replaced by either $\left(\sum_{x\in {\Lambda}_N}\right)$ or $\left(\sum_{x\in \mathring{{\Lambda}}_N}\right)$ as these changes affect only the partition function. In the case where ${\widehat}\phi \equiv 0$, we drop the corresponding superscript it from the notation. In the special case where ${\beta}=0$, we simply write ${{\ensuremath{\mathbf P}} }^{{\widehat}\phi}_{N,h}$ and $Z^{{\widehat}\phi}_{N,h}$ for the pinning measure and partition function (as they do not depend on ${\omega}$) respectively. This case is referred to as the *pure* (or homogeneous) model. When ${\beta}>0$, defines the pinning model with *quenched* disorder.
The free energy
---------------
The important properties of the system are given by the asymptotic behavior of the partition function, or more precisely by the free energy. The existence of quenched free energy for the disordered model has been proved in [@cf:CM1 Theorem 2.1]. We recall this result here together with some basic properties
\[freen\] The free energy $$\label{eq:freen}
{\textsc{f}}({\beta},h):=\lim_{N\to \infty} \frac{1}{N^d}{{\ensuremath{\mathbb E}} }\left[\log Z^{{\beta},{\omega}}_{N,h}\right]
\stackrel{{{\ensuremath{\mathbb P}} }({\mathrm{d}}{\omega})-a.s.}{=} \lim_{N\to \infty} \frac{1}{N^d}\log Z^{{\beta},{\omega}}_{N,h} \, ,$$ exists (and is self-averaging). It is a convex, nonnegative, nondecreasing function of $h$. Moreover there exists a $h_c({\beta})\in (0,\infty)$ which is such that $${\textsc{f}}({\beta},h)\begin{cases} =0 \text{ for } h\le h_c({\beta}),\\
>0 \text{ for } h> h_c({\beta}).
\end{cases}$$
Let us briefly explain why $h_c({\beta})$ marks a transition on the large scale behavior of $\phi$ under ${{\ensuremath{\mathbf P}} }^{{\beta},{\omega}}_{N,h}$. A simple computation gives $$\partial_h\left( \frac{1}{N^d} \log Z^{{\beta},{\omega}}_{N,h}\right)= \frac{1}{N^d}\sum_{x\in {\widetilde}{\Lambda}_N} {{\ensuremath{\mathbf E}} }^{{\beta},{\omega}}_{N,h} \left[ \delta_x \right].$$ Hence by convexity, we have $$\partial_h {\textsc{f}}({\beta},h)=\lim_{N\to \infty}\frac{1}{N^d}\sum_{x\in {\widetilde}{\Lambda}_N} {{\ensuremath{\mathbf E}} }^{{\beta},{\omega}}_{N,h} \left[ \delta_x \right],$$ for the $h$ for which ${\textsc{f}}({\beta},h)$ differentiable (for the hypothetical countable set where $\partial_h {\textsc{f}}({\beta},h)$ may not exist, we can replace $\lim$ by $\liminf$ resp. $\limsup$, $=$ by $\le$ resp. $\ge$ and consider the left- resp. right-derivative in the above equation).
For $h>h_c({\beta})$, we have $\partial_h {\textsc{f}}({\beta},h)>0$ by convexity and thus the expected number of point in contact with the substrate is asymptotically of order $N^d$. On the contrary when $h<h_c({\beta})$, the asymptotic expected contact fraction vanishes when $N$ tends to infinity.
Note that the whole model is perfectly defined for all $d\ge 1$. However, the case $d=1$, which is a variant of the random walk pinning model which as mentioned in the introduction was the object of numerous studies in the literature. However, the effect of disorder in dimension $1$ being quite different, in the remainder of the paper, we prove results for the case $d=2$ and discuss how they compare with those obtained in the more related case $d\ge 3$ [@cf:GL].
The pure model
--------------
In the case ${\beta}=0$, we simply write ${\textsc{f}}(h)$ for ${\textsc{f}}(0,h)$. In that case the behavior of the free energy is known in details (see [@cf:CM1 Fact 2.4] and also [@cf:GL Section 2.3 and Remark 7.10] for a full proof for $d\ge 3$). We summarize it below.
\[propure\] For all $d\ge 1$, we have $h_c(0)=0$ and moreover
- For $d=2$ $$\label{pure}
{\textsc{f}}(h)\stackrel{h\to 0+}{\sim} \frac{\sqrt{2} h}{\sqrt{|\log h|}},$$
- For $d\ge 3$ $$\label{pure3}
{\textsc{f}}(h)\stackrel{h\to 0+}{\sim} c_d h,$$ where $c_d:= {{\ensuremath{\mathbf P}} }[\sigma_d {{\ensuremath{\mathcal N}} }\in [-1,1]]$ and $\sigma_d$ is the standard deviation for the infinite volume free field in ${{\ensuremath{\mathbb Z}} }^d$.
To be more precise $\sigma_d:=\sqrt{G^0(x,x)}$ where $G^0$ is the Green function defined in . The result in dimension $2$ is well known folklore to people in the fields, but as to our knowledge, no proof of it is available in the literature. For this reason we present a short one in Appendix \[secpropure\].
The quenched/annealed free energy comparison
--------------------------------------------
Using Jensen’s inequality, we can for every ${\beta}\ge 0$, compare the free energy to that of the annealed system, which is the one associated to the averaged partition function ${{\ensuremath{\mathbb E}} }\left[ Z^{{\beta},{\omega}}_{N,h}\right]$, $$\label{anealed}
{\textsc{f}}({\beta},h)=\lim_{N\to \infty} \frac{1}{N^d}{{\ensuremath{\mathbb E}} }\left[ \log Z^{{\beta},{\omega}}_{N,h}\right] \le \lim_{N\to \infty} \frac{1}{N^d}\log {{\ensuremath{\mathbb E}} }\left[ Z^{{\beta},{\omega}}_{N,h}\right].$$ Our choice of parametrization implies $${{\ensuremath{\mathbb E}} }\left[ Z^{{\beta},{\omega}}_{N,h}\right]={{\ensuremath{\mathbf E}} }_N\left[ {{\ensuremath{\mathbb E}} }\left[e^{\sum_{x\in {\widetilde}{\Lambda}_N}( {\omega}_x-{\lambda}({\beta})+h)\delta_x}\right] \right]=
{{\ensuremath{\mathbf E}} }_N\left[ e^{\sum_{x\in {\widetilde}{\Lambda}_N}h\delta_x}\right]=
Z_{N,h},$$ and thus for this reason we have $$\label{annehilde}
{\textsc{f}}({\beta},h)\le {\textsc{f}}(h) \quad \text{and} \quad
h_c({\beta})\, \ge\, 0.$$ It is known that the inequality is strict: for $h>0$, we have ${\textsc{f}}({\beta},h)< {\textsc{f}}(h)$ in all dimensions (cf. [@cf:CM1]). However we can ask ourselves if the behavior of the model with quenched disorder is similar to that of the annealed one in several other ways
- Is the critical point of the quenched model equal to that of the annealed model (i.e. is $h_c({\beta})=0$)?
- Do we have a critical exponent for the free energy transition: do we have $${\textsc{f}}({\beta},h_c({\beta})+u)\stackrel{u\to 0+} \sim u^{\nu+o(1)},$$ and is $\nu$ equal to one, like for the annealed model (cf. Proposition \[propure\])?
This question has been almost fully solved in the case $d\ge 3$. Let us display the result here
For $d\ge 3$, for every ${\beta}\in [0,\bar\beta]$ we have
- $h_c({\beta})=0$ for all values of ${\beta}>0$.
- If ${\omega}$ is Gaussian, there exist positive constants $c_1({\beta})<c_2({\beta})$ such that for all $h\in (0,1)$. $$c_1({\beta}) h^2 \le {\textsc{f}}({\beta},h)\le c_2({\beta}) h^2.$$
- In the case of general ${\omega}$, for all there exist positive constants $c_1({\beta})<c_2({\beta})$ such that for all $h\in(0,1)$ $$c_1({\beta}) h^{66d}\le {\textsc{f}}({\beta},h)\le c_2({\beta}) h^2.$$
We strongly believe that the quadratic behavior holds for every ${\omega}$ as soon as ${\lambda}(2{\beta})<\infty$, and the Gaussian assumption is mostly technical. However, if ${\lambda}(2{\beta})=\infty$, we believe that the model is in a different universality class and the critical exponent depends on the tail of the distribution of the variable $\xi:=e^{{\beta}{\omega}_0}$.
The aim of the paper is to provide answers in the case of dimension $2$.
The main result
---------------
We present now the main achievement of this paper. We prove that similarly to the $d\ge 3$ case, the critical point $h_c({\beta})$ coincides with the annealed one for every value of ${\beta}$ (which is in contrast with the case $d=1$ where the critical points differs for every ${\beta}>0$ [@cf:GLT]). However, we are able to prove also that the critical behavior of the free energy is not quadratic, ${\textsc{f}}({\beta},h)$ is becomes smaller than any power of $h$ in a (positive) neighborhood of $h=0$. This indicates that the phase transition is of infinite order.
\[mainres\] When $d=2$, for every ${\beta}\in[0,\bar {\beta}]$ the following holds
- We have $h_c({\beta})=0$.
- We have $$\lim_{h\to 0+} \frac{\log {\textsc{f}}({\beta},h)}{\log h}=\infty.$$ More precisely, there exists $h_0({\beta})$ such that for all $h\in (0,h_0({\beta}))$ $$\label{breaks}
\exp\left(- h^{-20} \right)\le {\textsc{f}}({\beta},h) \le \exp\left( - | \log h|^{3/2} \right).$$
We do not believe that either bound in is sharp. However it seems to us that the strategy used for the lower-bound is closer to capture the behavior of the field. We believe that the true behavior of the free energy might be given by $${\textsc{f}}({\beta},h) \approx \exp( h^{-1+o(1)}).$$ While a lower bound of this type might be achieved by optimizing the proof presented in the present paper (but this would require some significant technical work), we do not know how to obtain a significant improvement on the upper-bound.
Co-membrame models in two dimension
-----------------------------------
Like in [@cf:GL], it worthwhile to notice that the proof of the results of the present paper can be adapted to a model for with a different localization mechanism. It is the analog of the model of a copolymer in the proximity of the interface between selective solvents, see [@cf:Bcoprev; @cf:coprev] and references therein. For this model given a realization of ${\omega}$ and two fixed parameters ${\varrho}, h>0$, the measure is defined via the following density $$\label{eq:modmeascop}
\frac{{\mathrm{d}}\check{{\ensuremath{\mathbf P}} }^{{\omega},{\varrho}}_{N,h}}{{\mathrm{d}}{{\ensuremath{\mathbf P}} }_N}\, \propto\, \exp\left( {\varrho}\sum_{x\in {\widetilde}{\Lambda}_N} ( {\omega}_x+h)\operatorname{\mathrm{sign}}\left(\phi_x \right)\right)\, ,$$ where we assume $\operatorname{\mathrm{sign}}(0)=+1$. A natural interpretation of the model is that the graph of $(\phi_x)_{x\in {\Lambda}_N}$ models a membrane lying between two solvents $A$ and $B$ which fill the upper and lower half-space respectively: for each point of the graph, the quantity ${\omega}_x+h$ describes the energetic preference for one solvent of the corresponding portion of the membrane (A if ${\omega}_x+h>0$ and B if ${\omega}_x+h<0$). As $h$ is positive and ${\omega}_x$ is centered, there is, on average, a preference for solvent A (by symetry this causes no loss of generality).
If ${{\ensuremath{\mathbb P}} }[({\omega}_x<-h)>0]$, there is a non-trivial competition between energy and entropy: the interaction with the solvent gives an incentive for the field $\phi$ to stay close to the interface so that its sign can match as much as possible that of ${\omega}+h$, but such a strategy might be valid only if the energetic rewards it brings is superior to the entropic cost of the localization.
A more evident analogy with the pinning measure can be made by observing that we can write $$\label{eq:modcop}
\frac{{\mathrm{d}}\check{{\ensuremath{\mathbf P}} }^{{\omega},{\varrho}}_{N,h}}{{\mathrm{d}}{{\ensuremath{\mathbf P}} }_N}\, =\, \frac1{\check Z^{{\omega},{\varrho}}_{N,h}} \exp\left( -2{\varrho}\sum_{x\in {\widetilde}{\Lambda}_N} ( {\omega}_x+h){\Delta}_x
\right)\, ,$$ where ${\Delta}_x:= (1- \operatorname{\mathrm{sign}}(\phi_x))/2$, that is ${\Delta}_x$ is the indicator function that $\phi_x$ is in the lower half plane. It is probably worth stressing that from to there is a non-trivial (but rather simple) change in energy. And in the form . In particular, the strict analog of Proposition \[freen\] holds – the free energy in this case is denoted by $\check{\textsc{f}}({\varrho}, h)$ – and, precisely like for the pinning case, one sees that $\check{\textsc{f}}({\varrho}, h)\ge 0$. We then set $\check h_c({\varrho}):= \inf\{h>0:\, \check{\textsc{f}}({\varrho}, h)=0\}$. Adapting the proof for the lower-bound in we can identify the value of $\check h_c({\varrho})$.
\[th:cop\] For $d= 2$, for any ${\varrho}\in(0, \bar {\beta}/2)$ we have $$\label{eq:cop}
\check h_c({\varrho})\, =\, \frac1{2{\varrho}} {\lambda}(-2{\varrho})\, .$$ Moreover , with ${\textsc{f}}({\beta},h)$ replaced by $\check{\textsc{f}}({\varrho}, h_c({\varrho})-h)$, holds true.
Note that while pure co-membrane model (i.e. with no disorder) displays a first order phase transition in $h$, the above result underlines that the transition becomes of infinite order in the presence of an arbitrary small quantity of disorder. Note that this result differs both from the one obtained in dimension $d\ge 3$ (for which the transition is shown to be quadratic at least for Gaussian environment [@cf:GL Theorem 2.5]), and that in dimension $1$: for the copolymer model based on renewals presented in [@cf:Bcoprev], $\frac1{2{\varrho}} {\lambda}(-2{\varrho})$ is in most cases a strict upper-bound on $\check h_c({\varrho})$ (see e.g. the results in [@cf:Tcopo]).
The proof of Theorem \[th:cop\] is not given in the paper but it can be obtained with straightforward modification, from that of Theorem \[mainres\].
Organization of the paper
-------------------------
The proof of the upper-bound and of the lower-bound on the free energy presented in Equation are largely independent. However some general technical results concerning the covariance structure of the free field are useful in both proofs, and we present these in Section \[toolbox\]. Most of the proofs for results presented in this section are in Appendix \[appendix\].
The proof of the upper-bound is developed in Section \[seclower\]. The proof of the lower-bound is spreads from Section \[finicrit\] to \[intelinside\]. In Section \[finicrit\] we present an estimate on the free energy in terms of a finite system with “stationary” boundary condition. In Section \[decompo\], we give a detailed sketch of the proof of the lower-bound based on this finite volume criterion, divided into several steps. The details of these steps are covered in Section \[liminouze\] and \[intelinside\].
For the proof of both the upper and the lower-bound, we need fine results on the structure of the free field. Although these results or their proof cannot directly be extracted from the existing literature, our proof (especially the techniques developed in Section \[intelinside\]) is largely based on tools that were developed in the numerous study on extrema and extremal processes of the two dimensional free field [@cf:BDG; @cf:BDZ; @cf:Dav; @cf:DZ2] and other $\log$-correlated Gaussian processes [@cf:A; @cf:AS; @cf:ABK; @cf:brams; @cf:Mad] (the list of references being far from being complete). In particular for the lower bound, we present an *ad-hoc* decomposition of the field in Section \[decompo\] and then exploit decomposition to apply a conditioned second moment technique, similarly to what is done e.g. in [@cf:AS].
For the upper-bound, we also make use a change of measure machinery inspired by a similar techniques developed in the study of disordered pinning model [@cf:BL; @cf:DGLT; @cf:GLT; @cf:GLT2] and adapted successfully to the study of other models [@cf:BT; @cf:BS1; @cf:BS2; @cf:L2; @cf:L3; @cf:YZ].
A toolbox {#toolbox}
=========
Notation and convention
-----------------------
Throughout the paper, to avoid a painful enumeration, we use $C$ to denote an arbitrary constant which is not allowed to depend on the value of $h$ or $N$ nor on the realization of ${\omega}$. Its value may change from one equation to another. For the sake of clarity, we try to write $C({\beta})$ when the constant may depend on ${\beta}$. When a constant has to be chosen small enough rather than large enough, we may use $c$ instead of $C$.
For $x=(x_1,x_2)\in {{\ensuremath{\mathbb Z}} }^2$ we let $|x|$ denote its $l_1$ norm. $$|x|:= |x_1|+|x_2|.$$ The notation $|\cdot|$ is also used to denote the cardinal of a finite set as this should yield no confusion.
If $A\subset {{\ensuremath{\mathbb Z}} }^2$ and $x\in {{\ensuremath{\mathbb Z}} }^2$ we set $$\label{distA}
d(x,A):=\min_{y\in A} |x-y|.$$ We use double brackets to denote interval of integers, that for $i<j$ in ${{\ensuremath{\mathbb Z}} }$ $${\llbracket}i,j{\rrbracket}:= [i,j]\cap {{\ensuremath{\mathbb Z}} }=\{i,i+1,\dots,j\}.$$ If $(A_i)_{i=1}^k$ is a finite family of events, we refer to the following inequality as *the union bound*. $${{\ensuremath{\mathbb P}} }(\cup_{i=1}^k A_i)\le \sum_{i=1}^k {{\ensuremath{\mathbb P}} }(A_i).$$ We let $(X_t)_{t\ge 0}$ denote continuous time simple random walk on ${{\ensuremath{\mathbb Z}} }^d$ whose generator ${\Delta}$ is the lattice Laplacian defined by $$\label{laplace}
{\Delta}f(x):= \sum_{y\sim x} \big( f(y)-f(x) \big)$$ and we let $P^x$ denote its law starting from $x\in {{\ensuremath{\mathbb Z}} }^d$. We let $P_t$ denote the associated heat-kernel $$\label{heat}
P_t(x,y)=P^x(X_t=y).$$ If $\mu$ denote a probability measure on a space ${\Omega}$, and $f$ a measurable function on ${\Omega}$ we denote the expectation of $f$ by $$\mu(f) = \int_{{\Omega}} f({\omega}) \mu( {\mathrm{d}}{\omega}),$$ with an exception where the probability measure is denoted by the letter $P$, in that case we use $E$ for the expectation.
If ${{\ensuremath{\mathcal N}} }(\sigma)$ is a Gaussian of standard deviation $\sigma$, it is well known that we have $$\label{gtail}
P\left[ {{\ensuremath{\mathcal N}} }(\sigma) \ge u \right] \le \frac{\sigma}{ \sqrt{2\pi}u}e^{-\frac{u^2}{2\sigma^2}}.$$ We refer to the Gaussian tail bound when we use this inequality.
The massive free field {#secmass}
----------------------
In this section we quickly recall the the definition and some basic properties of the massive free field. Given $m>0$, and a set ${\Lambda}\subset {{\ensuremath{\mathbb Z}} }^d$ and a function ${\widehat}\phi$, we define the law ${{\ensuremath{\mathbf P}} }^{m,{\widehat}\phi}_{{\Lambda}}$ of the massive free field on ${\Lambda}$ with boundary condition ${\widehat}\phi$ and mass $m$ as follows: it is absolutely continuous w.r.t ${{\ensuremath{\mathbf P}} }^{{\widehat}\phi}_{{\Lambda}}$ and $$\label{massivedensity}
\frac{{\mathrm{d}}{{\ensuremath{\mathbf P}} }^{m,{\widehat}\phi}_{{\Lambda}} }{{\mathrm{d}}{{\ensuremath{\mathbf P}} }^{{\widehat}\phi}_{{\Lambda}}}(\phi):= \frac{1}{{{\ensuremath{\mathbf E}} }^{{\widehat}\phi}_{{\Lambda}}
\left[ \exp\left(-m^2 \sum_{x\in \mathring {\Lambda}} \phi_x^2\right) \right] } \exp\left(-m^2 \sum_{x\in \mathring {\Lambda}} \phi_x^2\right).$$ We let ${{\ensuremath{\mathbf P}} }_N^{m,{\widehat}\phi}$ denote the law of the massive field on ${\Lambda}_N$. (in the special case ${\widehat}\phi_x\equiv 0$, ${\widehat}\phi$ is omitted in the notation).
We let ${{\ensuremath{\mathbf P}} }^m$ denote the law of the centered infinite volume massive free field ${{\ensuremath{\mathbb Z}} }^d$, which is the limit of ${{\ensuremath{\mathbf P}} }^{m}_{{\Lambda}}$ when ${\Lambda}\to {{\ensuremath{\mathbb Z}} }^d$ (see Section \[hkernel\] for a proper definition with the covariance function). We will in some cases have to choose the boundary condition ${\widehat}\phi$ itself to be random and distributed like an infinite volume centered massive free field (independent $\phi$), in which case we denote its law by ${\widehat}{{\ensuremath{\mathbf P}} }^m$ instead of ${{\ensuremath{\mathbf P}} }^m$.
Note that the free field and its massive version satisfy a Markov spatial property. In particular the law of $(\phi)_{x\in {\Lambda}_N}$ under ${\widehat}{{\ensuremath{\mathbf P}} }^{m} \times{{\ensuremath{\mathbf P}} }^{m,{\widehat}\phi}_N$ is the same as under the infinite volume measure ${{\ensuremath{\mathbf P}} }^{m}$.
Getting rid of the boundary condition {#grbc}
-------------------------------------
Even if the definition of the free energy given in Proposition \[freen\] is made in terms of the partition function with ${\widehat}\phi\equiv 0$ it turns out that our methods to obtain upper and lower bounds involve considering non-trivial boundary conditions (cf. Proposition \[scorpiorizing\] and Proposition \[th:finitevol\]).
However, it turns out to be more practical to work with a fixed law for the field and not one that depends on ${\widehat}\phi$. Fortunately, given a boundary condition ${\widehat}\phi$ the law of ${{\ensuremath{\mathbf P}} }^{m,{\widehat}\phi}_N$ can simply be obtained by translating the field with $0$ boundary condition by a function that depends only on ${\widehat}\phi$. This is a classical property of the free field but let us state it in details. As the covariance function of $\phi$ under ${{\ensuremath{\mathbf P}} }^{m,{\widehat}\phi}_N$ and ${{\ensuremath{\mathbf P}} }^{m}_N$ are the same, we have we have $$\label{transkix}
{{\ensuremath{\mathbf P}} }^{m,{\widehat}\phi}_N[\phi \in \cdot \ ]={{\ensuremath{\mathbf P}} }^{m}_N[\phi + H^{m,{\widehat}\phi}_N \in \cdot \ ],$$ where $$H^{m,{\widehat}\phi}_N(x):={{\ensuremath{\mathbf E}} }^{m,{\widehat}\phi}_N[\phi(x)].$$ It is not difficult to check that $H^{m,{\widehat}\phi}_N$ must be a solution of the system (recall ) $$\label{defH}\begin{cases}
H(x):= {\widehat}\phi(x), & x \in \partial {\Lambda}_N,\\
{\Delta}H(x) =m^2 H^{m,{\widehat}\phi}_N, \quad & x \in \mathring {\Lambda}_N.
\end{cases}$$ We simply write $H^{{\widehat}\phi}_N(x)$ when $m=0$. The solution of is unique and $H^{m,{\widehat}\phi}_N$ has the following representation: consider $X_t$ the simple random walk on ${{\ensuremath{\mathbb Z}} }^d$ and for $A\subset {{\ensuremath{\mathbb Z}} }^d$ let $\tau_A$ denotes the first hitting of $A$. We have $$\label{RWrepresent}
H^{{\widehat}\phi}_N(x):= E_x\left[ e^{-m^2 \tau_{\partial {\Lambda}_N}}{\widehat}\phi\left(X_{\tau_{\partial {\Lambda}_N}}\right)\right].$$ Given ${\widehat}\phi$ and $x\in {\widetilde}{\Lambda}_N$, we introduce the notation $$\label{deltaf}
\delta^{{\widehat}\phi}_x:= {\mathbf{1}}_{[-1,1]}(\phi_x+ H^{{\widehat}\phi}_N(x)).$$ In view of an alternative way of writing the partition function is $$\label{migrate}
Z^{{\beta},{\omega},{\widehat}\phi}_{N,h}= {{\ensuremath{\mathbf E}} }_N \left[ e^{\sum_{x\in {\widetilde}{\Lambda}_N} ({\beta}{\omega}_x-{\lambda}({\beta})+h) \delta^{{\widehat}\phi}_x} \right].$$ In some situation the above expression turns our to be handier than the definition .
Some estimates on Green functions and heat Kernels {#hkernel}
--------------------------------------------------
In this section we present some estimates on the covariance function of the free field and massive free field in dimension $2$, which will be useful in the course of the proof. These are not new results, but rather variants of existing estimates in the literature (see e.g [@cf:Dav Lemma 2.1]).
The covariance kernel of the infinite volume free field with mass $m>0$ in ${{\ensuremath{\mathbb Z}} }^2$ or $m\ge 0$ in ${\Lambda}_N$ is given by the Green function $G^m$ which is the inverse of ${\Delta}-m^2$ (this can in fact be taken as the definition of the infinite volume free field, requiring in addition that it is centered). The covariance function of the field under the measure ${{\ensuremath{\mathbf P}} }_N$ is $G^{m,*}$ which is the inverse of ${\Delta}-m^2$ with Dirichlet boundary condition on $\partial {\Lambda}_N$. Both of these functions can be represented as integral of the heat kernel , we have $$\label{greenff}
\begin{split}
{{\ensuremath{\mathbf E}} }^m[\phi(x)\phi(y)]&=\int_{0}^{\infty}e^{-m^2t}P_t(x,y){\mathrm{d}}t=:G^m(x,y),\\
{{\ensuremath{\mathbf E}} }^m_N[\phi(x)\phi(y)]&=\int_{0}^{\infty}e^{-m^2t}P^*_t(x,y){\mathrm{d}}t=:G^{m,*}(x,y),
\end{split}$$ where $P^*_t$ is the heat kernel on ${\Lambda}_N$ with Dirichlet boundary condition on $\partial {\Lambda}_N$, $$P^*_t(x,y):= P_x\left[ X_t=y \ ; \ \tau_{\partial {\Lambda}_N}<t\right].$$ We simply write $G^*$ in the case $m=0$.
Note that, because of the spatial Markov property (Section \[secmass\]) and of , when ${\widehat}\phi$ has law ${\widehat}{{\ensuremath{\mathbf P}} }^m$ and $\phi$ has law ${{\ensuremath{\mathbf P}} }_N$, $(H^{m,{\widehat}\phi}_N(x)+ \phi_x)_{x\in {\Lambda}_N}$ has the same law as the (marginal in ${\Lambda}_N$ of the) infinite volume field. Hence as a consequence $$\label{covH}
{\widehat}{{\ensuremath{\mathbf E}} }^m[H^{{\widehat}\phi}_N(x)H^{{\widehat}\phi}_N(y)]=G^m(x,y)-G^{m,*}(x,y)=\int_{0}^{\infty}e^{-m^2t}(P_t(x,y)-P^*_t(x,y)){\mathrm{d}}t.$$
Before giving more involved estimates, let us mention first a quantitative version of the Local Central Limit Theorem [@cf:LL Theorem 2.1.1] for the heat kernel which we use as an essential building brick to obtain them. There exists a constant $C$ such that for all $t\ge 1$, $$\label{lclt}
\left|P_t(x,x)-\frac{1}{4\pi t}\right|\le \frac{C}{t^{3/2}},$$ Let us recall the notation for the distance between a set and a point. The following two lemmas are proved in Appendix \[appendix\].
\[Greenesteem\] There exists a constant such that $C$
- For all $m\le 1$, for any $x\in {{\ensuremath{\mathbb Z}} }^2$ $$\label{eq:variance}
\left| G^m(x,x)+\frac{1}{2\pi} \log m\right|\le C$$
- For all $m\le 1$, for any $x\in {\Lambda}_N$ $$\label{eq:stimagreen}
\left| G^{m,*}_N(x,x)-\frac{1}{2\pi} \log \min(m^{-1},d(x,\partial {\Lambda}_N))\right|\le C.$$
\[lem:kerestimate\] The following assertions hold
- There exists a constant $C$ such that for all $t\ge 1$, $|x-y| \le \sqrt{t}$, we have $$\label{gradientas}\begin{split}
\left( P_t(x,x)- P_t(x,y)\right)&\le \frac{ C |x-y|^2 }{t^2},\\
\left( P^*_t(x,x)+P^*_t(y,y)-2P^*_t(x,y)\right)&\le \frac{ C |x-y|^2 }{t^2}.
\end{split}$$
- There exist a constant $C$ such that for all $t\ge 1$ and $x,y$ satifying $ |x-y| \le t$ we have $$\label{croco}
P_t(x,y)\le \frac{C}{t}e^{-\frac{|x-y|^2}{Ct}}.$$ and as a consequence $$\label{greensum}
\sum_{y\in {{\ensuremath{\mathbb Z}} }^2} G^{m}(x,y)\le Cm^{-2}.$$
- We have for all $x$ $$\label{ltest}
\frac{P^*_t(x,x)}{P_t(x,x)}\le C \frac{\left[d(x,\partial {\Lambda}_N)\right]^2}{t}.$$
- We have for all $x$ $$\begin{cases}\label{kilcompare}
P_t(x,x)-P^*_t(x,x)\le \frac{C}{t}e^{-\frac{d(x,\partial {\Lambda}_N)^2}{C t}}, \quad &\text{ for } t\ge d(x,\partial {\Lambda}_N),\\
P_t(x,x)-P^*_t(x,x)\le \frac{C}{t}e^{-\frac{1}{C}d(x,\partial {\Lambda}_N) \log \left(\frac{ d(x,\partial {\Lambda}_N)}{t} \right) }, \quad &\text{ for } t\le d(x,\partial {\Lambda}_N).
\end{cases}$$
Cost of positivity constraints for Gaussian random walks
--------------------------------------------------------
Finally we conclude this preliminary section with an estimate for the probability to remain above a line for Gaussian random walks. The statement is not optimal and the term $(\log k)$ could be replaced by $1$ but as the rougher estimate is sufficient for our purpose we prefer to keep the proof simpler. We include the proof in the Appendix \[appendix\] for the sake of completeness.
\[lem:bridge\] Let $(X_i)_{i=1}^k$ be arandom walk with independent centered Gaussian increments, each of which with variance bounded above by $2$ and such that the total variance satisfies ${\mathrm{Var}}(X_k)\ge k/2$. Then we have for all $x\ge 0$ $$1-e^{-\frac{x^2}{k}}\le {{\ensuremath{\mathbf P}} }\left[ \max_{i}X_i\le x \ | \ X_k=0 \right]\le \frac{ C(x+(\log k))^2}{k}.$$
The upper-bound on the free energy {#seclower}
==================================
Let us briefly discuss the structure of the proof before going into more details. The main idea is presented in Section \[changeofme\]: we introduce a function which penalizes some environments ${\omega}$ which are too favorable, and use it to get a bet annealed bound which penalizes the trajectories with clustered contact points in a small region (Proposition \[nonrandom\]).
However, to perform the coarse-graining step of the proof, we need some kind of control on $\phi$. For this reason, in Section \[restrictou\] we start the proof by showing that restricting the partition function to a set of uniformly bounded trajectory does not affect a lot the free energy.
Restricting the partition function {#restrictou}
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In this section, we show that restricting the partition function by limiting the maximal height of the field $\phi$ does not affect too much the free energy. This statement is to be used to control the boundary condition of each cell when performing a coarse-graining argument in Proposition \[scorpiorizing\]. Let us set $${{\ensuremath{\mathcal A}} }^h_N := \left\{ \forall x \in {\Lambda}_N, \ |\phi_x|\le |\log h|^2 \right\},$$ and write $$Z^{{\beta},{\omega}}_{N,h}({{\ensuremath{\mathcal A}} }^h_N):={{\ensuremath{\mathbf E}} }_N \left[ \exp \left(\sum_{x\in {\widetilde}{\Lambda}_N}(h+{\beta}{\omega}_x-{\lambda}({\beta}))\delta_x\right){\mathbf{1}}_{{{\ensuremath{\mathcal A}} }^h_N} \right].$$
\[boundtheprob\] There exists a constant $c$ such for any $h\in(0,h_0)$ and ${\beta}>0$ we have $$\liminf_{N\to \infty} \frac{1}{N^2} {{\ensuremath{\mathbb E}} }\log {{\ensuremath{\mathbf P}} }^{{\beta},{\omega}}_{N,h}\left[ {{\ensuremath{\mathcal A}} }^h_N \right]\ge -\exp\left(-c |\log h|^2\right).$$
As a consequence, we have $${\textsc{f}}({\beta},h)\le \liminf_{N\to \infty} \frac{1}{N^2} {{\ensuremath{\mathbb E}} }\log Z^{{\beta},{\omega}}_{N,h}({{\ensuremath{\mathcal A}} }^h_N)+\exp\left(-c |\log h|^2\right).$$
For practical purposes we introduce the two following events $$\begin{split}\label{evvents}
{{\ensuremath{\mathcal A}} }^{h,1}_N&:= \left\{\forall x \in \mathring {\Lambda}_N,\ \phi_x\le |\log h|^2\right\},\\
{{\ensuremath{\mathcal A}} }^{h,2}_N&:= \left\{\forall x \in \mathring {\Lambda}_N,\ \phi_x\ge -|\log h|^2\right\},\\
{{\ensuremath{\mathcal B}} }_N&:= \left\{\forall x \in \mathring {\Lambda}_N,\ \phi_x\ge 1\right\},\\
{{\ensuremath{\mathcal C}} }_N&:= \left\{\forall x \in \mathring {\Lambda}_N,\ \phi_x\le -1 \right\}.
\end{split}$$ We have $ {{\ensuremath{\mathcal A}} }^{h}_N= {{\ensuremath{\mathcal A}} }^{h,1}_N\cap {{\ensuremath{\mathcal A}} }^{h,2}_N$. In order to obtain a bound on the probability of ${{\ensuremath{\mathcal A}} }^{h}_N$ we need to use the FKG inequality for the Gaussian free field which we present briefly (we refer to [@cf:Notes Section B.1] for more details). We denote by $\le$ the natural order on the set of functions $\{\phi, \ {\Lambda}_N \to {{\ensuremath{\mathbb Z}} }^d\}$ defined by $$\left\{ \ \phi\le \phi' \ \right\} \quad \Leftrightarrow \quad \left\{\ \forall x \in {\Lambda}_N, \ \phi_x\le \phi'_x \ \right\}.$$ An event $A$ is said to be increasing if for $\phi\in A$ we have $$\phi'\ge \phi \Rightarrow \phi'\in A$$ and decreasing if its complement is increasing. Let us remark that all the events described in are either decreasing or increasing. A probability measure $\mu$ is said to satisfy the FKG inequality if for any pair of increasing events $A, B$ we have $\mu(A \cap B)\le \mu(A)\mu( B)$. Note that this yields automatically similar inequalities for any pairs of monotonic events which we also call FKG inequalities.
It is well know that ${{\ensuremath{\mathbf P}} }_N$ satisfies the FKG inequality: it is sufficient to check that Holley’s criterion [@cf:FKG; @cf:Holley] is satisfied by the Hamiltonian in . The same argument yields that ${{\ensuremath{\mathbf P}} }^{{\beta},{\omega}}_{N,h}$ as well as the conditionned measures ${{\ensuremath{\mathbf P}} }^{{\beta},{\omega}}_{N,h}\left( \cdot \ | \ {{\ensuremath{\mathcal A}} }^{h,1}_N\right)$ and ${{\ensuremath{\mathbf P}} }^{{\beta},{\omega}}_{N,h}\left( \cdot \ | \ {{\ensuremath{\mathcal B}} }^{h,1}_N\right)$ also satisfy the FKG inequality. Hence using the FKG inequality for ${{\ensuremath{\mathbf P}} }^{{\beta},{\omega}}_{N,h}$, we have $${{\ensuremath{\mathbf P}} }^{{\beta},{\omega}}_{N,h}({{\ensuremath{\mathcal A}} }^{h,1}_N) \ge {{\ensuremath{\mathbf P}} }^{{\beta},{\omega}}_{N,h}({{\ensuremath{\mathcal A}} }^{h,1}_N \ | \ {{\ensuremath{\mathcal B}} }_N ) ={{\ensuremath{\mathbf P}} }_{N}({{\ensuremath{\mathcal A}} }^{h,1}_N \ | \ {{\ensuremath{\mathcal B}} }_N ).$$ Then, using the FKG inequality for ${{\ensuremath{\mathbf P}} }^{{\beta},{\omega}}_{N,h}( \ \cdot \ | \ {{\ensuremath{\mathcal A}} }^{h,1}_N)$ and we have $${{\ensuremath{\mathbf P}} }^{{\beta},{\omega}}_{N,h}\left( {{\ensuremath{\mathcal A}} }^{h,2}_N \ | \ {{\ensuremath{\mathcal A}} }^{h,1}_N\right)\ge {{\ensuremath{\mathbf P}} }^{{\beta},{\omega}}_{N,h}\left({{\ensuremath{\mathcal A}} }^{h,2}_N \ | \ {{\ensuremath{\mathcal C}} }_N \right)
\ge {{\ensuremath{\mathbf P}} }_{N}\left({{\ensuremath{\mathcal A}} }^{h,2}_N \ | \ {{\ensuremath{\mathcal C}} }_N \right)={{\ensuremath{\mathbf P}} }_{N}({{\ensuremath{\mathcal A}} }^{h,1}_N \ | \ {{\ensuremath{\mathcal B}} }_N ),$$ where we used symmetry to get the last equality. Then we can conclude that $${{\ensuremath{\mathbf P}} }^{{\beta},{\omega}}_{N,h}({{\ensuremath{\mathcal A}} }^{h}_N)= {{\ensuremath{\mathbf P}} }^{{\beta},{\omega}}_{N,h}({{\ensuremath{\mathcal A}} }^{h,1}_N \cap {{\ensuremath{\mathcal A}} }^{h,2}_N)
\ge \left[{{\ensuremath{\mathbf P}} }_{N}({{\ensuremath{\mathcal A}} }^{h,1}_N \ | \ {{\ensuremath{\mathcal B}} }_N )\right]^2
\ge \left[{{\ensuremath{\mathbf P}} }_{N}({{\ensuremath{\mathcal A}} }^{h,1}_N \cap {{\ensuremath{\mathcal B}} }_N )\right]^2.$$ We are left with estimating the last term. Note that changing the boundary condition by a constant amount does not affect the leading order of the asymptotic thus to conclude it is sufficient to bound asymptotically the probability of the event $${{\ensuremath{\mathcal A}} }^{h,3}_N:= \left\{ \max_{x \in {\Lambda}_N} |\phi_x| \le \frac{|\log h|^2-1}{2} \right\},$$ which is a translated version of ${{\ensuremath{\mathcal A}} }^{h,1}_N \cap {{\ensuremath{\mathcal B}} }_N$. More precisely we have for an adequate constant $K_h$ $${{\ensuremath{\mathbf P}} }^{{\beta},{\omega}}_{N,h}({{\ensuremath{\mathcal A}} }^{h}_N)\ge \exp(-K_h N) \left[{{\ensuremath{\mathbf P}} }_{N}({{\ensuremath{\mathcal A}} }^{h,3}_N) \right]^2$$ To bound the probability of ${{\ensuremath{\mathcal A}} }^{h,3}_N$ we use the following result, whose proof is postponed to the end of the Section.
\[controlgrid\] There exists a constant $C$ such that for any $N$, and for any set ${\Gamma}\subset \mathring{\Lambda}_N$ which is such that ${\Gamma}\cup \partial {\Lambda}_N$ is connected, we have $${{\ensuremath{\mathbf P}} }_{N}\left[ \max_{x \in {\Gamma}} |\phi_x|\le 1 \right] \ge \exp(-C |{\Gamma}|).$$
We divide ${\Lambda}_N$ in cells of side-length $$N_0:=\exp(c|\log h|^2)$$ for some small constant $c$. We set $${\Lambda}(y,N_0):= yN_0+ {\Lambda}_{N_0}.$$ We apply Lemma \[controlgrid\] for the following set $${\Gamma}_N=\left(\bigcup_{y\in {{\ensuremath{\mathbb Z}} }^2} \partial{\Lambda}(y,N_0)\right)\cap \mathring{\Lambda}_N,$$ which is is a grid which splits ${\Lambda}_N$ in cells of side-length $N_0$. We obtain that $$\frac{1}{N^2} \log{{\ensuremath{\mathbf P}} }_{N}\left[ \max_{x \in {\Gamma}_N} |\phi_x|\le 1 \right] \ge \frac{2C}{N_0}= 2C \exp(-c|\log h|^2),$$ where we used the inequality $|{\Gamma}_N|\le 2 N^2/N_0$ valid for all $N$. To conclude we need to show that $$\label{fdfe}
\frac{1}{N^2} \log {{\ensuremath{\mathbf P}} }_{N}\left[ \max_{x \in {\Lambda}_N} |\phi_x| \le \frac{|\log h|^2-1}{2} \ \Big| \ \max_{x \in {\Gamma}_N} |\phi_x|\le 1 \right] \ge - (N_0)^{-2}.$$ To prove it is sufficient to remark that conditioned to $(\phi_x)_{x\in {\Gamma}_N}$, the variance of the field $(\phi_x)_{x\in{\Lambda}(y,N_0)}$ is uniformly bounded by $\frac{1}{2\pi} \log N_0+C$ (cf. for $m=0$). Thus, for any realization of $\phi$ satisfying $\max_{x \in {\Gamma}_N} |\phi_x|\le 1$, for any $z\in {\Lambda}_N \setminus {\Gamma}_N$, using the Gaussian tail bound we have for $h$ sufficiently small $${{\ensuremath{\mathbf P}} }_N \left[ |\phi_z| \ge \frac{|\log h|^2-1}{2} \ \Big| \ (\phi_x)_{x\in {\Gamma}_N} \right] \le \exp\left( -\frac{ \pi |\log h|^4}{ 4 \log N_0} \right)\le
\exp\left(-\frac{\pi}{4c} |\log h|^2\right).$$ Now with this in mind we can apply union bound in ${\Lambda}(y,N_0)$ and obtain $$\begin{gathered}
\label{ddsad}
{{\ensuremath{\mathbf P}} }_{N}\left[ \max_{z \in {\Lambda}(y,N_0)} |\phi_z| \le \frac{|\log h|^2-1}{2} \ \Big| \ (\phi_x)_{x\in {\Gamma}_N} \right]\\
\ge 1- (N_0-1)^2\exp\left(-\frac{\pi}{4c} |\log h|^2\right)
\ge e^{-1/2}.\end{gathered}$$ where the last inequality is valid provided the constant $c$ is chosen sufficiently small. As, conditioned to the realization of $(\phi_x)_{x\in {\Gamma}_N}$, the fields $(\phi_x)_{x\in{\Lambda}(y,N_0)}$ are independent for different values of $y$, we prove that the inequality holds by multiplying for all distinct ${\Lambda}(y,N_0)$ which fit (at least partially) in ${\Lambda}_N$ (there are at most $(N/N_0)^2$ full boxes, to which one must add at most $2N/N_0+1$ uncompleted boxes), and taking the expectation with respect to $(\phi_x)_{x\in {\Gamma}_N}$ conditioned on the event $\max_{x \in {\Gamma}_N} |\phi_x|\le 1$. This ends the proof of Proposition \[boundtheprob\].
We can prove it by induction on the cardinality of ${\Gamma}$. Assume that the result is valid for ${\Gamma}$ and let us prove it for ${\Gamma}\cup\{z\}$. $${{\ensuremath{\mathbf P}} }_{N}\left[ \phi_z\in [-1,1] \ | \ \max_{x\in {\Gamma}} |\phi_x|\le 1 \right]\ge \exp(-C).$$ Note that conditioned to $(\phi_x)_{x\in {\Gamma}}$, $\phi_z$ is a Gaussian variable. Its variance is given by $$E_x\left[\int^{\tau_{\partial {\Lambda}_N \cup {\Gamma}}}_0 {\mathbf{1}}_{\{X_t=z\}} {\mathrm{d}}t \right]\le 1.$$ The reason being that as by assumption $\partial {\Lambda}_N \cup {\Gamma}\cup \{z\}$, the walk $X$ is killed with rate one while it lies on $z$. In addition, if $\max_{x\in {\Gamma}} |\phi_x|\le 1$, then necessarily
$${{\ensuremath{\mathbf E}} }_N\big[ \phi_z \ | \ (\phi_x)_{x\in {\Gamma}} \big]\in[-1,1].$$
For this reason, the above inequality is valid if one chooses $$C:=-\max_{u\in[-1,1]} \log P\left ({{\ensuremath{\mathcal N}} }\in [-1+u,1+u] \right)= -\log P\left({{\ensuremath{\mathcal N}} }\in [0,2] \right),$$ where ${{\ensuremath{\mathcal N}} }$ is a standard normal.
Change of measure {#changeofme}
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To bound the expectation of ${{\ensuremath{\mathbb E}} }[ \log Z^{{\beta},{\omega}}_{N,h} ({{\ensuremath{\mathcal A}} }^h_N)]$ we use a “change of measure” argument . The underlying idea is that the annealed bound obtained by Jensen’s inequality is not sharp because some very atypical ${\omega}$’s (a set of ${\omega}$ of small probability) give the most important contribution to the annealed partition function. Hence our idea is to identify these bad environments and to introduce a function $f({\omega})$ that penalizes them. This idea originates from [@cf:GLT] where it was used to prove the non-coincidence of critical point for a hierarchical variant of the pinning model and was then improved many times in the context of pinning [@cf:BL; @cf:DGLT; @cf:GLT2] and found application for other models like random-walk pinning, directed polymers, random walk in a random environment or self-avoiding walk in a random environment [@cf:BT; @cf:BS1; @cf:BS2; @cf:L2; @cf:L3; @cf:YZ].
In [@cf:BL; @cf:DGLT; @cf:GLT2], we used the detailed knowledge that we have on the structure of the set of contact points, (which is simply a renewal process) in order to find the right penalization function $f({\omega})$.
Here we have a much less precise knowledge on the structure $(\delta_x)_{x\in {\Lambda}_N}$ under ${{\ensuremath{\mathbf P}} }_N$ (especially because we have to consider possibly very wild boundart condition), but we know that one typical feature of the two-dimensional free field is that the level sets tend to have a clustered structure. We want to perform a change of measure that has the consequence of penalizing these clusters of contact points: we do so by looking at the empirical mean of ${\omega}$ in some small regions and by giving a penalty when it takes an atypically high value.
Let us be more precise about what we mean by penalizing with a function $f({\omega})$. Using Jensen inequality, we remark that $${{\ensuremath{\mathbb E}} }\left[ \log Z^{{\beta},{\omega}}_{N,h}( {{\ensuremath{\mathcal A}} }^h_N )\right]= 2{{\ensuremath{\mathbb E}} }\left[ \log \sqrt{ Z^{{\beta},{\omega}}_{N,h}( {{\ensuremath{\mathcal A}} }^h_N )}\right]\le 2 \log {{\ensuremath{\mathbb E}} }\left[ \sqrt{Z^{{\beta},{\omega}}_{N,h}( {{\ensuremath{\mathcal A}} }^h_N )} \right]$$ If we let $f({\omega})$ be an arbitrary positive function of $({\omega}_x)_{x\in {\widetilde}{\Lambda}_N}$, we have by Cauchy-Schwartz inequality $${{\ensuremath{\mathbb E}} }\left[ \sqrt{Z^{{\beta},{\omega}}_{N,h}( {{\ensuremath{\mathcal A}} }^h_N )} \right]^2 \le {{\ensuremath{\mathbb E}} }[f({\omega})^{-1}] {{\ensuremath{\mathbb E}} }\left[ f({\omega})Z^{{\beta},{\omega}}_{N,h}( {{\ensuremath{\mathcal A}} }^h_N )\right] ,$$ and hence $$\label{Cauchyschwartz}
\frac{1}{N^2}{{\ensuremath{\mathbb E}} }\left[ \log Z^{{\beta},{\omega}}_{N,h} ( {{\ensuremath{\mathcal A}} }^h_N )\right]\le \frac{1}{N^2}\log {{\ensuremath{\mathbb E}} }[f({\omega})^{-1}]+
\frac{1}{N^2}\log {{\ensuremath{\mathbb E}} }\left[ f({\omega})Z^{{\beta},{\omega}}_{N,h}( {{\ensuremath{\mathcal A}} }^h_N )\right] .$$ Let us now present our choice of $f({\omega})$. Our idea is to perform some kind of coarse-graining argument: we divide ${\Lambda}_N$ into cells of fixed side-length $N_1$ $$\label{cellsize}
N_1(h):=h^{-1/4},$$ and perform a change of measure inside of each cell. We assume that $N_1$ is an even integer (the free energy being monotone this causes no loss of generality), and that $N=k N_1$ is a sufficiently large multiple of $N_1$. Given $y\in {{\ensuremath{\mathbb Z}} }^2$, we let ${\widetilde}{\Lambda}_{N_1}(y)$ denote the translation of the box ${\widetilde}{\Lambda}_{N_1}$ which is (approximately) centered at $yN_1$ (see Figure \[fig:structure\]) $${\widetilde}{\Lambda}_{N_1}(y):= N_1\left[y-\left(\frac{1}{2},\frac{1}{2}\right)\right]+{\widetilde}{\Lambda}_{N_1}.$$ In the case $y=(1,1)$ we simply write ${\widetilde}{\Lambda}'_{N_1}$ (note that it is not identical to ${\widetilde}{\Lambda}_{N_1}$). We define the event $${{\ensuremath{\mathcal E}} }_{N_1}(y):=
\left\{ \exists x\in {\widetilde}{\Lambda}_{N_1}(y), \!\!\!\!\!\! \sum_{\{ z\in {\widetilde}{\Lambda}_{N_1}(y) \ : \ |z-x|\le (\log N_1)^2 \}} \!\!\!\!\!\! \!\!\!\!\!\! {\omega}_z \quad \ge \frac{{\lambda}'({\beta}) (\log N_1)^3}{2} \right\}.$$ which is simply denoted by ${{\ensuremath{\mathcal E}} }_{N_1}$ in the case when $y=(1,1)$. Here ${\lambda}'({\beta})$ denotes the derivative of ${\lambda}$ defined in . Finally we set $$f({\omega}):= \exp\left(- 2\sum_{y\in {\llbracket}1, k-1{\rrbracket}} {\mathbf{1}}_{{{\ensuremath{\mathcal E}} }_{N_1}(y)}\right).$$ The effect of $f({\omega})$ is to give a penalty (multiplication by $e^{-2}$) for each cell in which one can find a region of ${\omega}$ with diameter $(\log N_1)^2$ and atypically high empirical mean.
Combining Proposition \[boundtheprob\] and , we have (provided that the limit exists) $$\label{uppbn}
{\textsc{f}}({\beta},h)\le e^{-c(\log h)^{2}}+ \lim_{k\to \infty} \frac{1}{N^2}\log {{\ensuremath{\mathbb E}} }[(f({\omega}))^{-1}]+
\liminf_{k\to \infty} \frac{1}{N}\log {{\ensuremath{\mathbb E}} }\left[ f({\omega})Z^{{\beta},{\omega}}_{N,h}({{\ensuremath{\mathcal A}} }^h_N)\right].$$
We can conclude the proof with the two following results, which evaluate respectively the cost and the benefit of our change of measure procedure.
\[cost\] There exists positive constants $c({\beta})$ and $h_0({\beta})$ and such that for all $h\in (0,h_0({\beta}))$ sufficiently small, for all $k$ $$\log {{\ensuremath{\mathbb E}} }\left[ (f({\omega}))^{-1}\right]\le (k-1)^2 e^{-c({\beta})(\log h)^2}.$$ As a consequence we have $$\frac{1}{N^2}\log {{\ensuremath{\mathbb E}} }[(f({\omega}))^{-1}]\le e^{-c({\beta})(\log h)^2}.$$
\[benefit\] There exists $h_0({\beta})>0$ such that for all $h\in (0,h_0({\beta}))$ $$\limsup_{k\to \infty} \frac{1}{N^2}\log {{\ensuremath{\mathbb E}} }\left[ f({\omega})Z^{{\beta},{\omega}}_{N,h}({{\ensuremath{\mathcal A}} }^h_N)\right]\le e^{-2|\log h|^{3/2}}.$$
As a consequence of and of the two propositions above, we obtain that for $h\in(0, h_0({\beta})),$ we have $${\textsc{f}}({\beta},h)\le e^{-|\log h|^{3/2}}.$$ The proof of Proposition \[cost\] is simple and short and is presented below. The proof Proposition \[benefit\] requires a significant amount of work. We decompose it in important steps in the next subsection.
Because of the product structure, we have $$\label{cgt}
{{\ensuremath{\mathbb E}} }\left[ (f({\omega}))^{-1}\right]= \big( {{\ensuremath{\mathbb E}} }\left[ \exp\left( 2{{\ensuremath{\mathcal E}} }_{N_1}\right) \right]\big)^{(k-1)^2}.$$ Hence it is sufficient to obtain a bound on $$\label{fds}
\log {{\ensuremath{\mathbb E}} }\left[ \exp\left( 2{\mathbf{1}}_{{{\ensuremath{\mathcal E}} }_{N_1}}\right) \right]\le (e^2-1) {{\ensuremath{\mathbb P}} }[{{\ensuremath{\mathcal E}} }_{N_1}(0)].$$ As an easy consequence of the proof of Cramérs Theorem (see e.g. [@cf:DZ Chapter 2]), there exists a constant $c({\beta})$ that any $x\in {\widetilde}{\Lambda}_{N_1}$ $${{\ensuremath{\mathbb P}} }\left[ \sum_{\{ z\in {\widetilde}{\Lambda}'_{N_1} \ : \ |z-x|\le (\log N_1)^2 \}} \!\!\!\!\!\! \!\!\!\!\!\! {\omega}_z \ge \, \frac{{\lambda}'({\beta}) (\log N_1)^3}{2} \ \right]
\le e^{-c({\beta})(\log N_1)^2},$$ and by union bound we obtain that ${{\ensuremath{\mathbb P}} }[{{\ensuremath{\mathcal E}} }_{N_1}]\le N^2_0\exp(-c(\log N_1)^2)$, which in view of and is sufficient to conclude
=10.5 cm \[c\]\[l\][$0$]{} \[c\]\[l\][ $N_1/2$]{} \[c\]\[l\][ $3N_1/2$]{} \[c\]\[l\][ $N=kN_1$]{} \[c\]\[l\][ $N$]{} \[c\]\[l\][${\widetilde}{\Lambda}_{N_1}(1,4)$]{} \[c\]\[l\][${\Lambda}_{2N_1}(1,4)$]{} \[c\]\[l\][${\widetilde}{\Lambda}_{2N_1}(4,2)$]{}
\[fig:structure\]
Decomposing the proof of Proposition \[benefit\]
------------------------------------------------
The proof is split in three steps, whose details are performed in Section \[proufin\], \[proufeux\] and \[proufoix\] respectively. In the first one we show that our averaged partition function ${{\ensuremath{\mathbb E}} }\left[ f({\omega}) Z^{{\beta},{\omega}}_{N,h}({{\ensuremath{\mathcal A}} }^h_N)\right]$, can be bounded from above by the partition of an homogenous system where an extra term is added in the Hamiltonian to penalize the presence of clustered contact in a small region (here a region of diameter $(\log N_1)^2$). We introduce the event ${{\ensuremath{\mathcal C}} }_{N_1}(y)$ which indicates the presence of such a cluster in ${\widetilde}{\Lambda}_{N_1}(y)$, $${{\ensuremath{\mathcal C}} }_{N_1}(y):= \left \{ \exists x\in {\widetilde}{\Lambda}_{N_1}(y), \!\!\!\!\!\! \!\! \sum_{\{ z\in {\widetilde}{\Lambda}_{N_1}(y) \ : \ |z-x|\le (\log N_1)^2 \}}
\!\!\!\!\!\! \!\!\!\!\!\! {\delta}_z \ge \ (\log N_1)^3 \right\}.$$ We simply write ${{\ensuremath{\mathcal C}} }_{N_1}$ for the case $y=(1,1)$.
\[nonrandom\] We have $$\begin{gathered}
\label{gteraf}
{{\ensuremath{\mathbb E}} }\left[ f({\omega}) Z^{{\beta},{\omega}}_{N,h}({{\ensuremath{\mathcal A}} }^h_N)\right]\\
\le
{{\ensuremath{\mathbf E}} }_N\left[ \exp\left( h\sum_{x \in {\widetilde}{\Lambda}_{N}} \delta_x - \sum_{y \in {\llbracket}1, k-1 {\rrbracket}^2} {\mathbf{1}}_{{{\ensuremath{\mathcal C}} }_{N_1}(y)} \right){\mathbf{1}}_{{{\ensuremath{\mathcal A}} }^h_N} \right]=:
{\widehat}Z(N,N_1,h).\end{gathered}$$
In the second step, we perform a factorization in order to reduce the estimate of ${\widehat}Z(N,N_1,h)$ to that of similar system with only one cell. Let us set (see Figure \[fig:structure\]) $${\Lambda}_{2N_1}(y):= N_1\big( y-(1,1)\big)+{\Lambda}_{2N_1}.$$ Note that for every for $y\in {\llbracket}1,k-1{\rrbracket}^2$ we have ${\Lambda}_{2N_1}(y)\subset {\Lambda}_N$ and that ${\Lambda}_{2N_1}((1,1))={\Lambda}_{2N_1}$.
\[scorpiorizing\] We have
$${\widehat}Z(N,N_1,h)\le e^{2 N_1 N h}\left(\max_{\{ {\widehat}\phi \ : \ |{\widehat}\phi|_{\infty}\le |\log h|^2\} }
{{\ensuremath{\mathbf E}} }^{{\widehat}\phi}_{2N_1}\left[e^{4h\sum\limits_{x \in {\widetilde}{\Lambda}'_{N_1}}
\delta_x -4{\mathbf{1}}_{{{\ensuremath{\mathcal C}} }_{N_1}}} \right] \right)^{\frac{(k-1)^2}{4}}$$
Let us notice two important features in our factorization which are present to reduce possible nasty boundary effects:
- There is a restriction on the boundary condition $|{\widehat}\phi|_{\infty}\le |\log h|^2$, which forbids wild behavior of the field. This restriction is directly inherited from the restriction to ${{\ensuremath{\mathcal A}} }^h_N$ in the partition function and brings some light on the role of Proposition \[boundtheprob\] in our proof.
- The Hamiltonian $$4h\sum\limits_{x \in {\widetilde}{\Lambda}'_{N_1}}
\delta_x- 4{\mathbf{1}}_{{{\ensuremath{\mathcal C}} }_{N_1}},$$ is a functional of $(\phi_x)_{x\in {\widetilde}{\Lambda}'_{N_1}}$ i.e. of the field restricted to a region which is distant from the boundary of the box $\partial {\Lambda}_{2N_1}$.
The final step of the proof consists in evaluating the contribution of one single cell to the partition function.
\[onecell\]
There exists a constant $c$ such that for all $h$ sufficiently small for all ${\widehat}\phi$ satisfying $|{\widehat}\phi|_{\infty}\le |\log h|^2$ we have $$\log {{\ensuremath{\mathbf E}} }^{{\widehat}\phi}_{2N_1}\left[e^{ 4h\sum\limits_{x \in {\widetilde}{\Lambda}_{N_1}}
\delta_x- 4{\mathbf{1}}_{{{\ensuremath{\mathcal C}} }_{N_1}} } \right]\le e^{-2(\log h)^{3/2}}.$$
Combining the three results presented above, we have $${{\ensuremath{\mathbb E}} }\left[ f({\omega}) Z^{{\beta},{\omega}}_{N,h}({{\ensuremath{\mathcal A}} }^h_N)\right]\le 2 N_1 N h+\frac{(k-1)^2}{4}e^{-2(\log h)^{3/2}},$$ and this is sufficient to conclude the proof of Proposition \[benefit\].
Proof of Proposition \[nonrandom\] {#proufin}
----------------------------------
Given a realization $\phi$, we let ${{\ensuremath{\mathbb P}} }^{\phi}$ be a probability law which is absolutely continuous with respect to ${{\ensuremath{\mathbb P}} }$ and whose the density is given by $$\frac{{\mathrm{d}}{{\ensuremath{\mathbb P}} }^{ \phi}}{{\mathrm{d}}{{\ensuremath{\mathbb P}} }}({\omega}):= \exp\left(\sum_{x\in {\widetilde}{\Lambda}_N} \left({\beta}{\omega}_x-{\lambda}({\beta})\right)\delta_x \right).$$ Under ${{\ensuremath{\mathbb P}} }^{ \phi}$, the variables $({\omega}_x)_{x\in {{\ensuremath{\mathbb Z}} }^d}$ are still independent but they are not IID, as the law of the ${\omega}_x$s for which $\delta_x=1$ have been tilted. In particular it satisfies $${{\ensuremath{\mathbb E}} }^{\phi}[{\omega}_x]= {\lambda}'({\beta})\delta_x \quad \text{ and } {\mathrm{Var}}_{{{{\ensuremath{\mathbb P}} }}^{\phi}}[{\omega}_x]=1+ ({\lambda}''({\beta})-1)\delta_x$$ where ${\lambda}'({\beta})$ and ${\lambda}''({\beta})$ denote the two first derivatives of ${\lambda}$ the function defined in . This notation gives us another way of writing the quantity that we must estimate $$\label{rewrite}
{{\ensuremath{\mathbb E}} }\left[ f({\omega})Z^{{\beta},{\omega}}_{N,h}({{\ensuremath{\mathcal A}} }^h_N)\right]= {{\ensuremath{\mathbf E}} }_N\left[ {{\ensuremath{\mathbb E}} }^{\phi}[f({\omega})] e^{h \sum_{x \in {\widetilde}{\Lambda}_N }\delta_x}{\mathbf{1}}_{{{\ensuremath{\mathcal A}} }^h_N}\right].$$ To conclude it is sufficient to prove that $${{\ensuremath{\mathbb E}} }^{\phi}[f({\omega})] \le \exp \left( -\sum_{y \in {\llbracket}0, k-1 {\rrbracket}} {\mathbf{1}}_{{{\ensuremath{\mathcal C}} }_{N_1}(y)} \right).$$ Note that because both ${{\ensuremath{\mathbb E}} }^{\phi}$ and $f({\omega})$ have a product structure, it is in fact sufficient to prove that for any $y\in {\llbracket}0, k-1 {\rrbracket}^2$ we have $$\label{unecase}
{{\ensuremath{\mathbb E}} }^{\phi}\left[e^{-2 {\mathbf{1}}_{{{\ensuremath{\mathcal E}} }_{N_1}(y)}}\right]\le e^{-{\mathbf{1}}_{{{\ensuremath{\mathcal C}} }_{N_1}(y)}}.$$ With no loss of generality we assume that $y=(1,1)$. The result is obvious when $\phi \notin {{\ensuremath{\mathcal C}} }_{N_1}$ hence we can also assume $\phi \in {{\ensuremath{\mathcal C}} }_{N_1}$. Let $x_0\in {\widetilde}{\Lambda}'_{N_1}$ be a vertex satisfying $$\sum_{\{z\in {\Lambda}'_{N_1} \ : \ |z-x_0|\le (\log N_1)^2\}} \delta_x \ge (\log N_1)^3,$$ (e.g. the smallest one for the lexicographical order). We have $$\begin{split}
{{\ensuremath{\mathbb E}} }^{\phi}\left[ \sum_{\{ z\in {\widetilde}{\Lambda}'_{N_1} \ : \ |z-x_0|\le (\log N_1)^2\}} \!\!\!\!\!\! \!\! {\omega}_z \ \right]&= {\lambda}'({\beta}) \!\!\!\!\!\! \!\!\!\!\!\!
\sum_{\{ z\in {\widetilde}{\Lambda}'_{N_1} \ : \ |z-x_0|\le (\log N_1)^2\}} \!\!\!\!\!\! \!\!\!\!\!\! {\delta}_z \, \ge \, {\lambda}'({\beta})(\log N_1)^3,\\
{\mathrm{Var}}_{{{\ensuremath{\mathbb P}} }^{\phi}}\left[ \sum_{\{ z\in {\widetilde}{\Lambda}'_{N_1} \ : \ |z-x_0|\le (\log N_1)^2\}} \!\!\!\!\!\! \!\! {\omega}_z \ \right]
&\le \left[2(\log N_1)^2+1\right]^2\max({\lambda}''({\beta}),1).
\end{split}$$ Hence in particular if $N_1$ is sufficiently large, Chebychev’s inequality gives $${{\ensuremath{\mathbb P}} }^{\phi} \left[{{\ensuremath{\mathcal E}} }_{N_1}\right]\le e^{-1}-e^{-2},$$ which implies .
Proof of Proposition \[scorpiorizing\] {#proufeux}
--------------------------------------
We start by taking care of the contribution of the contact points located near the boundary $\partial {\Lambda}_N$, as they are not included in any ${\widetilde}{\Lambda}_{N_1}(y)$. Assuming that all these points are contact points we obtain the following crude bound $$\sum_{x \in {\widetilde}{\Lambda}_N } \delta_x\le \left[N^2-(k-1)^2N_1^2\right] +\sum_{y \in {\llbracket}1, k-1 {\rrbracket}^2} \sum_{x \in {\widetilde}{\Lambda}_{N_1}(y)} \delta_x.$$ and the first term is smaller than $2NN_1$. Hence we have
$$\label{gteraf2}
{\widehat}Z(N,N_1,h)
\le e^{2N_1 N h} {{\ensuremath{\mathbf E}} }_N\left[ e^{\sum_{y \in {\llbracket}1,k-1{\rrbracket}^2} \left( h\sum_{x \in {\widetilde}{\Lambda}_{N_1}(y)} \delta_x -
{\mathbf{1}}_{{{\ensuremath{\mathcal C}} }_{N_1}(y)}\right)} {\mathbf{1}}_{{{\ensuremath{\mathcal A}} }^h_N}\right].$$
We partition the set of indices ${\llbracket}1, k-1 {\rrbracket}^2$ into $4$ subsets, according to the parity of the of the coordinates. If we let $\alpha_1(i)$ and $\alpha_2(i)$ denote the first and second diadic digits of $i-1$. We set $$\Xi(i):= \big\{ y=(y_1,y_2) \in {\llbracket}1,k-1{\rrbracket}^2 \ : \ \forall j \in \{1,2\}, \, y_j \stackrel{(\text{mod } 2)}{=}\alpha_j(i)\big\}.$$ Using Hölder’s inequality we have $$\begin{gathered}
{{\ensuremath{\mathbf E}} }_N\left[ e^{\sum_{y \in {\llbracket}1,k-1{\rrbracket}^2} \left( h\sum_{x \in {\widetilde}{\Lambda}_{N_1}(y)} \delta_x -
{\mathbf{1}}_{{{\ensuremath{\mathcal C}} }_{N_1}(y)}\right)} {\mathbf{1}}_{{{\ensuremath{\mathcal A}} }^h_N}\right]^{4}
\\
\le \prod_{i=1}^4
{{\ensuremath{\mathbf E}} }_N\left[ e^{4\sum_{y \in \Xi(i)} \left( h\sum_{x \in {\widetilde}{\Lambda}_{N_1}(y)} \delta_x -
{\mathbf{1}}_{{{\ensuremath{\mathcal C}} }_{N_1}(y)}\right)} {\mathbf{1}}_{{{\ensuremath{\mathcal A}} }^h_N}\right].\end{gathered}$$ For a fixed $i\in {\llbracket}1, 4 {\rrbracket}$, the interiors of the boxes ${\Lambda}_{2N_1}(y)$, $y\in \Xi(i)$ are disjoint (neighboring boxes overlap only on their boundary, we refer to Figure \[fig:structure\]). This gives us a way to factorize the exponential: let us condition the expectation to the realization of $(\phi_x)_{x\in {\Gamma}(i)}$ where $${\Gamma}(i):= \bigcup_{y \in \Xi(i)} \partial {\Lambda}_{2N_1}(y).$$ The spatial Markov property implies that conditionally on $(\phi_x)_{x\in {\Gamma}(i)}$, the restrictions $\left[(\phi_x)_{x\in {\Lambda}_{2N_1}(y)}\right]_{y \in \Xi(i)}$ are independent. Hence we can factorize the expectation and get $$\begin{gathered}
\label{yipendence}
{{\ensuremath{\mathbf E}} }_N\left[ e^{4\sum_{y \in \Xi(i)} \left( h\sum_{x \in {\widetilde}{\Lambda}_{N_1}(y)} \delta_x -
{\mathbf{1}}_{{{\ensuremath{\mathcal C}} }_{N_1}(y)}\right)} \ | \ (\phi_x)_{x\in {\Gamma}(i)} \right]\\
\le
\prod_{y\in \Xi(i)} {{\ensuremath{\mathbf E}} }_N\left[ e^{4 \left( h\sum_{x \in {\widetilde}{\Lambda}_{N_1}(y)} \delta_x -
{\mathbf{1}}_{{{\ensuremath{\mathcal C}} }_{N_1}(y)}\right)} \ | \ (\phi_x)_{x\in {\Gamma}(i)} \right].\end{gathered}$$ On the event $${{\ensuremath{\mathcal A}} }^h(i):= \{\max_{x\in {\Gamma}(i)} |\phi_x|\le |\log h|^2 \},$$ we have for any $y\in \Xi(i)$, by translation invariance, $$\begin{gathered}
{{\ensuremath{\mathbf E}} }_N\left[ e^{4 \left( h\sum_{x \in {\widetilde}{\Lambda}_{N_1}(y)} \delta_x -
{\mathbf{1}}_{{{\ensuremath{\mathcal C}} }_{N_1}(y)}\right)} \ | \ (\phi_x)_{x\in {\Gamma}(i)} \right]\\
\le
\max_{\{ {\widehat}\phi \ : \ \|{\widehat}\phi\|_{\infty}\le |\log h|^2 \}}
{{\ensuremath{\mathbf E}} }^{{\widehat}\phi}_{2N}\left[e^{4h\left( \sum_{x \in {\widetilde}{\Lambda}'_{N_1}}
\delta_x\right) - 4{\mathbf{1}}_{{{\ensuremath{\mathcal C}} }_{N_1}}}
\right].
\end{gathered}$$ and hence we can conclude by taking the expectation of restricted to the event ${{\ensuremath{\mathcal A}} }^h(i)$ (which includes ${{\ensuremath{\mathcal A}} }^h_N$).
Proof of Proposition \[onecell\] {#proufoix}
--------------------------------
Note that because of our choice of $N_1=h^{-1/4}$ we always have $$h \sum_{x \in {\widetilde}{\Lambda}'_{N_1}} \delta_x \le h N^2_1\le h^{1/2},$$ which is small. Hence for that reason, if $h$ is sufficiently small, the Taylor expansion of the exponential gives $$\begin{gathered}
\label{xfiles}
\log {{\ensuremath{\mathbf E}} }^{{\widehat}\phi}_{2N_1}\left[e^{4h\sum_{x \in {\widetilde}{\Lambda}'_{N_1}}
\delta_x- 4{\mathbf{1}}_{{{\ensuremath{\mathcal C}} }_{N_1}}}\right]
\le \log {{\ensuremath{\mathbf E}} }^{{\widehat}\phi}_{2N_1}\Big[1+ 5h\sum_{x \in {\widetilde}{\Lambda}'_{N_1}}
\delta_x- \frac{1}{2}{\mathbf{1}}_{{{\ensuremath{\mathcal C}} }_{N_1}} \Big]
\\ \le 5h {{\ensuremath{\mathbf E}} }^{{\widehat}\phi}_{2N_1}\Big[ \sum_{x \in {\widetilde}{\Lambda}'_{N_1}}
\delta_x \Big]- \frac{1}{2}{{\ensuremath{\mathbf P}} }^{{\widehat}\phi}_{2N_1}[{{\ensuremath{\mathcal C}} }_{N_1}]\\
\le 5 N_1^{-2} \max_{x\in {\widetilde}{\Lambda}'_{N_1}}{{\ensuremath{\mathbf P}} }^{{\widehat}\phi}_{2N_1}[\phi_x\in[-1,1]]-\frac{1}{2}{{\ensuremath{\mathbf P}} }^{{\widehat}\phi}_{2N_1}[{{\ensuremath{\mathcal C}} }_{N_1}].\end{gathered}$$ We have to prove that the r.h.s. is small. Before going into technical details let us quickly expose the main idea of the proof. For the r.h.s. of to be positive, we need $$\label{hpuissance}
\frac{\max_{x\in {\widetilde}{\Lambda}'_{N_1}}{{\ensuremath{\mathbf P}} }^{{\widehat}\phi}_{2N_1}\big(\phi_x\in[-1,1]\big)}{{{\ensuremath{\mathbf P}} }^{{\widehat}\phi}_{2N_1}[{{\ensuremath{\mathcal C}} }_{N_1}]}\ge \frac{N_1^2}{10}.$$ What we are going to show is that for this ratio to be large we need the boundary condition ${\widehat}\phi$ to be very high above the substrate (or below by symmetry), but that in that case the quantity $\left(\max_{x\in {\widetilde}{\Lambda}'_{N_1}}{{\ensuremath{\mathbf P}} }^{{\widehat}\phi}_{2N_1}[\phi_x\in[-1,1]]\right)$ itself has to be very small and this should allow ourselves to conclude.
To understand the phenomenon better we need to introduce quantitative estimates. Let $G^*$ denote the Green function in the box ${\Lambda}_{2N_1}$ with $0$ boundary condition, and set $$V_{N_1}:= \max_{x\in {\widetilde}{\Lambda}'_{N_1}} G^*(x,x).$$ We have from Lemma \[Greenesteem\] $$|V_{N_1}- \frac{1}{2\pi} \log N_1 | \le C.$$ Recall that from we have $${{\ensuremath{\mathbf P}} }^{{\widehat}\phi}_{2N_1}\big( \ \phi_x\in[-1,1] \ \big)\le {{\ensuremath{\mathbf P}} }_{2N_1}\left(\phi_x\in \left[-1-H^{{\widehat}\phi}_{2N_1}(x),1-H^{{\widehat}\phi}_{2N_1}(x)\right] \ \right).$$ With this in mind we fix $$u=u({\widehat}\phi, N_1):=\min_{x\in {\widetilde}{\Lambda}_N} |H^{{\widehat}\phi}_{2N_1}(x)|.$$ Hence using basic properties of the Gaussian distribution, we obtain (provided that $h$ is sufficiently small) $$\label{krix}
\max_{x\in {\widetilde}{\Lambda}'_{N_1}}{{\ensuremath{\mathbf P}} }^{{\widehat}\phi}_{2N_1}\big(\phi_x\in[-1,1]\big) \le e^{-\frac{(u-1)^2}{2V_{N_1}}}.$$ It requires a bit more work to obtain a good lower bound for ${{\ensuremath{\mathbf P}} }^{{\widehat}\phi}_{2N_1}[{{\ensuremath{\mathcal C}} }_{N_1}]$ which is valid for all values of $u$. Fortunately we only need a rough estimate as the factor $N^2_1$ in gives us a significant margin in the computation.
Recall that $P_t^*$ denotes the two-dimensional heat-kernel with zero boundary condition on $\partial {\Lambda}_{2N_1}$. Let us set $$V'_{N_1}:= \min_{x \in {\widetilde}{\Lambda}'_{N_1}} \int_{(\log N_1)^8}^{\infty} P^*_t(x,x) {\mathrm{d}}t.$$ From the estimates in Lemma \[lem:kerestimate\], we can deduce that $$\label{boundvn}
\left|V'_{N_1}- \frac{1}{2\pi}\left(\log N_1 - 4 \log \log N_1\right)\right|- C.$$ For instance we have $$\label{okless}
\left| \int_{0}^{(\log N_1)^8} P^*_t(x,x) {\mathrm{d}}t - \frac{2}{\pi} \log \log N_1 \right|\le \frac{C}{2}.$$ for some appropriate $C$ (the estimate is obtained using and ) so that the result can be deduced from the estimate in the Green-function .
\[boundforcluster\] For all $h$ sufficiently small, for all ${\widehat}\phi$ satisfying $|{\widehat}\phi|_{\infty}<|\log h|^2$, and all $u\in (0, (2\log N_1)^2)$ we have $${{\ensuremath{\mathbf P}} }^{{\widehat}\phi}_{2N_1}[{{\ensuremath{\mathcal C}} }_{N_1}]\ge c(\log N_1)^{-1} e^{-\frac{u^2}{2V'_{N_1}}},$$
Combining the above result with and We have
$$\begin{gathered}
\label{zorrib}
\max_{\{ {\widehat}\phi \ : \ |{\widehat}\phi|\le |\log h|^2 \}} \log
{{\ensuremath{\mathbf E}} }^{{\widehat}\phi}_{2N_1}\left[\exp\left( 4h\sum_{x \in {\widetilde}{\Lambda}_{N_1}}
\delta_x- 4{\mathbf{1}}_{{{\ensuremath{\mathcal C}} }_{N_1}} \right) \right]\\\le
\sup_{u>0}\left( 5N_1^{-2} e^{-\frac{(u-1)^2}{2V_{N_1}}}- c( 2\log N_1)^{-1}e^{-\frac{u^2}{2V'_{N_1}}}{\mathbf{1}}_{\{ u \le (\log N_1)^2\}} \right)
\\ = \sup_{u>0} \frac{5e^{-\frac{(u-1)^2}{2V_{N_1}}}}{N_1^2}\left[1- \frac{cN_1^2}{10(\log N_1)} e^{-
\frac{u^2(V_{N_1}-V'_{N_1})}{2V'_{N_1}V_{N_1}}-\frac{2u-1}{2V_{N_1}}} {\mathbf{1}}_{\{ u \le (\log N_1)^2\}}\right].\end{gathered}$$
Now note that for the second factor to be positive, we need one of the terms in the exponential to be at least of order $\log N_1$ in absolute value. Using the estimates we have for $V'_{N_1}$ and $V_{N_1}$, we realize that the exponential term is larger than $$c\exp\left(-\frac{cu^2(\log \log N_1)}{\log N_1}\right),$$ and hence the expression is negative if $u^2\le c (\log N_1)^3 (\log \log N_1)^{-1},$ for some small $c$. For the other values of $u$ we can just consider the first factor which already gives a satisfying bound, and we can conclude that the l.h.s. of is smaller than $$e^{-\frac{c (\log N_1)^2}{\log \log N_1}}
\le e^{-c |\log h|^{3/2}}.$$
Decomposing the proof of Proposition \[boundforcluster\]
--------------------------------------------------------
We show here how to split the proof the proposition into three lemmas which we prove in the next subsection. Set $$x_{\min}:=\operatorname*{\mathrm{argmin}}_{x\in {\widetilde}{\Lambda}'_N} |H^{{\widehat}\phi}_{2N_1}(x)|,$$ (it is not necessarily unique but in the case it is not we choose one minimizer in a deterministic manner) and $${\widehat}{\Lambda}:= \{ z \in {\widetilde}{\Lambda}'_{N_1} \ : \ |x_{\min}-z|\le (\log N_1)^2 \}$$ We bound from below the probability of ${{\ensuremath{\mathcal C}} }_{N_1}$ by only examining the possibility of having a cluster of contact around $x_{\min}$. Using we have $$\begin{gathered}
\label{kxmin}
{{\ensuremath{\mathbf P}} }^{{\widehat}\phi}_{2N_1}[ {{\ensuremath{\mathcal C}} }_{N_1}]\le {{\ensuremath{\mathbf P}} }^{{\widehat}\phi}_{2N_1}\left[ \sum_{z\in {\widehat}{\Lambda}} \delta_z \ge
(\log N_1)^3 \right]\\
= {{\ensuremath{\mathbf P}} }_{2N_1}\left[ \sum_{z \in {\widehat}{\Lambda}} {\mathbf{1}}_{[-1,1]}\left(\phi_z+H^{{\widehat}\phi}_{2N_1}(x)\right)\ge
(\log N_1)^3 \right].\end{gathered}$$
To estimate the last probability, we first remark that for $x \in {\widehat}{\Lambda}$, $H^{{\widehat}\phi}_{2N_1}(x)$ is very close to $H^{{\widehat}\phi}_{2N_1}(x_{\min})$ which we assume to be equal to $-u$ for the rest of the proof (the case $H^{{\widehat}\phi}_{2N_1}(x_{\min})=+u$ is exactly similar). The factor $\log N_1$ in the estimate is not necessary, but it yields a much simpler proof.
\[regular\] We have for all $x, y \in {\widetilde}{\Lambda}'_{N_1}$ $$\left| H^{{\widehat}\phi}_{2N_1}(x)-H^{{\widehat}\phi}_{2N_1}(y) \ \right| \le \frac{C |{\widehat}\phi|_{\infty}(\log N_1)|x-y|}{N_1}.$$ In particular if $h$ is sufficiently small, $|x-y|\le (\log N_1)^2$ and $|{\widehat}\phi|_{\infty}\le |\log h|^2$, we have $$|H^{{\widehat}\phi}_{2N_1}(x)-H^{{\widehat}\phi}_{2N_1}(y)|\le 1/4.$$
Then to estimate the probability for $\phi$ to form a cluster of point close to height $u$, we decompose the field $(\phi_x)_{x\in {\widehat}{\Lambda}}$ into a rough field $\phi_1$ which is almost constant on the scale $(\log N_1)^2$ and an independent field $\phi_2$ which accounts for the local variations of $\phi$. We set $$\begin{split}
Q^1(x,y):= \int_{(\log N)^8}^{\infty} P^*_t(x,y) {\mathrm{d}}t,\\
Q^2(x,y):= \int_0^{(\log N)^8} P^*_t(x,y) {\mathrm{d}}t.
\end{split}$$ We let $(\phi_1(x))_{x\in {\Lambda}_{2N_1}}$ and $(\phi_2(x))_{x\in {\Lambda}_{2N_1}}$ denote two independent centered fields with respective covariance function $Q^1$ and $Q^2$. By construction the law of $\phi_1+\phi_2$ has a law given by ${{\ensuremath{\mathbf P}} }_{2N_1}$, and thus we set for the remainder of the proof $$\phi:= \phi_1+\phi_2,$$ and use ${{\ensuremath{\mathbf P}} }_{2N_1}$ to denote the law of $(\phi_1, \phi_2)$. We have by standard properties of Gaussian variables that for every $u>0$, and for $h$ sufficiently small $$\label{onyva}
{{\ensuremath{\mathbf P}} }_{2N_1}\left[\phi_1(x_{\min})\in [u-1/4,u+1/4]\right]\ge \frac{1}{4 \sqrt{ 2\pi V_{N_1} }}e^{-\frac{u^2}{2V'_{N_1}}} \ge \frac{1}{5 \sqrt{\log N_1}}e^{-\frac{u^2}{2V'_{N_1}}}.$$ Now we have to check that the field $\phi_1$ remains around level $u$ on the whole box ${\widehat}{\Lambda}$.
\[locareg\] There exists a constant $c$ such that for all $h$ sufficiently small we have $${{\ensuremath{\mathbf P}} }_{2N_1}\left[ \exists y\in {\widehat}{\Lambda}, \ |\phi_1(y)-\phi_1(x_{\min})| >1/4 \right] \le e^{-c(\log N_1)^4}.$$
Finally we show that it is rather likely for $\phi_2$ to have a lot of points around level zero.
\[lastep\] There exists a constant $c$ such that for all $h$ sufficiently small we have $$\label{hooop}
{{\ensuremath{\mathbf P}} }_N\left[ \sum_{z\in {\widehat}{\Lambda}} {\mathbf{1}}_{\{ |\phi_2(z)|\le 1/4\} } \ge (\log N_1)^3 \right]\ge c(\log\log N_1)^{-1/2}.$$
We can now combine all these ingredient into a proof
According to Lemma \[regular\], if $|x_{\min}-z|\le (\log N_1)^2$ we have $$\begin{gathered}
\left\{ (\phi_z+H^{{\widehat}\phi}_{2N_1}(z))\in [-1,1] \right\} \supset \left\{ \phi_z \in [-3/4+u,3/4+u] \right\} \\
\supset
\left\{ \ \left |\phi_1(x_{\min})-u \right| \le 1/4 \right\}\cap \left\{ |\phi_1(x_{\min})-\phi_1(z)| \le 1/4 \right\} \cap \left\{ |\phi_2(z)| \le 1/4 \right\}.\end{gathered}$$ Thus we obtain as a consequence $$\begin{gathered}
\left\{ \sum_{z \in {\widehat}{\Lambda}} {\mathbf{1}}_{[-1,1]}(\phi_z+H^{{\widehat}\phi}_{2N_1}(x)) \ge
(\log N_1)^3\right\} \\
\subset \left\{ \ \left|\phi_1(x_{\min})-\phi_1(z) \right| \le 1/4 \right\} \cap \left\{ \forall z\in {\widehat}{\Lambda}, \ |\phi_1(x_{\min})-\phi_1(z)| \le 1/4 \right\}\\
\cap \Big\{ \sum_{z\in {\widehat}{\Lambda}} {\mathbf{1}}_{\{ |\phi_2(z)|\le (1/4)\} } \ge (\log N_1)^3 \Big\}.\end{gathered}$$ Using combined with Lemmas \[locareg\] and \[lastep\] and the independence of $\phi_1$ and $\phi_2$ we conclude that $$\begin{gathered}
{{\ensuremath{\mathbf P}} }_{2N_1} \left[ \sum_{z \in {\widehat}{\Lambda}} {\mathbf{1}}_{[-1,1]}(\phi_z+H^{{\widehat}\phi}_{2N_1}(x) \ge
(\log N_1)^3 \right] \\
\ge \left[ \frac{c}{\sqrt{ \log N_1 }}e^{-\frac{u^2}{2V'_{N_1}}} - e^{-c(\log N)^4} \right]c(\log N_1)^{-1/2} \ge \frac{c'}{(\log N_1)}e^{-\frac{u^2}{2V'_{N_1}}}.\end{gathered}$$ where the last inequality is holds if $u\le (\log N_1)^2$ and $h$ is sufficiently small. We can thus conclude using .
Proof of the technical lemmas
-----------------------------
Given $x, y \in {\widetilde}{\Lambda}'_{N_1}$, let $X^x$ and $X^y$ be two simple random walk starting from $x=(x_1,x_2)$ and $y=(y_1,y_2)$, and coupled as follows: the coupling is made as the product of two one-dimensional couplings, along each coordinate the walk are independent until the coordinate match, then they move together. Let $\tau_{x,y}$ be the time where the two walks meet and $\tau^x_{\partial{\Lambda}_{2N_1}}$ be the time when $X^x$ hits the boundary. Recalling we have $$\left|H^{{\widehat}\phi}_{2N_1}(y)- H^{{\widehat}\phi}_{2N_1}(x)\right| \le |{\widehat}\phi|_{\infty} P\left[ \tau_{x,y}< \tau^x_{\partial {\Lambda}_{2N_1}}\right].$$ We conclude by showing that $$P\left[ \tau_{x,y}< \tau^x_{\partial {\Lambda}_{2N_1}} \right]< \frac{C|x-y| (\log N_1) }{N_1}$$ By union bound, we can reduce to the one dimensional case. Let $Y^x$ and $Y^y$ denote the first coordinates of $X^x$ and $X^y$. Until the collision time, they are two independent one dimensional random walk in ${\llbracket}0 , 2N_1 {\rrbracket}$ with initial condition $x_1$ and $y_1$ in ${\llbracket}N_1/2 , 3N_1/2 {\rrbracket}$. Let $T_{x,y}$ and $T'$ denote respectively their collision time and the first hitting time of $\{0,2N_1\}$ for $Y^x$. We are going to show that $$P\left[ T'<T_{x,y} \right]< C \frac{|x_1-y_1|(\log N_1)}{N_1}$$ Note that before collision, $Y^x-Y^y$ is a nearest neighbor random-walk with jump rate equal to $2$ and for that reason we have for any $t$ $$P[ T_{x,y}>t]\le C |x_1-y_1| t^{-1/2}.$$ On the other hand, we have for any $t\ge N_1^2$ $$\label{sectr}
P[T\le t] \le 2 \exp\left(- \frac{cN_1^2}{t} \right).$$ We can conclude choosing $t=N^2_1 (\log N_1)^2$.
We obtain the result simply by performing a union bound on $y\in {\widehat}{\Lambda}$. Hence we only need to prove a bound on the variance
$$\begin{gathered}
{{\ensuremath{\mathbf E}} }_{2N_1} \left[ \left(\phi_1(y)-\phi_1(x_{\min})\right)^2 \right] \\
\le \int_{(\log N_1)^8}^{\infty} \left[P^*_t(x_{\min},x_{\min})- P^*_t(x_{\min},y)-P^*(y,y) \right]{\mathrm{d}}t \end{gathered}$$
Using , we obtain that for any $y\in {\widehat}{\Lambda}$ $${{\ensuremath{\mathbf E}} }_{2N_1} \left[ \left(\phi_1(y)-\phi_1(x_{\min})\right)^2 \right]\le C(\log N_1)^{-4},$$ and thus that $${{\ensuremath{\mathbf P}} }_{2N_1} \big[ \ |\phi_1(y)-\phi_1(x_{\min})|\ge 1/4 \big]\le |{\widehat}{\Lambda}| e^{- c(\log N_1)^4 },$$ which allows to conclude
We set $$J:= \sum_{\{z \ : \ |x_{\min}-z|\le (\log N_1)^2 \}} {\mathbf{1}}_{\{\phi_2(z)\in [-1/4,1/4]\}}.$$ Using the fact that the sum is deterministically bounded by $C(\log N_1)^4$, we have $$\label{chubby}
{{\ensuremath{\mathbf P}} }_{2N_1} \left[ J \ge \frac{{{\ensuremath{\mathbf E}} }_N[J]}{2} \right]\ge \frac{{{\ensuremath{\mathbf E}} }_N[J]}{2C (\log N_1)^4}.$$ From , we have for small $h$, $${\mathrm{Var}}(\phi_2(x))= Q^2(x,x)\le \log \log N_1.$$ Then as $\phi_2(x)$ are centered Gaussians, we have $${{\ensuremath{\mathbf E}} }_{2N_1} \left[ \sum_{\{z \ : \ |z-x_{\min}|\le (\log N_1)^2\}} {\mathbf{1}}_{\{\phi_2(z)\in [-1/4,1/4]\}} \right]\ge c (\log N_1)^{4} (\log \log N_1)^{-1/2},$$ which combined with allows to conclude.
Finite volume criteria: adding mass and changing the boundary condition {#finicrit}
=======================================================================
Let us remark that it seems technically easier to get a lower bound for $N^{-2}{{\ensuremath{\mathbb E}} }\left[\log Z^{{\beta},{\omega}}_{N,h}\right]$ for a given $N$ than to prove one directly for the limit. However as there is no obvious sub-additivity property which allows to compare the two.
In [@cf:GL], for $d\ge 3$ we introduced the idea of replacing the boundary condition by an infinite volume free field in order to recover sub-additivity . In dimension $2$, the infinite volume free field does not exists as the variance diverges with the distance to the boundary of the domain. A way to bypass the problem it to artificially introduce mass and then to find a comparison between the free energy of the system with massive free field and the original one. This is the method that we adopted in our previous paper (see [@cf:GL Proposition 7.1 and Lemma 7.2]). However our previous results turn out out to be a bit two rough for our proof. We present here an improvement of it (Proposition \[th:finitevol\]) on which we build the proof of Theorem \[mainres\].
A first finite volume criterion {#nonoptfinit}
-------------------------------
Let us recall the comparison used in [@cf:GL]. Even it is not sufficient for our purpose in this paper, it will help us to explain the improvement presented in Section \[optfinit\]. Given $u>0$ and $m>0$, we introduce the notation $$\delta^u_x:={\mathbf{1}}_{[u-1,u+1]}(\phi_x)$$ and set $$Z^{{\beta},{\omega},m}_{N,h,u}:= {{\ensuremath{\mathbf E}} }^m_N\left[\exp\left( \sum_{x\in {\widetilde}{\Lambda}_N} ({\beta}{\omega}_x-{\lambda}({\beta})+h)\delta^u_x\right)\right].$$ and $$\label{massfree}
{\textsc{f}}({\beta},h,m,u):=\lim_{N\to \infty}\frac{1}{N^d} \log Z^{{\beta},{\omega},m}_{N,h,u}.$$ The existence of the above limit is proved in [@cf:GL]. We can compare this free-energy to the original one using the following result.
\[massivecompa\] We have for every $u$ and $m$ $$\label{massivecompar}
{\textsc{f}}({\beta},h,m,u)\le {\textsc{f}}({\beta},h)+f(m).$$ where $$f(m):= \frac 1 2\int_{[0,1]^2} \log \left( 1+ \frac{m^2}{4 \left[\sin^2(\pi x/2)+ \sin^2(\pi y/ 2)\right]}\right) {\mathrm{d}}x {\mathrm{d}}y.$$ There exists $C>0$ such that for every $m\le 1$ we have $$\label{asymf}
\left| f(m)-\frac{1}{4\pi}m^2|\log m| \right| \le Cm^2.$$ Moreover for all $N$ we have $$\label{subbadd}
{\textsc{f}}({\beta},h,m,u)\ge \frac{1}{N^2} {\widehat}{{\ensuremath{\mathbf E}} }^m {{\ensuremath{\mathbb E}} }\left[ \log {{\ensuremath{\mathbf E}} }^{m,{\widehat}\phi}_N
\left[ \exp\left( \sum_{x\in {\widetilde}{\Lambda}_N} ({\beta}{\omega}_x-{\lambda}({\beta})+h)\delta^u_x\right)\right]\right].$$
The result is proved in [@cf:GL] (as Proposition 7.1 and Lemma 7.2) but let us recall briefly how it is done. For the first point, we have to remark that changing the height of the substrate (i.e. replacing $\delta_x$ by $\delta^u_x$ in ) for the original model does not change the value of the free energy, that is , $${\textsc{f}}({\beta},h,0,u)={\textsc{f}}({\beta},h,0,0), \quad \text{ for all values of } u.$$ Heuristically this is because the free field Hamiltonian is translation invariant but a proof is necessary to show that the boundary effect are indeed negligible (see [@cf:GL Proposition 4.1]). Note that for the massive free field, the limit really depends on $u$ because adding an harmonic confinement breaks this translation invariance.
Then we can compare the partition of the two free fields by noticing that the density of the massive field with respect to the original one (recall ) satisfies $$\label{reldensity}
\frac{{\mathrm{d}}{{\ensuremath{\mathbf P}} }^m_N}{{\mathrm{d}}{{\ensuremath{\mathbf P}} }_N}(\phi):=\frac{\exp\left( -\frac{m^2}{2}\sum_{x \in \mathring{\Lambda}_N} \phi^2_x\right)}{{{\ensuremath{\mathbf E}} }_N\left[\exp\left( -\frac{m^2}{2}\sum_{x \in \mathring{\Lambda}_N} \phi^2_x\right)\right]}
\le \frac{1}{{{\ensuremath{\mathbf E}} }_N\left[\exp\left( -\frac{m^2}{2}\sum_{x \in \mathring{\Lambda}_N} \phi^2_x\right)\right]},$$ and that $$\label{lmimw}
\lim_{N\to \infty}\frac{1}{N^2}\log {{\ensuremath{\mathbf E}} }_N\left[\exp\left( -\frac{m^2}{2}\sum_{x \in \mathring{\Lambda}_N} \phi^2_x\right)\right]=: \lim_{N\to \infty}\frac{1}{N^2}\log W^m_N =-f(m).$$ Equation then follows from of a sub-additive argument (see the proof of Proposition 4.2. in [@cf:GL] or that of below).
Note that Proposition \[massivecompa\] gives a bound on ${\textsc{f}}({\beta},h)$ which depends only on the partition function of a finite system . $$\label{finitevol1}
{\textsc{f}}({\beta},h)\ge \frac{1}{N^2} {\widehat}{{\ensuremath{\mathbf E}} }^m {{\ensuremath{\mathbb E}} }\left[ \log {{\ensuremath{\mathbf E}} }^{m,{\widehat}\phi}_N
\left[ \exp\left( \sum_{x\in {\widetilde}{\Lambda}_N} ({\beta}{\omega}_x-{\lambda}({\beta})+h)\delta^u_x\right)\right]\right]-f(m).$$ In particular we can prove Theorem \[mainres\], if for any $h>0$, ${\beta}\in(0,{\overline{{\beta}}})$ we can find values for $u$ and $m$ and $N$ such that the l.h.s. is positive. However it turns out that with our techniques, we cannot prove that the l.h.s is positive for very small $h$. This is mostly because of the presence of a $|\log m|$ factor in the asymptotic behavior of $f(m)$ around $0$. Therefore we need a better criterion in which the subtracted term is proportional to $m^2$.
A finer comparison {#optfinit}
------------------
To obtain a more efficient criterion, we want to restrict the partition function to a set of $\phi$ where $({\mathrm{d}}{{\ensuremath{\mathbf P}} }^m_N/{\mathrm{d}}{{\ensuremath{\mathbf P}} }_N)(\phi)$ is much smaller than $\exp(N^2f(m))$. We define ${{\ensuremath{\mathcal D}} }^0_N$ as a set where the density $({\mathrm{d}}{{\ensuremath{\mathbf P}} }^{m}_N/{\mathrm{d}}{{\ensuremath{\mathbf P}} }_N)(\phi)$ takes “typical” values (see Proposition \[thednproba\]). For some constant $K>0$, we set $$\label{dnzero}
{{\ensuremath{\mathcal D}} }^0_N:=\left\{ \sum_{x\in {\Lambda}_N} \phi(x)^2 \ge N^2\left( \frac{f(m)}{m^2}- K\right) \right\}.$$ Recall that ${\widehat}{{\ensuremath{\mathbf P}} }^{m}$ denotes the law of the infinite volume massive free field (see Section \[secmass\]) for the boundary condition ${\widehat}\phi$.
\[th:finitevol\] For any value of $N$, and $K$ and $m$ we have $$\label{finitesds}
{\textsc{f}}({\beta},h) \ge \frac{1}{N^2}{\widehat}{{\ensuremath{\mathbf E}} }^{m} {{\ensuremath{\mathbb E}} }\left[ \log {{\ensuremath{\mathbf E}} }^{m,{\widehat}\phi}_N \left[ \exp\left(\sum_{x\in {\widetilde}{\Lambda}_N} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{u}_x\right){\mathbf{1}}_{{{\ensuremath{\mathcal D}} }^0_N}\right] \right]
-Km^2.$$
With the idea of working with a measure that does not depend on the boundary condition, we set similarly to $$\delta^{{\widehat}\phi,u}_x:= {\mathbf{1}}_{[u-1,u+1]}(\phi(x)+H^{m,{\widehat}\phi}_N(x)),$$ and $$\label{defDn}
{{\ensuremath{\mathcal D}} }_N:= \left\{ \phi \ : \ \sum_{x\in {\widetilde}{\Lambda}_N} (\phi_x+H^{m,{\widehat}\phi}_N(x))^2 \ge N^2\left( \frac{2f(m)}{m^2}- K\right)\right\}.$$ With this notation and in view of the considerations of Section \[grbc\] the expected value in the l.h.s. in is equal to $$\label{otherexpr}
{\widehat}{{\ensuremath{\mathbf E}} }^m {{\ensuremath{\mathbb E}} }\left[ \log {{\ensuremath{\mathbf E}} }^{m}_N \left[ \exp\left(\sum_{x\in {\widetilde}{\Lambda}_N} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{u,{\widehat}\phi}_x\right){\mathbf{1}}_{{{\ensuremath{\mathcal D}} }_N} \right]\right].$$
Using the criterion
-------------------
Before giving a proof of Proposition \[th:finitevol\] let us show how we are going to use it to prove our lower bound on the free energy . for the remainder of the proof we set $$\label{parameters}\begin{split}
N_h&:= \exp(h^{-20}),\\
m_h&:= N^{-1}_h (\log N_h)^{1/4},\\
u_h&:=\sqrt{\frac{2}{\pi}}\log N_h-\frac{2+\alpha}{2\sqrt{2\pi}}\log \log N_h,
\end{split}$$ where $\alpha=3/4$ (we find that the computations are easier to follows with the letter $\alpha$ instead of a specific number, in fact any value in the interval $(11/20,1)$ would also work). With Proposition \[th:finitevol\], the proof of the lower bound in is reduced to the following statement, whose proof will be detailed in the next three sections.
\[mainproposition\] For any ${\beta}\le {\overline{{\beta}}}$, there exists $h_0({\beta})$ such that for any $h\in (0,h_0({\beta}))$
$${\widehat}{{\ensuremath{\mathbf E}} }^m
{{\ensuremath{\mathbb E}} }\left[ \log {{\ensuremath{\mathbf E}} }^{m_h,{\widehat}\phi}_{N_h} \left[ \exp\left(\sum_{x\in {\widetilde}{\Lambda}_{N_h}} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{u_h}_x\right)
{\mathbf{1}}_{{{\ensuremath{\mathcal D}} }^0_{N_h}} \right]\right]
-K(m_hN_h)^2\ge 1.$$
Indeed the result directly implies that $${\textsc{f}}({\beta},h)\ge (N_h)^{-1}.$$
Proof of Proposition \[th:finitevol\] {#homogg}
-------------------------------------
Let us start by setting $$\begin{gathered}
Z'_N({\widehat}\phi)=Z'_N:= {{\ensuremath{\mathbf E}} }^m_N \left[ \exp\left(\sum_{x\in {\widetilde}{\Lambda}_N} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{{\widehat}\phi,u}_x\right){\mathbf{1}}_{{{\ensuremath{\mathcal D}} }_N} \right]\\
={{\ensuremath{\mathbf E}} }^{m,{\widehat}\phi}_N \left[ \exp\left(\sum_{x\in {\widetilde}{\Lambda}_N} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{u}_x\right){\mathbf{1}}_{{{\ensuremath{\mathcal D}} }^0_N} \right].\end{gathered}$$ A simple computation (see below) is sufficient to show that for any $k\ge 0$ we have $$\label{subaddt}
{\widehat}{{\ensuremath{\mathbf E}} }^m {{\ensuremath{\mathbb E}} }\left[ \log Z'_{2^k N}\right]\ge 4^k {\widehat}{{\ensuremath{\mathbf E}} }^m {{\ensuremath{\mathbb E}} }\left[\log Z'_{N}\right].$$ Hence that it is sufficient to prove with $N$ replaced by $2^k N$ for an arbitrary integer $k$, or by the limit when $k$ tends to infinity.
Let us prove . We divide the box ${\Lambda}_{2N}$ into $4$ boxes, ${\Lambda}^i_N$, $i=1,\dots,4$. Set $$\begin{split}
{\Lambda}^i_N&:={\Lambda}_N+(\alpha_1(i),\alpha_2(i))N \\
{\widetilde}{\Lambda}^i_N&:={\widetilde}{\Lambda}_N+(\alpha_1(i),\alpha_2(i))N,
\end{split}$$ where $\alpha_j(i)\in\{0,1\}$ is the $j$-th digit of the dyadic development of $i-1$. Set $${{\ensuremath{\mathcal D}} }^{0,i}_N:= \left\{ \sum_{x\in {\widetilde}{\Lambda}^i_N} \phi(x)^2\ge \left( \frac{2f(m)}{m^2}-K\right)N^2 \right\}.$$ We notice that $$\label{dainclusion}
\bigcap_{i=1}^4 {{\ensuremath{\mathcal D}} }^{0,i}_{N}\subset{{\ensuremath{\mathcal D}} }^0_{2N}.$$ We define $${\Gamma}_{N}:=\left( \bigcup_{i=1}^{4} \partial {\Lambda}^i_N \right) \setminus \partial{\Lambda}_{2N}.$$ If we condition on the realization on $\phi$ in ${\Gamma}_{N}$, the partition functions of the system of size $2N$ factorizes into $4$ partition functions of systems of size $N$, whose boundary conditions are determined by ${\widehat}\phi$ and $\phi|_{{\Gamma}_N}$, and we obtain $$\begin{gathered}
{{\ensuremath{\mathbf E}} }^{m,{\widehat}\phi}_{2N} \left[ \exp\left( \sum_{x\in {\widetilde}{\Lambda}_{2N}} ({\beta}{\omega}_x-{\lambda}({\beta})+h)\delta^u_x\right){\mathbf{1}}_{\bigcap_{i=1}^4 {{\ensuremath{\mathcal D}} }^{0,i}_{N}} \ \Bigg| \ \phi|_{{\Gamma}_N} \right]
\\ = \prod_{i=1}^{4}{{\ensuremath{\mathbf E}} }^{m,{\widehat}\phi}_{2N}
\left[ \exp\left( \sum_{x\in {\widetilde}{\Lambda}^i_{N}} ({\beta}{\omega}_x-{\lambda}({\beta})+h)\delta^u_x\right){\mathbf{1}}_{{{\ensuremath{\mathcal D}} }^{0,i}_{N}} \ \Bigg| \ \phi|_{{\Gamma}_N} \right]
=: \prod_{i=1}^{4} {\widetilde}Z^i({\widehat}\phi , \phi|_{{\Gamma}_N} , {\omega}).\end{gathered}$$ By the spatial Markov property for the infinite volume field, each ${\widetilde}Z^i({\widehat}\phi , \phi|_{{\Gamma}_N} , {\omega})$ has the same distribution as $Z'_N$ (if ${\widehat}\phi$ and $\phi|_{{\Gamma}_N}$ have distribution ${\widehat}{{\ensuremath{\mathbf E}} }^m$ and ${{\ensuremath{\mathbf E}} }^{m,{\widehat}\phi}_{2N}$ respectively and the ${\omega}_x$s are IID). Using and Jensen’s inequality for $ {{\ensuremath{\mathbf E}} }^{m,{\widehat}\phi}_{2N} \left[ \ \cdot \ | \ \phi|_{{\Gamma}_N} \right]$ we have $$\begin{split}
{{\ensuremath{\mathbb E}} }{\widehat}{{\ensuremath{\mathbf E}} }^m \left[ \log Z'_{2N}\right]\ge
\sum_{i=1}^{4} {{\ensuremath{\mathbb E}} }{\widehat}{{\ensuremath{\mathbf E}} }^m {{\ensuremath{\mathbf E}} }^{m,{\widehat}\phi}_{2N}\left[ \log {\widetilde}Z^i({\widehat}\phi , \phi|_{{\Gamma}_N} \ ) \right]
=4 {{\ensuremath{\mathbb E}} }{\widehat}{{\ensuremath{\mathbf E}} }^m\left[ \log Z'_{N}\right],
\end{split}$$ which ends the proof of .
Now we set $M:=2^kN$ with $k$ large. In the computation, we write sometimes $H$ for $H^{m,{\widehat}\phi}_M$ for simplicity. We remark that for $\phi\in {{\ensuremath{\mathcal D}} }_M$ we have $$\begin{gathered}
\log \left( \frac{{\mathrm{d}}P^{m}_M}{{\mathrm{d}}P_M}(\phi) \right)\\
= \frac{m^2}{2} \left(
\sum_{x\in {\Lambda}_M} H^2(x)-\sum_{x\in {\Lambda}_M} (\phi_x-H(x))^2-2\sum_{x\in {\Lambda}_M} \phi_x H(x)\right) -\log W_M\\
\le\left[ M^2 \left(\frac{ m^2 K}{2}-f(m)\right)-\log W_M \right] + m^2 \left[ \sum_{x\in {\Lambda}_M} \phi_x H(x)+ \frac{1}{2} \sum_{x\in {\Lambda}_M} H^2(x)\right]\\
\le m^2 M^2 K+ m^2 \left[ \sum_{x\in {\Lambda}_M} \phi_x H(x)+ \frac{1}{2} \sum_{x\in {\Lambda}_M} H^2(x)\right],\end{gathered}$$ where the first inequality follows from the definition of ${{\ensuremath{\mathcal D}} }_M$ and the last one from and is valid provided $K$ is sufficiently large. From this inequality we deduce that $$\begin{gathered}
\label{rhss}
Z'_{M}\le e^{m^2K M^2} {{\ensuremath{\mathbf E}} }_M\left[ e^{\sum_{x\in {\widetilde}{\Lambda}_M} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{{\widehat}\phi,u}_x} e^{m^2 \sum_{x\in {\Lambda}_M} \left[H(x)\phi(x)
+\frac{H(x)^2}{2}\right]} \right]\\
:=e^{m^2K M^2} Z''_M.\end{gathered}$$ To conclude the proof, we must show that the r.h.s. is not affected, in the limit, by the presence of $H$ (which produces the two last terms and enters in the definition of $\delta^{{\widehat}\phi,u}_x$) i.e. that $$\label{zsecond}
\lim_{M\to \infty} \frac{1}{M^2} {{\ensuremath{\mathbb E}} }{\widehat}{{\ensuremath{\mathbf E}} }^m \left[ \log Z''_M \right]={\textsc{f}}({\beta},h).$$ We can replace $\delta^{{\widehat}\phi,u}_x$ by $\delta^u_x$ at the cost of a Girsanov-type term in the density. For computations, it is pratical to define $$H^0(x):= H(x){\mathbf{1}}_{\{x\in \mathring{{\Lambda}}_N\}}.$$ The distribution $\phi+H^0$ under ${{\ensuremath{\mathbf P}} }_M$ is absolutely continuous with respect to that of $\phi$. The density of its distribution ${\widetilde}{{\ensuremath{\mathbf P}} }_M$ with respect to ${{\ensuremath{\mathbf P}} }_M$ is given by $$\begin{gathered}
\label{densityxx}
\frac{{\mathrm{d}}{\widetilde}{{\ensuremath{\mathbf P}} }_M}{{\mathrm{d}}{{\ensuremath{\mathbf P}} }^0_M}(\phi)=\exp\bigg(\frac 1 2 \sum_{{\Lambda}_M} ({{\ensuremath{\nabla}} }\phi)^2-({{\ensuremath{\nabla}} }\phi -{{\ensuremath{\nabla}} }H^0)^2 \bigg)\\=
\exp\bigg(-\frac 1 2 \sum_{{\Lambda}_M} ({{\ensuremath{\nabla}} }H_0)^2+ \sum_{{\Lambda}_M}{{\ensuremath{\nabla}} }\phi{{\ensuremath{\nabla}} }H^0 \bigg)\\
=\exp\bigg( -m^2\sum_{x\in {\Lambda}_M} \left( H(x)\phi(x)- \frac{H^0(x)^2}{2} \right)\\
+ \sum_{x\in \partial{\Lambda}_M}{\sum_{\substack{y\in \partial^- {\Lambda}_M \\ y\sim x}}}\left( H(x)\phi(y)-\frac{H(x)H(y)}{2} \right)\bigg),\end{gathered}$$ where we used the notation $$\sum_{{\Lambda}_M} {{\ensuremath{\nabla}} }R {{\ensuremath{\nabla}} }T:= \frac{1}{2}{\sum_{\substack{x,y \in {\Lambda}_M \\ x\sim y}}} (R(x)-R(y))(T(x)-T(y)).$$ To obtain the second line in we have used the summation by part formula (which is valid without adding boundary terms since the functions we are integrating have zero boundary condition) and to obtain $$\begin{split}
\sum_{{\Lambda}_M} {{\ensuremath{\nabla}} }H {{\ensuremath{\nabla}} }\phi&=- \sum_{x\in \mathring{{\Lambda}}_M} {\Delta}H(x)\phi(x)= -m^2 \sum_{x\in {\Lambda}_M} H(x)\phi(x),\\
\sum_{{\Lambda}_M} {{\ensuremath{\nabla}} }H {{\ensuremath{\nabla}} }H^0&=- \sum_{x\in \mathring{{\Lambda}}_M} {\Delta}H(x)H^0(x)= -m^2 \sum_{x\in {\Lambda}_M} H^0(x)^2.
\end{split}$$ The substitution of $H$ by $H^0$ produces the second term (boundary effects). Hence the expectation in is equal to (assume $u>1$) $$\begin{gathered}
\label{Zseccc}
\exp\bigg(\sum_{x\in \partial {\Lambda}_M \cap {\widetilde}{\Lambda}_M}\left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right){\mathbf{1}}_{[u-1,u+1]}({\widehat}\phi(x))\\ +m^2\sum_{x\in {\widetilde}{\Lambda}_M}
\frac{H(x)^2+H^0(x)^2}{2}
-\sum_{x\in \partial{\Lambda}_M}{\sum_{\substack{y\in \partial^- {\Lambda}_M \\ y\sim x}}} \frac{H(x)H(y)}{2} \bigg)\\
\times
{{\ensuremath{\mathbf E}} }^0_M\left[ \exp\left(\sum_{x\in {\widetilde}{\Lambda}_M} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{u}_x-
{\sum_{\substack{x\in \partial{\Lambda}_M, y\in \partial^- {\Lambda}_M \\ y\sim x}}} H(x)\phi(y) \right)\right].\end{gathered}$$ Let us show first that the exponential term in front of the expectation in does not affect the limit of $M^{-2}\log Z''_M$. We have $$\begin{gathered}
\label{limitain}
\lim_{M\to \infty} {{\ensuremath{\mathbb E}} }{\widehat}{{\ensuremath{\mathbf E}} }^m \left|\frac{1}{M^2} \sum_{x\in \partial {\Lambda}_M \cap {\widetilde}{\Lambda}_M}\left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right){\mathbf{1}}_{[u-1,u+1]}({\widehat}\phi(x))\right|
\\ \le \lim_{M\to \infty} \frac{1}{M^2} {{\ensuremath{\mathbb E}} }\sum_{x\in \partial {\Lambda}_M \cap {\widetilde}{\Lambda}_M}| {\beta}{\omega}_x-{\lambda}({\beta})+h |=0.\end{gathered}$$ For the other terms, set $$\mathcal M_M:= \max_{x \in \partial {\Lambda}_M}{\widehat}\phi(x).$$ Being a maximum over $4M$ Gaussian variables of finite variance, it is not difficult to check that for all $M$ sufficiently large, $${\widehat}{{\ensuremath{\mathbf E}} }[{{\ensuremath{\mathcal M}} }^2_M]\le (\log M)^2.$$ Moreover from the definition of $H^{{\widehat}\phi,N}_M$ we gave for any $x\in \mathring {\Lambda}_{M}$ we have $$|H^{{\widehat}\phi,N}_M(x)|= \frac{1}{2d+m^2}\left|\sum_{y\sim x}H^{{\widehat}\phi,N}_M(y)\right|\le \frac{2d}{2d+m^2} \max_{y\sim x} |H^{{\widehat}\phi,N}_M(y)|.$$ This implies that the maximum of $H$ is attained on the boundary and that $$|H^{{\widehat}\phi,N}_M(x)|\le {{\ensuremath{\mathcal M}} }_M \left(\frac{2d}{2d+m^2}\right)^{d(x,\partial {\Lambda}_M)}.$$ This implies that $$\left|m^2\sum_{x\in {\widetilde}{\Lambda}_M}
\frac{H(x)^2+H^0(x)^2}{2}\\
-\sum_{x\in \partial{\Lambda}_M}{\sum_{\substack{y\in \partial^- {\Lambda}_M \\ y\sim x}}} \frac{H(x)H(y)}{2} \bigg)\right|\le C_m M {{\ensuremath{\mathcal M}} }^2_M.$$ In particular we have $$\label{limitdeux}
\lim_{M\to \infty} \frac{1}{M^2} {\widehat}{{\ensuremath{\mathbf E}} }^m \left |m^2\sum_{x\in {\widetilde}{\Lambda}_M}
\frac{H(x)^2+H^0(x)^2}{2}
-\sum_{x\in \partial{\Lambda}_M}{\sum_{\substack{y\in \partial^- {\Lambda}_M \\ y\sim x}}} \frac{H(x)H(y)}{2} \right|=0.$$ Hence from , and , Equation holds provided we can show that $$\label{statz}
\lim_{M\to \infty } \frac{1}{M^2}{{\ensuremath{\mathbb E}} }{\widehat}{{\ensuremath{\mathbf E}} }^m\log
{{\ensuremath{\mathbf E}} }_M\left[ \exp\left(\sum_{x\in {\widetilde}{\Lambda}_M} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{u}_x+ T({\widehat}\phi, \phi)\right)
\right]
={\textsc{f}}({\beta},h),$$ where we have used the notation $$T({\widehat}\phi, \phi):=\sum_{x\in \partial{\Lambda}_M}{\sum_{\substack{y\in \partial^-{{\Lambda}}_M \\ y\sim x}}}{\widehat}\phi(x)\phi(y).$$ This is extremely similar to the proof of [@cf:GL Proposition 4.2] but we include the main line of the computation for the sake of completeness. First we note that because of uniform integrability, holds if we prove the convergence in probability, $$\label{statz3}
\lim_{M\to \infty}\frac{1}{M^2}\log
{{\ensuremath{\mathbf E}} }_M\left[ \exp\left(\sum_{x\in {\widetilde}{\Lambda}_M} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{u}_x+ T({\widehat}\phi, \phi)\right)
\right]={\textsc{f}}({\beta},h).$$
Note that conditioned to ${\widehat}\phi$, $T({\widehat}\phi, \phi)$ is a centered Gaussian random variable. We show in fact ${\widehat}{{\ensuremath{\mathbf P}} }^m \otimes {{\ensuremath{\mathbb P}} }$ almost sure convergence for rather than convergence of the expectation of , but since $$\begin{gathered}
\left|M^{-2}\log
{{\ensuremath{\mathbf E}} }_M\left[ \exp\left(\sum_{x\in {\widetilde}{\Lambda}_M} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{u}_x+ T({\widehat}\phi, \phi)\right)\right] \right|\\
\le
M^{-2}\sum_{x\in {\widetilde}{\Lambda}_M} | {\beta}{\omega}_x-{\lambda}({\beta})+h|+ M^{-2} \log {{\ensuremath{\mathbf E}} }_M\left[e^{T({\widehat}\phi,\phi)}\right]\\
=M^{-2}\sum_{x\in {\widetilde}{\Lambda}_M} | {\beta}{\omega}_x-{\lambda}({\beta})+h|+ M^{-2}{\mathrm{Var}}_{{{\ensuremath{\mathbf P}} }_M} \left(T({\widehat}\phi,\phi) \right),\end{gathered}$$ and the sequence is uniformly integrable (cf. ), almost sure convergence implies convergence in $L_1$.
Now to prove , we set $${{\ensuremath{\mathcal M}} }_M({\widehat}\phi):= \max_{x\in \partial {\Lambda}_M} |{\widehat}\phi_x|.$$ As the covariance function of $\phi$ is positive, we have $$\label{variartz}
{{\ensuremath{\mathbf E}} }_M\left[ T({\widehat}\phi, \phi)^2\right]\le {{\ensuremath{\mathcal M}} }_M ^2 {{\ensuremath{\mathbf E}} }_M \left[ \left(\sum_{x\in \partial{\Lambda}_M}{\sum_{\substack{y\in \partial^- {\Lambda}_M \\ y\sim x}}}\phi(y)\right)^2\right]
=4(M-1){{\ensuremath{\mathcal M}} }_M ^2.$$ We define $$A_M:=\big\{\ |T({\widehat}\phi, \phi)|\le M^{7/4}{{\ensuremath{\mathcal M}} }_M \big\}$$ Combining our bound on the variance and standard Gaussian estimates, we obtain $$\label{deviats}
\begin{split}
{{\ensuremath{\mathbf P}} }_M\left[ A_M \right]&\le e^{-c M^{5/2}},\\
{{\ensuremath{\mathbf E}} }_M\left[ e^{T({\widehat}\phi, \phi)} {\mathbf{1}}_{A_M}\right]&\le e^{-c M^{5/2}}.
\end{split}$$ Combining the second line of with an annealed bound we obtain that $$\lim_{M\to \infty} \frac{1}{M^2} \log {{\ensuremath{\mathbf E}} }_M\left[ e^{\sum_{x\in {\widetilde}{\Lambda}_M} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{u}_x+
T({\widehat}\phi,\phi) } {\mathbf{1}}_{A^{\complement}_M}\right]=-\infty,$$ and hence is equivalent to $$\label{statz2}
\lim_{M\to \infty} \frac{1}{M^2} \log {{\ensuremath{\mathbf E}} }_M\left[ e^{\sum_{x\in {\widetilde}{\Lambda}_M} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{u}_x+
T({\widehat}\phi,\phi)} {\mathbf{1}}_{A_M} \right]={\textsc{f}}({\beta},h).$$ To prove , we first note using the first line of that implies that $$\lim_{M\to \infty} \frac{1}{M^2} \log {{\ensuremath{\mathbf E}} }_M\left[ \exp\left(\sum_{x\in {\widetilde}{\Lambda}_M} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{u}_x\right) {\mathbf{1}}_{A_M} \right]={\textsc{f}}({\beta},h).$$ By definition of $A_M$ we have $$\frac{1}{M^2}
\left| \log \frac{ {{\ensuremath{\mathbf E}} }_M\left[ e^{\sum_{x\in {\widetilde}{\Lambda}_M} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{u}_x+T({\widehat}\phi,\phi)}
{\mathbf{1}}_{A_M} \right]}{{{\ensuremath{\mathbf E}} }_M\left[ e^{\sum_{x\in {\widetilde}{\Lambda}_M} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{u}_x} {\mathbf{1}}_{A_M} \right]} \right|
\le M^{-1/4} {{\ensuremath{\mathcal M}} }_M.$$ Hence to conclude we just need to show that $$\lim_{M\to \infty} M^{-1/4} {{\ensuremath{\mathcal M}} }_M=0.$$ This follows from the definition of ${{\ensuremath{\mathcal M}} }_M$ and Borel-Cantelli’s Lemma. .
Decomposition of the proof of Proposition \[mainproposition\] {#decompo}
=============================================================
The overall idea for the proof is to restrict the partition function to a set of typical trajectories $\phi$ and to control the first two moments of the restricted partition function to get a good estimate for the expected $\log$. However the implementation of this simple idea requires a lot of care. We decompose the proof in three steps.
In Section \[sketch\], we briefly present these steps and combine them to obtain the proof and in Section \[propnineone\] we perform the first step of the proof, which is the simpler one. The two other steps need some detailed preparatory work which is only introduced in Section \[liminouze\].
Sketch of proof {#sketch}
---------------
The first step is to show that ${{\ensuremath{\mathcal D}} }_N$ is a typical event in order to ensure that our restriction to ${{\ensuremath{\mathcal D}} }_N$ in the partition function does not cost much.
\[thednproba\] We can choose $K$ in a way that for all $m\le 1$ sufficiently large, for all $N\ge m^{-1}|\log m|^{1/4}$, and for all realization of ${\widehat}\phi$ $${{\ensuremath{\mathbf P}} }_N[{{\ensuremath{\mathcal D}} }^{\complement}_N]\le C(\log N)^{-1/2}.$$
The result is not used directly in the proof of Proposition \[mainproposition\] but is a crucial input for the proof of Proposition \[prop:boundary\] below.
The aim of the second step is to show that at a moderate cost one can restrict the zone of the interaction to a sub-box ${\Lambda}'_N$ defined by $$\label{defglprim}
{\Lambda}'_N:= {{\ensuremath{\mathbb Z}} }^2 \cap [N(\log N)^{-1/8},N(1-(\log N)^{-1/8})]^2.$$ The reason for which we want to make that restriction is that it is difficult to control the effect of the boundary condition (i.e. of $H^{m,{\widehat}\phi}_N$) in $\partial {\Lambda}_N \setminus {\Lambda}'_N$. Inside ${\Lambda}'_N$ however, due to the choice of the relative values of $m$ and $N$ in , $H^{m,{\widehat}\phi}_N$ is very small and has almost no effect.
\[prop:boundary\] There exists an event ${{\ensuremath{\mathcal C}} }_N\subset {{\ensuremath{\mathcal D}} }_N$ satisfying $$\label{lecnepetit}
{{\ensuremath{\mathbf P}} }[{{\ensuremath{\mathcal C}} }^{{\complement }}_N]\le C (\log N)^{-1/16}$$ and a constant $C({\beta})$ such that $$\begin{gathered}
\label{primieq}
{{\ensuremath{\mathbf E}} }^m_N \left[ \exp\left(\sum_{x\in {\widetilde}{\Lambda}_N} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{{\widehat}\phi,u}_x\right){\mathbf{1}}_{{{\ensuremath{\mathcal D}} }_N} \right]\\
\ge {{\ensuremath{\mathbf E}} }^m_N \left[ \exp\left(\sum_{x\in {\Lambda}'_N} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{{\widehat}\phi,u}_x\right){\mathbf{1}}_{{{\ensuremath{\mathcal C}} }_N} \right]-
C({\beta}) (\log \log N)^4(\log N)^{\alpha-1/16}.
\end{gathered}$$
Finally we have to show that the expected $\log$ of the restricted partition function in the r.h.s. of is indeed sufficiently large to compensate for the second term. We actually only prove that this is the case for the set of good boundary conditions ${\widehat}\phi$ which have no significant influence in the bulk of the box $${\widehat}{{\ensuremath{\mathcal A}} }_N:= \{ \forall x\in {\Lambda}'_N, |H^{m,{\widehat}\phi}_N(x)|\le 1\},$$ and show that the contribution of bad boundary condition is irrelevant.
We have chosen $u_h$ in a way such that the density of expected density contact is very scarce (the total expected number of contact in the box is a power of $\log N$, see below), but the unlikely event that $\phi$ has a lot of contact is sufficient to make the second moment of the partition very large. Hence for our analysis to work, it is necessary to restrict the partition function to trajectories which have few contacts. We set $$\begin{split}\label{defbn}
L_N&:= \sum_{x\in {\Lambda}'_N} \delta^{{\widehat}\phi,u}_x,\\
{{\ensuremath{\mathcal B}} }_N&:={{\ensuremath{\mathcal C}} }_N\cap \left\{ L_N\le (\log N)^{\frac{\alpha+1}{2}}\right\}.
\end{split}$$
\[prop:inside\] We have
- For $N$ sufficiently large $$\label{bcondition}
{\widehat}{{\ensuremath{\mathbf P}} }^m[ {\widehat}{{\ensuremath{\mathcal A}} }^{\complement }_N] \le N^{-4}.$$
- For any ${\widehat}\phi \notin {\widehat}{{\ensuremath{\mathcal A}} }_N$ $$\label{binfluence}
{{\ensuremath{\mathbb E}} }{{\ensuremath{\mathbf E}} }^m_N \left[ \exp\left(\sum_{x\in {\Lambda}'_N} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{{\widehat}\phi,u}_x\right){\mathbf{1}}_{{{\ensuremath{\mathcal C}} }_N} \right]\ge -N^2 {\lambda}({\beta})-\log 2.$$
- There exists a constant $c>0$ such that for any ${\widehat}\phi \in {\widehat}{{\ensuremath{\mathcal A}} }_N$ $$\label{gcondition}
{{\ensuremath{\mathbb E}} }\log {{\ensuremath{\mathbf E}} }^m_N \left[ \exp\left(\sum_{x\in {\Lambda}'_N} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{{\widehat}\phi,u}_x\right){\mathbf{1}}_{{{\ensuremath{\mathcal B}} }_N} \right]\ge c h (\log N)^{\alpha}-\log 2.$$
Using Proposition \[prop:inside\], we have $$\begin{gathered}
{{\ensuremath{\mathbb E}} }{\widehat}{{\ensuremath{\mathbf E}} }^{m}\log {{\ensuremath{\mathbf E}} }^m_N \left[ \exp\left(\sum_{x\in {\Lambda}'_N} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{{\widehat}\phi,u}_x\right){\mathbf{1}}_{{{\ensuremath{\mathcal C}} }_N} \right]
\\
\ge -{\widehat}{{\ensuremath{\mathbf P}} }^m[ {\widehat}{{\ensuremath{\mathcal A}} }^{{\complement }}_N]\left(N^2 {\lambda}({\beta})+\log 2\right)\\
+ {{\ensuremath{\mathbb E}} }{\widehat}{{\ensuremath{\mathbf E}} }^{m}\left[ \log {{\ensuremath{\mathbf E}} }^m_N \left[ \exp\left(\sum_{x\in {\Lambda}'_N} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{{\widehat}\phi,u}_x\right){\mathbf{1}}_{{{\ensuremath{\mathcal B}} }_N}\right]
{\mathbf{1}}_{{\widehat}{{\ensuremath{\mathcal A}} }_N}\right]
\\ \ge c h (\log N)^{\alpha}-1.
\end{gathered}$$ Using Proposition \[prop:boundary\] and recalling our choice of parameters , we have, for $h$ sufficiently small $$\begin{gathered}
{{\ensuremath{\mathbf E}} }^m_N \left[ \exp\left(\sum_{x\in {\widetilde}{\Lambda}_N} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{{\widehat}\phi,u}_x\right){\mathbf{1}}_{{{\ensuremath{\mathcal D}} }_N} \right]- K(mN)^2\\
\ge
c h (\log N)^{\alpha}-C({\beta})(\log \log N)^4 (\log N)^{\alpha-\frac{1}{16}}- K (\log N)^{1/2}-1\\
\ge
(c/2) (\log N)^{\alpha-\frac{1}{20}},\end{gathered}$$ where in the last line we used that $\alpha-\frac{1}{20}>1/2$. This is sufficient to conclude.
Proof of Proposition \[thednproba\] {#propnineone}
-----------------------------------
Again in this proof simply write $H$ for $H^{m,{\widehat}\phi}_N$ The proof simply relies on computing the expectation and variance of $$\sum_{x\in {\widetilde}{\Lambda}_N} (\phi(x)+H(x))^2.$$ We have $$\label{espuin}
{{\ensuremath{\mathbf E}} }^m_N \left[ \sum_{x\in {\widetilde}{\Lambda}_N} [\phi(x)+H(x)]^2\right]= {{\ensuremath{\mathbf E}} }^m_N \left[ \sum_{x\in {\Lambda}_N} \phi(x)^2\right]+\sum_{x\in {\widetilde}{\Lambda}_N} H(x)^2.$$ From , for an appropriate choice of $C$, the following holds $$\label{espuout}
\frac{1}{N^2} {{\ensuremath{\mathbf E}} }^m_N \left[ \sum_{x\in {\Lambda}_N} \phi(x)^2\right] \ge \frac{1}{2\pi} | \log m | -C\ge \frac{2f(m)}{m^2}-C.$$ Now let us estimate the variance. With the cancellation of odd moments of Gaussians, the expansion of the products gives $$\begin{gathered}
\label{eq:decompp}
{{\ensuremath{\mathbf E}} }^m_N \left[ \left( \sum_{x\in {\widetilde}{\Lambda}_N} (\phi(x)+H(x))^2 \right)^2\right]-\left({{\ensuremath{\mathbf E}} }^m_N \left[ \sum_{x\in {\widetilde}{\Lambda}_N} (\phi(x)+H(x))^2\right]\right)^2\\
=
{{\ensuremath{\mathbf E}} }^m_N \left[ \left( \sum_{x\in {\Lambda}_N} \phi(x)^2 \right)^2\right] - \left({{\ensuremath{\mathbf E}} }^m_N \left[ \sum_{x\in {\Lambda}_N} \phi(x)^2\right]\right)^2\\
+
4{{\ensuremath{\mathbf E}} }^m_N\left[ \sum_{x,y \in {\widetilde}{\Lambda}_N} \phi(x)\phi(y)H(x)H(y) \right].\end{gathered}$$ We treat the last term separately and first concentrate on the two firsts which correspond to the zero boundary condition case. We have $${{\ensuremath{\mathbf E}} }^m_N\left[\phi(x)^2\phi(y)^2\right]- {{\ensuremath{\mathbf E}} }^m_N\left[\phi(x)^2\right]{{\ensuremath{\mathbf E}} }^m_N\left[\phi(y)^2\right]= 2 \left[ G^{m,*}(x,y)\right]^2,$$ and hence from we can deduce that $$\begin{gathered}
{{\ensuremath{\mathbf E}} }^m_N \left[ \left( \sum_{x\in {\Lambda}_N} \phi(x)^2 \right)^2\right]-\left({{\ensuremath{\mathbf E}} }^m_N \left[ \sum_{x\in {\Lambda}_N} \phi(x)^2\right]\right)^2
\\ = 2 \sum_{x, y\in {\Lambda}_N} G^{m,*}(x,y)
\le C N^2 m^{-2}.
\end{gathered}$$ Concerning the last term in , we bound it as follows $$\begin{gathered}
{{\ensuremath{\mathbf E}} }^m_N\left[ \sum_{x,y\in {\widetilde}{\Lambda}_N} \phi(x)\phi(y)H(x)H(y) \right]= \sum_{x,y\in {\widetilde}{\Lambda}_N} G^{m,*}(x,y)H(x)H(y)\\
\le \sum_{x\in{\widetilde}{\Lambda}_N} H(x)^2 \sum_{y\in {\Lambda}_N} G^{m,*}(x,y)\le C m^{-2}\sum_{x\in {\widetilde}{\Lambda}_N} H(x)^2,\end{gathered}$$ where in the last inequality we used . This gives $$\label{lavarr}
{\mathrm{Var}}_{{{\ensuremath{\mathbf P}} }^m_N}\left( \sum_{x\in {\widetilde}{\Lambda}_N} (\phi(x)+H(x))^2 \right) \le C m^{-2}\left( N^2+ \sum_{x\in {\widetilde}{\Lambda}_N} H(x)^2\right).$$ Hence, as long as $K$ is chosen sufficiently large, using and - we obtain $$\begin{gathered}
{{\ensuremath{\mathbf P}} }^m_N\left[\frac{1}{N^2}\sum_{x\in {\widetilde}{\Lambda}_N} (\phi(x)+H(x))^2 \le \frac{2 f(m)}{m^2}-K \right]\\
\le \frac{{\mathrm{Var}}_{{{\ensuremath{\mathbf P}} }^m_N}\left( \sum_{x\in {\widetilde}{\Lambda}_N} (\phi(x)+H(x))^2 \right)}{\left({{\ensuremath{\mathbf E}} }^m_N\left[\sum_{x\in {\widetilde}{\Lambda}_N} (\phi(x)+H(x))^2 \right]-N^2 \left[\frac{2 f(m)}{m^2}-K\right] \right)^2}
\\
\le \frac{ C m^{-2}\left( \sum_{x\in {\widetilde}{\Lambda}_N} H(x)^2+N^{2} \right)}{\left((K-C)N^2+\sum_{x\in {\widetilde}{\Lambda}_N} H(x)^2 \right)^2}
\le C m^{-2} N^{-2}.
\end{gathered}$$ The result thus follows for our choice for the range of $N$.
Preliminary work for the proofs of Propositions \[prop:boundary\] and \[prop:inside\] {#liminouze}
=====================================================================================
Both proofs require a detailed knowledge on the distribution of the number of contact in ${\Lambda}_N\setminus {\Lambda}'_N$ and in ${\Lambda}'_N$. The highly correlated structure of the field makes this kind of information difficult to obtain.
We have chosen $u$ quite high in order to obtain a very low empirical density of contact. For this reason our problem is quite related to that of the study of the maximum and of the extremal process of the $2$-dimensional free field, which has been the object of numerous studies in the past [@cf:BDG; @cf:BDZ; @cf:Dav; @cf:Mad] together with the related subject of Branching Random Walk [@cf:A; @cf:AS; @cf:HS] or Brownian Motion [@cf:ABK]. We borrow two key ideas from this literature:
- The Gaussian Free Field can be written as a sum of independent fields whose correlation spread on different scales. This makes the process very similar to the branching random walk.
- The number of point present at a height close to the expected the maximum of the field is typically much smaller than its expectation (that is: by a factor $\log N$) but this $\log$ factor disapears if one conditions to a typical event.
These two points are respectively developed in Section \[galefash\] and \[typic\].
Decomposing the free field in a martingale fashion {#galefash}
--------------------------------------------------
Let us decompose the massive free field into independent fields in order to separate the different scales in the correlation structure. The idea of decomposing the GFF is not new was used a lot to study the extremum and there are several possible choices (see [@cf:BDG] where a coarser decomposition is introduced or more recently [@cf:BDZ]). Our choice of decomposition is made in order to have a structure similar to that present in [@cf:Mad].
There are several possible choices for the decomposition. The advantage of the one we present below is that the kernel of all the fields are expressed in terms of the heat-kernel, for which we have good estimates (cf. Section \[hkernel\]). Set (recall \[greenff\]) $$k:=\lfloor G^m(x,x) \rfloor$$ (it does not depend on $x$ as $G$ is translation invariant). We perform the decomposition of $\phi$ into a sum $k$ subfield, each of which having (roughly) unit variance. With this construction, $\phi(x)$ is the final step of a centered Gaussian random walk with $k$ steps. With this in mind we define a decreasing sequence of times $t_i$, $i\in {\llbracket}0,k {\rrbracket}$ as follows $$\label{defti}\begin{cases}
t_0:=\infty,\\
\int_{t_1}^{\infty}e^{-m^2t} P_t(x,x) {\mathrm{d}}t:=1,\\
\int_{t_{i+1}}^{t_{i}}e^{-m^2t} P_t(x,x) {\mathrm{d}}t:=1, \quad i\in {\llbracket}1,k-2 {\rrbracket}, \\
t_k:=0.
\end{cases}$$ This definition implies that $$\int_{0}^{t_{k-1}}e^{-m^2t} P_t(x,x) {\mathrm{d}}t \in [1,2).$$ From the Local Central Limit Theorem we can deduce that there exists a constant $C>0$ such that $$\begin{split}\label{madga}
\sup_{i\in {\llbracket}1,k-1 {\rrbracket}} |\log t_i- 4\pi(k-i)|\le C,\\
\left |k+ \frac{1}{2\pi}\log m\right|\le C,
\end{split}$$ We define $(\xi_i)_{i\in{\llbracket}1,k {\rrbracket}}$ to be a sequence of centered Gaussian fields (we use ${{\ensuremath{\mathbf P}} }$ to denote their joint law) indexed by ${\Lambda}_N$, each with covariance functions given by $$Q^{*}_i(x,y):=\int_{t_{i}}^{t_{i-1}}e^{-m^2t} P^{*}_t(x,y) {\mathrm{d}}t,$$ and set $$\phi_i:=\sum_{j=1}^i \xi_i.$$ Note that the covariance of $\phi_k$ is given by $G^{m,*}_N$ and for this reason we simply set $\phi:=\phi_k$ and work from now on this extended probability space. For this reason we use simply ${{\ensuremath{\mathbf P}} }$ instead of ${{\ensuremath{\mathbf P}} }^m_N$ (this should bring no confusion as $m$ and $N$ a now fixed by ).
Note that the distribution of the field $\xi_i$ in the bulk of ${\Lambda}_N$ is “almost” translation invariant and its variance is very close to one. When $x$ is close to the boundary $Q^*_i(x,x)$ becomes smaller, and this effect starts at distance $\exp(2\pi (k-i))$ from the boundary. The distance $\exp(2\pi(k-1))$ is also the scale on which covariance function $Q^*_i(x,y)$ varies in the bulk. For this reason it is useful to set $$\label{defjx}
j(x):= \left ( k- \left\lceil \frac{1}{2\pi}\log d(x,\partial{\Lambda}_N) \right\rceil \right)_+.$$ As a consequence of , of the definition of $k$ and that $j(x)$, we have $$\label{kratz1}
|{{\ensuremath{\mathbf E}} }[\phi^2(x)]-(k-j(x))|\le C.$$ We can deduce from this an estimate of the variance of $\phi_i(x)$, up to a $O(1)$ correction: there exists a constant $C$ such that $$\label{eq:varj}
\forall x \in {\Lambda}'_N, \, \forall i\in \{0,\dots,k\}, \quad \left| {{\ensuremath{\mathbf E}} }[\phi^2_i(x)]- (i-j(x))_+ \right | \le C$$ Indeed from Lemma \[lem:kerestimate\] $(iii)$, we have $$\label{kratz2}
\int_{\infty}^{t_{j(x)}} P^*_t(x,x){\mathrm{d}}t\le C,$$ As the variance of $\xi_i(x)$ is bounded by $1$ (or $2$ when $i=k$) this implies $${{\ensuremath{\mathbf E}} }[\phi^2_i(x)]\le C+(i-j(x))_+.$$ Finally we obtain the other bound using the fact that, as the increments have variance smaller than one (ore two for the last one) we have $${{\ensuremath{\mathbf E}} }[\phi^2(x)]-{{\ensuremath{\mathbf E}} }[\phi^2_i(x)] \le k-i+1$$ and we conclude using .
The conditional expectation for the number of contact {#typic}
-----------------------------------------------------
Now we are going to use the decomposition in order to obtain finer results on the structure of the field $\phi$. The idea is to show that with high probability the trajectory of $(\phi_i(x))_{i\in {\llbracket}0,k {\rrbracket}}$ tend to stay below a given line, for all $x\in {\Lambda}_N$, and thus if $\phi(x)$ reaches a value close to the maximum of the field, then conditioned to its final point, $(\phi_i(x))_{i=0}^k$ look more like a Brownian excursion than like a Brownian bridge, as it “feels” a constraint from above. If one restricts to the typical event described above, this constraint yields a loss of a factor $k$ (hence $\log N$) in the probability of contact.
Note that for technical reasons, points near the boundary are a bit delicate to handle and thus we choose to prove a property in a sub-box ${\Lambda}''_N$ which excludes only a few points of ${\Lambda}_N$. We set $${\Lambda}''_N:= {{\ensuremath{\mathbb Z}} }^2 \cap \left[N(\log N)^{-2}, N(1-(\log N)^{-2})\right],$$ and $$\gamma:= 2\sqrt{2\pi},$$ and define the event $$\label{defan}
\mathcal A_N= \left\{ \forall x \in {\Lambda}''_N,\ \forall i\ge j(x),\, \phi_i(x)\le \gamma(i-j(x))+ 100\gamma\log \log N \right\}.$$ We show that this event is very typical. This is a crucial step to define the event ${{\ensuremath{\mathcal C}} }_N$ and to estimate the probability of ${{\ensuremath{\mathcal B}} }_N$.
\[th:condfirstmom\] We have $${{\ensuremath{\mathbf P}} }\left[ \mathcal A_N \right]\ge 1-(\log N)^{-99},$$
We define for $i=0,\dots,k$ $$M_i:=\frac{1}{|{\Lambda}''_N|}\sum_{x\in {\Lambda}''_N} \exp\left( \gamma \phi_i(x)-\frac{\gamma^2}{2} {{\ensuremath{\mathbf E}} }\left[\phi^2_i(x)\right] \right).$$ It is trivial to check that it is a martingale for the filtration $${{\ensuremath{\mathcal F}} }_i:=\sigma(\phi_j(x), j\le i, x\in {\Lambda}''_N).$$ Integrating the second inequality in on the interval $[t_i,\infty)$, we have for all $x,y \in {\Lambda}_N$ $$|x-y|\le e^{2\pi(k-i)} \Rightarrow {{\ensuremath{\mathbf E}} }\left[ \left( \phi_i(x)-\phi_i(y) \right)^2\right]\le C |x-y|^2 e^{-4\pi(k-i)}.$$ Using a union bound, this implies that for $N$ sufficiently large $$\label{eq:localvar}
{{\ensuremath{\mathbf P}} }\left[ \max_{i\in{\llbracket}0,k-1 {\rrbracket}}\max_{\left\{ (x,y)\in ({\Lambda}_N)^2 \ : \ |x-y| \le e^{2\pi(k- i)}(\log N)^{-1} \right\}} |\phi_i(x)-\phi_i(y)| > 1 \right]
\le \frac{1}{N}.$$ On the complement of this event, if for a fixed $x\in {\Lambda}''_N$ we have$$\phi_i(x)\ge \gamma (i-j(x)) + 100\log \log N,$$ then $$\label{ineqcro}
M_j\ge \frac{1}{|{\Lambda}''_N|}\sum_{\{ y \ : \ |y-x|\le e^{2\pi(k- i)}(\log N)^{-1} \} }
e^{ \gamma^2 (i-j(x))+ 100\gamma\log \log N - \frac{\gamma^2}{2} {{\ensuremath{\mathbf E}} }[\phi^2_i(y)] }.$$ Now as $i\ge j(x)$, we realize that in the range of $y$ which is considered $j(y)\ge j(x)-1$ and hence from we have $${{\ensuremath{\mathbf E}} }[\phi^2_i(y)] \le i-j(x)+C+1.$$ For this reason, if $N$ is sufficiently large, implies that. $$\begin{gathered}
M_i\ge \frac{1}{{\Lambda}''_N} \exp\left( 4\pi(k-i)+ \frac{\gamma^2}{2} (i-j(x))+100\gamma (\log \log N) \right)\\
\le
C e^{-4 \pi j(x)} (\log N)^{100\gamma}\le (\log N)^{-100}.
\end{gathered}$$ The last inequality is valid for $N$ sufficiently large, it is obtained by using the definition and the fact that $x\in {\Lambda}''_N$ (which implies that $j(x)\le \frac{1}{\pi} \log \log N + C$). Using and the fact that $M$ is a martingale, we conclude that $${{\ensuremath{\mathbf P}} }[{{\ensuremath{\mathcal A}} }_N]\le \frac{1}{N}+ {{\ensuremath{\mathbf P}} }\left[ \exists i, \, M_i\ge (\log N)^{100} \right] \le \frac{1}{N}+ (\log N)^{-100}.$$
To conclude this section, we note that conditioning on the event ${{\ensuremath{\mathcal A}} }_N$ the probability of having a contact drops almost by a factor $(\log N)$, in the bulk of the box. More precisely
\[probacont\] There exists a constant $C$ such that
- For all $x\in {\Lambda}_N$ we have $$\frac{1}{C}N^{-2}(\log N)^{1+\alpha} \le {\widehat}{{\ensuremath{\mathbf E}} }{{\ensuremath{\mathbf E}} }\left[ \delta^{{\widehat}\phi,u}_x\right]\le CN^{-2}(\log N)^{1+\alpha}.$$
- For all $x\in {\Lambda}''_N$, we have $$\label{greluche}
{{\ensuremath{\mathbf E}} }\left[ \delta^{{\widehat}\phi,u}_x {\mathbf{1}}_{{{\ensuremath{\mathcal A}} }_N} \right]
\le C N^{-2} (\log N)^{\alpha} \left[ H(x)^2+(\log \log N)^2\right] \exp\left(\gamma H(x)- \frac{\gamma^2}{2}j(x) \right).$$ In particular $$\label{grelot}
{\widehat}{{\ensuremath{\mathbf E}} }^m {{\ensuremath{\mathbf E}} }\left[ \delta^{{\widehat}\phi,u}_x {\mathbf{1}}_{{{\ensuremath{\mathcal A}} }_N} \right]\le C N^{-2} (\log N)^{\alpha} (\log \log N)^2.$$
For the first point we notice that under law ${\widehat}{{\ensuremath{\mathbf P}} }^m\otimes {{\ensuremath{\mathbf P}} }$, $\phi_x+H(x)$ is distributed like an infinite volume free field and hence has covariance $G^{m}(x,x)\in[k,k+1)$. For this reason if $u\ge 1$ we have $${\widehat}{{\ensuremath{\mathbf E}} }{{\ensuremath{\mathbf E}} }\left[ \delta^{{\widehat}\phi,u}_x\right]= \int_{u-1}^{u+1}\frac{{\mathrm{d}}t}{\sqrt{2\pi G^m(x,x)}} e^{-\frac{-t^2}{2G^{m}(x,x)}} \le \frac{2}{\sqrt{2\pi G^m(x,x)}} e^{-\frac{-(u-1)^2}{2G^{m}(x,x)}},$$ and the result (the upper bound, but the lower bound is proved similarly) follows by replacing $u$ by its value, and $G^{m}(x,x)$ by the asymptotic estimate $\frac{1}{2\pi} \log N+O(1)$.
Let us now focus on the second point. First we note that the result is completely obvious is $H(x)\ge 4u/5$ (the l.h.s. of is larger than one). Hence we assume $H(x)\le 4u/5$. Then note that $$\begin{gathered}
{{\ensuremath{\mathbf E}} }\left[ \delta^{{\widehat}\phi,u}_x {\mathbf{1}}_{{{\ensuremath{\mathcal A}} }_N} \right]\le
{{\ensuremath{\mathbf P}} }\big[ \forall i\in {\llbracket}j(x),k {\rrbracket},\ \phi_i(x)\le \gamma (i-j(x))+100(\log \log N)\ ; \\
\phi(x)+H(x)\in [u-1,u+1] \big]\end{gathered}$$ A first step is to show that $$\label{asdas}
{{\ensuremath{\mathbf P}} }\big[ \phi_k(x)+H(x)\in [u-1,u+1] \big]\le C N^{-2}(\log N)^{\alpha+1} \exp\left(\gamma H(x)- \frac{\gamma^2}{2}i(x) \right).$$ Using the Gaussian tail estimate and we have $${{\ensuremath{\mathbf P}} }\big[ \phi_k(x)+H(x)\in [u-1,u+1] \big]\le \frac{C\sqrt{k}}{u-H(x)}\exp\left(-\frac{\left(u-1-H(x)\right)^2}{2(k-j(x)+C)} \right).$$ Note that the factor in front of the exponential is smaller than $C(\log N)^{-1/2}$ when $H(x)\le 4u/5$. Concerning the exponential term, notice that $$\begin{gathered}
\frac{\left(u-1-H(x)\right)^2}{2(k-j(x)+C)}=\frac{u^2}{2k}+ \frac{u^2(j(x)-C)}{2k(k-j(x)+C)}- \frac{(1+H(x))u}{k-j(x)+C}+ \frac{(1+H(x))^2}{2(k-j(x)+C)}
\\
\ge 2\log N-(\alpha+3/2)(\log \log N)+\frac{\gamma^2}{2}j(x)-\gamma H(x)-C'.\end{gathered}$$ This yields . To conclude the proof we need to show that for all $t\in [u-H(x)-1,u-H(x)+1]$ $$\begin{gathered}
{{\ensuremath{\mathbf P}} }\big[ \forall i\in \{ 0,\dots, k\}, \phi_i(x)\le \gamma (i-j(x))_+ +100(\log \log N)\ | \ \phi(x)=t \big]\\
\le C(\log N)^{-1}\left( H(x)^2+(\log \log N)^2\right).\end{gathered}$$ We use Lemma \[lem:bridge\], for the re-centered walk $$\phi_i(x)-{{\ensuremath{\mathbf E}} }[\phi_i(x) \ | \ \phi(x)=t].$$ Let $V_i=V_i(x)$ denote the variance of $\phi_i(x)$ and $V=V(x)$ that of $\phi(x)$. We have by standard properties of Gaussian variables $${{\ensuremath{\mathbf E}} }[\phi_i(x) \ | \ \phi(x)=t]=(V_i/V) t.$$ Using the bound , for all the considered values of $t$ we have $$\gamma (i-j(x))_+ +100(\log \log N)-(V_i/V)t\le 200 (\log \log N)+ |H(x)|.$$ Hence we have $$\begin{gathered}
{{\ensuremath{\mathbf P}} }\left[ \forall i\in {\llbracket}0, k{\rrbracket}, \phi_i(x)\le \gamma (i-j(x))_+ +100(\log \log N)\ | \ \phi(x)=t \right]\\
\le {{\ensuremath{\mathbf P}} }\big[ \ \forall i\in \{ 0,\dots, k\}, \phi_i(x)\le 200 (\log \log N)+ |H(x)| \ | \ \phi(x)=0 \ \big],\end{gathered}$$ and we conclude using Lemma \[lem:bridge\].
Proof of Proposition \[prop:boundary\] {#boundarry}
--------------------------------------
We are now ready to define the event ${{\ensuremath{\mathcal C}} }_N$. We set $${{\ensuremath{\mathcal C}} }_N:={{\ensuremath{\mathcal D}} }_N\cap {{\ensuremath{\mathcal C}} }'_N,$$ where $${{\ensuremath{\mathcal C}} }'_N:=\left\{ \left(
\sum_{x\in {\widetilde}{\Lambda}_N\setminus {\Lambda}'_N}\delta^{{\widehat}\phi,u}_x\right)\le (\log N)^{1/16}{{\ensuremath{\mathbf E}} }\left[\sum_{x\in {\widetilde}{\Lambda}_N\setminus {\Lambda}'_N}\delta^{{\widehat}\phi,u}_x \ |
\ {{\ensuremath{\mathcal A}} }_N \right] \right\}.$$ From Markov’s inequality, it is obvious that $${{\ensuremath{\mathbf P}} }[{{\ensuremath{\mathcal C}} }^{{\complement }}_N \ | \ {{\ensuremath{\mathcal A}} }_N] \le (\log N)^{-1/16},$$ and we can conclude (provided that $N$ is large enough) by using Proposition \[th:condfirstmom\], that holds.
Let us turn to the proof of . We want to get rid of the environment outside ${\Lambda}'_N$. The reader can check (by computing the second derivative that can be expressed as a variance) $$\begin{gathered}
{\beta}_2 \mapsto{{\ensuremath{\mathbb E}} }\left[ \log {{\ensuremath{\mathbf E}} }\left[ \exp\left( \sum_{x\in {\Lambda}'_N} ({\beta}{\omega}_x+h-{\lambda}({\beta})) \delta^{{\widehat}\phi,u}_x\right.\right.\right. \\+
\left.\left.\left. \sum_{x\in {\widetilde}{\Lambda}_N\setminus {\Lambda}'_N}({\beta}_2 {\omega}_x+h-{\lambda}({\beta}))\delta^{{\widehat}\phi,u}_x \right) {\mathbf{1}}_{{{\ensuremath{\mathcal D}} }_N} \right] \right]\end{gathered}$$ is convex in ${\beta}_2$ and has zero derivative at $0$. Hence reaches its minimum when ${\beta}_2$ equals zero, and $$\begin{gathered}
{{\ensuremath{\mathbb E}} }\left[ \log {{\ensuremath{\mathbf E}} }\left[ \exp\left(\sum_{x\in {\widetilde}{\Lambda}_N} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{{\widehat}\phi,u}_x\right) {\mathbf{1}}_{{{\ensuremath{\mathcal D}} }_N} \right] \right]\\
\ge {{\ensuremath{\mathbb E}} }\left[ \log {{\ensuremath{\mathbf E}} }\left[ \exp\left( \sum_{x\in {\Lambda}'_N} ({\beta}{\omega}_x+h-{\lambda}({\beta})) \delta^{{\widehat}\phi,u}_x -{\lambda}({\beta})
\sum_{x\in {\widetilde}{\Lambda}_N\setminus {\Lambda}'_N}\delta^{{\widehat}\phi,u}_x\right){\mathbf{1}}_{{{\ensuremath{\mathcal D}} }_N} \right] \right]\\
\ge
{{\ensuremath{\mathbb E}} }\left[ \log {{\ensuremath{\mathbf E}} }\left[ \exp\left( \sum_{x\in {\Lambda}'_N} ({\beta}{\omega}_x+h-{\lambda}({\beta})) \delta^{{\widehat}\phi,u}_x\right){\mathbf{1}}_{{{\ensuremath{\mathcal C}} }_N} \right] \right] \\
- (\log N)^{1/16}{\lambda}({\beta}){{\ensuremath{\mathbf E}} }\left[\sum_{x\in {\widetilde}{\Lambda}_N\setminus {\Lambda}'_N}\delta^{{\widehat}\phi,u}_x \ | \ {{\ensuremath{\mathcal A}} }_N \right],\end{gathered}$$ where the last line is obtained by restricting the expectation to ${{\ensuremath{\mathcal C}} }_N$ in order to bound $(\sum_{x\in {\widetilde}{\Lambda}_N\setminus {\Lambda}'_N}\delta^{{\widehat}\phi,u}_x)$ from below. Finally, using Lemma \[probacont\] and the definition of ${\Lambda}'_N$ we obtain that $${\widehat}{{\ensuremath{\mathbf E}} }^m{{\ensuremath{\mathbf E}} }\left[\sum_{x\in {\widetilde}{\Lambda}_N\setminus {\Lambda}'_N}\delta^{{\widehat}\phi,u}_x \ | \ {{\ensuremath{\mathcal A}} }_N\right]
\le C(\log \log N)^4(\log N)^{\alpha-1/8},$$ which is sufficient to conclude.
Proof of Proposition \[prop:inside\] {#intelinside}
====================================
Control of bad boundary conditions: Proof of and
-------------------------------------------------
We start with the easy part of the proposition: showing that the probability of a bad boundary condition is scarce , and that for this reason, a quite rough bound is sufficient to bound their contribution to the total expectation.
To prove , we use Lemma \[lem:kerestimate\]. For a fixed $x\in {\Lambda}'_N$, we set in the next equation $d:= d(x,\partial {\Lambda}_N)$. We have $$\begin{gathered}
\label{smass}
{\widehat}{{\ensuremath{\mathbf E}} }^m [ (H^{m,{\widehat}\phi}_N(x))^2 ]= \int^{\infty}_0 e^{-m^2t}[ P_t(x,x)-P^*_t(x,x) ]{\mathrm{d}}t\\
\le
\int^{\infty}_0 \frac{C}{t} e^{-m^2t} \exp \left( -C^{-1}\min\left(\frac{d^2}{t},d\log[(d/t)+1] \right)\right){\mathrm{d}}t\\
\le e^{-c'dm} \le \exp\left(-c' (\log N)^{1/8} \right).\end{gathered}$$ We have used in the last inequality that $d(x,{\Lambda}_N)\ge N(\log N)^{-1/8}$ for $x\in {\Lambda}'_N$. Hence we have for any $x\in {\Lambda}'_N$ $${\widehat}{{\ensuremath{\mathbf P}} }^m\left[|H^{m,{\widehat}\phi}_N(x)|\ge 1 \right]\le \exp\left(-e^{c(\log N)^{1/8}}\right),$$ and we can conclude using a union bound.
To prove , we use Jensen’s inequality and obtain $$\begin{gathered}
{{\ensuremath{\mathbb E}} }\log {{\ensuremath{\mathbf E}} }\left[ \exp\left(\sum_{x\in {\Lambda}'_N} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{{\widehat}\phi,u}_x\right) \ \big| \ {{\ensuremath{\mathcal C}} }_N \right]\\
\ge {{\ensuremath{\mathbb E}} }{{\ensuremath{\mathbf E}} }\left[\sum_{x\in {\Lambda}'_N} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{{\widehat}\phi,u}_x \ \big| \ {{\ensuremath{\mathcal C}} }_N \right]
\ge -{\lambda}({\beta})N^2.
\end{gathered}$$ Hence the conclusion follows from ${{\ensuremath{\mathbf P}} }[{{\ensuremath{\mathcal C}} }_N]\ge 1/2$.
Decomposing the proof of
-------------------------
Proving that good boundary conditions give a good contribution to the expected $\log$ partition function , is the most delicate point. We divide the proof in several steps. First we want to show that conditioned on the event ${{\ensuremath{\mathcal B}} }_N$, the expected $\log$ partition function is close to the corresponding annealed bound (obtained by moving the expectation w.r.t. ${\omega}$ inside the $\log$). This result is obtained by a control of the second moment of the restricted partition function.
\[consad\] For any ${\widehat}\phi\in {\widehat}{{\ensuremath{\mathcal A}} }_N$ we have $$\label{consaf1}
\log {{\ensuremath{\mathbf E}} }\left[ \exp\left(\sum_{x\in {\Lambda}'_N} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{{\widehat}\phi,u}_x\right) \ | \ {{\ensuremath{\mathcal B}} }_N \right]
\ge h {{\ensuremath{\mathbf E}} }\left[L_{N} \ | \ {{\ensuremath{\mathcal B}} }_N \right]- 1.$$
The second point is to show that ${{\ensuremath{\mathbf E}} }\left[L_{N} \ | \ {{\ensuremath{\mathcal B}} }_N \right]$ is large. What makes this difficult is that $L_N$ typically does not behave like its expectation ${{\ensuremath{\mathbf E}} }_N[L_N]$ (cf. Lemma \[probacont\]) We are going to prove that conditioned to ${{\ensuremath{\mathcal A}} }_N$, $L_N$ almost behaves like its expectation. To prove such a statement, we will impose a restriction to the trajectories which is slightly stronger than ${{\ensuremath{\mathcal A}} }_N$, as this makes computation easier.
\[sdasda\] We have for any ${\widehat}\phi \in {\widehat}{{\ensuremath{\mathcal A}} }_N$ $$\label{condexp}
{{\ensuremath{\mathbf E}} }\left[L_{N} \ | \ {{\ensuremath{\mathcal B}} }_N \right]\ge c(\log N)^\alpha.$$
Combining and We have for ${\widehat}\phi\in {\widehat}{{\ensuremath{\mathcal A}} }_N$, $$\begin{gathered}
{{\ensuremath{\mathbb E}} }\log {{\ensuremath{\mathbf E}} }\left[ \exp\left(\sum_{x\in {\Lambda}'_N} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{{\widehat}\phi,u}_x\right){\mathbf{1}}_{{{\ensuremath{\mathcal C}} }_N}\right]\\
\ge {{\ensuremath{\mathbb E}} }\log {{\ensuremath{\mathbf E}} }\left[ \exp\left(\sum_{x\in {\Lambda}'_N} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{{\widehat}\phi,u}_x\right) \ | \ {{\ensuremath{\mathcal B}} }_N\right]
+ \log {{\ensuremath{\mathbf P}} }\left[ {{\ensuremath{\mathcal B}} }_N \right]\\
\ge h{{\ensuremath{\mathbf E}} }\left[ L_N \ | \ {{\ensuremath{\mathcal B}} }_N \right] -1
\ge c h (\log N)^{\alpha}-1.
\end{gathered}$$
Proof of Lemma \[consad\]
-------------------------
First let us get a rough estimate on the probability of ${{\ensuremath{\mathcal B}} }_N$, valid for $N$ sufficiently large $$\label{kkk}
{{\ensuremath{\mathbf P}} }[{{\ensuremath{\mathcal B}} }^{\complement }_N]\le C(\log N)^{-\frac{1-\alpha}{4}}.$$ According to , for all ${\widehat}\phi\in {\widehat}{{\ensuremath{\mathcal A}} }_N$ $${{\ensuremath{\mathbf E}} }\left[ L_N {\mathbf{1}}_{{{\ensuremath{\mathcal A}} }_N}\right]\le C(\log N)^{\alpha}(\log \log N)^2.$$ Hence using the Markov inequality and the definition of ${{\ensuremath{\mathcal B}} }_N$ , we have $${{\ensuremath{\mathbf P}} }[ {{\ensuremath{\mathcal B}} }^{\complement }_N \ | \ {{\ensuremath{\mathcal A}} }_N] \ge {{\ensuremath{\mathbf P}} }[ L_N\ge (\log N)^{\frac{1+\alpha}{2}} \ | \ {{\ensuremath{\mathcal A}} }_N] \le C(\log N)^{-\frac{1-\alpha}{2}}(\log \log N)^2.$$ We deduce from the above equation, using the fact that ${{\ensuremath{\mathbf P}} }[{{\ensuremath{\mathcal A}} }_N]$ tends to one very fast (Proposition \[th:condfirstmom\]).
We continue the proof by setting, $$Y_N:={{\ensuremath{\mathbf E}} }\left[ \exp\left(\sum_{x\in {\Lambda}'_N} \left( {\beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{{\widehat}\phi,u}_x\right) {\mathbf{1}}_{{{\ensuremath{\mathcal B}} }_N} \right],$$ and $\zeta:=Y_N/{{\ensuremath{\mathbf E}} }[Y_N]$. We can bound the first term from below using Jensen’s inequality as follows $${{\ensuremath{\mathbb E}} }\left[ \log Y_N \right]= \log {{\ensuremath{\mathbb E}} }\left[ Y_N \right]+ {{\ensuremath{\mathbb E}} }\log[\zeta].$$ We have $$\log {{\ensuremath{\mathbb E}} }[Y_N]=\log {{\ensuremath{\mathbf E}} }\left[ \exp\left( h L_{N}\right){\mathbf{1}}_{{{\ensuremath{\mathcal B}} }_N} \right]
\ge h {{\ensuremath{\mathbf E}} }\left[L_{\alpha} \ | \ {{\ensuremath{\mathcal B}} }_N \right]+ \log {{\ensuremath{\mathbf E}} }[{{\ensuremath{\mathcal B}} }_N].$$ By , the second term is larger than $-\log 2$. To estimate ${{\ensuremath{\mathbb E}} }\log[\zeta]$ we simply compute the second moment of $\zeta$. We have $${{\ensuremath{\mathbb E}} }[\zeta^2]=
{\widetilde}{{\ensuremath{\mathbf E}} }_h^{\otimes 2}\left[\exp\left(\sum_{x\in {\Lambda}'_N}\chi({\beta})\delta^{(1)}_x \delta^{(2)}_x \right) \right],$$ where $\chi({\beta}):= {\lambda}(2{\beta})-2{\lambda}({\beta})$ and $$\frac{{\mathrm{d}}{\widetilde}{{\ensuremath{\mathbf P}} }_h}{{\mathrm{d}}{{\ensuremath{\mathbf P}} }}(\phi):= \frac{1}{{{\ensuremath{\mathbb E}} }[Y_N]}\exp\left( hL_{N} \right) {\mathbf{1}}_{{{\ensuremath{\mathcal B}} }_N}.$$ Note that as a consequence of the definition of ${{\ensuremath{\mathcal B}} }_N$ for $N$ sufficiently large, the density is bounded from above as follows $$\frac{{\mathrm{d}}{\widetilde}{{\ensuremath{\mathbf P}} }_h}{{\mathrm{d}}{{\ensuremath{\mathbf P}} }}(\phi)\le \frac{1}{{{\ensuremath{\mathbf P}} }[{{\ensuremath{\mathcal B}} }_N]}\exp\left( h (\log N)^{\frac{1+\alpha}{2}} \right)\le N^{1/4}.$$ Using the inequality $$\exp\left(\chi X\right)\le 1+[e^{\chi K}-1]X$$ valid for $X\in [0,K]$, we obtain $$\begin{gathered}
\label{hophop}
{{\ensuremath{\mathbb E}} }[\zeta^2]\le 1+e^{\chi({\beta})(\log N)^{\frac{1+\alpha}{2}}}{\widetilde}{{\ensuremath{\mathbf E}} }_h^{\otimes 2}[\delta^{(1)}_x \delta^{(2)}_x]\\
\le 1+N^{1/2}e^{\chi({\beta})(\log N)^{\frac{1+\alpha}{2}}}\sum_{x\in {\Lambda}'_N} ({{\ensuremath{\mathbf E}} }[\delta^{{\widehat}\phi}_x])^2\le 1+N^{3/4}\sum_{x\in {\Lambda}'_N} ({{\ensuremath{\mathbf E}} }[\delta^{{\widehat}\phi}_x])^2.\end{gathered}$$ Note that from and our choice for $m$ , the variance of $\phi$ satisfies $$\label{danslaboit}
\forall x \in {\Lambda}'_N, \quad \left| G^{*,m}(x,x)+ \frac{1}{2\pi} \log m \right| \le C.$$ Thus using our assumption on $|H(x)|\le 1$, and replacing $u$ by its value we obtain that for all $x\in {\Lambda}'_N$ $${{\ensuremath{\mathbf E}} }[\delta^{{\widehat}\phi,u}_x]^2\le \left[ \frac{2}{\sqrt{2\pi G^{*,m}(x,x)}} \exp\left(- \frac{(u-1-H(x))^2}{2G^{*,m}(x,x)} \right) \right]^2\le C N^{-4}(\log N)^{2(1+\alpha)}.$$ Thus we deduce from that $$\label{varixx}
{{\ensuremath{\mathbb E}} }[\zeta^2]-1\le N^{-1}.$$ This ensures that $\zeta$ is close to one with a large probability. However to estimate ${{\ensuremath{\mathbb E}} }[\log \zeta]$, we also need some estimate on the right-tail distribution of $\log \zeta$. We use a rather rough one $$\label{aaaa}
|\log \zeta|\le \max_{x\in {\Lambda}'_N}|{\beta}{\omega}_x-{\lambda}({\beta})|.$$ To conclude we note that for $\zeta\ge 1/2$ we have $$\log (\zeta)+1-\zeta \ge -(\zeta-1)^2,$$ and hence that $${{\ensuremath{\mathbb E}} }[\log \zeta]={{\ensuremath{\mathbb E}} }[\log(\zeta)+1-\zeta]\ge -{{\ensuremath{\mathbb E}} }[(\zeta-1)^2]+{{\ensuremath{\mathbb E}} }\left[ (\log (\zeta) +1-\zeta){\mathbf{1}}_{\{ \zeta\le 1/2\}} \right].$$ The first term in the r.h.s. can be controlled using . By Cauchy-Schwartz, the second term is smaller in absolute value than $$({{\ensuremath{\mathbb P}} }[\zeta\le 1/2])^{1/2}\left({{\ensuremath{\mathbb E}} }\left[ (\log \zeta+1-\zeta)^2 {\mathbf{1}}_{\{ \zeta\le 1/2\}} \right] \right)^{1/2}\le ({{\ensuremath{\mathbb P}} }[\zeta\le 1/2])^{1/2}\left({{\ensuremath{\mathbb E}} }\left[ (\log \zeta)^2\right] \right)^{1/2} .$$ Using Chebychev inequality together with , we get that $${{\ensuremath{\mathbb P}} }[\zeta\le 1/2]\le 4N^{-1}.$$ Using and the fact that ${\omega}$ have exponential tails (cf. assumption ), we have $${{\ensuremath{\mathbb E}} }\left[ (\log \zeta)^2 \right]\le C (\log N)^2.$$ Altogether we obtain that $$\log {{\ensuremath{\mathbb E}} }[Y_N]\ge h {{\ensuremath{\mathbf E}} }\left[L_{N} \ | \ {{\ensuremath{\mathcal B}} }_N \right]+ \log {{\ensuremath{\mathbf E}} }[{{\ensuremath{\mathcal B}} }_N]- CN^{-1/2}(\log N),$$ and we can conclude using .
Proof of Lemma \[sdasda\]
-------------------------
Instead of counting all the contacts, we decide to consider only a subset of them: those for which the trajectory $(\phi_i(x))_{i\in{\llbracket}0,k {\rrbracket}}$ stays below a given line. We choose the restriction to be a bit stronger than the one used in the definition of the event ${{\ensuremath{\mathcal A}} }_N$ . We set $$\begin{split}
\delta'_x&:={\mathbf{1}}_{\big\{ \left(\phi(x)-u+H(x)\right) \in [-1,1], \, \forall i\in {\llbracket}1,k{\rrbracket}, \, \phi_i(x)\le \frac{u i}{k}+10\big \}},\\
L'_N&:= \sum_{x\in {\Lambda}'_N} \delta'_x.
\end{split}$$ Let us first show how to reduce the proof of Lemma \[sdasda\] to a control on the two first moment of $L'_N$. We have $$\begin{gathered}
\label{dsaddsadcz}
{{\ensuremath{\mathbf E}} }\left[L_{N} {\mathbf{1}}_{{{\ensuremath{\mathcal B}} }_N} \right]\ge {{\ensuremath{\mathbf E}} }\left[L'_{N} {\mathbf{1}}_{{{\ensuremath{\mathcal B}} }_N}\right]
= {{\ensuremath{\mathbf E}} }\left[L'_{N}\right]- {{\ensuremath{\mathbf E}} }\left[ L'_{N} {\mathbf{1}}_{{{\ensuremath{\mathcal B}} }^{\complement}_N}\right]\\
\ge {{\ensuremath{\mathbf E}} }\left[L'_{N}\right]- \sqrt{ {{\ensuremath{\mathbf E}} }\left[ (L'_{N})^2 \right]} \sqrt{ {{\ensuremath{\mathbf P}} }\left[ {{\ensuremath{\mathcal B}} }^{\complement}_N \right] }. \end{gathered}$$ Thus we can conclude provided that one can prove the two following bounds on the expectation and variance of $L'_N$ $$\label{lnprime}\begin{split}
{{\ensuremath{\mathbf E}} }[L'_N] &\ge c(\log N)^\alpha,\\
{{\ensuremath{\mathbf E}} }[(L'_N)^2]&\le C(\log N)^{2\alpha}(\log \log N)^{8}.
\end{split}$$ It is then sufficient to combine these results with and . Hence we need to prove the two following results.
\[lesperance\] For all $x\in {\Lambda}'_N$ and ${\widehat}\phi\in {\widehat}{{\ensuremath{\mathcal A}} }_N$, we have $$\label{grominet}
{{\ensuremath{\mathbf E}} }[\delta'_x]\ge c N^{-2} (\log N)^{\alpha}.$$
\[covariancee\] We have for all $x, y\in {\Lambda}'_N$ and ${\widehat}\phi\in {\widehat}{{\ensuremath{\mathcal A}} }_N$, $$\label{secondmoment}
{{\ensuremath{\mathbf E}} }[\delta'_{x}\delta'_{y}]\le \frac{CN^{-4} (\log N)^{2\alpha+3}(\log \log N)^8}{(j(x,y)+1)^{3/2}(k-j(x,y)+1)^3}e^{\frac{j(x,y) u^2}{2k}}.$$ where $$j(x,y):= \left\lceil k-\frac{1}{2\pi}\log|x-y|_+ \right\rceil.$$
The quantity $j(x,y)$ can be interpreted as the step around which the increments of $(\phi_i(x))_{i=1}^k$ and $(\phi_j(x))_{i=1}^k$ decorrelate.
Before giving the details of these lemmas, let us prove . The bound on the expectation follows immediately from . Concerning the bound on the variance, as for a fixed $l=1$, we have $$\#\{ \ (x,y)\in ({\Lambda}'_N)^2 \ : \ j(x,y)=l \} \le C N^2e^{4\pi(k-l)}=C N^4 (\log N)^{-1/2} e^{-4\pi l},$$ and a trivial bound of $N^4$ for the case $l=0$. Hence we have $$\begin{gathered}
{{\ensuremath{\mathbf E}} }[(L'_N)^2]= \sum_{x,y\in {\Lambda}'_N} {{\ensuremath{\mathbf E}} }[\delta'_{x}\delta'_{y}]\\
\le C (\log N)^{2\alpha+3}(\log \log N)^8 \left[ (\log N)^{-3}+
(\log N)^{-1/2} \sum_{l=1}^k \frac{e^{-l \left(4\pi-\frac{u^2}{2k}\right)}}{(l+1)^{3/2}(k-l+1)^3} \right].
\end{gathered}$$ We must then control the above sum. Note that from and we deduce that $$\frac{u^2}{2k}-4\pi=2\pi\left(1+\alpha\right)\frac{\log \log N}{\log N}+O((\log N)^{-1}),$$ and hence that $$\sum_{j=1}^k \frac{e^{-j \left(4\pi-\frac{u^2}{2k}\right)}}{(j+1)^{3/2}(k-j+1)^3}\le C ( \log N)^{-\min(3, \frac{5}{2}+\alpha)}.$$ This implies .
Proof of Lemma \[lesperance\]
-----------------------------
If $(u-H(x)-1)\ge 0$ (which is satisfied if $h$ is small enough because as ${\widehat}\phi\in \mathcal A_N$ we have $|H(x)|\le 1$), we obtain from the expression of the Gaussian density $${{\ensuremath{\mathbf P}} }\big[\phi(x)\in [-1,1]+u-H(x) \big]\ge \frac{2}{\sqrt{2\pi G^{*,m}(x,x)}}\exp\left( -\frac{(u-H(x)+1)^2}{2G^{*,m}(x,x)} \right).$$ Using again that $H(x)\in[-1,1]$, we obtain, using $${{\ensuremath{\mathbf P}} }\left[\phi(x)\in [-1,1]+u-H(x)\right]\ge cN^{-2}(\log N)^{1+\alpha}.$$ Now we can conclude provided we show that for all $t$ in the interval $[u-1-H(x),u+1-H(x)]$, we have $$\label{condprob}
{{\ensuremath{\mathbf P}} }\left[ \forall i\in {\llbracket}1,k{\rrbracket}, \, \phi_i(x)\le \frac{u i}{k}+10 \ | \ \phi(x)=t \right]\ge \frac{c}{\log N}.$$ Let us recall the notation of Section \[typic\]: $V_i=V_i(x)$ denotes the variance of $\phi_i(x)$. For $i\le k-1$, we have $$V_i(x)=\int_{t_i}^{\infty} e^{-m^2t}P_t(x,x){\mathrm{d}}t= i- \int_{t_i}^{\infty} e^{-m^2t}\left[ P_t(x,x)-P^*_t(x,x)\right]{\mathrm{d}}t.$$ Hence from we have $$\label{bvar}
\forall x \in {\Lambda}'_N, \forall i \in {\llbracket}1,k {\rrbracket}, \quad V_i(x)\in[i-1,i+1].$$ We can check that and $t\in[u-2,u+2]$ implies $$\frac{u i}{k}+10-(V_i/V) t\ge 1,$$ To prove , we use simply Lemma \[lem:bridge\] $(ii)$ for the re-centered process $\phi_i(x)-(V_i/V)t$. We have $$\begin{gathered}
{{\ensuremath{\mathbf P}} }\left[ \forall i\in {\llbracket}1,k{\rrbracket}, \, \phi_i(x)\le \frac{u i}{k}+10 \ | \ \phi(x)=t \right]\\
\ge {{\ensuremath{\mathbf P}} }\left[ \forall i\in {\llbracket}1,k{\rrbracket}, \, \phi_i(x)\le 1 \ | \ \phi(x)=t \right] \ge \frac{C}{k}. \end{gathered}$$
A simplified version of Lemma \[covariancee\]
---------------------------------------------
We replace $(\phi_i(x))_{i=1}^k$ and $(\phi_i(y))_{i=1}^k$ and their intricate correlation structure by a simplified picture. Let $(X^{(1)}_i)_{i=1}^k $, $(X^{(2)}_i)_{i=1}^k$ be two walks, with IID standard Gaussian increments which are totally correlated until step $j$ and independent afterwards. More formally the covariance structure is given by $$\label{struct}
\begin{split}
&{{\ensuremath{\mathbf E}} }[X^{(1)}_{i_1} X^{(2)}_{i_2}]:= \min(i_1,i_2,j),\\
{{\ensuremath{\mathbf E}} }[X^{(1)}_{i_1} & X^{(1)}_{i_2}]={{\ensuremath{\mathbf E}} }[X^{(2)}_{i_1} X^{(2)}_{i_2}]:= \min(i_1,i_2).
\end{split}$$ For $i\le j$ we set $X_i=X^{(1)}_{i}=X^{(2)}_{i}$. The simplified version of we are going to prove is the following $$\begin{gathered}
{{\ensuremath{\mathbf P}} }\left[ \forall l\in \{1,2\}, \forall i \in {\llbracket}1 ,,k{\rrbracket}, \ X^{(l)}_i \le \left( \frac{iu}{k}+10\right), \ X^{(l)}_k \in [u-2,u+2] \right]
\\ \le \frac{C(\log \log N)^8}{(j+1)^{3/2}(k-j+1)^3}\exp\left( -\frac{(k+j)u^2}{2k^2} \right).\end{gathered}$$ Note that we replaced the interval $[u-H(x)-1,u-H(x)+1]$ and $[u-H(y)-1,u-H(y)+1]$ by $[u-2,u+2]$, and we also do so in the true proof of Lemma \[covariancee\]. This is ok since we are looking for an upper bound and as ${\widehat}\phi \in {\widehat}{{\ensuremath{\mathcal A}} }_N$, the latter inverval includes the other two.
The strategy is to first evaluate the probability $${{\ensuremath{\mathbf P}} }\left[ X_j \in {\mathrm{d}}t \ ; \ \forall l\in \{1,2\},\ X^{(l)}_k \in [u-2,u+2] \right],$$ and then compute the cost of the constraint $X^{(l)}_i \le \frac{iu}{k}+10$ using Lemma \[lem:bridge\] and the fact that conditioned to $X_j$, $X^{(1)}_k$ and $X^{(2)}_k$, the processes $(X_i)_{i=1}^j$, $(X^{(1)}_i)_{i=j}^k$ and $(X^{(2)}_i)_{i=j}^k$ are three independent Brownian bridges. For the first step, notice that we have $$\begin{gathered}
{{\ensuremath{\mathbf P}} }\left[ X_j \in {\mathrm{d}}t,\ X^{(1)}_k \in {\mathrm{d}}s_1,\ X^{(2)}_k \in {\mathrm{d}}s_2 \right] \\
= \frac{1}{(2\pi)^{3/2}(k-j)\sqrt{j}}
\exp\left(-\frac{t^2}{2j}-\frac{(s_1-t)^2+(s_2-t)^2}{2(k-j)}\right) {\mathrm{d}}t {\mathrm{d}}s_1 {\mathrm{d}}s_2.\end{gathered}$$ With the constraint $s_1,s_2\in [u-2,u+2]$ and $t\le \left( \frac{ju}{k}+10\right)$, at the cost of loosing a constant factor we can replace $s_1$ and $s_2$ by $u-2$. We obtain, after integrating over $s_1$ and $s_2$, $$\begin{gathered}
\label{struct1}
{{\ensuremath{\mathbf P}} }\left[ X_j \in {\mathrm{d}}t,\ X^{(1)}_k, X^{(2)}_k \in [u-2,u+2] \right]\le \frac{C}{(k-j)\sqrt{j}}\exp\left(-\frac{t^2}{2j}-\frac{(u-2-t)^2}{k-j}\right) {\mathrm{d}}t \\
\le \frac{C}{(k-j)\sqrt{j}} \exp\left( -\frac{(2k-j)u^2}{2k^2}-\left(\frac{u}{k}-\frac{2}{k-j}\right) \left(\frac{uj}{k}-t\right) \right) {\mathrm{d}}t.\end{gathered}$$ Note that due to our choice for $u$ and value of $k$ we have $\left(\frac{u}{k}-\frac{2}{k-j}\right)\in [\gamma/2,\gamma]$ provided that $h$ is sufficiently small and $k-j$ is sufficiently large (and hence the term can be replaced by $\gamma/2$ at the cost of changing the value of $C$).
Now using Lemma \[lem:bridge\] (after re-centering the process), we obtain that $$\begin{gathered}
\label{bridjun}
{{\ensuremath{\mathbf P}} }\left[ X_i \le \left(\frac{ui}{k}+10\right), \ \forall i\in{\llbracket}0,j{\rrbracket}\ | \ X_j=t \right]\\
= {{\ensuremath{\mathbf P}} }\left[ X_i \le \left(\frac{ui}{k}+10\right)- \frac{i t}{j},\ \forall i\in{\llbracket}0,j{\rrbracket}\ | \ X_j=0 \right]
\\ \le C j^{-1}\left( \left(\frac{uj}{k}-t\right)^2+ (\log j)^2 \right),\end{gathered}$$ where we have used that for $t\le \left(\frac{uj}{k}+10\right)$ and $i\le j$ $$\left(\frac{ui}{k}+10\right)- \frac{i t}{j}= \frac{i}{j}\left(\frac{uj}{k}-t \right)+10\le \left(\frac{uj}{k}-t \right)+20.$$ In the same manner we obtain that for $l\in \{1,2 \}$ $$\begin{gathered}
\label{bridj2}
{{\ensuremath{\mathbf P}} }\left[ X^{(l)}_i \le \frac{ui}{k}+10, \ \forall i\in{\llbracket}j,k {\rrbracket}\ | \ X_j=t, X^{(l)}_k\in[u-2,u+2] \right]\\
\le C(k-j)^{-1}\left( \left(\frac{uj}{k}-t\right)^2+ (\log (k-j))^2\right).\end{gathered}$$ Hence using -- and conditional independence we obtain that $$\begin{gathered}
{{\ensuremath{\mathbf P}} }\left[ \forall l\in \{1,2\}, \ \forall i \in {\llbracket}1,k{\rrbracket}, \ X^{(l)}_i \le \frac{iu}{k}+10\ ; \ X^{(l)}_k \in [u-2,u+2]\ ; \ X_j\in {\mathrm{d}}t \right] \\
\le C (k-j)^{-3} j^{-3/2} (\log k)^6 \exp\left( -\frac{(2k-j)u^2}{2k^2}- (\gamma/2)\left(\frac{uj}{k}-t\right) \right) {\mathrm{d}}t,\end{gathered}$$ which after integration over $t\le (\frac{uj}{k}+10)$ gives $$\begin{gathered}
{{\ensuremath{\mathbf P}} }\left[ \forall l\in \{1,2\} \forall i \in {\llbracket}1,k{\rrbracket}, \ X^{(l)}_i \le \frac{iu}{k}+10, X^{(l)}_k \in [u-2,u+2] \right]\\ \le
C (k-j)^{-3} j^{-3/2} (\log k)^6\exp\left( -\frac{ (2k-j)u^2}{2k^2} \right).\end{gathered}$$
Proof of Lemma \[covariancee\]
------------------------------
Now, we are ready to handle the case were $X^{(1)}_i$ and $X^{(2)}_i$ are replaced by $\phi_i(x)$ and $\phi_i(y)$. Some adaptation are needed since the increments of $\phi_i(x)$ and $\phi_i(y)$ have a less simple correlation structure but the method presented above is hopefully robust enough to endure such mild modifications. Given $x$ and $y$ set $$Z_i(x,y)=Z_i:=\frac{\phi_i(x)+\phi_i(y)}{2} \quad \text{and} \quad U_i:= {{\ensuremath{\mathbb E}} }\left[ Z_i \right]^2.$$ Let us prove that there exists a constant $C$ such that $$\label{tborne}\begin{cases}
\left|U_i-i \right|\le C, \quad \forall i \in {\llbracket}0,j{\rrbracket},\\
\left|U_i-\frac{i+j}{2} \right|\le C, \quad \forall i\in {\llbracket}j,k{\rrbracket}.
\end{cases}$$ To see this it is sufficient to remark that $$\begin{gathered}
U_i:= \frac{1}{4}\int_{t_i}^{\infty} e^{-m^2t}\left[P^*_t(x,x)+P^*_t(y,y)+2P^*_t(x,y)\right] {\mathrm{d}}t\\
=
\frac{1}{2}\int_{t_i}^{\infty} e^{-m^2t} \left[ P_t(x,x)+P_t(x,y)\right] {\mathrm{d}}t- r_i(x,y).\end{gathered}$$ where $$r_i(x,y):= \frac{1}{4}\int_{t_i}^{\infty} e^{-m^2t}\left[(P_t-P^*_t)(x,x)+(P_t-P^*_t)(y,y)+2(P_t-P^*_t)(x,y)\right] {\mathrm{d}}t.$$ Using , we see that $P^*_t$ can be replaced by $P_t$ at the cost of a small correction i.e. that $r_i$ is small. Using the definition of $t_i$ We have for $i\in {\llbracket}0,j{\rrbracket}$ $$\label{truit}
\frac{1}{2}\int_{t_i}^{\infty} e^{-m^2t}\left[ P_t(x,x)+P_t(x,y) \right] {\mathrm{d}}t=
i- \int_{t_i}^{\infty} e^{-m^2t}\left[ P_t(x,x)-P_t(x,y)\right] {\mathrm{d}}t,$$ while for $i\in {\llbracket}j,k{\rrbracket}$ we have $$\begin{gathered}
\label{struit}
\frac{1}{2}\int_{t_i}^{\infty} e^{-m^2t} \left[P_t(x,x)+P_t(x,y)\right] {\mathrm{d}}t\\
= \frac{i+j}{2}- \int_{t_j}^{\infty}
e^{-m^2t} \left[ P_t(x,x)-P_t(x,y) \right] {\mathrm{d}}t+ \int^{t_j}_{t_i} e^{-m^2t} P_t(x,y) {\mathrm{d}}t.
\end{gathered}$$ The kernel estimates and then allow to conclude that the integrals in the r.h.s of and are bounded by a constant and thus that hold. Similarly to , we are first going to show that we have, for all $t\le (\frac{uj}{k}+10)$, $$\begin{gathered}
\label{crig}
{{\ensuremath{\mathbf P}} }\big[ Z_j \in {\mathrm{d}}t \; \ \phi(x), \phi(y) \in [u-2,u+2] \big]\\
\le
\frac{C}{(k-j)\sqrt{j}} \exp\left( -\frac{(2k-j)u^2}{2k^2}- (\gamma/2) \left(\frac{uj}{k}-t\right) \right) {\mathrm{d}}t.\end{gathered}$$ Using the independence of $Z_j$ and $Z_k-Z_j$ and the fact that, up to correction of a constant order their respective variance are respectively equal to $j$ and $(k-j)/2$ (cf -), we can obtain (provided that $(k-j)$ is large enough), similary to that $$\begin{gathered}
\label{creg}
{{\ensuremath{\mathbf P}} }\left[ Z_j \in {\mathrm{d}}t\ ;\ Z_k \in [u-2,u+2] \right]\\
\le
\frac{C}{\sqrt{j(k-j)}} \exp\left( -\frac{(2k-j)u^2}{2k^2}- (\gamma/2) \left(\frac{uj}{k}-t\right) \right) {\mathrm{d}}t.\end{gathered}$$ Now, on top of that, we want to show that $$\label{crog}
{{\ensuremath{\mathbf P}} }\big[ \ (\phi(x)-\phi(y))\in[-4,4] \ | \ Z_j \in {\mathrm{d}}t, Z_k \in [u-2,u+2] \ \big]\le C(k-j)^{-1/2}.$$ As $(\phi(x)-\phi(y))$ is a Gaussian we can prove by showing that $$\label{varparlba}
{\mathrm{Var}}_{{{\ensuremath{\mathbf E}} }[\cdot \ | \ Z_j, Z_k ]} \left[ \phi(x)-\phi(y) \right]\ge c(k-j),$$ at least when $(k-j)$ is large: it implies that conditional density is bounded by $(2\pi c(k-j))^{-1/2}$ and thus that holds. In fact we prove this bound for the variance conditioned to $\phi_j(x), \phi_j(y)$ and $Z_k$ (which is smaller as the conditioning is stronger) as it is easier to compute.
If one sets $$Z'_i=\phi_i(x)-\phi_i(y),$$ one can remark, first using the fact that the increments of $(Z,Z')$ are independent and then using the usual formula for the conditional variance of Gaussian variable, that $$\label{fromul}
{\mathrm{Var}}_{{{\ensuremath{\mathbf E}} }[\cdot \ | \ \phi_j(x), \phi_j(y), Z_k ]} = {{\ensuremath{\mathbf E}} }[ (Z'_k-Z'_j)^2]-\frac{\left({{\ensuremath{\mathbf E}} }\left[(Z'_k-Z'_j)(Z_k-Z_j)\right]\right)^2}{{{\ensuremath{\mathbf E}} }[(Z_k-Z_j)^2]}.$$ Using (to replace $P^*_t$ by $P_t$) and (to control the term $P^*_t(x,y)$) we have $$\begin{gathered}
{{\ensuremath{\mathbf E}} }[ (Z'_k-Z'_j)^2]= \int^{t_j}_0 e^{-m^2t} \left[ P^*_{t}(x,x)+P^*_t(y,y)-2P^*_t(x,y)\right]{\mathrm{d}}t \\
\ge \int^{t_j}_0 e^{-m^2t} \left[P_{t}(x,x)+P_t(y,y)\right]{\mathrm{d}}t - C\ge 2(k-j)-C.\end{gathered}$$ Obviously ${{\ensuremath{\mathbf E}} }\left[(Z_k-Z_j)^2\right]$ is of the same order, and from again. $$|{{\ensuremath{\mathbf E}} }\left[(Z'_k-Z'_j)(Z_k-Z_j)\right]|= \left| \frac{1}{2}\int^{t_j}_0 e^{-m^2t} (P^*_{t}(x,x)-P^*_t(y,y)) {\mathrm{d}}t \right|\le 1.$$ Hence combining these inequalities in we obtain that holds. To conclude the proof we need to show that $$\label{pstnfut}
{{\ensuremath{\mathbf P}} }\left[ \forall i\in [0,j], Z_i\le \frac{uj}{k}+10 \ | \ Z_{j}=t \right] \le C \left[ \left(\frac{uj}{k}-t\right)^2+(\log j)^2 \right] j^{-1}$$ and $$\begin{gathered}
\label{pstnfut2}
{{\ensuremath{\mathbf P}} }\left[ \ \forall i\in [j,k], \phi_i(x), \phi_i(y)\le \frac{uj}{k}+10 \ | \ Z_{j}=t, \ \phi(x),\phi(y)\in[u-2,u+2] \ \right] \\
\le \left[ \left(\frac{uj}{k}-t\right)^2+(\log j)^2 \right]^2 (k-j)^{-2}.\end{gathered}$$ Indeed using conditional independence we can multiply the inequalities and with to obtain $$\begin{gathered}
{{\ensuremath{\mathbf P}} }[ \delta'_x\delta'_y, \ Z_{j}\in {\mathrm{d}}t] \\
\le \frac{C\left[ \left(\frac{uj}{k}-t\right)^2+(\log k)^2 \right]^3}{(k-j)^{3}j^{3/2}}
\exp\left( -\frac{(2k-j)u^2}{2k^2}- (\gamma/2) \left(\frac{uj}{k}-t\right) \right) {\mathrm{d}}t,\end{gathered}$$ and conclude by integrating over $t$. The proof of is quite similar to that of . $$\begin{gathered}
{{\ensuremath{\mathbf P}} }\left[ \forall i\le j,\ Z_i\le \frac{u i}{k}+10 \ | \
Z_j=t \right]\\ ={{\ensuremath{\mathbf P}} }\left[ \forall i\le j,\ Z_i\le \frac{u i}{k}+10- (U_i/U_j)t \ | \
Z_j=0 \right].\end{gathered}$$ We use to obtain for all $i\in {\llbracket}0, j {\rrbracket}$, $$\frac{u i}{k} - \frac{ U_i t}{U_j}\le \frac{U_i}{U_j}\left(\frac{u j}{k}-t \right)+C\le
\left(t- \frac{u i}{k} \right)+C'$$ and apply Lemma \[lem:bridge\], we obtain $$\label{croco2}
{{\ensuremath{\mathbf P}} }\left[ \forall i\le j,\ Z_i\le \frac{u i}{k}+10 \ | \
Z_j=t \right]\le C j^{-1}\left( \left(\frac{uj}{k}-t\right)^2+ (\log j)^2 \right).$$ To prove we have to be more careful as the increments of $\phi(x)$ and $\phi(y)$ are correlated. It is more practical in the computation to condition to the constraint $(\phi_j(x),\phi_j(y))=(t_1,t_2)$ than to $Z_j=t$. To obtain a bound we then take the maximum over the constraint $(t_1+t_2)=2t$. We consider only the case $\phi(x)=\phi(y)=u-2$ in the conditioning as the others can be deduced by monotonicity (which follows from positive correlations in the Gaussian processes that are considered).
We can consider without loss of generality that $$\label{restrict}
u-C(k-j)\le t_1 , t_2 \le \frac{ju}{k}+10,$$ the upper bound is due to the conditioning, and if the lower-bound is violated, $t$ is so small that the r.h.s. of is larger than one. Similarly to , using to control the value of $V_i$ we can prove $$\begin{gathered}
\label{rick}
{{\ensuremath{\mathbf P}} }\left[ \forall i\in[j,k],\ \phi_i(x)\le \frac{u i}{k}+10 \ | \ \phi_j(x)=t_1,\ ; \ \phi(x)\in[u-2,u+2] \right] \\ \le
C (k-j)^{-1}\left( \left(\frac{uj}{k}-t_1\right)^2+ (\log (k- j))^2 \right).\end{gathered}$$ Now the challenge lies in estimating the cost of the constraint $\phi_i(y)\le (\frac{u i}{k}+10)$, on the segment ${\llbracket}j,k {\rrbracket}$, knowing $\phi(y)$, $\phi_j(y)$ and $\phi_i(x)$, $i\in{\llbracket}1,k {\rrbracket}$. After conditioning to $\phi_j(y)$ and $(\phi_i(x))_{i\in {\llbracket}1,k{\rrbracket}}$, note that $\left(\phi_i(y)\right)_{i\in {\llbracket}j,k{\rrbracket}}$ is still a process with independent increments. Hence we can apply Lemma \[lem:bridge\] provided we get to know the expectation and variance of these increments. Let $V_i$ denote the conditional variance of $\phi_i(y)$ knowing $(\phi_r(x))_{r\in{\llbracket}0,k {\rrbracket}}$. For a sequence $f_i$ (random or deterministic) indexed by the integers, we set $$\label{defnabla}
\nabla f_i: = f_i-f_{i-1}.$$ Let $T_i$ measure the correlation between $\nabla \phi_i(x)$ and $\nabla\phi_i(y)$. We have $$\begin{split}
\nabla V_i&= {{\ensuremath{\mathbf E}} }[ (\nabla \phi_i(y))^2]- {{\ensuremath{\mathbf E}} }[\nabla \phi_i(x)\nabla\phi_i(y)],\\
T_i&:= \frac{{{\ensuremath{\mathbf E}} }[\nabla \phi_i(x)\nabla\phi_i(y)]}{{{\ensuremath{\mathbf E}} }[ (\nabla \phi_i(y))^2]}.
\end{split}$$ Note that from we have ${{\ensuremath{\mathbf E}} }[ (\nabla \phi_i(y))^2]\ge 1/2$, and thus we deduce from that $$\sum_{i=j+1}^k T_i \le 2 \int^{t_j}_{t_i} P_t(x,y){\mathrm{d}}t\le C.$$ Also using we obtain that for all $i\in{\llbracket}j,k {\rrbracket}$ $$\label{approx}
\left| V_i- V_j -(i-j) \right|\le C.$$ The conditional expectation of $\phi_i(y)$, $i\ge j$ given $\phi_j(y)$ and $(\phi_r(x))_{r\in{\llbracket}0,k{\rrbracket}}$ is given by $${{\ensuremath{\mathbf E}} }\big[ \phi_i(y)-\phi_j(y) \ | \ (\phi_r(x))_{r\in{\llbracket}0,k{\rrbracket}}\big]= \sum_{r=j+1}^k T_r \nabla \phi_r(x).$$ In particular this is smaller (in absolute value) than $C\log (k-j)$ on the event $${{\ensuremath{\mathcal H}} }(j,N,x)={{\ensuremath{\mathcal H}} }:=\left\{ |\nabla \phi_i(x)|\le \log (k-j), \ \forall i\in{\llbracket}j+1,k {\rrbracket}\ \right\}.$$ Note that ${{\ensuremath{\mathcal H}} }$ is a very likely event. We have, uniformly in $t_1$ satisfying $${{\ensuremath{\mathbf P}} }\left[ {{\ensuremath{\mathcal A}} }^{{\complement }} \ | \ \phi_j(x)=t_1\ ; \ \phi(x)=u-2 \right] \le \exp\left(-c (\log (k-j))^2 \right).$$ Indeed, after conditioning, the increments $\nabla \phi_i(x)$ are Gaussian variables of variance smaller than $1$ (or $2$ for $i=k$) and their mean, equal to $(V_i-V_j)(u-2-t_1)/V_i$, is bounded by a uniform constant, due to the restriction . If one add the conditioning to $\phi(y)$ and $\phi_j(y)$ one obtains, for all $(\phi_i(x))_{i\in {\llbracket}0,k {\rrbracket}} \in {{\ensuremath{\mathcal H}} }$ $$\begin{gathered}
{{\ensuremath{\mathbf E}} }\left[ \phi_i(y) \ | \ (\phi_r(x))_{r\in{\llbracket}0,k{\rrbracket}}, \phi_j(y)=t_2 \ ; \ \phi(y)=u-2 \right] \\ \ge
t_2+ \left(\frac{V_i-V_j}{V_k-V_j}\right)(u-2-t_2) + \sum_{r=j+1}^k T_r \nabla \phi_r(x)
\\ \ge t_2 + \left(\frac{V_i-V_j}{V_k-V_j}\right) \frac{(k-j)u}{k}- C ( \log(k-j)+1) \end{gathered}$$ where to obtain the last inequality we used and and the definition of ${{\ensuremath{\mathcal H}} }$. We have $$\begin{gathered}
\frac{iu}{k}- t_2 - \left(\frac{V_i-V_j}{V_k-V_j}\right) \frac{(k-j)u}{k}\\
= \left(\frac{ju}{k}- t_2 \right)+ \frac{u}{k} \left( \frac{(V_i-V_j(k-j)}{V_k-V_j}- (i-j) \right)
\ge \left(\frac{ju}{k}- t_2 \right)- C.\end{gathered}$$ Hence, using Lemma \[lem:bridge\], after the necessary re-centering for the bridge conditioned to $(\phi_r(x))_{r\in{\llbracket}0,k{\rrbracket}}$ we obtain that if $(\phi_r(x))_{r\in{\llbracket}0,k{\rrbracket}}\in {{\ensuremath{\mathcal H}} }$ and is satisfied we have $$\begin{gathered}
\label{rock}
{{\ensuremath{\mathbf E}} }[ \forall i \in {\llbracket}j,k {\rrbracket}, \ \phi_i(y)\le u \ | \ (\phi_r(x))_{r\in{\llbracket}0,k{\rrbracket}} \ ; \ \phi_j(y)=t_2 \ ; \ \phi(y)=u-2 ] \\
\le C (k-j)^{-1}\left[ \left(\frac{ju}{k}- t_2 \right)^2+ C \log(k-j) \right]^2.\end{gathered}$$ Using and , we obtain that $$\begin{gathered}
{{\ensuremath{\mathbf P}} }\Big[ \forall i\in {\llbracket}j,k {\rrbracket}, \phi_i(x), \phi_i(y)\le \frac{uj}{k}+10 \\
\big| \ \phi_j(x)=t_1,\ ; \ \phi_j(y)=t_2 \ ; \ \phi(x),\phi(y)\in[u-2,u+2]\Big]\\
\\ \le C (k-j)^{-2}\left[ \left(\frac{ju}{k}- t_1 \right)^2+ C \log(k-j) \right]\left[ \left(\frac{ju}{k}- t_2 \right)^2+ C \log(k-j)^2 \right]\\
+ {{\ensuremath{\mathbf P}} }\left[ {{\ensuremath{\mathcal H}} }^{{\complement }} \ | \ \phi_j(x)=t_1, \phi(x)=u-2 \right].
\end{gathered}$$ The last term is negligible when compared to the first and taking the maximum over $t_1+t_2=2t$ satisfying , this concludes the proof of .
[**Acknowldedgements:**]{} The author would like to express his gratitude to Jian Ding, Giambattista Giacomin and Thomas Madaule for various enlightening discussions.
Estimates on heat-kernels and random walks {#appendix}
==========================================
Proof of Lemma \[Greenesteem\]
------------------------------
To estimate the Green Function of the massive field we use a bit of potential theory. We let $a$ denote the potential Kernel of ${\Delta}$ in ${{\ensuremath{\mathbb Z}} }^2$ i.e. $$a(x):=\lim_{T\to \infty} \int_{0}^T \left( P_t(0,0)-P_t(x,0)\right){\mathrm{d}}t.$$ From [@cf:LL Theorem 4.4.4] we have $$\label{potentas}
a(x):=\frac{1}{2\pi}\log |x| + O(1).$$ Set $a(x,y):=a(x-y)$. Now recall that $X$ is a continuous time random-walk on ${{\ensuremath{\mathbb Z}} }^2$ with generator ${\Delta}$ and that $P^x$ denote is law when the initial condition is $x\in {{\ensuremath{\mathbb Z}} }^2$, and $\tau_{A}$ denote the hitting time of $A$. Let $T_m$ be a Poisson variable of mean $m^{-2}$ which is independent of $X$.
By adapting the proof of [@cf:LL Proposition 4.6.2(b)] we obtain that $$\label{sdasdad}\begin{split}
G^{m,*}(x,y)=E^x\left[ a\left(X_{\tau_{\partial{\Lambda}_N}\wedge T_m},y\right)\right]-a(x,y),\\
G^{m}(x,y)=E^x\left[ a\left(X_{T_m},y\right)\right]-a(x,y).
\end{split}$$ Considering the case $y=x$ and when there is no boundary, it is not difficult to see that $$\label{stima}
G^{m}(x,x)=E^x\left[ a\left(X_{T_m},x\right)\right]:= -\frac{1}{2\pi}\log m+O(1).$$ In the case $x=y$ with boundary, this is more delicate. On one side it is easy to deduce from that for some appropriate $C>0$, $$\frac{1}{2\pi}\log \left( \min\left( d(x,\partial {\Lambda}_N),m^{-1}\right)\right)-C \le G^{m,*}(x,x)\le -\frac{1}{2\pi}\log m+C.$$ What remain to prove is that $\frac{1}{2\pi}\log d(x,\partial {\Lambda}_N)$ is an upper-bound (which is a concern only if $d(x,{\Lambda}_N)\le m^{-1}$).
Note that the Green Function with Dirichlet boundary condition is an increasing function of the domain and a decreasing function of $m$. Hence to obtain an upper-bound on $G^{m,*}$, we can compare it with the the variance of the massless free field in the half plane ${{\ensuremath{\mathbb Z}} }_+\times {{\ensuremath{\mathbb Z}} }$ at the point $$\label{ddxx}
x_d:= (d(x,\partial {\Lambda}_N),0)$$ that is given by $$E^{x_d}\left[ a\left(X_{\tau_{\{0\}\times {{\ensuremath{\mathbb Z}} }}},x_d\right)\right]\ge G^{m}(x,x) .$$ Now note that $\tau_{\{0\}\times {{\ensuremath{\mathbb Z}} }}$ is simply the hitting time of zero by one dimensional simple random walk starting from $d(x,\partial {\Lambda}_N)$. Hence $$P^{x_d}[\tau_{\{0\}\times {{\ensuremath{\mathbb Z}} }}\ge t]\le C d(x,\partial {\Lambda}_N) t^{-1/2}.$$ As the second coordinate of $X_{\tau_{\{0\}\times {{\ensuremath{\mathbb Z}} }}}$ is simply the value of an independent random walk evaluated at $\tau$ we get that for some constant $C'$ all $u>0$ $$P^{x_d}[ |X_{\tau_{\{0\}\times {{\ensuremath{\mathbb Z}} }}}-x_d|\ge u ]\le C (u/d).$$ This tail estimate, together with is sufficient to conclude that $$E^{x_d}\left[ a\left(X_{\tau_{\{0\}\times {{\ensuremath{\mathbb Z}} }}},x_d\right)\right]\le \frac{1}{2\pi}\log d(x,\partial {\Lambda}_N)+C.$$
Proof of Lemma \[lem:kerestimate\]
----------------------------------
Let us start with $(i)$ The first inequality in can be deduced from [@cf:LL Theorem 2.3.6] which is a fine estimate for $P_t(x,x)- P_t(x,y)$ in discrete time.
For the second one, we notice that we can reduce the problem to proving that for any $u,v\in[0,N]$ $$\left( p^*_t(u,u)+p^*_t(v,v)-2p^*_t(u,v)\right)\le \frac{ C |u-v|^2 }{t^{3/2}},$$ where $p^*_t$ is the heat-kernel associated with the simple random-walk on ${\llbracket}0,N {\rrbracket}$ with Dirichlet boundary condition. Indeed if $x$ and $y$ differ by only one coordinate, say $x_1=y_1$ we can factorize the l.h.s of by the common coordinate and obtain $$\begin{gathered}
p^*_t(x_1,x_1)\left[p^*_t(x_2,x_2)+p^*_t(y_2,y_2)-2p^*_t(x_2,y_2)\right ] \\
\le \frac{C}{\sqrt{t}}\left[p^*_t(x_2,x_2)+p^*_t(y_2,y_2)-2p^*_t(x_2,y_2)\right ].\end{gathered}$$ If the two coordinates of $x$ and $y$ differ, then if we let $\varphi$ be a field with covariance function $P^*_t$, the l.h.s of can be rewritten as $${{\ensuremath{\mathbb E}} }[\left(\varphi_x-\varphi_y\right)^2]\le 2\left({{\ensuremath{\mathbb E}} }[\left(\varphi_x-\varphi_z\right)^2]+{{\ensuremath{\mathbb E}} }[\left(\varphi_y-\varphi_z\right)^2]\right)$$ and we reduce to the first case by choosing $z=(x_1,y_2)$.
Now, by Fourier decomposition of the kernel, we have $$\left( p^*_t(u,u)+p^*_t(v,v)-2p^*_t(u,v)\right)=\frac{2}{N}
\sum_{i=1}^{N-1} e^{-{\lambda}_i t}\left [\sin\left( \frac{i\pi u}{N}\right)- \sin\left( \frac{i\pi v}{N}\right) \right]^2.$$ where ${\lambda}_i:= 2\left(1-\cos\left(\frac{i\pi}{N}\right)\right)$. The sum can obviously be bounded by $$\frac{C|v-u|^2}{N^3} \sum_{i=1}^{N-1} e^{-{\lambda}_i t} i^2,$$ It is a simple exercise to show that this sum is of order $k^2 t^{-3/2}$.
For $(ii)$ we can just use large deviations estimates for $|x-y|\ge C\sqrt{t \log t}$ with $C$ chosen sufficiently large, and use the local central limit Theorem [@cf:LL Theorem 2.1.1] to cover the case $|x-y|\le C\sqrt{t \log t}$. For $(iii)$ we can compare to the half-plane case where $x=x_d$ (recall that from the argument presented before this gives an upper bound). In that case we have
$$P^*_t(x,x)=P[ X_t=0; \forall s\in[0,t],\ X_s\le d].$$
where $X$ is the simple random walk on ${{\ensuremath{\mathbb Z}} }^2$ starting from zero. By a reflexion argument we have $$P[ X_t=0; \forall s\in[0,t],\ X_s< d]= P[ X_t=0]-P[X_t=2d]=P_t(0,0)-P_t(0,2d{{\mathbf e}}_1).$$ The later quantity can be estimated with [@cf:LL Theorem 2.3.6], and shown to be smaller than $2d^2/t$. For $(iv)$ we have
$$\begin{gathered}
P_t(x,x)- P^*_t(x,x)= P[ X_t=0; \exists s \in[0,t],\ X_s+x \in \partial{\Lambda}_N ]\\
\le P\left[ X_t=0; \max_{s \in[0,t]}|X_s|\ge d \right].
\end{gathered}$$
The right-hand side is smaller than $$4 P\left[ X_t=0; \max X^{(1)}_s \ge d \right]\le 4 P_t(2d{{\mathbf e}}_1)$$ The later quantity can be estimated with the LCLT for large $t$ [@cf:LL Theorem 2.3.6], or with large deviation estimates for small $t$.
Proof of Lemma \[lem:bridge\]
-----------------------------
Let $V_i$ denote the variance of $X_i$ (without conditioning), $\nabla V_i=(V_i-V_{i-1})$ and set $V:=V_k$ ($V\in[k/2.k]$). After conditioning to $X_k:=0$, the process $(X_i)^k_{i=1}$ remains Gaussian and centered but the covariance structure is given by $${{\ensuremath{\mathbf P}} }\left[ X_i X_j \ | \ X_k=0 \right]= \frac{V_i(V-V_j)}{V} \quad
\quad 0\le i\le j\le k.$$ We denote by ${\widetilde}{{\ensuremath{\mathbf P}} }$ the law of the conditioned process. We can couple this process with a Brownian Motion conditioned to $B_V=0$: a centered Brownian bridge $(B_t)_{t\in[0,V]}$, by setting $X_i:=B_{V_i}$. Note that we have (by applying standard reflexion argument at the first hitting time of $x$) $$\label{reflex}
{\widetilde}{{\ensuremath{\mathbf P}} }\left[ \max_{t\in [0,V]} B_t \ge x \right]= 1 - e^{-\frac{x^2}{2V}}.$$ As the max of $B$ is larger than that of $X$ this gives the lower bound. To prove $(i)$, by monotonicity, we can restrict the proof to the case $x\ge (\log k)$. Two estimate the difference between and the probability we have to estimate, we let let $B^i$ denote the brownian bridges formed by $B$ between the $X_i$, $$(B^i_s)_{s\in [V_{i-1},V_{i}]}:= B_s- \frac{(s-V_{i-1})B_{V_{i-1}}+(V_{i}-s)B_{V_{i}}}{V}.$$ We have $$\begin{gathered}
{\widetilde}{{\ensuremath{\mathbf P}} }\left[\max_{i\in{\llbracket}1,k-1{\rrbracket}} X_i\le x\right]\le {\widetilde}{{\ensuremath{\mathbf P}} }\left[\max_{t\in[0,V]} B_t\le 2x\right]+
\sum_{i=1}^k {\widetilde}{{\ensuremath{\mathbf P}} }\left[\min_{s\in [V_{i},V_{i+1}]}B^i_s\le -x\right]\\
=\left( 1-e^{\frac{-2x^2}{V}}\right)+\sum_{i=1}^k \exp\left(-\frac{x^2}{2 \nabla V_i} \right)\end{gathered}$$ where in the last line we used for $B$ and $B^{i}$. This is smaller than $C x^2/k$ for some well chosen $C$.
Proof of Proposition \[propure\] {#secpropure}
================================
We use Proposition \[massivecompa\] to prove the lower-bound in the asymptotic, and then briefly explain how to obtain a matching upper-bound. First note that using for ${\beta}=0$ and $u=0$ we obtain
$$\label{JKJK}
{\textsc{f}}(h)\ge \lim_{N\to \infty} \frac{1}{N^2} \log {{\ensuremath{\mathbf E}} }^{m}_N \left[ e^{\sum_{x\in {\widetilde}{\Lambda}_N} h\delta_x}\right] -f(m).$$
Now, using Jensen’s inequality we have $$\frac{1}{N^2} \log {{\ensuremath{\mathbf E}} }^{m}_N \left[ e^{\sum_{x\in {\widetilde}{\Lambda}_N} h\delta_x}\right]\ge \frac{h}{N^2} {{\ensuremath{\mathbf E}} }^{m}_N \left[ \sum_{x\in {\widetilde}{\Lambda}_N} \delta_x \right]
\ge h P[ {{\ensuremath{\mathcal N}} }(\sigma_m) \in [-1,1] ]$$ where $\sigma_m:= \sqrt{G^m(x,x)}$ denote the standard deviation of the infinite volume massive free field and ${{\ensuremath{\mathcal N}} }(\sigma_m)$ is a centered normal variable with standard deviation $\sigma_m$. As the variance grows when $m$ tends to zero we obtain that for arbitrary ${\varepsilon}>0$ for $m\le m_{{\varepsilon}}$ we have $${\textsc{f}}(h)\ge (1-{\varepsilon})\frac{h}{\sqrt{2\pi}\sigma_m}-f(m).$$ Using the above inequality for $m=\frac{\sqrt{h}}{|\log h|}$, using to estimate $\sigma_m$ and for $f(m)$ we obtain that for any ${\varepsilon}$, for $h\le h_{{\varepsilon}}$ sufficiently small we have $${\textsc{f}}(h)\ge h P[ \sigma_m {{\ensuremath{\mathcal N}} }\in [-1,1] ] - f(m)\ge \frac{h}{\sqrt{(1/2)\log h}}(1-{\varepsilon}).$$
Concerning the upper-bound, we can show as in [@cf:GL Equation (2.20)] that $\sup_{{\widehat}\phi} Z_{N,h}^{{\widehat}\phi}$ is a sub-multiplicative function and thus that we have for every $N\ge 1$ we have $${\textsc{f}}(h)\le \sup_{{\widehat}\phi} \frac{1}{N^2} \log Z^{{\widehat}\phi}_{N,h}.$$ We use this inequality for $$N=h^{-1/2} |\log h|^{-1}.$$ In that case, the Taylor expansion of the exponential in the partition function gives $$Z^{{\widehat}\phi}_{N,h}\le 1+ e^{(\log h)^{-2}} {{\ensuremath{\mathbf E}} }^{{\widehat}\phi}_N\left[ \sum_{x\in {\widetilde}{\Lambda}_N} \delta_x\right]\le e^{(\log h)^{-2}} {{\ensuremath{\mathbf E}} }_N\left[
\sum_{x\in {\widetilde}{\Lambda}_N} \delta_x\right],$$ where in the last inequality we used that the probability for a Gaussian of a given variance to be in $[-1,1]$ is maximized if its mean is equal to zero. Using then to estimate the probability, it is a simple exercise to check that for any ${\varepsilon}>0$ and $N$ large enough, we have $${{\ensuremath{\mathbf E}} }_N[ \sum_{x\in {\widetilde}{\Lambda}_N} \delta_x]\le \frac{(1+{\varepsilon}) 2N^2}{ \sqrt{\log N}}.$$ Combining all these inequality, we obtain that for $h$ sufficiently small $${\textsc{f}}(h) \le \frac{(1+2{\varepsilon}) \sqrt{2} h}{ \sqrt{\log h}}.$$
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|
---
abstract: 'In this communication, we propose a model to study the non-equilibrium process by which actin stress fibers develop force in contractile cells. The emphasis here is on the non-equilibrium thermodynamics, which is necessary to address the mechanics as well as the chemistry of dynamic cell contractility. In this setting we are able to develop a framework that relates (a) the dynamics of force generation within the cell and (b) the cell’s response to external stimuli to the chemical processes occurring within the cell, as well as to the mechanics of linkage between the stress fibers, focal adhesions and extra-cellular matrix.'
author:
- |
M. Maraldi$^1$, K. Garikipati$^{1, 2}$[^1]\
\
bibliography:
- 'biblio.bib'
title: 'The mechano-chemistry of cytoskeletal force generation'
---
Introduction {#intro}
============
Actin stress fibers can be found in cells such as fibroblasts and muscle cells and play an important role in cell locomotion, cell adhesion and wound healing, due to their ability to generate forces over the entire extent of the cell. They are constituted by several bundles of actin filaments held together by the actin binding protein $\alpha$-actinin, and others. Myosin proteins make the actin filaments slide past each other and confer on the stress fiber the ability to generate contractile force [@pellegrin:2007]; in fact, it is well-established that a cell’s traction force development is associated with stress fiber formation [@chrzanowska:1996; @ingber:2003]. Besides, the kinetics of stress fiber formation and disassembly [@pollard:2003] appears itself to be modulated by force: traction force promotes the binding of stress fiber proteins [@colombelli:2009; @hirata:2008], which in turn enhances acto-myosin contractile activity, establishing a “feed-forward” process for stress fibers growth.
Stress fibers are anchored to the extra-cellular matrix (ECM) through focal adhesions
[@chrzanowska:1996; @geiger:2001]. Focal adhesion proteins including zyxin, paxillin, talin, vinculin and many others, connect integrin (the trans-membrane adhesion protein) to F-actin bundles of stress fibers [@geiger:2001; @zamir:2001]. The kinetics of binding/unbinding of focal adhesion proteins is also modulated by force; moreover, it has been shown that focal adhesions exhibit this mechano-sensitivity regardless of whether the force is imposed by acto-myosin contractility of stress fibers, or is externally-imposed and merely transmitted by the stress fibers [@balaban:2001; @riveline:2001].
In recent years, a large number of experimental studies have been conducted to investigate the behavior and the properties of the cytoskeletal system: cell traction force [@tan:2003], cell response and differentiation induced by changes in the substrate stiffness [@chan:2008; @engler:2006],stress fiber orientation parallel to [@franke:1984] or away from the direction of cyclic stretching [@kaunas:2005], cell contractility stimulated by treatment with calyculin [@peterson:2004] and stress fiber relaxation upon severance by a laser [@kumar:2006] among others. Concurrently, a number of studies concerned with the modeling of the mechano-chemical response of stress fibers have appeared. Some of these studies do not explicitly consider focal adhesions [@besser:2011; @besser:2007; @kaunas:2010; @kaunas:2008; @kruse:2000; @stachowiak:2008; @stachowiak:2009], some others focus solely on modeling the focal adhesions [@shemesh:2005; @besser:2006; @olberding:2010], while others do indeed model the coupled system of stress fibers and focal adhesions [@deshpande:2006; @deshpande:2008; @harland:2011; @walcott:2010]. The diversity of experimental studies, and theoretical models motivated by these experiments, have begun to shed some light on the role of stress fibers and focal adhesions in the development of cell traction. However, despite these studies there is not yet a complete understanding of how the dynamic mechano-chemistry of stress fibers affects that of focal adhesions, and vice-versa.
In this paper, we propose a model able to describe the interaction between stress fibers, focal adhesions and the ECM, and how such interaction regulates the development of force in the cytoskeletal system. Toward this end, we have developed the model to be grounded in non-equilibrium thermodynamics, accounting for the mechano-chemistry of focal adhesions, stress fibers and ECM. This paper’s focus is on the development of the model itself and of the theoretical framework for analyzing the mechano-chemistry of such biological systems. A few simulations results are provided in this communication in order to demonstrate the model at work. A comprehensive study of the model’s response needs to be rather extensive because of its complexity, and is beyond the scope of this mainly theoretical paper. Such a study is being provided elsewhere [@maraldi:2013].
An archetypal model for the cytoskeletal system {#sect:model-intro}
===============================================
We consider a minimal system consisting of a stress fiber connected to a focal adhesion at each end, with each focal adhesion being attached to the ECM underlying the cell, as depicted in Fig. \[fig:model\]; the system also includes the cytosolic reservoir supplying proteins to the stress fiber and the focal adhesions.
![[]{data-label="fig:model"}](System-cell.pdf){width="0.9\columnwidth"}
We represent the focal adhesions as one-dimensional clusters of generic complexes (each complex being an ensemble of proteins consisting of a ligand, an integrin molecule and associated focal adhesion proteins). The stress fiber is modeled as a one-dimensional, tension bearing structure composed of a number of generic protein complexes, each incorporating, among others, actin molecules, myosin molecules and ATP molecules to fuel myosin contractility. The system may be subjected to a mechanical load applied to the ECM in order to reproduce the effects of external perturbations that, in a living cell, may come from a direct action on the ECM itself, from the action of surrounding cells, or from other such stress fiber-focal adhesion systems within the same cell.
It is to be expected that a system formed by coupling two mechano-chemically dependent sub-systems (stress fiber and focal adhesions) will gain in complexity. In this case, complexity is enhanced beyond the mechano-sensitivity discussed above because of the rate-dependent response of the stress fiber that will be discussed in Sect. \[sect:SF-int\].
System kinematics resulting from protein addition {#sect:kinematics}
-------------------------------------------------
A comprehensive model of the cytoskeletal system has to account for the experimentally observed interplay between mechanical and chemical phenomena; the model proposed here highlights this connection in that protein binding to (or unbinding from) the stress fiber and the focal adhesions results in a change in the geometry of the system, which in turn affects its ability to generate contractile force and to sustain external loads. It also accounts for the fact that the chemistry of both the stress fiber and the focal adhesions is influenced by the force within the system, as will be explained in the sections to follow.
In general, force is non-uniformly distributed along the focal adhesion. However, for the sake of simplicity, in the following we will assume a uniform force distribution over the focal adhesion. This choice allows us to carry out the calculations by considering only the overall force acting on the focal adhesion and the consequent overall deformation. The hypothesis of uniform force distribution along the focal adhesion is also supported by the results of previous computations [@olberding:2010], which showed that the behavior of the focal adhesion model we adopt in this paper is not significantly affected by the choice of the force distribution. In a similar fashion, for the sake of simplicity but while retaining the most relevant physics, we will model the stress fiber as a 1-D subsystem. While accounting for its diameter, the force and deformation will be considered uniformly distributed over the stress fiber’s cross section; i.e. the same force in each actin filament. Hence, again, only the overall force and deformation will matter and the stress fiber will be modeled as a rod-like element. The number of proteins in the stress fiber, $N_\mathrm{sf}$, is related to its geometry as: $$\label{eq:radius}
r_\mathrm{sf} = \sqrt{ \frac{N_\mathrm{sf} V_\mathrm{act}}{\pi x_\mathrm{sf}^0} },$$ where $r_\mathrm{sf}$ is the stress fiber radius, $V_\mathrm{act}$ is the volume of a single actin monomer and $x_\mathrm{sf}^0$ is the unstretched stress fiber length. Another quantity relevant to the geometry of the system is the number of actin filaments in the stress fiber, which is given by: $$\label{eq:Nfil}
N_\mathrm{fil} = \frac{N_\mathrm{sf} L_\mathrm{actmon}}{x_\mathrm{sf}^0},$$ where $L_\mathrm{actmon}$ is the length of an actin monomer. Note that as the system evolves and proteins bind to or unbind from the stress fiber, its radius (and, likewise, the number of actin filaments) evolves.
![F[]{data-label="fig:fa-bind"}](FA-kinetics.pdf)
Each complex in the focal adhesion has length $\lambda$. Complexes can bind and unbind anywhere along the focal adhesion. In our model we track the concentration of these complexes, $c_\mathrm{fa}$, which is a number density per unit length along the focal adhesion. However, we only need to track binding and unbinding at the proximal and distal ends to compute the change in focal adhesion length. This has been depicted in Fig. \[fig:fa-bind\]. Accordingly, the focal adhesion ends move at the following velocities [@olberding:2010]:
\[eq:edges-speed\] $$\begin{aligned}
& \dot{x}_\mathrm{p} = \lambda^2 \dot{c}_\mathrm{fa}^\mathrm{p} \\
& \dot{x}_\mathrm{d} = - \lambda^2 \dot{c}_\mathrm{fa}^\mathrm{d},\end{aligned}$$
where ${x}_\mathrm{p}$ and ${x}_\mathrm{d}$ are the positions of the proximal and distal ends, respectively, and $\dot{c}_\mathrm{fa}^\mathrm{p}$, $\dot{c}_\mathrm{fa}^\mathrm{d}$ are the corresponding rates of change of concentration of focal adhesion complexes due to protein binding/unbinding. Knowing ${x}_\mathrm{p}$ and ${x}_\mathrm{d}$, the focal adhesion length is: $$\hat{x}_\mathrm{fa} = \vert {x}_\mathrm{p} - {x}_\mathrm{d}\vert.$$ When the velocities of the two ends have the same direction, focal adhesion *translation* is observed. It is best quantified as a motion of the focal adhesion’s centroid. If $\tilde{x}_\mathrm{fa}$ is the resultant translation of the centroid relative to the reference (initial) position $\tilde{x}_\mathrm{fa} = 0$ at time $t = 0$, we write: $$\label{eq:fa-cent-transl}
\dot{\tilde{x}}_\mathrm{fa} = \frac{1}{2}{\left( {\dot{x}_\mathrm{p} + \dot{x}_\mathrm{d}} \right)}
= \frac{\lambda^2}{2}{\left( {\dot{c}_\mathrm{fa}^\mathrm{p} - \dot{c}_\mathrm{fa}^\mathrm{d}} \right)}.$$ Focal adhesion translation is likely associated with protein *treadmilling* through the focal adhesion and cytosol.[^2] As we will see below, focal adhesion translation provides a mechanism for force relaxation within the system.
\[t\] ![[]{data-label="fig:system-len"}](model-kinematics.pdf "fig:")
As depicted in Fig. \[fig:system-len\], under the action of mechanical load, the focal adhesion deforms elastically as well as translates; therefore, the distance between the current position of the focal adhesion’s centroid and its initial position is given by: $$\label{eq:fa-observable}
x_\mathrm{fa} = x_\mathrm{fa}^\mathrm{e} + \tilde{x}_\mathrm{fa},$$ where $x_\mathrm{fa}^\mathrm{e}$ is the magnitude of the centroid’s displacement due to elastic deformation of the focal adhesion. For more details on the additive decomposition implied by Eq. see Fig. \[fig:system-len\]b.
Stress fiber internal variables {#sect:SF-int}
-------------------------------
An accurate description of the mechanical response of the stress fiber has to account for its sarcomeric structure, due to which anti-parallel actin filaments can: (a) actively slide past each other due to the stepping action of myosin molecules and (b) passively slide past each other on account of the tension-dependent breakage of bonds between $\alpha$-actinin and actin. These two effects endow the stress fibers with a rate-dependent response, which can be modeled by means of active stress and viscoelastic models, respectively. A classical work on the force-dependent kinetics of myosin stepping on actin filaments is that by @hill:1938, while the viscoelastic response of stress fibers has been demonstrated by @kumar:2006.
In this treatment we are only concerned with the force at the ends of the stress fiber and its overall deformation (see Sects.\[thermo:basis\]–\[sect:nonequilibrium\]). For this reason, we are not modeling explicitly the individual sarcomeric units; rather, we account for their collective effect. Following the structural model proposed by @pellegrin:2007, we represent the stress fiber rheology using a purely elastic element in parallel with a Maxwell viscoelastic element and a contractile element (Fig. \[fig:rehological\]).
For a complete description of the state of the stress fiber it is necessary to introduce a number of internal variables, namely the displacement of the purely elastic element $\xi^\mathrm{e}$, the displacement of the elastic part of the Maxwell element $\xi^\mathrm{ve}$, the displacement of the viscous element $\xi^\mathrm{v}$, and the contraction of the active element $\xi^\mathrm{ac}$. The internal force conjugate to $\xi^\mathrm{e}$ is the elastic force $P^\mathrm{e}_\mathrm{sf}$, that conjugate to $\xi^\mathrm{ve}$ and $\xi^\mathrm{v}$ is the visco-elastic force $P^\mathrm{ve}_\mathrm{sf}$, and that conjugate to $\xi^\mathrm{ac}$ is the active force $P^\mathrm{ac}_\mathrm{sf}$. According to the schematic representation in Fig. \[fig:rehological\], the stress fiber internal variables can be related to one another and to the current length of the stress fiber $x_\mathrm{sf}$ (the observable variable) as: $$\label{eq:SF-const-displ}
d x_\mathrm{sf} = d\xi^\mathrm{e} = d\xi^\mathrm{ve} + d\xi^\mathrm{v} = d\xi^\mathrm{ac},$$ whereas the force acting on the stress fiber is related to the internal forces through the following equation: $$\label{eq:SF-const-force}
P_\mathrm{sf} = P_\mathrm{sf}^\mathrm{e} + P_\mathrm{sf}^\mathrm{ve} + P_\mathrm{sf}^\mathrm{ac}.$$ Contraction of the active element happens only upon ATP hydrolysis, which releases ADP; the reaction occurs under the following chemical constraint on the number of ATP and ADP molecules: $$dN_\mathrm{ATP} + dN_\mathrm{ADP} = 0,
\label{atp-adp-constraint}$$ which accounts for the stoichiometry of the ATP-to-ADP conversion reaction.
![[]{data-label="fig:rehological"}](model-mech.pdf)
The thermodynamic basis {#thermo:basis}
=======================
In the current Section and in Sect. \[sect:equilibrium\], we review standard equilibrium thermodynamics as applied to the model in order to build the necessary background for the non-equilibrium thermodynamic treatment of Sect. \[sect:nonequilibrium\]. In turn, the non-equilibrium treatment will pose restrictions on constitutive equations and rate laws in Sects. \[constforcesect\]–\[sect:rate-laws\].
For modeling purposes, the system is split into four sub-systems: the two focal adhesions, the stress fiber, the cytosol and the ECM. Mechanics and chemistry are both relevant to the stress fiber and focal adhesion sub-systems; the cytosolic reservoir is a purely chemical sub-system; the ECM is a purely mechanical sub-system. For symmetry reasons (Fig. \[fig:system-len\]), the two focal adhesions are identical in geometry and response.
We consider internal energy functions with the following parameterizations for the different sub-systems: $$\label{intengies}
\begin{aligned}
&U_\mathrm{sf} = U_\mathrm{sf} {\left( {N_\mathrm{sf}, S_\mathrm{sf}, \xi^\mathrm{e}_\mathrm{sf}, \xi^\mathrm{ve}_\mathrm{sf}, N_\mathrm{ATP}, N_\mathrm{ADP}} \right)}\\
&U_\mathrm{fa} = U_\mathrm{fa} {\left( {x^\mathrm{e}_\mathrm{fa}, N_\mathrm{fa}, S_\mathrm{fa}} \right)} \\
&U_\mathrm{cyt} = U_\mathrm{cyt} {\left( {\widehat{N}_\mathrm{sf}, \widehat{N}_\mathrm{fa}, S_\mathrm{cyt}} \right)} \\
&U_\mathrm{ecm} = U_\mathrm{ecm} {\left( {x^\mathrm{e}_\mathrm{ecm}, S_\mathrm{ecm}} \right)}
\end{aligned}$$ where $U_{\alpha}$ is the internal energy and $S_{\alpha}$ is the entropy of sub-system $\alpha$. The variables $N_\mathrm{sf}$ and $N_\mathrm{fa}$ are, respectively, the numbers of generic proteins in the stress fiber and of complexes in the focal adhesion, while $\widehat{N}_\mathrm{sf}$ and $\widehat{N}_\mathrm{fa}$ are, respectively, the numbers of generic stress fiber proteins and focal adhesion complexes in the cytosol. Note that $\xi^\mathrm{e}_\mathrm{sf}$ and $\xi^\mathrm{ve}_\mathrm{sf}$ are the only internal variables responsible for internal energy storage in the stress fiber (in the form of elastic energy). Finally, $x^\mathrm{e}_\mathrm{ecm}$ is the length of the ECM subjected to elastic deformation (see Fig. \[fig:system-len\] and the explanation below).
Given Eq. , the entropy functions are parametrized as: $$\label{entropies}
\begin{aligned}
& S_\mathrm{sf} = S_\mathrm{sf} {\left( {N_\mathrm{sf}, U_\mathrm{sf}, \xi^\mathrm{e}, \xi^\mathrm{ve}, N_\mathrm{ATP}, N_\mathrm{ADP}} \right)}\\
&S_\mathrm{fa} = S_\mathrm{fa} {\left( {x^\mathrm{e}_\mathrm{fa}, N_\mathrm{fa}, U_\mathrm{fa}} \right)} \\
&S_\mathrm{cyt} = S_\mathrm{cyt} {\left( {\widehat{N}_\mathrm{sf}, \widehat{N}_\mathrm{fa}, U_\mathrm{cyt}} \right)} \\
&S_\mathrm{ecm} = S_\mathrm{ecm} {\left( {x^\mathrm{e}_\mathrm{ecm}, U_\mathrm{ecm}} \right)}.
\end{aligned}$$ Assuming that the system is adiabatic leads to the following constraint on internal energies: $$\label{eq:int-nrg}
dU = dU_\mathrm{sf} + 2 \ dU_\mathrm{fa} + dU_\mathrm{cyt} + dU_\mathrm{ecm} = P_\mathrm{ext} dx^\mathrm{e}_\mathrm{ecm},$$ which means that the system’s energy can only be changed by performing mechanical work on the ECM. As stated in Sect. \[sect:model-intro\], the agents for this mechanical work could be neighboring cells, other stress fiber - focal adhesions systems within the same cell, or other agents acting through the ECM.
The geometry of the system imposes a kinematic constraint on the system’s mechanical extensive variables; with reference to Fig. \[fig:system-len\] we can write: $$\label{eq:kin-constr}
x_\mathrm{sf} + 2 \ x_\mathrm{fa} = x_\mathrm{ecm},$$ where $x_\mathrm{ecm}$ is the length of the ECM that initially underlies the stress fiber – focal adhesions assembly. Considering Eq. and observing that $x_\mathrm{ecm} = x^\mathrm{e}_\mathrm{ecm} + 2\tilde{x}_\mathrm{fa}$, Eq. can be reduced to: $$x_\mathrm{sf} + x_\mathrm{fa}^\mathrm{e} = x_\mathrm{ecm}^\mathrm{e},$$ or, in incremental form: $$\label{eq:kin-constr-dlta}
dx_\mathrm{sf} + 2 \ dx_\mathrm{fa}^\mathrm{e} - dx_\mathrm{ecm}^\mathrm{e} = 0.$$
In addition, the following chemical constraints hold, since the total numbers of generic stress fiber and focal adhesion proteins are conserved: $$\label{eq:chem-constr}
\begin{aligned}
\widehat{N}_\mathrm{sf} + N_\mathrm{sf} &= N_\mathrm{sf}^\mathrm{max}, \\
\widehat{N}_\mathrm{fa} + 2 \ N_\mathrm{fa} &= N_\mathrm{fa}^\mathrm{max},
\end{aligned}$$ where $N_\mathrm{sf}^\mathrm{max}$ and $N_\mathrm{fa}^\mathrm{max}$ are the maximum numbers of proteins available to the stress fiber and to a single focal adhesion, respectively. Eq. can also be written in incremental form as:
$$\label{eq:chem-constr-dlta}
\begin{aligned}
d\widehat{N}_\mathrm{sf} + dN_\mathrm{sf} &= 0, \\
d\widehat{N}_\mathrm{fa} + 2 \ dN_\mathrm{fa} &= 0.
\end{aligned}$$
Thermodynamic equilibrium and its applicability to the modeled processes {#sect:equilibrium}
========================================================================
Motivated by the rapid decay of transient heat fluxes across the cell and by the rapid propagation of elastic waves throughout the system, we assume that the system is at thermal and mechanical equilibrium at each instant of its evolution. In this section we also consider equilibrium with respect to chemistry and the stress fiber’s internal variables, in order to obtain constitutive relations rigorously from the thermodynamics. Proper non-equilibrium conditions with respect to chemistry and stress fiber internal variables will be discussed in Sect. \[sect:nonequilibrium\]. The framework that we lay out in this section has been arrived at by applying classical equilibrium thermodynamics to this system following the steps laid out in [@callen:1985].
Equilibrium is characterized by a maximum of the system entropy [@callen:1985]: $$\label{eq:entropy-max}
dS = dS_\mathrm{sf} + 2 \ dS_\mathrm{fa} + dS_\mathrm{cyt} + dS_\mathrm{ecm} = 0.$$ Differentiating the entropy function of each sub-system with respect to its natural variables listed in Eq. and using the constraints represented by Eq. , and (and considering Eq. for the stress fiber rheology), Eq. becomes: $$\label{eq:entr-max-verbose}
\small
\begin{aligned}
dS \!&= \!{\left( {\frac{\!\partial S_\mathrm{sf}}{\partial U_\mathrm{sf}}
\!-\! \frac{\partial S_\mathrm{ecm}}{\partial U_\mathrm{ecm}}\!} \right)} \! dU_\mathrm{sf} + \!
2 {\left( {\frac{\partial S_\mathrm{fa}}{\partial U_\mathrm{fa}}
- \frac{\partial S_\mathrm{ecm}}{\partial U_\mathrm{ecm}}} \right)} \! dU_\mathrm{fa} +
\!{\left( {\!\frac{\partial S_\mathrm{cyt}}{\partial U_\mathrm{cyt}}
\!-\! \frac{\partial S_\mathrm{ecm}}{\partial U_\mathrm{ecm}}\!} \right)} \! dU_\mathrm{cyt} + \\
&+ {\left( {\!\frac{\partial S_\mathrm{sf}}{\partial N_\mathrm{sf}}
\!-\! \frac{\partial S_\mathrm{cyt}}{\partial \widehat{N}_\mathrm{sf}}\!} \right)} \! dN_\mathrm{sf} +
2 {\left( {\frac{\partial S_\mathrm{fa}}{\partial N_\mathrm{fa}}
- \frac{\partial S_\mathrm{cyt}}{\partial \widehat{N}_\mathrm{fa}}} \right)} \! dN_\mathrm{fa} +
\!{\left( {\frac{\partial S_\mathrm{sf}}{\partial N_\mathrm{ATP}}
- \frac{\partial S_\mathrm{sf}}{\partial N_\mathrm{ADP}}} \right)} \! dN_\mathrm{ATP} + \\
&+ 2 {\left( {\frac{\partial S_\mathrm{fa}}{\partial x^\mathrm{e}_\mathrm{fa}}
+ \frac{\partial S_\mathrm{ecm}}{\partial x^\mathrm{e}_\mathrm{ecm}}
+ \frac{\partial S_\mathrm{ecm}}{\partial U_\mathrm{ecm}} P_\mathrm{ext}} \right)} \! dx^\mathrm{e}_\mathrm{fa} +
\!{\left( {\frac{\partial S_\mathrm{sf}}{\partial x_\mathrm{sf}}
+ \frac{\partial S_\mathrm{ecm}}{\partial x^\mathrm{e}_\mathrm{ecm}}
+ \frac{\partial S_\mathrm{ecm}}{\partial U_\mathrm{ecm}} P_\mathrm{ext}} \right)} \! dx_\mathrm{sf} + \\
&+ \frac{\partial S_\mathrm{sf}}{\partial \xi^\mathrm{ve}} d\xi^\mathrm{ve} = 0.
\end{aligned}$$ The condition expressed in has to be satisfied for any generic perturbations of the independent variables of the sub-system entropies; this leads to the condition of thermal equilibrium: $$\frac{\partial S_\mathrm{sf}}{\partial U_\mathrm{sf}} =
\frac{\partial S_\mathrm{fa}}{\partial U_\mathrm{fa}} =
\frac{\partial S_\mathrm{cyt}}{\partial U_\mathrm{cyt}} =
\frac{\partial S_\mathrm{ecm}}{\partial U_\mathrm{ecm}},
\label{eq:therm-eq}$$ to the condition of mechanical equilibrium: $$\frac{\partial S_\mathrm{sf}}{\partial x_\mathrm{sf}} =
\frac{\partial S_\mathrm{fa}}{\partial x_\mathrm{fa}} =
- {\left( {\frac{\partial S_\mathrm{ecm}}{\partial x^\mathrm{e}_\mathrm{ecm}}
+ \frac{\partial S_\mathrm{ecm}}{\partial U_\mathrm{ecm}} P_\mathrm{ext}} \right)},
\label{eq:mech-eq}$$ and to the conditions of chemical equilibrium of stress fiber and focal adhesion proteins: $$\begin{aligned}
\frac{\partial S_\mathrm{sf}}{\partial N_\mathrm{sf}} &= \frac{\partial S_\mathrm{cyt}}{\partial \widehat{N}_\mathrm{sf}} \\
\frac{\partial S_\mathrm{fa}}{\partial N_\mathrm{fa}} &= \frac{\partial S_\mathrm{cyt}}{\partial \widehat{N}_\mathrm{fa}}.
\end{aligned}
\label{eq:chem-eq}$$ It also gives the condition of equilibrium with respect to the stress fiber’s internal variables: $$\frac{\partial S_\mathrm{sf}}{\partial \xi^\mathrm{ve}} = 0
\label{eq:int-eq1}$$ and the condition for the equilibrium state at which ATP does not undergo hydrolysis: $$\frac{\partial S_\mathrm{sf}}{\partial N_\mathrm{ATP}} = \frac{\partial S_\mathrm{sf}}{\partial N_\mathrm{ADP}}.
\label{eq:atp-adp-eq}$$ We note that at chemical equilibrium no net protein exchange between the cytosol and the focal adhesions occurs; therefore, according to Eq. , focal adhesion translation also vanishes ($d\tilde{x}_\mathrm{fa} = 0$). This is a mechanical manifestation of chemical equilibrium. In a similar manner, if the ATP-to-ADP conversion does not occur, active force cannot be generated in the stress fiber; hence Eq. implies, via a constitutive relation: $$\label{eq:ac-force-eq}
P_\mathrm{sf}^\mathrm{ac} = 0.$$ At equilibrium, the fundamental relations of thermodynamics can be stated for the separate sub-systems: $$\begin{aligned}
&T_\mathrm{sf} dS_\mathrm{sf} = dU_\mathrm{sf} - P_\mathrm{sf}^\mathrm{e} d\xi^\mathrm{e} - P_\mathrm{sf}^\mathrm{ve} d\xi^\mathrm{ve} - \mu_\mathrm{sf} dN_\mathrm{sf} + {\left( {\mu_\mathrm{ATP} - \mu_\mathrm{ADP}} \right)} dN_\mathrm{ADP} \\
&T_\mathrm{fa} dS_\mathrm{fa} = dU_\mathrm{fa} - P_\mathrm{fa} dx^\mathrm{e}_\mathrm{fa} - \mu_\mathrm{fa} dN_\mathrm{fa} \\
&T_\mathrm{cyt} dS_\mathrm{cyt} = dU_\mathrm{cyt} - \mu^\mathrm{sf}_\mathrm{cyt} d\widehat{N}_\mathrm{sf} - \mu^\mathrm{fa}_\mathrm{cyt} d\widehat{N}_\mathrm{fa} \\
&T_\mathrm{ecm} dS_\mathrm{ecm} = dU_\mathrm{ecm} - P_\mathrm{ecm} dx^\mathrm{e}_\mathrm{ecm},
\end{aligned}
\label{eq:fundrel}$$ where $T_\alpha$ is the temperature and $\mu_\alpha$ the chemical potential of the corresponding sub-system $\alpha$, and $\mu_\mathrm{cyt}^\beta$ for $\beta = \mathrm{sf}, \, \mathrm{fa}$, is the chemical potential of the $\beta$ proteins in the cytosol.
By differentiating the entropy functions with respect to their natural variables as expressed in Eq. and comparing them with the fundamental relations , the equilibrium constitutive relations are obtained: $$\label{eq:const-rel-eq}
\begin{aligned}
&\frac{\partial S_\alpha}{\partial U_\alpha} = \frac{1}{T_\alpha} \\
&\frac{\partial S_\alpha}{\partial N_\alpha} = -\frac{\mu_\alpha}{T_\alpha}, \quad\quad \alpha = \mathrm{sf, fa} \\
&\frac{\partial S_\mathrm{cyt}}{\partial \widehat{N}_\alpha} = -\frac{\mu^\mathrm{cyt}_\alpha}{T_\mathrm{cyt}}, \quad \ \alpha = \mathrm{sf, fa} \\
&\frac{\partial S_\alpha}{\partial x^\mathrm{e}_\alpha} = -\frac{P_\alpha}{T_\alpha},\quad\quad \alpha = \mathrm{fa, ecm} \\
&\frac{\partial S_\mathrm{sf}}{\partial \xi^\mathrm{e}} =
\frac{\partial S_\mathrm{sf}}{\partial x_\mathrm{sf}} = -\frac{P^\mathrm{e}_\mathrm{sf}}{T_\mathrm{sf}}, \quad \frac{\partial S_\mathrm{sf}}{\partial \xi^\mathrm{ve}} = -\frac{P^\mathrm{ve}_\mathrm{sf}}{T_\mathrm{sf}}\\
&\frac{\partial S_\mathrm{sf}}{\partial N_\mathrm{ATP}} = -\frac{\mu_\mathrm{ATP}}{T_\mathrm{sf}},\quad\quad \ \frac{\partial S_\mathrm{sf}}{\partial N_\mathrm{ADP}} = -\frac{\mu_\mathrm{ADP}}{T_\mathrm{sf}}.
\end{aligned}$$ Substituting in (\[eq:therm-eq\]–\[eq:atp-adp-eq\]) and accounting for Eq. and for the stress fiber rheology defined in Eq. , the equations of equilibrium reduce to:
$$\begin{aligned}
&T_\mathrm{sf} = T_\mathrm{fa} = T_\mathrm{cyt} = T_\mathrm{ecm} := T \label{eq:temp-equil} \\
&\mu_\mathrm{sf} = \mu^\mathrm{cyt}_\mathrm{sf},\quad \mu_\mathrm{fa} = \mu^\mathrm{cyt}_\mathrm{fa} \label{eq:chempot-equil}\\
&P_\mathrm{sf} = P_\mathrm{fa} = P_\mathrm{ext} - P_\mathrm{ecm} := P \label{eq:force-equil}\\
&P^\mathrm{ve}_\mathrm{sf} = 0 \label{int-force-equil}\\
&\mu_\mathrm{ATP} = \mu_\mathrm{ADP}. \label{atp-adp-equil}\end{aligned}$$
Non-equilibrium processes and restrictions to the model {#sect:nonequilibrium}
=======================================================
We continue to consider thermal and mechanical equilibrium to hold. Therefore, Eq. and Eq. remain valid. On the other hand, we do not seek chemical equilibrium with respect to stress fiber and focal adhesion protein binding, or to ATP-to-ADP conversion, as the kinetics of these phenomena occur on much longer time scales. We also do not seek equilibrium with respect to the mechanical processes internal to the stress fiber. However, as is commonly done in non-equilibrium thermodynamics we do assume that the constitutive relations (\[eq:const-rel-eq\]), rigorously obtained at equilibrium, continue to hold far from equilibrium [@degrootmazur1984]. In this setting the entropy rate is: $$\label{eq:entropy-rate}
\dot S = -(\mu_\mathrm{sf} - \mu_\mathrm{cyt}^\mathrm{sf}) \dot{N}_\mathrm{sf} - 2 \ (\mu_\mathrm{fa} - \mu_\mathrm{cyt}^\mathrm{fa}) \dot{N}_\mathrm{fa}
+ P_\mathrm{sf}^\mathrm{ve} \dot{\xi}^\mathrm{v} + P_\mathrm{sf}^\mathrm{ac} \dot{x}_\mathrm{sf} + {\left( {\mu_\mathrm{ATP} - \mu_\mathrm{ADP}} \right)} \dot{N}_\mathrm{ADP}.$$ The second law of thermodynamics requires that this rate be non-negative, i.e. $\dot S \ge 0$, hence: $$\label{eq:second-law}
- (\mu_\mathrm{sf} - \mu_\mathrm{cyt}^\mathrm{sf}) \dot{N}_\mathrm{sf}
- 2 \ (\mu_\mathrm{fa} - \mu_\mathrm{cyt}^\mathrm{fa}) \dot{N}_\mathrm{fa}
+ P_\mathrm{sf}^\mathrm{ve} \dot{\xi}^\mathrm{v}
+ P_\mathrm{sf}^\mathrm{ac} \dot{x}_\mathrm{sf}
+ {\left( {\mu_\mathrm{ATP} - \mu_\mathrm{ADP}} \right)} \dot{N}_\mathrm{ADP} \ge 0.$$ Compliance with the second law of thermodynamics in the form must be ensured for every admissible process, i.e. for every generic choice of the process vector $$\Gamma = {\left( {\dot{N}_\mathrm{sf},\dot{N}_\mathrm{fa},\dot{\xi}^\mathrm{v},\dot{x}_\mathrm{sf},\dot{N}_\mathrm{ADP}} \right)}.$$ The corresponding sufficient conditions are that the following relations are satisfied:
\[secondlawsuff\] $$\begin{aligned}
&\mathrm{sgn}(\dot{N}_\mathrm{sf}) = \mathrm{sgn}(\mu_\mathrm{cyt}^\mathrm{sf} - \mu_\mathrm{sf}) \label{stress fiber-suff}\\
&\mathrm{sgn}(\dot{N}_\mathrm{fa}) = \mathrm{sgn}(\mu_\mathrm{cyt}^\mathrm{fa} - \mu_\mathrm{fa}) \label{focal adhesion-suff}\\
&\mathrm{sgn}(\dot{N}_\mathrm{ADP}) = \mathrm{sgn}(\mu_\mathrm{ATP} - \mu_\mathrm{ADP}) \label{ATP-suff} \\
&P_\mathrm{sf}^\mathrm{ac} \dot{x}_\mathrm{sf} + {\left( {\mu_\mathrm{ATP} - \mu_\mathrm{ADP}} \right)} \dot{N}_\mathrm{ADP} \ge 0 \label{active-suff} \\
&P_\mathrm{sf}^\mathrm{ve} \ \dot{\xi}^\mathrm{v} \ge 0. \label{visc-suff}\end{aligned}$$
Eq. , Eq. and Eq. follow from considering processes in which, respectively, only $\dot{N}_\mathrm{sf}$, $\dot{N}_\mathrm{fa}$ and $\dot{N}_\mathrm{ADP}$ are non-zero. The same procedure leads to Eq. , by considering a particular process in which only $\dot{\xi}^\mathrm{v}$ is non-zero.
We observe that active generation of tension in the stress fiber due to actomyosin contractility implies $P_\mathrm{sf}^\mathrm{ac} \ge 0$ and $\dot{x}_\mathrm{sf} \le 0$. But this requires $ {\left( {\mu_\mathrm{ATP} - \mu_\mathrm{ADP}} \right)} \dot{N}_\mathrm{ADP} \ge -P_\mathrm{sf}^\mathrm{ac} \dot{x}_\mathrm{sf}$, a condition that can be satisfied by hydrolysis of ATP to ADP, giving $\dot{N}_\mathrm{ADP} \ge 0$ when $\mu_\mathrm{ATP} \ge \mu_\mathrm{ADP}$. The reverse reaction, in which ADP is converted into ATP, does not produce active force; hence, $P_\mathrm{sf}^\mathrm{ac} = 0$ holds and inequality reduces to Eq. . Notably, in the case of active force generation, the mechano-chemical coupling requires that ${\left( {\mu_\mathrm{ATP} - \mu_\mathrm{ADP}} \right)} \dot{N}_\mathrm{ADP}$ exceeds a finite, positive number, not merely that ${\left( {\mu_\mathrm{ATP} - \mu_\mathrm{ADP}} \right)} \dot{N}_\mathrm{ADP} \ge 0$ (see inequality ). This emphasizes the importance of considering the full extent of mechano-chemical coupling via inequality in satisfying .
Taken together, the conditions in represent the restrictions posed by thermodynamics on the constitutive nature of the different sub-systems and on the kinetics of the modeled processes. In the following sections we will exploit these restrictions to formulate suitable constitutive equations for the mechanical forces, the chemical potentials and for the kinetics of protein binding/unbinding.
Constitutive equations for mechanical forces {#constforcesect}
============================================
We begin with constitutive relations for all the components of the stress fiber force appearing in Eq. . The active component of the stress fiber force is written as: $$P_\mathrm{sf}^\mathrm{ac} = \frac{P_\mathrm{sf}^\mathrm{stl}}{\dot{\varepsilon}_\mathrm{sf}^\mathrm{con}} {\left( {\dot{\varepsilon}_\mathrm{sf}^\mathrm{con} - \frac{\dot{x}_\mathrm{sf}}{x_\mathrm{sf}^0}} \right)},
\label{eq:con-force}$$ where $\dot{\varepsilon}_\mathrm{sf}^\mathrm{con} < 0$ is the maximum contractile strain rate induced in the stress fiber by myosin molecules stepping along the actin filaments at their maximum velocity, and $P_\mathrm{sf}^\mathrm{stl}$ is the tensile stall force at which the stress fiber cannot contract. This linear force-velocity relationship has been simplified from Hill’s original work, which suggested a hyperbolic form [@hill:1938]. It also has been extended to allow stress fiber slippage for $P^\mathrm{ac}_\mathrm{sf} > P_\mathrm{sf}^\mathrm{stl}$, as reported by @debold:2005.
Noting that the myosin proteins act collectively against a tensile force, we take $P_\mathrm{sf}^\mathrm{stl}$ to be proportional to the number of myosin molecules in the stress fiber, which in turn we assume to be proportional to the number of actin monomers [@wu:2005]. This leads to the relation $$P_\mathrm{sf}^\mathrm{stl} = P_\mathrm{myos}^\mathrm{stl} \beta N_\mathrm{sf},$$ where $P_\mathrm{myos}^\mathrm{stl} $ is the stall force of a single myosin, and $\beta$ is the constant of proportionality relating myosin and actin molecule numbers. Consistently with this model, we assume that the number of myosin molecules does not affect the maximum contractile strain rate, which is dictated by the stepping velocity of a single myosin molecule: $$\dot{\varepsilon}_\mathrm{sf}^\mathrm{con} = \frac{\dot{x}_\mathrm{myos}^\mathrm{con}}{x_\mathrm{sf}^0}.$$ Under these constitutive assumptions on the stress fiber active force, condition becomes: $$\frac{P_\mathrm{myos}^\mathrm{stl} \beta N_\mathrm{sf}}{\dot{x}_\mathrm{myos}^\mathrm{con}} {\left( {\dot{x}_\mathrm{myos}^\mathrm{con} - \dot{x}_\mathrm{sf}} \right)} \dot{x}_\mathrm{sf}
+ {\left( {\mu_\mathrm{ATP} - \mu_\mathrm{ADP}} \right)} \dot{N}_\mathrm{ADP} \ge 0.
\label{active-suff-final}$$ Eq. must hold together with for every value of $\dot{x}_\mathrm{sf}$ and $\dot{N}_\mathrm{ADP}$. This translates into the requirement that:[^3] $$\dot{N}_\mathrm{ADP} {\left( {\mu_\mathrm{ATP} - \mu_\mathrm{ADP}} \right)} \ge \frac{1}{4} P_\mathrm{myos}^\mathrm{stl} \beta {\left\vert {\dot{x}_\mathrm{myos}^\mathrm{con}} \right\vert} N_\mathrm{sf}.
\label{adp-rate}$$ We do not pursue a law for $\dot{N}_\mathrm{ADP}$ here, but assume that holds.
For the elastic component of the stress fiber force, we assume the following: $$P_\mathrm{sf}^\mathrm{e} = \pi r_\mathrm{sf}^2 E_\mathrm{sf} \gamma_\mathrm{e} {\left( {\frac{\xi^\mathrm{e}}{x_\mathrm{sf}^0} - 1} \right)},$$ where $E_\mathrm{sf}$ is the stress fiber’s overall Young’s modulus and $\gamma_\mathrm{e}$ the non-dimensional elastic modulus for the elastic element in Fig. \[fig:rehological\]. Furthermore, satisfaction of condition is ensured by the following constitutive relation for the viscous force $P_\mathrm{sf}^\mathrm{ve}$: $$P_\mathrm{sf}^\mathrm{ve} = \frac{\pi r_\mathrm{sf}^2}{x_\mathrm{sf}^0} \ \eta \ \dot{\xi}^\mathrm{v},$$ where $\eta \ge 0$ is the intrinsic viscosity of stress fiber slippage. For the stress fiber constitutive model outlined in Sect. \[sect:SF-int\], the viscous force also can be expressed as: $$\label{eq:Pve-el}
P_\mathrm{sf}^\mathrm{ve} = \pi r_\mathrm{sf}^2 E_\mathrm{sf} \gamma_\mathrm{ve} {\left( {\frac{\xi^\mathrm{ve}}{x_\mathrm{sf}^0} - 1} \right)},$$ where $\gamma_\mathrm{ve}$ is the non-dimensional elastic modulus for the elastic part of the Maxwell element in Fig. \[fig:rehological\].
Finally, the viscous force component also can be re-written as a functional of the observable variable $x_\mathrm{sf}(t)$, via a history-dependent integral, yielding the following expression for the overall stress fiber force: $$P_\mathrm{sf} = \frac{P_\mathrm{sf}^\mathrm{stl}}{\dot{\varepsilon}_\mathrm{sf}^\mathrm{con}} {\left( {\dot{\varepsilon}_\mathrm{sf}^\mathrm{con} - \frac{\dot{x}_\mathrm{sf}}{x_\mathrm{sf}^0}} \right)}
+ \pi r_\mathrm{sf}^2 E_\mathrm{sf} \gamma_\mathrm{e} {\left( {\frac{x_\mathrm{sf}}{x_\mathrm{sf}^0} - 1} \right)}
+ \pi r_\mathrm{sf}^2 E_\mathrm{sf} \gamma_\mathrm{ve} \int\limits_{0}^{t} \frac{\dot{x}_\mathrm{sf}{\left( {s} \right)}}{x_\mathrm{sf}^0} e^{- {\left( {t - s} \right)}/\tau} \, ds \ ,
\label{eq:mech-const}$$ where $\tau = \eta / {\left( {E_\mathrm{sf} \gamma_\mathrm{ve}} \right)}$ is the viscous relaxation time. Note that, because we have adopted the standard solid viscoelastic model for the description of the stress fiber viscoelastic behavior, Eq. can be inverted to yield the stress fiber’s current length $x_\mathrm{sf}$ as a function of the force. This is unlike the Kelvin-Voigt model used in some studies [@besser:2011; @besser:2007; @kumar:2006; @stachowiak:2009], which results in a non-physical, unbounded force for strain-controlled test cases in which time-discontinuous stretches are applied.
The constitutive equation for the force in the focal adhesion is arrived at by modeling the focal adhesion as a linearly elastic structure: $$P_\mathrm{fa} =\frac{ \overline{E}_\mathrm{fa} \hat{x}_\mathrm{fa} b}{h} x_\mathrm{fa}^\mathrm{e},
\label{eq:const-fa}$$ where $h$ and $b$ are the height and width of the focal adhesion, respectively, and $\overline{E}_\mathrm{fa}$ is the focal adhesion’s effective elastic modulus, estimated as follows: $$\overline{E}_\mathrm{fa} = E_\mathrm{fa} \frac{c_\mathrm{fa}}{c^\mathrm{max}_\mathrm{fa}},
\label{eq:maxEfa}$$ where $c^\mathrm{max}_\mathrm{fa} = 1/\lambda$ is the maximum attainable concentration, whereas $E_\mathrm{fa}$ is the elastic modulus of a single protein complex. See [@olberding:2010] for details.
The constitutive equation for the force in the ECM also is arrived at by modeling it as a uniform, linearly elastic structure: $$P_\mathrm{ecm} = K_\mathrm{ecm} {\left( {x^\mathrm{e}_\mathrm{ecm} - x_\mathrm{ecm}^0} \right)},
\label{eq:const-ecm}$$ where $K_\mathrm{ecm}$ is its stiffness, which depends on the Young’s modulus of the ECM material, $E_\mathrm{ecm}$.
Constitutive equations for the chemical potentials {#sect:chem-pots}
==================================================
The chemical potentials regulating the evolution of the stress fiber are:
\[eq:sf-chem-pots\] $$\begin{aligned}
&\mu^\mathrm{sf}_\mathrm{cyt} = H_\mathrm{cyt}^\mathrm{sf} + k_B T \ \mathrm{ln}{\left[ {\widehat{N}_\mathrm{sf}/ {\left( {N_\mathrm{sf}^\mathrm{max} - \widehat{N}_\mathrm{sf}} \right)} } \right]} \label{eq:chem-pots_cyt_sf} \\
&\mu_\mathrm{sf} = \frac{1}{2} \frac{{\left( {P_\mathrm{sf} x^0_\mathrm{sf}} \right)}^2}{E_\mathrm{sf} N_\mathrm{sf}^2 V_\mathrm{act}} + U_\mathrm{sf}^\mathrm{conf} - \frac{P_\mathrm{sf}}{N_\mathrm{fil}} \ d_\mathrm{sf}. \label{eq:chem-pots_sf}\end{aligned}$$
The terms on the right-hand side of are respectively, the enthalpy of formation of the generic stress fiber proteins, and their mixing entropy in the cytosol. We assume here that the stress fiber proteins are compartmentalized in the cytosol and that only a fraction of the total proteins is available to a given stress fiber, proportional to its initial length with constant of proportionality $c_\mathrm{sf}^\mathrm{max}$: $$N_\mathrm{sf}^\mathrm{max} = c_\mathrm{sf}^\mathrm{max} x_\mathrm{sf}^0.$$ In the first term on the right-hand side is the strain energy in the stress fiber, the second term is the change in internal energy due to any conformational changes during the binding of stress fiber proteins, and the third term arises due to the work done by the force displacing through $d_\mathrm{sf}$ during the conformational change.
The chemical potentials regulating the evolution of the focal adhesions are:
\[eq:fa-chem-pots\] $$\begin{aligned}
&\mu^\mathrm{fa}_\mathrm{cyt} = H_\mathrm{cyt}^\mathrm{fa} + k_B T \ \mathrm{ln}{\left[ {\widehat{N}_\mathrm{fa}/ {\left( {N_\mathrm{fa}^\mathrm{max} - \widehat{N}_\mathrm{fa}} \right)} } \right]} \label{eq:chem-pots_cyt_fa}\\
&\mu_\mathrm{fa}^\mathrm{d} = \frac{1}{2} \frac{P_\mathrm{fa}^2 h}{\overline{E}_\mathrm{fa} c_\mathrm{fa} \hat{x}^2_\mathrm{fa} b} \!+\! \frac{1}{2} B \kappa^2 \lambda + U_\mathrm{fa}^{conf} \!-\! P_\mathrm{fa} \!{\left( {\!d_\mathrm{fa} \!+\! \frac{\lambda}{2}\!} \right)} \label{eq:chem-pots_fa_d} \\
&\mu_\mathrm{fa}^\mathrm{p} = \frac{1}{2} \frac{P_\mathrm{fa}^2 h}{\overline{E}_\mathrm{fa} c_\mathrm{fa} \hat{x}^2_\mathrm{fa} b} \!+\! \frac{1}{2} B \kappa^2 \lambda + U_\mathrm{fa}^{conf} \!-\! P_\mathrm{fa} \!{\left( {\!d_\mathrm{fa} \!-\! \frac{\lambda}{2}\!} \right)} \label{eq:chem-pots_fa_p}\end{aligned}$$
The expression for the chemical potential of focal adhesion proteins differs depending on whether it is evaluated at the distal end or at the proximal end of the focal adhesion, Eq. and Eq. , respectively. The detailed arguments behind the various mechano-chemical contributions to $\mu_\mathrm{cyt}^\mathrm{fa}$, $\mu_\mathrm{fa}^\mathrm{d}$ and $\mu_\mathrm{fa}^\mathrm{p}$ have been laid out in @olberding:2010. We provide a summary below.
The terms on the right-hand side of are, respectively, the enthalpy of formation and the mixing entropy of the focal adhesion protein complexes in the cytosol. In Eq. and Eq. , the first two terms on the right-hand side are, respectively, due to the strain energy of the focal adhesion and the strain energy involved in the bending of the cell membrane. In the second term, $B$ is the bending modulus of the cell membrane and $\kappa$ is its curvature. This term arises because the binding and unbinding of focal adhesion proteins can change the local curvature of the cell membrane. The third term is the change in internal energy due to any conformational changes during the binding of focal adhesion proteins, and the fourth term arises due to the work done by the focal adhesion force displacing through $d_\mathrm{fa}$ during the conformational change. Finally, the fifth term arises due to the work done by the force when the focal adhesion center of mass effectively translates by binding/unbinding of proteins at its ends.
Rate laws for the chemical species {#sect:rate-laws}
==================================
To complete the model, rate laws for the kinetics of stress fiber and focal adhesion proteins have to be defined. The following expressions comply with the thermodynamic restrictions posed by Eq. and Eq. :
\[eq:chem-evos\]
where binding occurs for the sub-system $\alpha$ if $\mu_\mathrm{\alpha} - \mu^\mathrm{\alpha}_\mathrm{cyt} \leq 0$, whereas unbinding occurs if $\mu_\mathrm{\alpha} - \mu^\mathrm{\alpha}_\mathrm{cyt} > 0$. In Eq. , $k^\mathrm{b}_\alpha$, $k^\mathrm{u}_\alpha$ are, respectively, the binding and unbinding rates for sub-system $\alpha$, $k_B$ is the Boltzmann constant and $\chi_{\alpha} = \chi_{\alpha} {\left( {P} \right)}$ is a force-dependent exponent regulating the rapid dissociation of molecular bonds [@bell:1978] that is equal to the force-dependent part of the chemical potential for the sub-system $\alpha$. Eq. is evaluated at both the distal and the proximal ends of the focal adhesion and allows for the determination of $\dot{c}_\mathrm{fa}^\mathrm{d}$ and $\dot{c}_\mathrm{fa}^\mathrm{p}$ (Eq. ). In [@olberding:2010] it was shown that the form of the rate laws in can be obtained from Transition State Theory. While that derivation was carried out for focal adhesion dynamics, we have adopted the general form here for stress fiber dynamics, also.
The force-dependent exponents that drive the rapid disassociation of the stress fiber and focal adhesion are:
\[eq:chi\] $$\begin{aligned}
\chi_\mathrm{sf} &= \frac{1}{2} \frac{{\left( {P_\mathrm{sf} x^0_\mathrm{sf}} \right)}^2}{E_\mathrm{sf} N_\mathrm{sf}^2 V_\mathrm{act}}
- \frac{P_\mathrm{sf}}{N_\mathrm{fil}} \ d_\mathrm{sf} \label{eq:chi-sf} \\
\chi_\mathrm{fa} &= \begin{cases}
\frac{1}{2} \frac{P_\mathrm{fa}^2 h}{\overline{E}_\mathrm{fa} c_\mathrm{fa} \hat{x}^2_\mathrm{fa} b}
- P_\mathrm{fa} {\left( {d_\mathrm{fa} + \frac{\lambda}{2}} \right)} \hspace{0.18cm} \textit{(proximal)} \\
\frac{1}{2} \frac{P_\mathrm{fa}^2 h}{\overline{E}_\mathrm{fa} c_\mathrm{fa} \hat{x}_\mathrm{fa}^2 b}
- P_\mathrm{fa} {\left( {d_\mathrm{fa} - \frac{\lambda}{2}} \right)} \hspace{0.18cm}\textit{(distal).}
\end{cases} \label{eq:chi-fa}\end{aligned}$$
Critical loads for subsystems growth or resorption {#sect:crit-loads}
==================================================
The mechano-chemical nature of the cytoskeletal system was considered for the development of the model and is manifested in the chemical potentials and being functions of the force developed within the system (Fig. \[fig:chem-pots\]). By comparing the force in the system with suitable critical values it can be established whether a given part of the system (the stress fiber or the focal adhesions) undergoes growth or disassembly.
![[]{data-label="fig:chem-pots"}](chem-pots-plot.pdf)
It is important to recognize, however, that the values of such critical loads vary, as they depend upon the geometry of the system, which is itself subject to change as the system chemically evolves and proteins are recruited to (or disassembled from) the system.
Fig. \[fig:chem-pots\] is a schematic of the chemical potential of a generic sub-system $\alpha$ (the stress fiber, or the focal adhesion at its distal or proximal end) as a function of the force in the system. Two critical loads are labeled, which identify regimes over which the sub-system undergoes growth or disassembly. The specific values of such critical loads for the stress fiber are:
\[eq:P\_cr\_sf\] $$\begin{aligned}
P_\mathrm{cr, sf}^1 &= \frac{N_\mathrm{sf} E_\mathrm{sf} V_\mathrm{act}}{x_\mathrm{sf}^0}
{\left[ { \frac{d_\mathrm{sf}}{L_\mathrm{actmon}} - f_\mathrm{sf}{\left( {N_\mathrm{sf}} \right)} } \right]} \\
P_\mathrm{cr, sf}^2 &= \frac{N_\mathrm{sf} E_\mathrm{sf} V_\mathrm{act}}{x_\mathrm{sf}^0}
{\left[ { \frac{d_\mathrm{sf}}{L_\mathrm{actmon}} + f_\mathrm{sf}{\left( {N_\mathrm{sf}} \right)} } \right]},\end{aligned}$$
where: $$f_\mathrm{sf}{\left( {N_\mathrm{sf}} \right)} = \sqrt{ \frac{d_\mathrm{sf}^2}{L_\mathrm{actmon}^2}
- \frac{2 {\left[ {U_\mathrm{sf}^\mathrm{conf} - H_\mathrm{cyt}^\mathrm{sf} - k_B T \ ln{\left( {\frac{N_\mathrm{sf}^\mathrm{max}}{N_\mathrm{sf}} - 1} \right)}} \right]}}{E_\mathrm{sf} V_\mathrm{act}} }\ .$$ For the focal adhesion distal end, the critical loads are:
\[eq:P\_cr\_fad\] $$\begin{aligned}
P_\mathrm{cr,fa,d}^1 &= \frac{\hat{x}_\mathrm{fa}^2 b \ \overline{E}_\mathrm{fa} c_\mathrm{fa}}{h}
{\left[ { d_\mathrm{fa} - \frac{\lambda}{2} - f_\mathrm{fa,d}{\left( {\hat{x}_\mathrm{fa}} \right)} } \right]} \\
P_\mathrm{cr,fa,d}^2 &= \frac{\hat{x}_\mathrm{fa}^2 b \ \overline{E}_\mathrm{fa} c_\mathrm{fa}}{h}
{\left[ { d_\mathrm{fa} - \frac{\lambda}{2} + f_\mathrm{fa,d}{\left( {\hat{x}_\mathrm{fa}} \right)} } \right]},\end{aligned}$$
whereas for the proximal end they are:
\[eq:P\_cr\_fap\] $$\begin{aligned}
P_\mathrm{cr,fa,p}^1 &= \frac{\hat{x}_\mathrm{fa}^2 b \ \overline{E}_\mathrm{fa} c_\mathrm{fa}}{h}
{\left[ { d_\mathrm{fa} + \frac{\lambda}{2} - f_\mathrm{fa,p}{\left( {\hat{x}_\mathrm{fa}} \right)} } \right]} \\
P_\mathrm{cr,fa,p}^2 &= \frac{\hat{x}_\mathrm{fa}^2 b \ \overline{E}_\mathrm{fa} c_\mathrm{fa}}{h}
{\left[ { d_\mathrm{fa} + \frac{\lambda}{2} + f_\mathrm{fa,p}{\left( {\hat{x}_\mathrm{fa}} \right)} } \right]},\end{aligned}$$
where: $$f_\mathrm{fa,d}{\left( {\hat{x}_\mathrm{fa}} \right)} = \sqrt{ d_\mathrm{fa}^2 + \frac{\lambda^2}{4} - \lambda d_\mathrm{fa}
- \frac{2 h {\left[ {U_\mathrm{fa}^\mathrm{conf} + \frac{1}{2} B \kappa^2 \lambda - H_\mathrm{cyt}^\mathrm{fa} - k_B T \ ln{\left( {\frac{N_\mathrm{fa}^\mathrm{max}}{N_\mathrm{fa}} - 1} \right)}} \right]}}{\hat{x}_\mathrm{fa} b \ \overline{E}_\mathrm{fa} c_\mathrm{fa}} }\ ,$$ $$f_\mathrm{fa,p}{\left( {\hat{x}_\mathrm{fa}} \right)} = \sqrt{ d_\mathrm{fa}^2 + \frac{\lambda^2}{4} + \lambda d_\mathrm{fa}
- \frac{2 h {\left[ {U_\mathrm{fa}^\mathrm{conf} + \frac{1}{2} B \kappa^2 \lambda - H_\mathrm{cyt}^\mathrm{fa} - k_B T \ ln{\left( {\frac{N_\mathrm{fa}^\mathrm{max}}{N_\mathrm{fa}} - 1} \right)}} \right]}}{\hat{x}_\mathrm{fa} b \ \overline{E}_\mathrm{fa} c_\mathrm{fa}} }\ .$$
If a choice of the model parameters is made such that $0 \le P_\mathrm{cr, \alpha}^1$, the dynamics of the system unfold as follows:
- for $P < P_\mathrm{cr, \alpha}^1$, the corresponding chemical potential is positive and the stress fiber undergoes disassembly; however, the associated force-dependent exponent, Eq. , is negative, causing unbinding to be force-penalized. This regime may be considered as a *slow disassembly* regime for the sub-system;
- for $P_\mathrm{cr, \alpha}^1 < P < P_\mathrm{cr, \alpha}^2$, the corresponding chemical potential is negative and proteins are recruited. This regime may be considered as a *growth* regime for the sub-system;
- for $P > P_\mathrm{cr, \alpha}^2$, the corresponding chemical potential is positive and the sub-system undergoes disassembly. The associated force-dependent exponent can be negative or positive; in the latter case, the sub-system undergoes a *force-boosted* (rapid) *disassembly*.
The complex mechano-chemical behavior of the cytoskeletal system can thus be modeled as the result of the interplay between the dynamics of all its sub-systems.
The ordinary differential equations governing the evolution of the system
=========================================================================
We now summarize the equations representing the model. Recall that due to the symmetry of the system (Fig. \[fig:system-len\]), the two focal adhesions are identical in geometry and response, and only one of them needs to be followed.
Our previous computations with a full reaction-diffusion model for focal adhesion dynamics reveal that $c_\mathrm{fa} = c^\mathrm{max}_\mathrm{fa}$ holds almost everywhere, except for a narrow regime immediately behind the proximal edge of a translating focal adhesion. The condition does hold uniformly for a focal adhesion that is growing at both ends. Both these results are available as supporting information in @olberding:2010 (see SI Movies 10 and 11). For this reason, we assume that the focal adhesion elastic modulus attains its maximum value, $\overline{E}_\mathrm{fa} = E_\mathrm{fa}$, corresponding to $c_\mathrm{fa} = c^\mathrm{max}_\mathrm{fa} = 1 / \lambda$ \[see Eq. (\[eq:maxEfa\])\].
Under these assumptions, the equations to be solved are:
\[eq:model-ode\] $$\begin{aligned}
\hspace{-0.1cm} \dot{x}_\mathrm{d} &= -\lambda^2 \!
\begin{cases}
k_\mathrm{fa}^\mathrm{b} {\left[ { 1 - \exp {\left( {\frac{\mu_\mathrm{fa}^\mathrm{d}-\mu^\mathrm{fa}_\mathrm{cyt}}{k_B T}} \right)} } \right]} \hspace{1.45cm} \textit{(binding)}\\
k_\mathrm{fa}^\mathrm{u} \ \exp {\left( {\chi_\mathrm{fa}} \right)} {\left[ { \exp \! {\left( {-\frac{\mu_\mathrm{fa}^\mathrm{d}-\mu^\mathrm{fa}_\mathrm{cyt}}{k_B T}} \right)} - 1} \right]} \hspace{0.0cm} \textit{(unbind.) ,}
\end{cases} \hspace{-1.1cm} \label{eq:ode-FAd} \\
\hspace{-0.1cm} \dot{x}_\mathrm{p} &= \lambda^2
\begin{cases}
k_\mathrm{fa}^\mathrm{b} {\left[ { 1 - \exp {\left( {\frac{\mu_\mathrm{fa}^\mathrm{p}-\mu^\mathrm{fa}_\mathrm{cyt}}{k_B T}} \right)} } \right]} \hspace{1.65cm} \textit{(binding)}\\
k_\mathrm{fa}^\mathrm{u} \ \exp {\left( {\chi_\mathrm{fa}} \right)} \! {\left[ { \exp {\left( {-\frac{\mu_\mathrm{fa}^\mathrm{p}-\mu^\mathrm{fa}_\mathrm{cyt}}{k_B T}} \right)} - 1} \right]} \hspace{0.2cm} \textit{(unbind.) ,}
\end{cases} \hspace{-1.1cm} \label{eq:ode-FAp} \\
\hspace{-0.1cm} \dot{N}_\mathrm{sf} &= \!
\begin{cases}
k_\mathrm{sf}^\mathrm{b} {\left( {N^\mathrm{max}_\mathrm{sf} \!-\! N_\mathrm{sf}} \right)}\! {\left[ { 1 - \exp \! {\left( {\frac{\mu_\mathrm{sf}-\mu^\mathrm{sf}_\mathrm{cyt}}{k_B T}} \right)}} \right]} \hspace{0.4cm} \textit{(binding)}\\
k_\mathrm{sf}^\mathrm{u} \ \exp {\left( {\chi_\mathrm{sf}} \right)} {\left[ { \exp {\left( {-\frac{\mu_\mathrm{sf}-\mu^\mathrm{sf}_\mathrm{cyt}}{k_B T}} \right)} -1 } \right]} \hspace{0.6cm} \textit{(unbind.) .}
\end{cases} \hspace{-0.7cm} \label{eq:ode-SF}
$$
The chemical potentials appearing in the system are, as discussed in Sect. \[sect:crit-loads\], functions of the force $P$ within the system, which is provided at every time $t$ by the solution of Eq. , Eq. , Eq. , Eq. and Eq. .
The system is constituted of ordinary differential equations and gives as a solution the functions $x^\mathrm{d}(t)$, $x^\mathrm{p}(t)$ and $N_\mathrm{sf}(t)$ (i.e. the time evolution of the FA distal and proximal end positions and of the number of generic stress fiber proteins in the stress fiber) provided that the following initial conditions are specified:
\[eq:model-ic\] $$\begin{aligned}
x_\mathrm{d} (0) &= x_\mathrm{d}^0
\label{eq:FAd-ic} \\
x_\mathrm{p} (0) &= x_\mathrm{p}^0
\label{eq:FAp-ic} \\
N_\mathrm{sf} (0) &= \frac{x_\mathrm{sf}^0}{L_\mathrm{actmon}}.
\label{eq:SF-ic}\end{aligned}$$
The initial focal adhesion length is therefore, $$\hat{x}_\mathrm{fa}^0 = {\left\vert {x_\mathrm{p}^0 - x_\mathrm{d}^0} \right\vert}.$$ For condition we assumed that the stress fiber initially consists of only one filament, in order to model the minimal precursor system, i.e., a single actin filament.
A test case: the influence of the ECM stiffness on cytoskeletal force generation and focal adhesion dynamics {#sect:results}
============================================================================================================
The model’s mechano-chemical response was studied by varying the Young’s modulus of the ECM, $E_\mathrm{ecm}$. The parameter values used in the computations are provided as Supporting Information.
For the sake of generality, results are provided in a non-dimensional form. We chose the set ${\left\{ {\lambda, E_\mathrm{sf}, k_\mathrm{sf}^\mathrm{b}, c_\mathrm{sf}^\mathrm{max}} \right\}}$ as the dimensional basis; consequently, $L^\ast = L/\lambda$, $t^\ast = t \ k_\mathrm{sf}^\mathrm{b}$, $P^\ast = P/(\lambda^2 E_\mathrm{sf})$, $N^\ast_\mathrm{sf} = N_\mathrm{sf}/(c_\mathrm{sf}^\mathrm{max} \lambda)$ and $E^\ast = E/E_\mathrm{sf}$ are, respectively, the non-dimensional length, time, force, number of proteins and elastic modulus.
\[h\] 
The detailed dynamics of the system are depicted in Fig. \[fig:dynamics-un\]. It can be observed that the force within the system (Fig. \[fig:dynamics-un\]a) first increases to a maximum value. Then, after a time that depends on the value of the ECM stiffness, it attains a near-plateau with a slightly negative slope. Our simulations show that increasing the ECM stiffness leads to an increase in the maximum force developed within the system; in this respect, our model provides results in agreement with experimental observations [@califano:2010; @ghibaudo:2008; @kong:2005; @lo:2000]. The force appears to be in direct correlation with the development of the stress fiber: as Fig. \[fig:dynamics-un\]b shows, a higher ECM stiffness results in more proteins being recruited by the stress fiber.
Pushing the model to its limit, we found that for very stiff ECMs ($E^\ast_\mathrm{ecm} = 80 E_0$) the force drops abruptly after having reached a maximum value, due to the collapse of the focal adhesion. The inset in Fig. \[fig:dynamics-un\]c shows that in this case the focal adhesion length falls quickly to zero, meaning that the focal adhesion is fully resorbed. For lower values of $E^\ast_\mathrm{ecm}$, i.e. in the range $E_0$–$20 E_0$, a low value of the stiffness gives rise to focal adhesions that are smaller in size (Fig. \[fig:dynamics-un\]c) but more motile (as seen in the evolving position of the focal adhesion centroid in Fig. \[fig:dynamics-un\]d), whereas if the ECM stiffness increases, the focal adhesions become more static and bigger in size. This trend seems to be in agreement with the experiments reported in literature which show that a compliant substrate promotes more dynamic focal adhesions, as opposed to the more elongated and stable focal adhesions developed in cells sitting on stiff substrates [@lo:2000; @pelham:1997]. However, our results do not follow this trend for very high values of the ECM stiffness, as the high value of the force developed within the system under this condition limits the growth of the focal adhesion. Our simulations also show that a regime of ECM stiffness does exist in which the focal adhesions’ size and position do not change dramatically from the initial values. This is seen for ECM stiffness in the range of $40E_0$–$60E_0$: as depicted in Fig. \[fig:dynamics-un\]c, the focal adhesion length grows over the time interval considered for ECM stiffness of $40E_0$, while it initially grows, and then begins to decrease, but somewhat gradually for $60E_0$. Besides, the focal adhesion translation shows evidence of leveling off for ECM stiffness $40E_0$, while it increases for $60E_0$, but at a lower rate than for $80E_0$, (Fig. \[fig:dynamics-un\]d). In this regime, the stress fiber-focal adhesion system is not exactly at equilibrium, but the system parameters ($N^\ast_\mathrm{sf}$, $L^\ast_\mathrm{fa}$ and $x^{c^\ast}_\mathrm{fa}$ ) remain within certain bounds over the non-dimensional times considered here. This could be representative of the typically limited times of $\sim 1$ hour over which the stress fiber-focal adhesion system has been tracked in the experimental literature [@riveline:2001; @mannetal:2012]. From the currently available experimental literature it is not clear whether the focal adhesions ever attain true equilibrium. Also, it remains difficult, if not impossible, to visualize stress fiber dynamics in live cells, leaving the question of equilibrium open.
Concluding remarks
==================
The model we have presented is a step toward a thermodynamically consistent treatment of the mechano-chemistry of cytoskeletal force generation, and dynamics. We note that mechano-chemical interactions dictate the primacy of the thermodynamic treatment to ensure consistency. In this setting, it is possible to access many aspects of cytoskeletal force generation and dynamics that have been reported in the experimental literature. While our focus is on the theoretical, non-equilibrium thermodynamic treatment, we also have demonstrated the model’s prediction of stress fiber–focal adhesion dynamics on ECMs of varying stiffnesses.
In the model, while the stress fibers do form a stable sub-system, the cytoskeletal system as a whole is never truly at equilibrium, but slowly evolves, mainly due to the dynamic focal adhesions. In discussing the corresponding results in Sect. \[sect:results\] we have emphasized that experiments have not been carried out for longer times than $\sim 1$ hour, and that over this interval, our model’s results are representative of the experimental studies.
While we have noted that the experimental literature currently leaves it unclear whether the stress fiber–focal adhesion system ever attains true equilibrium, we do consider this question in light of the particulars of the computations we have presented here: The non-equilibrium character of the focal adhesions is partly a consequence of a simplifying modeling assumption that we have made: that the force is uniformly distributed over the focal adhesion. As a result, the condition that leads to a focal adhesion at equilibrium in our model (i.e. both $\mu^\mathrm{p}_\mathrm{fa} = 0$ and $\mu^\mathrm{d}_\mathrm{fa} = 0$) can never be met. Exact mechano-chemical equilibrium can be achieved, however, with a non-uniform force distribution in our model. See [@olberding:2010] in this regard. In this case, the maintenance of mechanical equilibrium requires a non-uniform force through the radius of the stress fiber to balance a non-uniform force distribution over the focal adhesion. This can be attained by explicitly modeling the single actin filaments, each subjected to a different force. Separate rate laws of the form would be written for each filament. Besides, as the force acting on one filament would be equal to the force over some interval of the focal adhesion, protein binding/unbinding would be tracked at every point of the focal adhesion itself using . Alternatively, a continuum-level model could be developed in which concentrations of stress fiber and focal adhesion proteins would be tracked, instead of the reduced-order structural models used here. These developments are beyond the scope of this communication where we have sought to focus on the essential physics with simpler models, but will be presented later. A more comprehensive investigation on the behavior of the model is the subject of a separate work, where we have focused on specific applications to make further and deeper connection with the experimental literature [@maraldi:2013].
Parameters’ values
==================
[^1]: correspondence: krishna@umich.edu
[^2]: In [@olberding:2010] the term *treadmilling* has been applied to the mechanism of focal adhesion translation.
[^3]: Note that the right-hand side of inequality is positive, in compliance with condition .
|
---
abstract: 'For geometries with a closed three-form we briefly overview the notion of multi-moment maps. We then give concrete examples of multi-moment maps for homogeneous hypercomplex and nearly Kähler manifolds. A special role in the theory is played by Lie algebras with second and third Betti numbers equal to zero. These we call (2,3)-trivial. We provide a number of examples of such algebras including a complete list in dimensions up to and including five.'
---
IMADA Preprint 2010\
CP^3^-ORIGINS: 2010-52
**Homogeneous spaces, multi-moment maps and (2,3)-trivial algebras**
Thomas Bruun Madsen and Andrew Swann
Introduction
============
Symplectic geometry is a geometrisation of the theory of mechanical systems. A symplectic structure on a manifold $ M $ is defined by a closed $ 2 $-form $ \omega $ that may be expressed locally as $
\omega = dq_1 \wedge dp_1 + \dots + dq_n \wedge dp_n $, where $ \dim
M = 2n $. In this context the concepts of linear and angular momentum are captured by the notion of moment map corresponding to a group of symmetries $ G $ of $ M $ preserving $ \omega $. It is an equivariant map $ \mu\colon M\to {\operatorname{\mathfrak{g}}}^* $, to the dual of the Lie algebra $ {{\operatorname{\mathfrak{g}}}}$ of $ G $, characterised by the equation $
d{\inpd{\mu,{{\mathsf{X}}}}} = X {\mathop{\lrcorner}}\omega $, for each $ {{\mathsf{X}}} \in {\operatorname{\mathfrak{g}}}
$. Here $ X $ denotes the corresponding vector field on $ M $ generated by $ {{\mathsf{X}}} \in {\operatorname{\mathfrak{g}}} $.
Developments in string and other field theories with Wess-Zumino terms [@Michelson-S:conformal; @Strominger:superstrings; @Gates-HR:twisted; @Baez-HR:string] have highlighted the importance of geometries associated to closed $
3 $-forms $ c $. Geometric aspects of these theories are not so well developed. In [@Madsen-S:multi-moment], developing [@Madsen:torsion], we introduce a notion of multi-moment map adapted to such geometries. The important features of our definition are that the target space depends only on the symmetry group $ G $ and not the underlying manifold $ M $, in contrast to [@Carinena-CI:multisymplectic], and that existence of such maps are guaranteed in many circumstances. After reviewing the definition and basic properties, the rest of the paper is devoted to giving examples from hypercomplex and nearly Kähler geometry and to classifications of a special class of Lie algebras that arises.
We say the $ 3 $-form $ c \in \Omega^3(M) $ defines a *strong geometry* if $ c $ is closed, meaning $ dc = 0 $. The lack of simple canonical descriptions for $ 3 $-forms means that in general a non-degeneracy assumption is not appropriate. However, if $ X{\mathop{\lrcorner}}c = 0 $ occurs only for $ X = 0 \in T_xM $, we will say that the structure is *2-plectic*, following [@Baez-HR:string].
Suppose $ G $ acts on a strong geometry $ (M,c) $ preserving $ c
$. Then for each $ {{\mathsf{X}}} \in {{\operatorname{\mathfrak{g}}}}$, we have from the Cartan formula that $ X {\mathop{\lrcorner}}c $ is a closed $ 2 $-form. Similarly, if $ {{\mathsf{X}}} $ and $ {{\mathsf{Y}}} $ are two commuting elements of $ {{\operatorname{\mathfrak{g}}}}$, then $ (X\wedge Y) {\mathop{\lrcorner}}c = c(X,Y,\cdot) $ is a closed $ 1
$-form. If this can be integrated to a function $ \nu_{X\wedge Y}
$, then we have a multi-moment map for the $ {\mathbb R}^2 $ action generated by $ X $ and $ Y $.
In general, the space of commuting elements in $ {{\operatorname{\mathfrak{g}}}}$ forms a complicated variety. We therefore introduce the *Lie kernel* $
{\mathcal P_{{{\operatorname{\mathfrak{g}}}}}}$ as the kernel of the map $ \Lambda^2{\operatorname{\mathfrak{g}}} \to {\operatorname{\mathfrak{g}}} $ induced by the Lie bracket $ [\cdot,\cdot] $. A calculation shows that for $ {{\mathsf{p}}} = \sum_i X_i \wedge Y_i \in {\mathcal P_{{{\operatorname{\mathfrak{g}}}}}}$, the one-form $
p{\mathop{\lrcorner}}c $ is closed. This leads to
[@Madsen-S:multi-moment] A *multi-moment map* for the action of a group $ G $ of symmetries of a strong geometry $ (M,c) $ is an equivariant map $ \nu\colon M \to {\mathcal P_{{{\operatorname{\mathfrak{g}}}}}}^*
$ satisfying $ d{\inpd{\nu,{{\mathsf{p}}}}} = p{\mathop{\lrcorner}}c $, for all $ {{\mathsf{p}}}
\in {\mathcal P_{{{\operatorname{\mathfrak{g}}}}}}$.
The following result summarises the important existence results for multi-moment maps.
Multi-moment maps exist for the action of $ G $ on $ (M,c) $ if either
\[item:M-compact\] $ M $ is compact with $ b_1(M) = 0
$,
\[item:G-compact\] $ G $ is compact, $ b_1(M) = 0 $ and $ G $ preserves a volume form on $ M $,
\[item:G-exact\] $ c = d b $ with $ b \in \Omega^2(M) $ preserved by $ G $, or
\[item:2-3-trivial\] $ b_2({{\operatorname{\mathfrak{g}}}}) = 0 = b_3({{\operatorname{\mathfrak{g}}}}) $.
In this paper, we will give examples related to cases (\[item:M-compact\]–\[item:G-exact\]) in hypercomplex and nearly Kähler geometries. Underlying the nearly Kähler examples is an analogue of the Kostant-Kirillov-Souriau construction, which we prove in [@Madsen-S:multi-moment]. This construction is also relevant for the Lie algebras satisfying the conditions of case (\[item:2-3-trivial\]); these we have dubbed as *(2,3)-trivial*. This new class replaces that of semi-simple algebras in the theory of symplectic moment maps, since semi-simple algebras $ {{\operatorname{\mathfrak{h}}}}$ are characterised by $ b_1({{\operatorname{\mathfrak{h}}}}) = 0 = b_2({{\operatorname{\mathfrak{h}}}}) $. We have found the following structure theorem:
\[thm:2-3-trivial\] A Lie algebra $ {{\operatorname{\mathfrak{g}}}}$ is $
(2,3) $-trivial if and only if $ {{\operatorname{\mathfrak{g}}}}$ is solvable, with derived algebra $ {{\operatorname{\mathfrak{k}}}}= {{\operatorname{\mathfrak{g}}}}' $ of codimension one and satisfying $
H^1({{\operatorname{\mathfrak{k}}}})^{{{\operatorname{\mathfrak{g}}}}} = 0 = H^2({{\operatorname{\mathfrak{k}}}})^{{{\operatorname{\mathfrak{g}}}}} = H^3({{\operatorname{\mathfrak{k}}}})^{{{\operatorname{\mathfrak{g}}}}} $.
In this paper we use this result to give full lists of $ (2,3)
$-trivial algebras in dimensions at most five and to show that every nilpotent algebra $ {{\operatorname{\mathfrak{k}}}}$ of dimension at most six occurs in the above theorem. We also give some constructions of infinite families of $ (2,3) $-trivial Lie algebras.
A hypercomplex example {#sec:exhc}
======================
An *[<span style="font-variant:small-caps;">hkt</span>]{}structure* on a manifold is given by three complex structures $ I $, $ J $, $ K = IJ = -JI $ with common Hermitian metric such that $ Id\omega_I = Jd\omega_J = Kd\omega_K $. By [@Cabrera-S:aqH-torsion Prop. 6.2] it is unnecessary to check integrability of $ I $, $ J $ and $ K $. An example of a homogeneous [<span style="font-variant:small-caps;">hkt</span>]{}manifold is the compact simple Lie group $ {{\operatorname{\textsl{SU}}}}(3)
$. In fact it admits a left-invariant [<span style="font-variant:small-caps;">shkt</span>]{}structure, meaning that $ c = -Id\omega_I $ is closed. As a consequence of left-invariance, we may think of the [<span style="font-variant:small-caps;">hkt</span>]{}structure on $ {{\operatorname{\textsl{SU}}}}(3) $ in terms of a corresponding algebraic structure on its Lie algebra. Write $ E_{pq}
$ for the elementary $ 3\times 3 $-matrix with a $ 1 $ at position $ (p,q) $. Then the following $ 8 $ complex matrices constitute a basis for $ {{\operatorname{\mathfrak{su}}}}(3) $: $ A_1=i(E_{11}{-}E_{22}) $, $
A_2=i(E_{22}{-}E_{33}) $, $ B_{pq}=E_{pq}{-}E_{qp} $, $
C_{pq}=i(E_{pq}{+}E_{qp}) $, for $ p=1,2<q=2,3 $. We write $
a_1,\dots,c_{23} $ for the dual basis. Using the formula $
d\alpha(X,Y)=-\alpha([X,Y]) $, one finds that $$\label{eq:extdi}
\begin{gathered}
da_1 = - 2b_{12}c_{12} - 2b_{13}c_{13},\
da_2 = - 2b_{13}c_{13} - 2b_{23}c_{23},\\
db_{12} = (2a_1 - a_2)c_{12} + b_{13}b_{23} + c_{13}c_{23},\
db_{13} = (a_1 + a_2)c_{13} - b_{12}b_{23} + c_{12}c_{23},\\
db_{23} = (-a_1 + 2a_2)c_{23} + b_{12}b_{13} + c_{12}c_{13},\
dc_{12} = (-2a_1 + a_2)b_{12} - b_{13}c_{23} - b_{23}c_{13},\\
dc_{13} = (-a_1 - a_2)b_{13} - b_{12}c_{23} + b_{23}c_{12},\
dc_{23} = (a_1 - 2a_2)b_{23} + b_{12}c_{13} + b_{13}c_{12},
\end{gathered}$$ where $ \wedge $ signs have been omitted.
A metric is provided by minus the Killing form on $ {{\operatorname{\mathfrak{su}}}}(3) $; here we may take $ g $ to be the map $ (X,Y)\mapsto-\operatorname{tr}(XY) $: $ g
= 2a_1^2 - a_1a_2 + 2( a_2^2 + b_{12}^2 + b_{13}^2 + b_{23}^2 +
c_{12}^2 + c_{13}^2 + c_{23}^2) $.
In [@Joyce:hypercomplex Thm. 4.2] Joyce proved the existence of hypercomplex structures on certain compact Lie groups. For $ {{\operatorname{\textsl{SU}}}}(3)
$, Joyce’s hypercomplex structure comes from taking a highest root $
{{\operatorname{\mathfrak{su}}}}(2) $, e.g., the span of $ A_1 $, $ B_{12} $, $ C_{12} $ and the complement $ {\mathbb H}+{\mathbb R}$, $ {\mathbb H}\cong \Span{ B_{13}, C_{13},
B_{23}, C_{23} } $ and $ {\mathbb R}\cong \Span{A_1{+}2A_2} $. Concretely, let us write $ \mathbf I=A_1 $, $ \mathbf J=B_{12} $ and $ \mathbf K=C_{12} $. Then define $ I $ on $ {\mathbb H}$ to be $
\operatorname{ad}_{\mathbf I} $. Similarly $ J $ and $ K $ act on $ {\mathbb H}$ by $ \operatorname{ad}_{\mathbf J} $ and $ \operatorname{ad}_{\mathbf K} $, respectively. On $
{\mathbb R}+ {{\operatorname{\mathfrak{su}}}}(2)$ the actions of $ I $, $ J $ and $ K $ are given by $ IV = \mathbf I $, $ JV = \mathbf J $ and $ KV=\mathbf K $, where $ V=(A_1 + 2A_2)/\sqrt 3 $. The complex structures $ I $, $ J $ and $ K $ are now determined completely by the requirement that they square to $ -1 $. It is straightforward to check that $
IJ = K = -JI $. Computations show that $ I $, $ J $ and $ K $ are compatible with the metric, meaning $ g(X,Y)=g(IX,IY) $, etc., for all $ X,Y\in{{\operatorname{\mathfrak{su}}}}(3) $. Defining $ 2a_1'=2a_1- a_2 $ and $
2a_2'= \sqrt3 a_2 $, we find that the non-degenerate $ 2 $-forms $
\omega_I = g(I\cdot,\cdot) $, $ \omega_J $ and $ \omega_K $ are given by $$\begin{gathered}
\omega_I = -a'_1 a'_2 + b_{12} c_{12} + b_{13} c_{13} - b_{23}
c_{23},\ \omega_J = a_2' b_{12} - a_1' c_{12} - b_{13}
b_{23} - c_{13} c_{23},\\
\omega_K = a_2' c_{12}+a_1' b_{12} + b_{13} c_{23} + b_{23} c_{13}.\end{gathered}$$ Using it then follows that $$\begin{aligned}
d\omega_I&= - \sqrt{3}a_1'(b_{13}c_{13} + b_{23}c_{23}) +
a_2'(2b_{12}c_{12} + b_{13}c_{13} - b_{23}c_{23}){\\&\qquad}-
b_{12}b_{13}c_{23} - b_{12}b_{23}c_{13} - b_{13}b_{23}c_{12} -
c_{12}c_{13}c_{23},\\
d\omega_J&=2a_1'a_2'c_{12} + a_1'(b_{13}c_{23} + b_{23}c_{13}) -
a_2'(b_{13}b_{23} + c_{13}c_{23}){\\&\qquad}-
\sqrt{3}b_{12}b_{13}c_{13} - \sqrt{3}b_{12}b_{23}c_{23} +
b_{13}c_{12}c_{13} - b_{23}c_{12}c_{23},\\
d\omega_K&= - 2a_1'a_2'b_{12} + a_1'(b_{13}b_{23} + b_{23}c_{13}) +
a_2'(b_{13}c_{23} + b_{23}c_{13}){\\&\qquad}+
\sqrt{3}b_{13}c_{12}c_{13} + \sqrt{3}b_{23}c_{12}c_{23} +
b_{12}b_{13}c_{13} - b_{12}b_{23}c_{23}.\end{aligned}$$ From these formulae and the actions of $ I $, $ J $ and $ K $, we verify the $ {\textsc{hkt}\xspace}$ condition: $$\begin{split}
Id\omega_I&=a_1(2b_{12}c_{12} + b_{13}c_{13} - b_{23}c_{23}) -
a_2(b_{12}c_{12} - b_{13}c_{13} - 2b_{23}c_{23})\\
&\quad-b_{23}c_{12}c_{13} - b_{13}c_{12}c_{23} -
b_{12}c_{13}c_{23} - b_{12}b_{13}b_{23}\\
&= Jd\omega_J = Kd\omega_K.
\end{split}$$ It is easy to check that $ dc=0 $, and thus $ ({{\operatorname{\textsl{SU}}}}(3),g,I,J,K) $ is indeed an $ {\textsc{shkt}\xspace}$ manifold as claimed.
Unfortunately the multi-moment map for the left action of $ {{\operatorname{\textsl{SU}}}}(3) $ on the strong geometry $ ({{\operatorname{\textsl{SU}}}}(3),c) $ is trivial. However, we may instead turn our attention towards the multi-moment maps $ \nu_I $, $ \nu_J $ and $ \nu_K $ associated with the $ 3 $-forms $
d\omega_I $, $ d\omega_J $ and $ d\omega_K $ on $ {{\operatorname{\textsl{SU}}}}(3) $. As an example let us consider the multi-moment map $ \nu_I\colon {{\operatorname{\textsl{SU}}}}(3)
\to {\mathcal P_{{{\operatorname{\mathfrak{su}}}}(3)}}^* $, $ {\inpd{\nu,{{\mathsf{p}}}}} = \omega_I(p) $. The commutation relations for the chosen $ {{\operatorname{\mathfrak{su}}}}(3) $-basis may be used to establish a basis for $ {\mathcal P_{{{\operatorname{\mathfrak{su}}}}(3)}} \leqslant \Lambda^2{{\operatorname{\mathfrak{su}}}}(3) $ whilst the exterior derivative, via equations , gives a basis for the submodule $ {{\operatorname{\mathfrak{su}}}}(3)\leqslant\Lambda^2{{\operatorname{\mathfrak{su}}}}(3) $. We may now decompose $ \omega_I $ at the identity: $ \omega_I =
\omega_I^{{{\operatorname{\mathfrak{su}}}}(3)} + \omega_I^{{\mathcal P_{\relax}}} = 2\bigl(b_{12}c_{12} +
b_{13}c_{13}\bigr) + \bigl( - {\sqrt{3}a_1a_2}/2 - (b_{12}c_{12} +
b_{23}c_{23} - b_{13}c_{13})\bigr) $. It follows that $$\operatorname{Ad}_{g^{-1}}^*\nu_I(g) = -\tfrac{\sqrt3}2a_1a_2 - (b_{12}c_{12} +
b_{23}c_{23} - b_{13}c_{13}).$$ The image of $ \nu_I $ is the orbit of $ \beta_I = \nu_I(e) $ under $ {{\operatorname{\textsl{SU}}}}(3) $. This element is preserved by a maximal torus, invariant under $ I $, and its orbit is a copy of $ F_{1,2}({\mathbb C}^3)
$ inside $ {\mathcal P_{{{\operatorname{\mathfrak{su}}}}(3)}}^* $. At the algebraic level this is easily verified: $$\begin{split}
\ker (\nu_I)_*&= \ker d\nu_I = \bigl\{\, A\in{{\operatorname{\mathfrak{su}}}}(3) :
d\nu_I(p,A)=0 \text{ for all }
{{\mathsf{p}}} \in{\mathcal P_{{{\operatorname{\mathfrak{su}}}}(3)}} \,\bigr\} \\
&= \bigl\{\, A\in{{\operatorname{\mathfrak{su}}}}(3) : c(Ip,IA)=0 \text{ for all }
{{\mathsf{p}}} \in {\mathcal P_{{{\operatorname{\mathfrak{su}}}}(3)}} \,\bigr\}\\
&= I\bigl\{\, A\in{{\operatorname{\mathfrak{su}}}}(3) : g([Ip],A)=0\text{ for all } {{\mathsf{p}}} \in
{\mathcal P_{{{\operatorname{\mathfrak{su}}}}(3)}} \,\bigr\} = [I{\mathcal P_{{{\operatorname{\mathfrak{su}}}}(3)}}]^\bot = \Span{A_1,V}.
\end{split}$$ Similarly the images of $ \nu_J $ and $ \nu_K $ are full flags in $ {\mathbb C}^3 $. We find $$\begin{aligned}
{3}
&\begin{aligned}[b] \operatorname{Ad}^*_{g^{-1}}\nu_J(g)&= \tfrac{\sqrt3}2
\left(2(a_1+a_2)b_{12} -
(b_{23}c_{13}+b_{13}c_{23})\right)\\
&\quad
+\tfrac1{14}\left(2(2a_1-a_2)c_{12}-5(b_{13}b_{23}+c_{13}c_{23})\right),
\end{aligned}&&\ker(\nu_J)_*=\Span{V,B_{12}},
\\
&\begin{aligned}[b] \operatorname{Ad}^*_{g^{-1}}\nu_K(g)&= \tfrac{\sqrt3}{14}
\left((3a_1+2a_2)c_{12}
-2(b_{13}b_{23}+c_{13}c_{23})) \right)\\
&\quad -\tfrac12
\left(2(8a_1+5a_2)b_{12}-11(b_{13}c_{23}+b_{23}c_{13})\right),
\end{aligned}
&\quad&\ker(\nu_K)_*=\Span{V,C_{12}}.\end{aligned}$$ Putting these together, we get an equivariant map $ \underline{\nu} =
(\nu_I,\nu_J,\nu_K) \colon {{\operatorname{\textsl{SU}}}}(3) \to ({\mathcal P_{{{\operatorname{\mathfrak{su}}}}(3)}}^*)^3 $. The image is the Aloff-Wallach space $ A_{1,1}={{\operatorname{\textsl{SU}}}}(3)/{T^1_{1,1}} $. The relatively high dimension of this image indicates that multi-moment maps could be a useful tool to study homogeneous hyperHermitian structures.
Six-dimensional nearly Kähler manifolds
=======================================
A *nearly Kähler structure*, briefly an [<span style="font-variant:small-caps;">nk</span>]{}structure, on a six-dimensional manifold may be specified by a $ 2 $-form $ \sigma
$ and a $ 3 $-form $ \psi^+ $ whose common pointwise stabiliser in $ {{\operatorname{\textsl{GL}}}}(6,{\mathbb R}) $ is isomorphic to $ {{\operatorname{\textsl{SU}}}}(3) $. The [<span style="font-variant:small-caps;">nk</span>]{}condition is then $ d\sigma = \psi^+ $ and $ d\psi^- = -\tfrac23\sigma^2 $, where $ \psi^++i\psi^- \in \Lambda^{3,0} $. We will indicate how each homogeneous strict [<span style="font-variant:small-caps;">nk</span>]{}six-manifold $ G/H = F_{1,2}({\mathbb C}^3) $, $ \operatorname{{\mathbb C}P}(3) $, $ S^3\times S^3 $ and $ S^6 $, as classified by @Butruille:nK, may be realised as a 2-plectic orbit $
G\cdot\beta $ in $ {\mathcal P_{{{\operatorname{\mathfrak{g}}}}}}^* $. Let $ {d_{\mathcal P}}\colon {\mathcal P_{{{\operatorname{\mathfrak{g}}}}}}^* \to
\Lambda^3{{\operatorname{\mathfrak{g}}}}^* $ be the map induced by $ d $. Then our realisation is such that $ \Psi = {d_{\mathcal P}}\beta $ induces $ \psi^+ $ via $
{\inpd{\Psi,{{\mathsf{X}}}\wedge{{\mathsf{Y}}}\wedge{{\mathsf{Z}}}}}=\psi^+(X,Y,Z) $ and $
\sigma(X,Y) = \beta({{\mathsf{X}}},{{\mathsf{Y}}}) $.
In each case the element $ \beta\in{\mathcal P_{{{\operatorname{\mathfrak{g}}}}}}^* $ must be chosen with some care. For instance neither of the $ 3 $ copies $ F_{1,2}({\mathbb C}^3)
\subset {\mathcal P_{{{\operatorname{\mathfrak{su}}}}(3)}}^* $ from section \[sec:exhc\] behaves in the manner described above. On the other hand $ {{\operatorname{\textsl{SU}}}}(3) $ acting on the element $ \beta_1 = b_{12}c_{12} + c_{13}b_{13} + b_{23}c_{23} \in
{\mathcal P_{{{\operatorname{\mathfrak{su}}}}(3)}}^* $ gives a copy of the full flag with forms $
{d_{\mathcal P}}\beta_1 $ and $ \beta_1 $ inducing the [<span style="font-variant:small-caps;">nk</span>]{}structure. The associated almost complex structure $ J $ is given by $ J(B_{12}) =
C_{12} $, $ J(C_{13}) = B_{13} $, $ J(B_{23}) = C_{23} $.
To obtain $ \operatorname{{\mathbb C}P}(3) $, we let $ Sp(2) $ act on $ \beta_2=a_1b_{11}
+ b_{12}r + c_{12}q $ in $ {\mathcal P_{{{\operatorname{\mathfrak{sp}}}}(2)}}^* $. The chosen basis for $ {{\operatorname{\mathfrak{sp}}}}(2) $ consists of the $ 10 $ complex matrices $ A_1 =
i(E_{11} - E_{33}) $, $ A_2 = i(E_{22} - E_{44}) $, $ Q = E_{12} -
E_{21} + E_{34} - E_{43} $, $ R = i(E_{12} + E_{21} - E_{34} -
E_{43}) $, $ B_{k\ell } = E_{k,2+\ell } + E_{\ell ,2+k} -
E_{2+k,\ell } - E_{2+\ell ,k} $, $ C_{k\ell } = i(E_{k,2+\ell } +
E_{\ell ,2+k} + E_{2+k,\ell } + E_{2+\ell ,k}) $, $ 1\leqslant
k\leqslant \ell \leqslant 2 $. One easily verifies that $
\operatorname{stab}_{{{\operatorname{\mathfrak{sp}}}}(2)}\beta_1 = {{\operatorname{\mathfrak{u}}}}(1)\oplus{{\operatorname{\mathfrak{su}}}}(2) $, so that, up to discrete coverings, the orbit of $ \beta_2 $ is $ \operatorname{{\mathbb C}P}(3) $. We have $$\begin{gathered}
da_1 = - 2(4b_{11}c_{11}+b_{12}c_{12}+qr),\
db_{11} = 2a_1c_{11}+b_{12}q-c_{12}r,\\
db_{12} = (a_1+a_2)c_{12} + 2(-b_{11}+b_{22})q -
2(c_{11}+c_{22})r,\\
dc_{12} = - (a_1 + a_2)b_{12} + 2(b_{11}+b_{22})r +
2(-c_{11}+c_{22})q,\\
dq = (a_1 - a_2)r + 2(b_{11}-b_{22})b_{12} +
2(c_{11}-c_{22})c_{12},\\
dr = (-a_1 + a_2)q + 2 (c_{11}+c_{22})b_{12} -
2(b_{11}+b_{22})c_{12}.
\end{gathered}$$ Computations now show that $ \beta_2 $ and $ {d_{\mathcal P}}\beta_2 $ determine a [<span style="font-variant:small-caps;">nk</span>]{}structure which has almost complex structure given by $ J(A_1) = \tfrac12 B_{11} $, $ J(B_{12}) = R $ and $ J(C_{12}) =
Q $.
The homogeneous [<span style="font-variant:small-caps;">nk</span>]{}structure on $ S^3\times S^3 $ is obtained on the orbit of $ \beta_3 = e_1f_1 + e_2f_2 + e_3f_3 \in
{\mathcal P_{{{\operatorname{\mathfrak{su}}}}(2)\oplus{{\operatorname{\mathfrak{su}}}}(2)}}^* $. Here $ e_i $, $ f_i $ denotes a cyclic basis for $ {{\operatorname{\mathfrak{su}}}}(2)^*\oplus{{\operatorname{\mathfrak{su}}}}(2)^* $, meaning $ de_i =
e_{i+1}e_{i+2} $ and $ df_i = f_{i+1}f_{i+2} $ for $ i \in {\mathbb Z}/3
$. One may verify that the associated almost complex structure is given by $ J(E_i) = (E_i+2F_i)/\sqrt3 $.
Finally we obtain $ S^6 $ as the $ G_2 $-orbit of the element $
\beta_4 = b_1c_1 + b_3c_3 + c_4b_4 \in {\mathcal P_{{{\operatorname{\mathfrak{g}}}}_2}}^* $. The chosen basis for $ {{\operatorname{\mathfrak{g}}}}_2 $ is $ A_1 = iH_1 $, $ A_2 = iH_2 $, $ B_a =
X_a - Y_a $, $ C_a = i(X_a + Y_a) $, $ (1\leqslant a\leqslant 6)
$, with the elements $ \{H_1,\dots,Y_6\} $ defined in [@Fulton-Harris:rep Table 22.1]. Now we have $$\begin{gathered}
db_1 = (2a_1-a_2)c_1 + b_3b_2 + c_3c_2 + 2(b_4b_3 + c_4c_3) +
b_4b_5 + c_4c_5,\\
dc_1 = (-2a_1+a_2)b_1 + c_3b_2 + c_2b_3 + 2(c_4b_3 + c_3b_4) +
b_4c_5 + b_5c_4,\\
db_3 = (-a_1+a_2)c_3 + b_2b_1 + c_1c_2 + 2(b_1b_4 + c_1c_4) +
b_4b_6 + c_4c_6,\\
dc_3 = (a_1-a_2)b_3 + c_2b_1 + b_2c_1 + b_4c_6 + 2(b_1c_4 +b_4c_1)
+ b_4c_6 + b_6c_4,\\
db_4 = a_1c_4 + 2(b_3b_1 + c_1c_3) + b_5b_1+c_5c_1 + c_6c_3 +
b_6b_3,\\
dc_4 = b_4a_1 + 2(b_1c_3 + b_3c_1) + c_5b_1 + c_1b_5 + c_6b_3 +
c_3b_6,
\end{gathered}$$ and it can be verified that $ \beta_4 $ and $ {d_{\mathcal P}}\beta_4 $ induce a [<span style="font-variant:small-caps;">nk</span>]{}structure on $ S^6 $. In this case one has $ J(B_1) = C_1 $, $ J(B_3) = C_3 $ and $ J(C_4) = B_4 $.
Motivated by these examples, it makes sense to study the multi-moment map formalism for [<span style="font-variant:small-caps;">nk</span>]{}manifolds with less symmetry.
[LCCR]{} G & & [d\_[P]{}]{}& =G\
(3) & b\_[12]{}c\_[12]{}[+]{}c\_[13]{}b\_[13]{}[+]{}b\_[23]{}c\_[23]{} & 3(b\_[12]{}(b\_[13]{}c\_[23]{}[+]{}b\_[23]{}c\_[13]{})[+]{}c\_[12]{}(b\_[13]{}b\_[23]{}[+]{}c\_[13]{}c\_[23]{})) & F\_[1,2]{}([C]{}\^3)\
(2) & a\_1b\_[11]{}[+]{}b\_[12]{}r[+]{}c\_[12]{}q & [-]{}3(a\_1(b\_[12]{}q-c\_[12]{}r)+2b\_[11]{}(b\_[12]{}c\_[12]{}+qr)) & (3)\
(2)\^2 & e\_1f\_1[+]{}e\_2f\_2[+]{}e\_3f\_3 & e\_[12]{}f\_3+e\_[23]{}f\_1+e\_[31]{}f\_2-e\_1f\_[23]{}-e\_2f\_[31]{}-e\_3f\_[12]{} & S\^3S\^3\
G\_2 & b\_1c\_1[+]{}b\_3c\_3[+]{}c\_4b\_4 & 6(b\_1(b\_3c\_4[-]{}c\_3b\_4)[-]{}c\_1(b\_3b\_4+c\_3c\_4)) & S\^6\
Positive gradings of nilpotent algebras {#sec:posgr}
=======================================
A Lie algebra $ {{\operatorname{\mathfrak{k}}}}$ is *positively graded* if there is a vector space direct sum decomposition $ {{\operatorname{\mathfrak{k}}}}={{\operatorname{\mathfrak{k}}}}_1+\dots+{{\operatorname{\mathfrak{k}}}}_r $ with $ [{{\operatorname{\mathfrak{k}}}}_i,{{\operatorname{\mathfrak{k}}}}_j]\subseteq{{\operatorname{\mathfrak{k}}}}_{i+j} $ for all $ i $, $ j $. A grading of an $ n $-dimensional Lie algebra $ {{\operatorname{\mathfrak{k}}}}$ may be specified in terms of an $ n $ positive integers, see Example \[ex:posgrad\]. We have
Any nilpotent Lie algebra of dimension at most six admits a positive grading. From dimension seven and above, there are nilpotent Lie algebras, which do not admit a positive grading.
The nilpotent Lie algebras of dimension at most six and primitive positive gradings are listed in Table \[tab:posgrad\]. Example \[ex:posgrad\] illustrates how gradings are found. In [@Madsen-S:multi-moment] we show that there are examples of nilpotent algebras $ {{\operatorname{\mathfrak{k}}}}$ of dimension seven that can not arise as the derived algebra of any $ (2,3) $-trivial algebra $ {{\operatorname{\mathfrak{g}}}}$. It follows that such $ {{\operatorname{\mathfrak{k}}}}$ can not admit a positive grading.
Positive gradings can be used to generate $ (2,3) $-trivial algebras:
\[cor:extposgr\] Each of the $ 50 $ Lie algebras listed in Table \[tab:posgrad\] is the derived algebra of a completely solvable $ (2,3) $-trivial Lie algebra.
Let $ {{\operatorname{\mathfrak{g}}}}=\Span{A}+{{\operatorname{\mathfrak{k}}}}$, where $ {{\operatorname{\mathfrak{k}}}}$ is one of the algebras of Table \[tab:posgrad\] and $ \operatorname{ad}_A $ acts as multiplication by $
i $ on $ {{\operatorname{\mathfrak{k}}}}_i $. Then $ {{\operatorname{\mathfrak{g}}}}$ is a solvable algebra. Moreover $ (\Lambda^s{{\operatorname{\mathfrak{k}}}})^{{{\operatorname{\mathfrak{g}}}}}=\{0\} $ for $ s\geqslant1 $, so that $
H^1({{\operatorname{\mathfrak{k}}}})^{{{\operatorname{\mathfrak{g}}}}}=\{0\}=H^2({{\operatorname{\mathfrak{k}}}})^{{{\operatorname{\mathfrak{g}}}}}=H^3({{\operatorname{\mathfrak{k}}}})^{{{\operatorname{\mathfrak{g}}}}} $. Thus, by Theorem \[thm:2-3-trivial\], $ {{\operatorname{\mathfrak{g}}}}$ is $ (2,3) $-trivial. Since $ \operatorname{ad}_X $ has real eigenvalues for each $ X\in{{\operatorname{\mathfrak{g}}}}$ the Lie algebra is completely solvable.
\[ex:posgrad\] A Lie algebra $ {{\operatorname{\mathfrak{k}}}}$ may be specified in terms of the action of $ d\colon {{\operatorname{\mathfrak{k}}}}^*\to \Lambda^2{{\operatorname{\mathfrak{k}}}}^* $ on a basis for $ {{\operatorname{\mathfrak{k}}}}^* $. By $ (0^2,12,13,14{+}23,24{+}15) $ we thus denote the nilpotent Lie algebra $ {{\operatorname{\mathfrak{k}}}}$ which has a basis $ e^1,\dots,e^6 $ for $
{{\operatorname{\mathfrak{k}}}}^* $ satisfying $ de^1=0=de^2 $, $ de^3=e^1e^2 $, …, $
de^6 = e^2e^4 + e^1e^5 $. Assigning weights can now be formulated schematically as follows: $ e^1\to a $, $ e^2\to b $, $ e^3\to
a+b $, $ e^4\to2a+b $, $ e^5\to 3a+b=a+2b $, $
e^6\to2(a+b)=2(a+b) $. In particular we see that $ 2a=b $, so that a grading may be defined by $ {{\operatorname{\mathfrak{k}}}}={{\operatorname{\mathfrak{k}}}}_1\oplus\dots\oplus{{\operatorname{\mathfrak{k}}}}_6
$, where $ {{\operatorname{\mathfrak{k}}}}_i=\Span{e_i} $. Choosing $ a=1 $, this weight decomposition is denoted by $ 123456 $. Following the proof of Corollary \[cor:extposgr\], we may now define a $ (2,3)
$-trivial extension of $ {{\operatorname{\mathfrak{k}}}}$: $
(0,12,2.13,3.14{+}23,4.15{+}24,5.16{+}25{+}34,6.17{+}24{+}26) $.
[L@R]{} &\
(0) & 1\
(0\^2) & 1\^2\
(0\^3) & 1\^3\
(0\^2,12) & 1\^22\
(0\^4) & 1\^4\
(0\^3,12) & 1\^32\
(0\^2,12,13) & 1\^223\
(0\^5) & 1\^5\
(0\^4,12),(0\^4,12[+]{}34) & 1\^42\
(0\^3,12,13) & 1\^32\^2\
(0\^3,12,14) & 1\^323\
(0\^3,12,13[+]{}24) & 1\^22\^23\
(0\^2,12,13,23) & 1\^223\^2\
(0\^2,12,13,14) & 1\^2234\
(0\^2,12,13,14[+]{}23) & 12345\
(0\^6) & 1\^6\
(0\^5,12),(0\^5,12+34) & 1\^52\
[L@R]{} &\
(0\^4,12,13),(0\^4,13[+]{}42,14[+]{}23), &\
(0\^4,12,34),(0\^4,12,14[+]{}23) & 1\^42\^2\
(0\^4,12,15) & 1\^423\
(0\^3,12,13,23) & 1\^32\^3\
(0\^4,12,14[+]{}25),(0\^4,12,15[+]{}34), &\
(0\^3,12,13,14),(0\^3,12,23,1435), &\
(0\^3,12,13,24),(0\^3,12,13,14[+]{}35) & 1\^32\^23\
(0\^3,12,14,24) & 1\^323\^2\
(0\^3,12,14,15) & 1\^3234\
(0\^3,12,13[+]{}14,24),(0\^3,12,13[+]{}42,14[+]{}23) &\
(0\^3,12,13,14[+]{}23),(0\^3,12,14,13[+]{}42) & 1\^22\^23\^2\
(0\^3,12,14[-]{}23,15[+]{}34) & 1\^22\^234\
(0\^2,12,13,23,1425),(0\^2,12,13,23,14) & 1\^223\^24\
(0\^2,12,13,14,15),(0\^2,12,13,14,34[+]{}52) & 1\^22345\
(0\^3,12,14,15[+]{}23) & 1\^23234\
(0\^3,12,14,15[+]{}24) & 121345\
(0\^3,12,14,15[+]{}23[+]{}24) & 123\^245\
(0\^2,12,13,14+23,24+15) & 123456\
(0\^2,12,13,14[+]{}23,34[+]{}52) & 123457\
(0\^2,12,13,14,23[+]{}15) & 134567\
Families of (2,3)-trivial algebras {#sec:clf23}
==================================
While the method of positive gradings provides an effective tool for constructing examples of $(2,3)$-trivial algebras, it is inadequate if one aims for a general understanding of the $(2,3)$-trivial class. Therefore we now turn to give a complete list of such algebras in dimensions up to and including five.
In dimension one, the only Lie algebra is Abelian and is automatically $ (2,3) $-trivial. In dimension two a Lie algebra is either Abelian or isomorphic to the $ (2,3) $-trivial algebra $ (0,21) $. These first two examples are uninteresting from the point of view of multi-moment maps since they have $ {\mathcal P_{{{\operatorname{\mathfrak{g}}}}}}=\{0\} $. In next dimensions we have:
\[prop:solb345\] The $ (2,3) $-trivial Lie algebras in dimensions three, four and five are listed in the Tables \[tab:34sp\] and \[tab:5sp\].
We shall now give a proof of Proposition \[prop:solb345\]. Note that we do not discuss inequivalence of the algebras; imposing inequivalence would put further restrictions on the parameters, see for instance [@Andrada-BDO:four Theorem 1.1, 1.5].
[LCR]{} \_3 & (0,21[+]{}31,31) &\
\_[3,]{} & (0,21,.31) & 1,0\
’\_[3,]{} & (0,.21 [+]{}31,[-]{}21[+]{}.31) & 0\
\_4 & (0,21[+]{}31,31[+]{}41,41) &\
\_[4,]{} & (0,21,.31[+]{}41,.41) & 1,[-]{}12,0\
\_[4,(2)]{} & (0,21,\_1.31,\_2.41) & \_i,\_1[+]{}\_21,0\
’\_[4,(2)]{} & (0,\_1.21,\_2.31[+]{}41,-31[+]{}\_2.41) & \_10,\_2 ,0\
\_[4,]{} & (0,21,.31,(1[+]{}).41[+]{}32) & 2,[-]{}1,[-]{}12,0\
’\_[4,]{} & (0,.21[+]{}31,[-]{}21[+]{}.31,2.41[+]{}32) & 0\
\_4 & (0,21[+]{}31,31,2.41[+]{}32) &\
[LLL]{} \_5 & (0,21[+]{}31,31[+]{}41,41[+]{}51,51) &\
\_[5(1),]{} & (0,21,.31[+]{}41,.41[+]{}51,.51) & 1,[-]{}12,0\
\_[5(2),]{} & (0,21[+]{}31,31,.41[+]{}51,.51) & -2,-1,-12,0\
\_[5,(2)]{} & (0,21,\_1.31,\_2.41[+]{}51,\_2.51) & \_i-1,0 ; \_1[+]{}\_20,[-]{}1 ;\
& & 1[+]{}2\_2,\_1[+]{}2\_20\
\_[5,(3)]{} & (0,21,\_1.31,\_2.41,\_3.51) & \_i-1,0 ; \_1[+]{}\_2[+]{}\_30;\
&& \_i[+]{}\_j1,0 (ij)\
’\_[5,(2)]{} & (0,\_1.21[+]{}31,\_1.31,\_2.41[+]{}51,-41[+]{}\_2.51) & \_i,\_1[+]{}2\_20\
’\_[5,(3)]{} & (0,\_1.21,\_2.31,\_3.41[+]{}51,-41[+]{}\_3.51) & \_i 0 ; \_1 [-]{}\_2; \_1,\_2 [-]{} 2\_3\
”\_[5,]{} & (0,.21[+]{}31[+]{}41,[-]{}21[+]{}.31[+]{}51, .41[+]{}51,[-]{}41[+]{}.51) & 0\
”\_[5,(3)]{} & (0,\_1.21[+]{}31,[-]{}21[+]{}\_1.31, \_2.41[+]{}\_3.51, [-]{}\_3.41[+]{}\_2.51) & \_i0\
\_[5(1)]{} & (0,21,21[+]{}31,31[+]{}41,2.51[+]{}32) &\
\^\_[5(2)]{} & (0,21,21[+]{}31,2.41,2.5141[+]{}32) &\
\_[5(1),]{} & (0,21,.31,(1[+]{}).41,(1[+]{}).51[+]{}32[+]{}41) & 2,[-]{}32,[-]{}1,[-]{}23,[-]{}12,0\
\_[5(2),]{} & (0,21,21[+]{}31,.41,2.51[+]{}32) & -3,-1,0\
\_[5,(2)]{} & (0,21,\_1.31,\_2.41,(1[+]{}\_1).51[+]{}32) & \_12,[-]{}12,[-]{}1,0 ; \_20,[-]{}1 ;\
& & \_1[+]{}\_22,0 ; \_2[+]{}2\_11\
\_[5(3),]{} & (0,.21,31,31[+]{}41,(1[+]{}).51[+]{}32) & 3,[-]{}2,[-]{}1,[-]{}12,0\
’\^\_[5,]{} & (0,.21[+]{}31,-21[+]{}.31,2.41,2.5141[+]{}32) & 0\
’\_[5,(2)]{} & (0,\_1.21[+]{}31,-21[+]{}\_1.31,\_2.41,2\_1.51) & \_1,\_20\
\_5 & (0,21,21[+]{}31,2.41[+]{}32,3.51[+]{}42) &\
\_[5,]{} & (0,21,.31,(1[+]{}).41[+]{}32,(2[+]{}).51[+]{}42) & 3,[-]{}2,[-]{}1,[-]{}12,0\
The starting point for our analysis is Theorem \[thm:2-3-trivial\] which gives $ {{\operatorname{\mathfrak{g}}}}={\mathbb R}A+{{\operatorname{\mathfrak{k}}}}$, where $ {{\operatorname{\mathfrak{k}}}}={{\operatorname{\mathfrak{g}}}}' $ is nilpotent. The element $ A $ acts on $ H^i({{\operatorname{\mathfrak{k}}}}) $ as endomorphism with determinant $ a_i $. Now $ (2,3) $-triviality of $ {{\operatorname{\mathfrak{g}}}}$ may be rephrased as the non-vanishing of $ a_1 $, $ a_2 $ and $ a_3 $.
#### Dimension three
Let $ {{\operatorname{\mathfrak{g}}}}$ be a $ (2,3) $-trivial algebra of dimension three. Then $ {{\operatorname{\mathfrak{k}}}}$ is nilpotent and two-dimensional, so $ {{\operatorname{\mathfrak{k}}}}\cong{\mathbb R}^2
$. The element $ A $ acts on $ {\mathbb R}^2 $ invertibly and the induced action on $ H^2({\mathbb R}^2) \cong \Lambda^2{\mathbb R}^2 \cong {\mathbb R}$ is also invertible. So either $ A $ is diagonalisable over $ {\mathbb C}$ with non-zero eigenvalues whose sum is non-zero, giving cases $
{{\operatorname{\mathfrak{r}}}}_{3,\lambda\neq-1,0} $ and $ {{\operatorname{\mathfrak{r}}}}'_{3,\lambda\neq0} $, or $ A $ acts with Jordan normal form $ \left(\begin{smallmatrix}
\lambda&1\\0&\lambda \end{smallmatrix}\right) $, $ \lambda\ne0 $, giving case $ {{\operatorname{\mathfrak{r}}}}_3 $.
#### Dimension four
For $ {{\operatorname{\mathfrak{g}}}}$ of dimension four we have $ {{\operatorname{\mathfrak{k}}}}\cong {\mathbb R}^3 $ or the Heisenberg algebra $ {{\operatorname{\mathfrak{h}}}}_3 = (0^2,21) $. The former gives the algebras from the $ {{\operatorname{\mathfrak{r}}}}$- and $ {{\operatorname{\mathfrak{r}}}}' $-series. Derivations of $
{\mathbb R}^3 $ are just linear endomorphisms; therefore the relevant list of extensions of $ {\mathbb R}^3 $ may be obtained from considerations of invertible $ 3\times 3 $ matrices in normal form: $ A_1 =
\left(\begin{smallmatrix}
1&0&0\\0&\lambda_1&0\\0&0&\lambda_2
\end{smallmatrix}\right)
$, $ A_2 =
\left(\begin{smallmatrix}
1&0&0\\0&\lambda&1\\0&0&\lambda
\end{smallmatrix}\right)
$, $ A_3 =
\left(\begin{smallmatrix}
1&1&0\\0&1&1\\0&0&1
\end{smallmatrix}\right)
$ and $ A_4 =
\left(\begin{smallmatrix}
\lambda_1&0&0\\0&\lambda_2&1\\0&-1&\lambda_2
\end{smallmatrix}\right)
$. The element $ A_1 $ gives the family $ {{\operatorname{\mathfrak{r}}}}_{4,\lambda(2)} $, and the induced action on $ H^1({\mathbb R}^3)\cong{\mathbb R}^3 $ is, up to sign, multiplication by the transpose of $ A_1 $. Using this observation one easily finds the induced actions on $ H^2({\mathbb R}^3)\cong{\mathbb R}^3 $ and $ H^3({\mathbb R}^3)\cong{\mathbb R}$. We deduce that $ (2,3) $-triviality holds if and only if the determinants $ a_1 = \lambda_1\lambda_2 $, $ a_2
= (1 + \lambda_1)(1 + \lambda_2)(\lambda_1 + \lambda_2) $ and $ a_3
= 1 + \lambda_1 + \lambda_2 $ do not vanish. The matrix $ A_2 $ gives us the algebra $ {{\operatorname{\mathfrak{r}}}}_{4,\lambda} $. In this case $ a_1 =
\lambda^2 $, $ a_2 = 2\lambda(1 + \lambda)^2 $, $ a_3 = 1 +
2\lambda $, giving the restrictions on parameters in Table \[tab:34sp\]. The algebra $ {{\operatorname{\mathfrak{r}}}}_4 $ corresponds to $ A_3
$. Finally, $ A_4 $ occurs when the action has $ 2 $ complex eigenvalues. The corresponding family is $ {{\operatorname{\mathfrak{r}}}}'_{4,\lambda(2)} $, where $ \lambda_1,\lambda_2 $ are restricted by $ a_i\ne0 $ for $
a_1 = \lambda_1(1 + \lambda_2^2) $, $ a_2 = 2\lambda_2(1 +
(\lambda_1 + \lambda_2)^2) $ and $ a_3 = \lambda_1 + 2\lambda_2 $.
The Heisenberg algebra $ {{\operatorname{\mathfrak{h}}}}_3 $ has $ H^1({{\operatorname{\mathfrak{h}}}}_3) \cong
\Span{e^1,e^2} $, $ H^2({{\operatorname{\mathfrak{h}}}}_3) \cong \Span{e^{13},e^{23}} $, $
H^3({{\operatorname{\mathfrak{h}}}}_3) \cong \Span{e^{123}} $. The action of $ A $, being a derivation, is represented by a matrix of the form $ \left(\begin{smallmatrix}
B&0\\\underline{b}&\operatorname{tr}{B} \end{smallmatrix}\right) $ with $ B $ a real $ 2\times 2 $-matrix and $ \underline{b} = (b_1,b_2)\in{\mathbb R}^2
$. To see this, write $ \operatorname{ad}_A(e_i) = \sum_kb^k_ie_k $ and consider the relation $ \operatorname{ad}_A(e_3) = \operatorname{ad}_A[e_1,e_2] = [\operatorname{ad}_A(e_1),e_2] +
[e_1,\operatorname{ad}_A(e_2)] $. After the transformation $ A\to A-b_2 e_1+b_1
e_2 $ we may assume $ \underline{b} = 0 $, so that the algebras are distinguished by the normal form of $ B $. The family $
{{\operatorname{\mathfrak{d}}}}_{4,\lambda} $ arises when $ B = \operatorname{diag}(1,\lambda) $. Restrictions on $ \lambda $ now follow from the determinants $ a_1
= \lambda $, $ a_2 = (2+\lambda)(1+2\lambda) $ and $ a_3 =
2(1+\lambda) $ being non-zero. If $ B = \left(\begin{smallmatrix}
1&1\\0&1 \end{smallmatrix}\right) $ one has the algebra $ {{\operatorname{\mathfrak{h}}}}_4 $. Finally the action may have complex eigenvalues so that $ B
= \left(\begin{smallmatrix} \lambda&1\\-1&\lambda \end{smallmatrix}\right) $, corresponding to the family $ {{\operatorname{\mathfrak{d}}}}'_{4,\lambda} $. One finds $ a_1
= 1+\lambda^2 $, $ a_2 = 1+9\lambda^2 $ and $ a_3 = 4\lambda $, so we must have $ \lambda\neq0 $.
#### Dimension five
A five-dimensional $ (2,3) $-trivial Lie algebra has $
{{\operatorname{\mathfrak{k}}}}\cong{\mathbb R}^4 $, $ (0^3,21) $ or $ (0^2,21,31) $. In the Abelian case $ H^1({\mathbb R}^4)\cong{\mathbb R}^4 $, $ H^2({\mathbb R}^4)\cong{\mathbb R}^6 $, $
H^3({\mathbb R}^4)\cong{\mathbb R}^4 $. The solvable extensions are found by taking invertible matrices in the normal forms $ A_1,\dots,A_9 $: $$\begin{gathered}
\left(\begin{smallmatrix}
1&0&0&0\\
0&\lambda_1&0&0\\
0&0&\lambda_2&0\\
0&0&0&\lambda_3
\end{smallmatrix}\right)
,\quad
\left(\begin{smallmatrix}
1&0&0&0\\
0&\lambda_1&0&0\\
0&0&\lambda_2&1\\
0&0&0&\lambda_2
\end{smallmatrix}\right)
,\quad
\left(\begin{smallmatrix}
1&1&0&0\\
0&1&0&0\\
0&0&\lambda&1\\
0&0&0&\lambda
\end{smallmatrix}\right)
,\quad
\left(\begin{smallmatrix}
1&0&0&0\\
0&\lambda&1&0\\
0&0&\lambda&1\\
0&0&0&\lambda
\end{smallmatrix}\right)
,\quad
\left(\begin{smallmatrix}
1&1&0&0\\
0&1&1&0\\
0&0&1&1\\
0&0&0&1
\end{smallmatrix}\right),\\
\left(\begin{smallmatrix}
\lambda_1&0&0&0\\
0&\lambda_2&0&0\\
0&0&\lambda_3&1\\
0&0&-1&\lambda_3
\end{smallmatrix}\right)
,\quad
\left(\begin{smallmatrix}
\lambda_1&1&0&0\\
0&\lambda_1&0&0\\
0&0&\lambda_2&1\\
0&0&-1&\lambda_2
\end{smallmatrix}\right)
,\quad
\left(\begin{smallmatrix}
\lambda_1&1&0&0\\
-1&\lambda_1 &0&0\\
0&0&\lambda_2&\lambda_3\\
0&0&-\lambda_3&\lambda_2
\end{smallmatrix}\right)
,\quad
\left(\begin{smallmatrix}
\lambda&1&1&0\\
-1&\lambda&0&1\\
0&0&\lambda&1\\
0&0&-1&\lambda
\end{smallmatrix}\right).\end{gathered}$$ The matrix $ A_1 $ gives the family $ {{\operatorname{\mathfrak{r}}}}_{5,\lambda(3)} $ with restrictions on $ \lambda_i $ following from non-vanishing of $ a_1
= \lambda_1\lambda_2\lambda_3 $, $ a_2 = \prod_i
(1+\lambda_i)\prod_{i<j}(\lambda_i + \lambda_j) $, $ a_3 =
(\lambda_1+\lambda_2+\lambda_3)\prod_{i<j}(1 + \lambda_j + \lambda_k)
$. The form $ A_2 $ corresponds to the family $
{{\operatorname{\mathfrak{r}}}}_{5,\lambda(2)} $ and the determinants $ a_1 =
\lambda_1\lambda_2^2 $, $ a_2 = 2\lambda_2 (1+\lambda_1)
(1+\lambda_2)^2 (\lambda_1+\lambda_2)^2 $, and $ a_3 =
(1+\lambda_1+\lambda_2)^2 (1+2\lambda_2) (\lambda_1+2\lambda_2) $ should be non-zero. From $ A_3 $ we obtain the family $
{{\operatorname{\mathfrak{r}}}}_{5(2),\lambda} $ with parameter value constrained by $ a_1 =
\lambda^2 $, $ a_2 = 4\lambda(1+\lambda)^4 $, $ a_3 =
(1+2\lambda)^2(2+\lambda)^2 $ being non-zero. The matrix $ A_4 $ corresponds to $ {{\operatorname{\mathfrak{r}}}}_{5(1),\lambda} $ with $ \lambda $ constrained by $ a_i\ne0 $ for $ a_1 = \lambda^3 $, $ a_2 =
8\lambda^3(1+\lambda)^3 $, $ a_3 = 3\lambda(1+2\lambda)^3 $. The algebra $ {{\operatorname{\mathfrak{r}}}}_5 $ is from $ A_5 $. Members of the $ {{\operatorname{\mathfrak{r}}}}' $- and $ {{\operatorname{\mathfrak{r}}}}'' $-series occur when $ \operatorname{ad}_A $ has $ 2 $ or $ 4 $ complex eigenvalues. The algebra $ {{\operatorname{\mathfrak{r}}}}'_{5,\lambda(3)} $ corresponds to $ A_6 $; the conditions are that $ a_1 =
\lambda_1\lambda_2(1 + \lambda_3^2) $, $ a_2 = 2\lambda_3(\lambda_1
+ \lambda_2)(1 + (\lambda_1 + \lambda_3)^2)(1 + (\lambda_2 +
\lambda_3)^2) $ and $ a_3 = (\lambda_1 + 2\lambda_3)(\lambda_2 +
2\lambda_3)(1 + (\lambda_1 + \lambda_2 + \lambda_3)^2) $ are non-zero. The form $ A_7 $ gives the family $ {{\operatorname{\mathfrak{r}}}}'_{5,\lambda(2)}
$ with $ \lambda_1 $, $ \lambda_2 $ constrained by $ a_1 =
\lambda_1^2(1+\lambda_2^2) $, $ a_2 =
4\lambda_1\lambda_2(1+(\lambda_1+\lambda_2)^2)^2 $, $ a_3 =
(\lambda_1+2\lambda_2)^2(1+(2\lambda_1+\lambda_2)^2) $ being non-zero. The matrix $ A_8 $ has $ \lambda_3\ne0 $ and corresponds to the family $ {{\operatorname{\mathfrak{r}}}}''_{5,\lambda(3)} $; restrictions on parameter values follow from non-zero values for $ a_1 =
(1+\lambda_1^2) (\lambda_2^2+\lambda_3^2) $, $ a_2 =
4\lambda_1\lambda_2 ((\lambda_1+\lambda_2)^2 + (1+\lambda_3)^2)
((\lambda_1+\lambda_2)^2 + (1-\lambda_3)^2) $ and $ a_3 =
(\lambda_3^2 + (2\lambda_1+\lambda_2)^2) (1 +
(\lambda_1+2\lambda_2)^2) $. Finally $ A_9 $ gives the algebra $
{{\operatorname{\mathfrak{r}}}}''_{5,\lambda} $. The determinants $ a_1 = (1+\lambda^2)^2 $, $ a_2 = 64\lambda^4(1+\lambda^2) $ and $ a_3 = (1+9\lambda^2)^2 $ must be non-zero.
To analyse the cases $ (0^3,21) $ and $ (0^2,21,31) $ we follow and modify arguments given in [@Mubarakzjanov:lie5]. First consider $ {{\operatorname{\mathfrak{k}}}}\cong (0^3,21) $ which has $ H^1({{\operatorname{\mathfrak{k}}}}) \cong
\Span{e^1,e^2,e^3} $, $ H^2({{\operatorname{\mathfrak{k}}}}) \cong
\Span{e^{13},e^{14},e^{23},e^{24}} $ and $ H^3 \cong
\Span{e^{124},e^{134},e^{234}} $. Write $ A(e_i) = a^k_ie_k $ for $ i = 1,2,3,4 $. From the relations $ \operatorname{ad}_A(e_4) =
[\operatorname{ad}_A(e_1),e_2] + [e_1,\operatorname{ad}_A(e_2)] $, $ 0 = \operatorname{ad}_A[e_i,e_3] =
[\operatorname{ad}_A(e_i),e_3] + [e_i,\operatorname{ad}_A(e_3)] $, $ i=1,2 $, we deduce that $
a^4_4 = a^1_1+a^2_2 $ and $ a^1_4 = 0 = a^2_4 = a^3_4 = a^2_3 =
a^1_3 $. After the transformation $ A\to A-a^4_2e_1+a^4_1e_2 $ we may assume $ a^4_1 = a^4_2 = 0 $. The restriction $ B = (b^k_i) $ of $ \operatorname{ad}_A $ to the subspace $ \Span{e_1,e_2,e_3} $ has $ b^1_3 =
0 = b^2_3 $. This may be transformed to Jordan form via $ e_1\to
ae_1+be_2+ce_3 $, $ e_2\to pe_1+qe_2+re_3 $, $ e_3\to se_3 $ with $ aq-bp\ne0 $ and $ s\ne0 $. Excluding degenerate matrices, we may therefore take $ B = B_i $ to be one of: $
\left(\begin{smallmatrix}
1&0&0\\0&\lambda_1&0\\0&0&\lambda_2
\end{smallmatrix}\right)
$, $
\left(\begin{smallmatrix}
1&0&0\\1&1&0\\0&1&1
\end{smallmatrix}\right)
$, $
\left(\begin{smallmatrix}
1&0&0\\1&1&0\\0&0&\lambda
\end{smallmatrix}\right)
$, $
\left(\begin{smallmatrix}
\lambda_1&1&0&\\-1&\lambda_1&0\\0&0&\lambda_2
\end{smallmatrix}\right)
$, $
\left(\begin{smallmatrix}
\lambda&0&0&\\0&1&0\\0&1&1
\end{smallmatrix}\right)
$. Consider first the case $ B_1 = \operatorname{diag}(1,\lambda_1,\lambda_2) $. If $ \lambda_2\ne1+\lambda_1 $ we may assume, making a change $
e_3\to e_3+\alpha e_4 $ if necessary, that $ a^4_3 = 0 $. This gives us the family $ {{\operatorname{\mathfrak{d}}}}_{5,\lambda(2)} $. The determinants $ a_1
= \lambda_1\lambda_2 $, $ a_2 =
(1+\lambda_2)(2+\lambda_1)(\lambda_1+\lambda_2)(1+2\lambda_1) $, and $ a_3 = 2(1+\lambda_1)(2+\lambda_2+\lambda_1)(1+2\lambda_1+\lambda_2)
$ must be non-zero. Turning next to the case $ \lambda_2 =
1+\lambda_1 $, let us assume $ a_3^4\ne0 $, otherwise we get a member of the family $ {{\operatorname{\mathfrak{d}}}}_{5,\lambda(2)} $. After rescaling $
e_i\to \abs{a^4_3}^{1/2}e_i $, for $ i = 1,2 $, $ e_4\to
\abs{a^4_3}e_4 $, we obtain the families $ {{\operatorname{\mathfrak{d}}}}^{\pm}_{5(1),\lambda}
$ given by $ (0,21,\lambda.31,(1 + \lambda).41, (1 +
\lambda).51+32\pm41) $. Scaling $ (e_1,e_4,e_5) $ by factors $
(\lambda,\lambda^{-1},-1) $ and interchanging $ e_2 $ and $ e_3 $ gives $ {{\operatorname{\mathfrak{d}}}}^+_{5(1),\lambda} \cong {{\operatorname{\mathfrak{d}}}}^-_{5(1),1/\lambda} $, so there is only one family $ {{\operatorname{\mathfrak{d}}}}_{5(1),\lambda}: = {{\operatorname{\mathfrak{d}}}}^+_{5(1),\lambda}
$. Restrictions on $ \lambda $ follow from non-vanishing of $ a_1
= \lambda(1+\lambda) $, $ a_2 = (2+\lambda)^2(1+2\lambda)^2 $ and $ a_3 = 2(1+\lambda)(3+2\lambda)(2+3\lambda) $. For the matrix $
B_2 $ we may assume that $ a^4_3 = 0 $, so that we have the $
(2,3) $-trivial algebra $ {{\operatorname{\mathfrak{d}}}}_{5} $. The algebra $
{{\operatorname{\mathfrak{d}}}}_{5(2),\lambda} $ corresponds to the matrix $ B_3 $ with $
a^4_3 = 0 $. The following determinants $ a_1 = \lambda $, $ a_2
= 9(1+\lambda)^2 $, and $ a_3 = 4(3+\lambda)^2 $ must be non-zero. For $ B_3 $ with $ a_3^4\ne0 $ we obtain the algebra $
{{\operatorname{\mathfrak{d}}}}_{5(2)}^{\pm} $; this requires a rescaling $ e_i\to
\abs{a^4_3}^{1/2}e_i $, for $ i = 1,2 $, $ e_4\to \abs{a^4_3}e_4
$. From $ B_4 $ we obtain $ {{\operatorname{\mathfrak{d}}}}'_{5,\lambda(2)} $ when $ a_3^4
= 0 $. The requirement that $ a_1 = \lambda_2(1+\lambda_1^2) $, $
a_2 = (1+9\lambda_1^2)(1+(\lambda_1+\lambda_2)^2) $, $ a_3 =
4\lambda_1(1+(3\lambda_1+\lambda_2)^2) $ are non-zero enforces restrictions on the $ \lambda_i $’s. When $ a_3^4\ne0 $ we find, after appropriate rescaling, that $ B_4 $ corresponds to the family $ {{\operatorname{\mathfrak{d}}}}'^{\pm}_{5,\lambda} $. The determinants $ a_1 =
2\lambda(1+\lambda^2) $, $ a_2 = (1+9\lambda^2)^2 $ and $ a_3 =
4\lambda(1+25\lambda^2) $ must be non-zero. For the matrix $ B_5 $ we must have $ a_3^4 = 0 $ and so we get the family $
{{\operatorname{\mathfrak{d}}}}_{5(3),\lambda} $. The allowed values for $ \lambda $ are deduced from the determinants $ a_1 = \lambda $, $ a_2 =
2(1+\lambda)(1+2\lambda)(2+\lambda) $, $ a_3 =
4(1+\lambda)^2(3+\lambda) $ being non-zero.
Finally, for $ {{\operatorname{\mathfrak{k}}}}\cong(0^2,21,31) $ we have $ H^1({{\operatorname{\mathfrak{k}}}}) =
\Span{e^1,e^2} $, $ H^2({{\operatorname{\mathfrak{k}}}})\cong\Span{e^{14},e^{23}} $, $
H^3({{\operatorname{\mathfrak{k}}}})\cong\Span{e^{134},e^{234}} $. As above, write $ A(e_i) =
a^k_ie_k $. Considering the relations $ 0 = \operatorname{ad}_A[e_2,e_3] =
[\operatorname{ad}_A(e_2),e_3] + [e_2,\operatorname{ad}_A(e_3)] $, $ \operatorname{ad}_A(e_3) =
[\operatorname{ad}_A(e_1),e_2] + [e_1,\operatorname{ad}_A(e_2)] $ and $ \operatorname{ad}_A(e_4) =
[\operatorname{ad}_A(e_1),e_3] + [e_1,\operatorname{ad}_A(e_3)] $, one finds $ a^1_2 = a^1_3 =
a^1_4 = a^2_3 = a^2_4 = a^3_4 = 0 $, $ a^3_2 = a^4_3 $ and $ a^4_4
= a^1_1 + a^3_3 $, $ a^3_3 = a^1_1 + a^2_2 $. After the transformation $ A\to A-a^3_2e_1+a^3_1e_2+a^4_1e_3 $ we may assume that $ \operatorname{ad}_A $ takes the form $ \operatorname{diag}(p,q,p+q,2p+q) + A' $, where $ A' $ only has non-zero entries $ {a'}^2_1 = a^2_1 $ and $
{a'}^4_2 = a^4_2 $, below the diagonal. We then obtain $
{{\operatorname{\mathfrak{p}}}}_{5,\lambda} $ and $ {{\operatorname{\mathfrak{p}}}}_5 $ as follows. As $ {{\operatorname{\mathfrak{k}}}}= {{\operatorname{\mathfrak{g}}}}' $ one has $ p \ne 0 $ and we may to rescale $ \operatorname{ad}_A $ by $ 1/p $. If $ q\ne p $ we may consider the transformation $ e_1\to e_1+ {a^2_1
e_2}/(p-q) $. After appropriate transformations $ e_1\to e_1+ae_4
$, $ e_2\to e_2+be_4 $, we obtain the algebra $ {{\operatorname{\mathfrak{p}}}}_{5,\lambda} $ with $ \lambda = q/p $. For this family we have determinants $ a_1
= \lambda $, $ a_2 = (1+2\lambda)(3+\lambda) $ and $ a_3 =
6(1+\lambda)(2+\lambda) $, so that $ a_i\ne0 $ enforces $ \lambda
$ to be as specified in Table \[tab:5sp\]. Consider now $ q = p
$ and note that we may assume $ a^2_1\ne0 $, since otherwise we end with $ {{\operatorname{\mathfrak{p}}}}_{5,\lambda} $. Making a change $ e_i\to a^2_1 e_i $ for $ i = 2,3,4 $ and then transforming $ e_2\to e_2+c e_4 $, we get the algebra $ {{\operatorname{\mathfrak{p}}}}_5 $.
This concludes the proof of Proposition \[prop:solb345\].
#### Unimodular
The lists of $(2,3)$-trivial algebras in dimensions up to and including five reveal that algebraic properties of this class are not fully reflected in low-dimensional examples. A Lie algebra $ {{\operatorname{\mathfrak{g}}}}$ is called *unimodular* if the homomorphism $ \chi\colon {{\operatorname{\mathfrak{g}}}}\to{\mathbb R}$ given by $ \chi(x) = \operatorname{tr}(\operatorname{ad}(x)) $ has trivial image. By direct inspection we observe that the $ (2,3) $-trivial Lie algebras of dimensions two, three and four are not unimodular. On the other hand there are infinitely many five-dimensional algebras with this property:
The unimodular $ (2,3) $-trivial Lie algebras of dimension up to and including five are $ {\mathbb R}$, $ {{\operatorname{\mathfrak{r}}}}_{5(1),-1/3} $, $
{{\operatorname{\mathfrak{r}}}}_{5,\lambda,-(1+\lambda)/2} $, $
{{\operatorname{\mathfrak{r}}}}_{5,\lambda,\mu,-(1+\lambda+\mu)} $, $
{{\operatorname{\mathfrak{r}}}}'_{5,\lambda,-\lambda} $, $ {{\operatorname{\mathfrak{r}}}}''_{5,\lambda,-\lambda,\mu} $, $ {{\operatorname{\mathfrak{r}}}}'_{5,\lambda,\mu,-(\lambda+\mu)/2} $, $ {{\operatorname{\mathfrak{d}}}}_{5(2),-4} $, $
{{\operatorname{\mathfrak{d}}}}_{5,\lambda,-2(1+\lambda)} $, $ {{\operatorname{\mathfrak{d}}}}_{5(3),-3/2} $, $
{{\operatorname{\mathfrak{d}}}}'_{5,\lambda,-4\lambda} $ and $ {{\operatorname{\mathfrak{p}}}}_{5,-4/3} $, where parameters satisfy the conditions in Table \[tab:5sp\].
#### Higher dimensions
The quest for higher-dimensional examples is easily met. Indeed, one may construct infinite families of $ (2,3) $-trivial Lie algebras following the methods invoked in the proof of Proposition \[prop:solb345\]. In fact all the families appearing in dimension five have higher dimensional generalisations. Let us show how to obtain the following examples:
$ {{\operatorname{\mathfrak{r}}}}_n $: $ (0,21{+}31,31{+}41,\dots,(n{-}1)1{+}n1,n1) $,
$ {{\operatorname{\mathfrak{r}}}}_{n(k-1),\lambda} $: $ (0, 21{+}31, \dots,
(k{-}1)1+k1,k1, \lambda.(k{+}1)1+(k{+}2)1, \dots, \lambda.(n{-}1)1 +
n1,\allowbreak \lambda.n1) $, with $ k > 2 $ and $ \lambda \ne
0,{-}1,-2,-1/2 $,
$ {{\operatorname{\mathfrak{r}}}}_{n,\lambda(k)} $: $
(0,21,\lambda_1.31,\dots,\lambda_{k{-}1}.(k{+}1)1,\lambda_k.(k{+}2)1
+ (k{+}3)1,\dots,\lambda_k.(n{-}1)1 + n1,\allowbreak \lambda_k.n1)
$, with $n > k + 2 $ and non-zero $ \lambda_i $, $
1{+}\lambda_i $, $ \lambda_i{+}\lambda_j $, $ 1{+}2\lambda_k $, $ \lambda_i{+}2\lambda_k $, $ 1{+}\lambda_i{+}\lambda_j $ $
(i<j) $ and $ \lambda_i{+}\lambda_j{+}\lambda_{\ell} $ $
(i<j<\ell) $,
$ {{\operatorname{\mathfrak{d}}}}_{n,\lambda(n-3)} $: $ (0, 21,\lambda_1.31, \dots,
\lambda_{n{-}3}.(n{-}1)1, (1{+}\lambda_1).n1{+}32) $, with $\lambda_i \ne 0,{-}1$ for all $i$, $\lambda_1\ne {-2}, {-}1/2,
{-}\lambda_i, {-}1/2(1+\lambda_i), {-}2{-}\lambda_i,
{-}\lambda_i{-}\lambda_j$ for $1<i$, $1<i<j$ and non-zero $
\lambda_i+\lambda_j $, $ 1{+}\lambda_i+\lambda_j $ $(1<i<j)$, $ \lambda_i{+}\lambda_j{+}\lambda_k $ $ (1<i<j<k) $.
The members of the $ {{\operatorname{\mathfrak{r}}}}$-series have $ {{\operatorname{\mathfrak{k}}}}\cong{\mathbb R}^{n-1} $ and $ \operatorname{ad}_A $ belongs to the list $ J(n-1,1) $, $ J(k-1,1) \oplus
J(n-k,\lambda) $, $ \operatorname{diag}(1,\lambda_1,\dots,\lambda_{k-1}) \oplus
J(n-k-1,\lambda_k) $, where $ J(m,a) $ is an $ m\times m $-Jordan block with $ a $ on the diagonal and $ 1 $ immediately above the diagonal. The first matrix corresponds to $ {{\operatorname{\mathfrak{r}}}}_n $, the second corresponds to the families $ {{\operatorname{\mathfrak{r}}}}_{n(k-1),\lambda} $ and the third one gives $ {{\operatorname{\mathfrak{r}}}}_{n,\lambda(k)} $. For the latter two cases the requirement that $ A $ must act invertibly on cohomology enforces some restrictions on parameters. As $ A $ acts on $
H^1({\mathbb R}^{n-1})\cong{\mathbb R}^{n-1} $ by a lower triangular matrix, these restrictions are easily determined: the sum of one, two or three diagonal elements must be non-zero.
The family $ {{\operatorname{\mathfrak{d}}}}_{n,\lambda(n-3)} $ has $ {{\operatorname{\mathfrak{k}}}}\cong(0^{n-2},21) $ and $ \operatorname{ad}_A $ is $ \operatorname{diag}(1,\lambda_1, \dots, \lambda_{n-3},
1+\lambda_1) $. Now $ A $ acts diagonally on $ {{\operatorname{\mathfrak{k}}}}^* $, and restrictions on parameters may therefore be read off directly from the cohomology groups $ H^1({{\operatorname{\mathfrak{k}}}}) \cong {{\operatorname{\mathfrak{k}}}}^* \ominus \Span{e^{n-1}}$, $H^2({{\operatorname{\mathfrak{k}}}}) \cong \Lambda^2{{\operatorname{\mathfrak{k}}}}^* \ominus \Span{e^{12} ,
e^{i(n-1)}\colon\, i>2}$, $H^3({{\operatorname{\mathfrak{k}}}}) \cong \Lambda^3{{\operatorname{\mathfrak{k}}}}^* \ominus
\Span{e^{12i},e^{jk(n-1)}\colon\, 2<i<n-1,2<j<k}$.
An alternative way of constructing infinite families of $ (2,3)
$-trivial algebras goes via positive gradings of infinite families:
$ {{\operatorname{\mathfrak{f}}}}^1_n $: $ (0, 21, 31, 2.41 + 32, 3.51 + 42, \dots, (n -
2).n1 + (n - 1)2) $,
$ {{\operatorname{\mathfrak{f}}}}^2_n $: $ (0, 21, 2.31, 3.41 + 32, 4.51 + 42, 5.61 + 52
+ 43, \dots, (n - 1).n1+(n - 1)2+{(n - 2)3})$,
$ {{\operatorname{\mathfrak{f}}}}^3_n $: $ (0, 21, 31, 2.41 + 32, \dots, (n - 3).(n - 1)1
+ (n - 2)2, (n - 2).n1 + (n - 1)2 - {(n - 1)3} +(n - 2)3 -
\dotsb)$.
Here $ ({{{\operatorname{\mathfrak{f}}}}^1_n})'=(0^2,21,\dots,(n-2)1) $ has positive grading $
1^22\dotsb (n-2) $. The derived algebra $ ({{{\operatorname{\mathfrak{f}}}}^2_n})' =
(0^2,21,31,41+32,\dots,(n-2)1+(n-3)2) $ admits grading $ 12\dotsb
(n-1) $ and $ ({{{\operatorname{\mathfrak{f}}}}^3_n})' = (0^2, 21, 31, \dots, (n-3)1,
(n-2)1-(n-2)2+(n-3)3 -\dots -(-1)^k(k+1)k) $ with $ n=2k+1 $ has positive grading $ 1^223\dotsb (n-2) $.
In conclusion, the above exposition shows that the class of $(2,3)$-trivial algebras is quite rich. Yet, because of Theorem \[thm:2-3-trivial\], this class is much more accessible than the larger class of solvable algebras.
#### Acknowledgements {#acknowledgements .unnumbered}
We gratefully acknowledge financial support from <span style="font-variant:small-caps;">ctqm</span> and <span style="font-variant:small-caps;">geomaps</span>. AFS is also partially supported by the Ministry of Science and Innovation, Spain, under Project <span style="font-variant:small-caps;">mtm</span>[ 2008-01386]{} and thanks <span style="font-variant:small-caps;">nordita</span> for hospitality.
[13]{} \[1\][\#1]{} \[1\][`#1`]{} urlstyle\[1\][doi: \#1]{}
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T.B.Madsen & A.F.Swann
Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
*and*
CP^3^-Origins, Centre of Excellence for Particle Physics Phenomenology, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark.
*E-mail*: [tbmadsen@imada.sdu.dk](tbmadsen@imada.sdu.dk), [swann@imada.sdu.dk](swann@imada.sdu.dk)
|
---
abstract: 'We consider the discretization of parabolic initial boundary value problems by finite element methods in space and a Runge-Kutta time stepping scheme. Order optimal a-priori error estimates are derived in an energy-norm under natural smoothness assumptions on the solution and without artificial regularity conditions on the parameters and the domain. The key steps in our analysis are the careful treatment of time derivatives in the $H^{-1}$-norm and the the use of an $L^2$-projection in the error splitting instead of the Ritz projector. This allows us to restore the optimality of the estimates with respect to smoothness assumptions on the solution and to avoid artificial regularity requirements for the problem, usually needed for the analysis of the Ritz projector, which limit the applicability of previous work. The wider applicability of our results is illustrated for two irregular problems, for which previous results can either not by applied or yield highly sub-optimal estimates.'
address: 'Department of Mathematics, TU Darmstadt, Germany'
author:
- Herbert Egger
title: Energy norm error estimates for finite element discretization of parabolic problems
---
> [ parabolic problems, Galerkin distretization, finite element method, backward Euler scheme, energy-norm estimates]{}
> [ 35K20, 65L06, 65L70, 65M15, 65M20, 65M60]{}
Introduction
============
Let $\Omega \subset \RR^d$ be a bounded domain and $T>0$ denote a time horizon. We consider the numerical solution of parabolic initial boundary value problems of the form
\[eq:1\] $$\begin{aligned}
&&&& u'(t) + A(t) u(t) &= f(t), && \text{in } \Omega, \quad 0<t<T, &&&&\label{eq:1a}\\
&&&& u(t) &= 0, && \text{on } \partial\Omega, \ 0<t<T, &&&&\label{eq:1b}\\
&&&& u(0) &= u_0, && \text{in } \Omega. &&&&\label{eq:1c} \end{aligned}$$
Here $u'$ is the time derivative and $Au=-\div(a \nabla u) + b \cdot \nabla u + c u$ is a second order differential operator with coefficients $a,b,c \in L^\infty$ and $a(x,t) \ge \underline{a}>0$, such that $A(t)$ is uniformly elliptic for every point in time. It is well known, that for any $u_0 \in L^2(\Omega)$ and any $f \in L^2(0,T;H^{-1}(\Omega))$ there exists a unique weak solution in the *energy-space* $$\begin{aligned}
\label{eq:W}
W(0,T) = \{ u \in L^2(0,T;H_0^1(\Omega) : u' \in L^2(0,T;H^{-1}(\Omega))\},\end{aligned}$$ which can be bounded a-priori by $$\begin{aligned}
\label{eq:apriori}
\|u\|_{W(0,T)}
&=\|u\|_{L^2(0,T;H^1_0(\Omega))} + \| u'\|_{L^2(0,T;H^{-1}(\Omega))} \\
&\le C \big( \|f\|_{L^2(0,T;H^{-1}(\Omega))} + \|u_0\|_{L^2(\Omega)}\big).\end{aligned}$$ Moreover, the constant $C$ only depends on the domain and the bounds for the coefficients. We will refer to $\|\cdot\|_{W(0,T)}$ as the *energy-norm* of the problem. If the coefficients and the data are sufficiently smooth, the solution of - can be expected to be more regular: For instance, if one assumes that $a,b,c \in W^{1,\infty}$, then $$\begin{aligned}
\label{eq:reg1}
% \|u'\|_{W(0,T)}=
&\|u'\|_{L^2(0,T;H_0^1(\Omega))} + \|u''\|_{L^2(0,T;H^{-1}(\Omega))} \\
&\qquad \qquad \qquad \qquad
\le C \big( \|f'\|_{L^2(0,T;H^{-1}(\Omega))} + \|f(0) - A(0) u_0\|_{L^2(\Omega)}\big), \notag\end{aligned}$$ whenever the right hand side is bounded. If, in addition, also the domain is sufficiently regular and $u_0 \in H_0^1(\Omega)$, then $$\begin{aligned}
\label{eq:reg2}
% \|\nabla u\|_{W(0,T)} \le
&\|u\|_{L^2(0,T;H^2(\Omega))} + \|u'\|_{L^2(0,T;L^2(\Omega))}
% \\ &\qquad \qquad \qquad \qquad
\le C \big( \|f\|_{L^2(0,T;L^2(\Omega))} + \|u_0\|_{H^1_0(\Omega)}\big).
% \qquad \qquad \qquad \notag
% \qquad \qquad \quad \end{aligned}$$ We refer to [@Evans] for details and proofs of these estimates and further results.
Let us emphasize at this point that the two estimates and also characterize the basic regularity spaces for parabolic problems. In particular, time derivatives typically lack two order of spatial regularity compared to the solution itself.
In this paper, we study the numerical solution of - by finite element discretization in space and Runge-Kutta time stepping schemes. Such approximations have been investigated intensively in the literature, see e.g. [@Dupont74; @DouglasDupont70; @DouglasDuponWheeler; @Thomee74; @Varga; @Wheeler73]; we refer to [@Thomee] for a comprehensive collection of results and further references. Our main goal here is to derive order optimal a-priori error estimates in the energy-norm $\| \cdot \|_{W(0,T)}$. These can be obtained under minimal regularity assumptions on the coefficients and the domain, and we only require natural smoothness conditions for the solution. To keep the notation simple, we consider here in detail only the lowest order approximation by piecewise linear finite elements in space and the backward Euler method in time. The generalization of our arguments to approximations of higher order is however straight forward.
To put our results into perspective, let us shortly recall some of the standard results for the Galerkin approximation of parabolic problems [@Thomee; @Varga; @Wheeler73] and compare them with the main contributions of our manuscript:
As a first step, we will consider the semi-discretization in space by piecewise linear finite elements. The standard lowest order error estimate for parabolic problems then reads $$\begin{aligned}
\label{eq:ee1}
\|u-u_h\|_{L^\infty(0,T;L^2(\Omega))}
\le C h \big( \|u_0\|_{H^1(\Omega)} + \|u'\|_{L^1(0,T;H^1(\Omega))} \big).\end{aligned}$$ Here $u_h$ is the finite element approximation and $h$ the meshsize. This estimate amounts to [@Thomee Theorem 1.2] with $r=1$. Assuming sufficient regularity of the domain, the parameters, and the solution, also second order convergence can be obtained here. The basic step in the proof of this error estimate is the decomposition of the error into $$\begin{aligned}
\label{eq:decomp}
u-u_h = (u-R_h u) + (R_h u - u_h), \end{aligned}$$ where $R_h$ denotes the Ritz projection associated to the operator $A$; see [@Varga; @Wheeler73]. The proof of the error bound then relies on a discrete energy estimate for the semi-discrete problem and certain properties of the Ritz projection, in particular, on a bound $$\begin{aligned}
\label{eq:ritz}
\|u - R_h u\|_{L^2(\Omega)} \le C h \|u\|_{H^1(\Omega)},\end{aligned}$$ for the $L^2$-norm error, which can be obtained by the usual duality arguments [@Aubin67; @Nitsche68]. Without some additional regularity assumptions on the parameters and the domain, the estimate for the Ritz projection is however not valid, and therefore the validity of the bound for irregular problems cannot be granted by the proofs given in [@Thomee; @Varga; @Wheeler73]. Also note that the estimate is somwhat sub-optimal concerning the regularity requirements for the solution: In fact, it should suffice to assume that $u \in L^\infty(0,T;H^1(\Omega))$, which is already valid if $u \in L^2(0,T;H^2(\Omega)) \cap H^1(0,T;L^2(\Omega))$. Let us emphasize that no spatial regularity for the time derivative seems to be required. Also, in view of the a-priori estimate , this latter condition would be a natural regularity assumption.
As a replacement for , we will derive an error estimate in the energy-norm $\|\cdot\|_{W(0,T)}$, which arises in the a-priori estimate for the solution. We will show that $$\begin{aligned}
\label{eq:res1}
\|u-u_h\|_{W(0,T)}
&=\|u-u_h\|_{L^2(0,T;H^1(\Omega))} + \| u' - u'_h\|_{L^2(0,T;H^{-1}(\Omega))} \notag \\
&\le C h \big( \|u\|_{ L^2(0,T;H^2(\Omega))} + \| u'\|_{L^2(0,T;L^2(\Omega))} \big). %\notag\end{aligned}$$ By continuous embedding, one has $\|u-u_h\|_{L^\infty(0,T;\Omega)} \le C \|u - u_h\|_{W(0,T)}$, which yields the corresponding estimate also for the error in the $L^2$-norm at every point in time. One can easily see that the estimate is optimal with respect to the approximation properties of the finite element spaces and also with respect to the smoothness requirements on the solution, which are natural for the problem under investigation. Also note that we require no additional regularity of the domain or the parameters for the proof of this result.
As a second step, we will then investigate the time discretization by the backward Euler method. The standard error estimate for the fully discrete approximation reads $$\begin{aligned}
\label{eq:ee2}
\max_{0 \le t^n \le T} \|u(t^n) - u_h^n\|_{L^2(\Omega)} \le C h \big( \|u_0\|_{H^1(\Omega)} + \|u_t\|_{L^1(0,T;H^1(\Omega))} + \tau \|u_{tt}\|_{L^1(0,T;L^2)}\big),\end{aligned}$$ see [@Thomee Theorem 1.5] with $r=1$. Here $\tau = t^{n} - t^{n-1}$ is the time-step size, and $u_h^n$ denotes the $n$-th iterate of the Euler method. Again, we observe a certain sub-optimality concerning the regularity requirments for the solution. Moreover, the proof given in [@Thomee], see also [@Wheeler73; @Varga], is only valid under additional restrictive regularity assumptions on the parameters and the domain, which strongly limit the applicability of the results. As a replacement for , we will derive the energy-norm estimate $$\begin{aligned}
\label{eq:res2}
\|u - \tilde u_h\|_{W(0,T)}
&=\|u-\tilde u_h\|_{L^2(0,T;H^1(\Omega))} + \| u' - \tilde u_h'\|_{L^2(0,T;H^{-1}(\Omega))} \\
&\le C \big(
h \| u\|^2_{L^2(0,T;H^2(\Omega))} + h \|u'\|^2_{L^2(0,T;L^2(\Omega))}
\notag \\ & \qquad \qquad \qquad %\qquad \qquad
+ \tau \|u'\|_{L^2(0,T;H^1(\Omega))} + \tau \|u''\|_{L^2(0,T;H^{-1}(\Omega))}
% h \| u\|^2_{L^2(0,T;H^2)} + h \|u'\|^2_{L^2(0,T;L^2)}
% + \tau \|u'\|_{L^2(0,T;H^1)} + \tau \|u''\|_{L^2(0,T;H^{-1})}
\big).\notag\end{aligned}$$ Here $\tilde u_h$ denotes the function obtained from $u_h^n$ by piecewise linear interpolation in time. The bound for $u_h^n$ again follows easily by embedding of $W(0,T)$ into $L^\infty(0,T;L^2(\Omega))$. As before, the estimate can be seen to be optimal with respect to the approximation properties of the discretization, and, in view of the a-priori bounds and , the regularity requirements for the solution seem natural. Moreover, no artificial regularity of the domain and the coefficients will be required for the proof of this estimate.
One key step in the derivation of our results will be the careful estimation of time derivatives in the $H^{-1}$-norm. This seems natural in view of the definition of the energy-norm and its role in the a-priori estimates. The importance of such estimates has already been observed in the context of a-posteriori error estimation [@MakridakisNochetto03]. Let us also mention [@DouglasDupont77; @Wheeler75], where $H^{-1}$-Galerkin methods for the solution of parabolic initial boundary value problems have been investigated.
A second difference to previous investigations is, that we use here a somewhat different error splitting as usually employed, namely $$\begin{aligned}
u - u_h = (u - \pi_h u) + (\pi_h u - u_h), \end{aligned}$$ where $\pi_h$ is the $L^2$-projection onto the finite element space. The $L^2$-projection $\pi_h$ has important advantages in comparison to the Ritz-projector $R_h$: First, the estimate corresponding to and further approximation properties can be proven without regularity assumptions on the domain or the parameters. In addition, $\pi_h$ commutes with the time derivative, even for problems with time dependent parameters. At several places in our analysis, we will rely on the $H^1$-stability of the $L^2$-projection [@BankYserentant14; @BramblePasciakSteinbach02]. Morover, we use approximation error estimates for the $L^2$-projection in various norms, in particular including estimates in the $H^{-1}$-norm.
The remainder of the manuscript is organized as follows: In Section \[sec:prelim\], we formally present the problem to be investigated and we discuss the basic assumptions on the domain and the coefficients. In Section \[sec:fem\], we introduce the finite element spaces and summarize some estimates for the $L^2$-projection that are required later in our analysis. Section \[sec:semi\] is then concerned with the derivation of the estimate for the semi-discretization. In Section \[sec:time\], we investigate the time discretization by the backward Euler method and we derive the second estimate . For illustration of the wider applicability of our results, we discuss in Section \[sec:num\] two irregular test proplems, for which, due to lack of regularity, the standard bounds and cannot be applied directly. Due to the weaker requirements for our estimates and , the optimal convergence in the energy-norm can however still be guaranteed also theoretically. We close with a short discussion of our results and highlight some possibilities for generalization and future investigations.
Preliminaries {#sec:prelim}
=============
Let $\Omega \subset \RR^d$, $d=2,3$, be some bounded domain and $T>0$. We use standard notation for function spaces, see e.g. [@Evans]; in particular, $H_0^1(\Omega)$ denotes the sub-space of functions in $H^1(\Omega)$ with vanishing traces on $\partial \Omega$, and $H^{-1}(\Omega) = (H_0^1(\Omega))'$ is the space of bounded linear functional on $H^1_0(\Omega)$; the corresponding duality product is denoted by $\langle \cdot, \cdot\rangle_{H^{-1}(\Omega) \times H_0^1(\Omega)}$. Of particular importance for our analysis is the energy-space $$\begin{aligned}
W(0,T) = \{u \in L^2(0,T;H_0^1(\Omega)) : u' \in L^2(0,T;H^{-1}(\Omega))\},\end{aligned}$$ which is equipped with the norm $\|u\|_{W(0,T)}=\|u\|_{L^2(0,T;H^1(\Omega))} + \|u'\|_{L^2(0,T;H^{-1}(\Omega))}$. For later reference, let us recall the following embedding result [@Evans Sec. 5.9].
\[lem:embedding\] Let $u \in W(0,T)$. Then $u \in C([0,T];L^2(\Omega))$ and $$\sup_{0 \le t \le T}\|u(t)\|_{L^2(\Omega)} \le C \|u\|_{W(0,T)}.$$
As a consequence, all estimates derived in the energy-norm $\|\cdot\|_{W(0,T)}$ automatically yield corresponding bounds in $\|\cdot\|_{L^\infty(0,T;L^2(\Omega))}$, i.e., pointwise in time. Here and below, we denote by $C$ some generic constant which may have different values in different occasions.
Let us now turn to the initial-boundary value problem -. We assume that the operator $A$ has the form $A u = -\div (a \nabla u) + b \cdot \nabla u + c u$. In order to proof well-posedness of the initial boundary value problem, we assume that the parameters satisfy
- $a,c \in L^\infty(\Omega \times (0,T))$ and $b \in L^\infty(\Omega \times (0,T))^d$, and
- $a(x,t) \ge \underline a$ for some constant $\underline a > 0$ and a.e. $(x,t) \in \Omega \times (0,T)$.
Because of the first assumption, $A$ defines a bounded linear operator from $L^2(0,T;H_0^1(\Omega))$ to $L^2(0,T;H^{-1}(\Omega))$. For a.e. $t \in (0,T)$ we then define an associated bilinear form by $$\begin{aligned}
a(u,v;t) = \int_\Omega a(x,t) \nabla u(x) \nabla v(x) + b(x,t) \cdot \nabla u(x) v(x) + c(x,t) u(x) v(x) \ dx.\end{aligned}$$ The weak formulation of the initial boundary value problem - now reads
$ $ \[prob:weak\]\
Given $f \in L^2(0,T;H^{-1}(\Omega))$ and $u_0 \in L^2(\Omega)$, find $u \in W(0,T)$ such that $u(0) = u_0$ and $$\begin{aligned}
\langle u'(t), v \rangle_{H^{-1}(\Omega) \times H_0^1(\Omega)} + a(u(t),v;t) &= \langle f(t) , v\rangle_{H^{-1}(\Omega) \times H_0^1(\Omega)}\end{aligned}$$ holds for all test functions $v \in H_0^1(\Omega)$ and for a.e. $t \in (0,T)$.
Under assumptions (A1)-(A2), Problem \[prob:weak\] has a unique solution $u \in W(0,T)$ and there holds $\|u\|_{W(0,T)} \le C \big( \|u_0\|_{L^2(\Omega)} + \|f\|_{L^2(0,T;H^{-1}(\Omega)} \big)$ with a constant $C$ that depends only on the bounds for the coefficients and on the domain; see [@Evans] for details. The proof relies on a Gronwall argument and the following properties of the bilinear form.
$ $ \[lem:garding\]\
Let (A1) and (A2) hold. Then there exist constants $C_a,\alpha,\eta > 0$ such that $$\begin{aligned}
\label{eq:garding}
a(u,v;t) \le C_a \|u\|_{H^1(\Omega)} \|v\|_{H^1(\Omega)}
\quad \text{and} \quad
a(u,u;t) + \eta \|u\|^2_{L^2(\Omega)} \ge \alpha \|u\|_{H^1(\Omega)}^2,\end{aligned}$$ and these estimates hold uniformly for a.e. $t \in (0,T)$ and all $u,v \in H^1(\Omega)$.
The continuity follows from the Cauchy-Schwarz inequality. Using the lower and upper bounds for the coefficients, we get $$\begin{aligned}
a(u,u;t) \ge \underline a \|\nabla u\|^2_{L^2(\Omega)} + \|b(t)\|_{L^\infty} \|\nabla u\|_{L^2(\Omega)} \|u\|_{L^2(\Omega)} + \|c(t)\|_{L^\infty} \|u\|_{L^2(\Omega)}^2. \end{aligned}$$ The G[å]{}rding inequality then follows by Young’s inequality and choosing the coefficients, for instance, as $\alpha = \underline a/2$ and $\eta=\underline{a}/2 + \|b\|_{L^\infty}^2/(2 \underline a) + \|c\|_{L^\infty}$.
Properties of the $L^2$-projection onto finite element spaces {#sec:fem}
=============================================================
For the semi-discretization in space, we will employ a standard finite element method. To avoid technical difficulties, we assume that $\Omega$ is polyhedral and that it can be partitioned into a set $\Th=\{T\}$ of simplicial elements $T$. More precisely, we require that
- $\Th$ is a regular simplicial partition of $\Omega$, i.e., the intersection of two different elements either empty, or a vertex, an entire edge, (an entire face) of both elements;
- $\Th$ is locally quasi-uniform, i.e., there exists a $\gamma>0$ such that the diameter $h_T$ of an element $T$ and the radius $\rho_T$ of the largest ball that can be inscribed in $T$ are related by $\gamma h_T \le \rho_T \le h_T$ for all elements $T$.
Given such a mesh $\Th$, we consider the standard finite element space $$\begin{aligned}
V_h = \{v \in H_0^1(\Omega) : v|_{T} \in P_1(T) \text{ for all } T \in \Th\},\end{aligned}$$ of piecewise linear continuous functions that vanish on the boundary. Furthermore, we denote by $\pi_h : L^2(\Omega) \to V_h$ the $L^2$-orthogonal projection defined by $$\begin{aligned}
(\pi_h u, v_h)_{L^2(\Omega)} = (u, v_h)_{L^2(\Omega)} \qquad \text{for all } v_h \in V_h.\end{aligned}$$ Obviously, $\|\pi_h u\|_{L^2(\Omega)} \le \|u\|_{L^2(\Omega)}$, i.e., the $L^2$-projection is stable (a bounded linear operator) on $L^2(\Omega)$. Under the assumption (A3)-(A4), it is however also stable on $H^1_0(\Omega)$.
\[lem:h1stability\] Let (A3)-(A4) hold. Then $$\begin{aligned}
\|\pi_h u\|_{H^1(\Omega)} \le C \|u\|_{H^1(\Omega)} \qquad \text{for all } u \in H_0^1(\Omega),\end{aligned}$$ and the constant $C$ only depends on the domain and the regularity constants of the mesh.
For globally quasi-uniform meshes, the result follows directly from the Bramble-Hilbert Lemma and an inverse inequality. The proof for locally quasi-uniform meshes has been given in [@BramblePasciakSteinbach02]; see [@BankYserentant14] for generalizations including higher order approximations.
We will also require the following approximation error estimates.
\[lem:approximation\] Let (A3)-(A4) hold. Then $$\begin{aligned}
\|u - \pi_h u\|_{H^s(\Omega)} \le C h^{k-s} \|u\|_{H^{k}(\Omega)}
% \quad \text{for all } u \in H_0^{k}, \quad -1 \le s \le \min\{1,k\}, \ 0 \le k \le 2. \end{aligned}$$ for all $u \in H_0^{k}$ with $0 \le k \le 2$ and $-1 \le s \le \min\{1,k\}$.
For completeness, we sketch the main steps. The case $s=0$ and $0 \le k \le 2$ is well known and follows from the Bramble-Hilbert lemma and scaling arguments. To show the estimate for $s=1$ and $1 \le k \le 2$, let us denote by $\pi^1_h : H_0^1(\Omega) \to V_h$ the $H^1$-orthogonal projection defined by $$\begin{aligned}
(\pi_1^h u, v_h )_{H^1(\Omega)} = (u,v_h)_{H^1(\Omega)} \qquad \text{for all } v_h \in V_h.\end{aligned}$$ Recall that $\|\pi^1_h u - u\|_{H^1(\Omega)} \le C' h^{k-1} \|u\|_{H^k(\Omega)}$ for $1 \le k \le 2$, which is the usual finite element error estimate [@BrennerScott]. We can then proceed by $$\begin{aligned}
\|u - \pi_h u \|_{H^1(\Omega)}
&\le \|u - \pi_h \pi_h^1 u\|_{H^1(\Omega)} + \|\pi_h (u - \pi_h^1 u)\|_{H^1(\Omega)} \\
&\le (1 + C) \|u - \pi^1_h u\|_{H^1(\Omega)} \le (1+C) C' h \|u\|_{H^2(\Omega)},\end{aligned}$$ where we used the projection property and the $H^1$-stability of $\pi_h$, and the approximation properties of $\pi^1_h$ in the last two steps. Now assume that $u \in H^k(\Omega)$ with $0 \le k \le 2$. Then $$\begin{aligned}
\|u - \pi_h u\|_{H^{-1}(\Omega)}
&= \sup_{v \in H_0^1(\Omega)} (u - \pi_h u, v)_{L^2(\Omega)} /\|v\|_{H^1(\Omega)} \\
&= \sup_{v \in H_0^1(\Omega)} (u, v - \pi_h v)_{L^2(\Omega)} /\|v\|_{H^1(\Omega)}
\le C h^{k+1} \|u\|_{H^k(\Omega)}.\end{aligned}$$ Here we used the approximation property of $\pi_h$ for $s=0$. This yields the estimate for $s=-1$ and $0 \le k \le 2$ and completes the proof.
Corresponding estimates for real valued $s$ and $k$ can be obtained by interpolation arguments. Also note that the dependence on $h$ in the above estimates can be localized.
Galerkin semi-discretization {#sec:semi}
============================
As a first step in the approximation process, let us investigate the semi-discretization in space by finite elements. Proceeding in a standard fashion, we define
$ $ \[prob:semi\]\
Find $u_h \in H^1(0,T;V_h)$ such that $u_h(0)=\pi_h u_0$ and $$\begin{aligned}
\label{eq:semi}
(u_h'(t), v_h)_{L^2(\Omega)} + a(u_h(t),v_h;t) &= \langle f(t), v_h \rangle_{H^{-1}(\Omega) \times H_0^1(\Omega)} \end{aligned}$$ for all test functions $v_h \in V_h$ and a.e. $t \in (0,T)$.
The first term could also be written as $\langle u_h'(t), v_h \rangle_{H^{-1}(\Omega) \times H_0^1(\Omega)}$. By choosing a basis for $v_h$, the semi-discrete problem yields an ordinary differential equation, and existence and uniqueness follow by the Picard-Lindelöf theorem. More precicely, we have
$ $ \[lem:discrete\_apriori\]\
Let (A1)–(A4) hold. Then Problem \[prob:semi\] has a unique solution $u_h \in H^1(0,T;V_h)$ and $$\begin{aligned}
\|u_h\|_{W(0,T)} \le C (\|u_0\|_{L^2(\Omega)} + \|f\|_{L^2(0,T;H^{-1}(\Omega))}).\end{aligned}$$ The constant $C$ in this estimate depends only on the bounds for the coefficients, the domain, and the constants characterizing the regularity of the mesh.
It remains to verify the a-priori estimate. By testing the discrete variational problem in the usual way with $v_h = u_h(t)$, one can show that $$\begin{aligned}
\frac{1}{2} \frac{d}{dt}\|u_h(t)\|_{L^2(\Omega)}^2 + \frac{\alpha}{2} \|u_h(t)\|^2_{H^1(\Omega)}
&\le \eta \|u\|^2_{L^2(\Omega)} + \frac{1}{2\alpha}\|f(t)\|_{H^{-1}(\Omega)}^2.\end{aligned}$$ Here we only used the G[å]{}rding inequality and some elementary manipulations. Integrating over time and applying a Gronwall argument then yields the energy estimate $$\begin{aligned}
\|u\|_{L^\infty(0,T;L^2(\Omega))}^2 +\|u\|_{L^2(0,T;H^1(\Omega))}^2
\le C \big(\|u_0\|_{L^2(\Omega)}^2 + \|f\|_{L^2(0,T;H^{-1}(\Omega))}^2 \big). \end{aligned}$$ To obtain the remaining bound for the time derivative, observe that the $H^{-1}$-norm of a discrete function can be expressed as $$\begin{aligned}
\|u_h'(t)\|_{H^{-1}(\Omega)}
&= \sup_{v \in H_0^1(\Omega)} (u_h'(t), v)_{L^2(\Omega)} / \|v\|_{H^1(\Omega)} \\
&= \sup_{v \in H_0^1(\Omega)} (u_h'(t), \pi_h v)_{L^2(\Omega)} / \|v\|_{H^1(\Omega)}. \end{aligned}$$ Using the discrete variational problem with $v_h = \pi_h v$ further yields $$\begin{aligned}
(u_h'(t), \pi_h v)_{L^2(\Omega)}
&= \langle f_h(t), \pi_h v\rangle_{H^{-1}(\Omega) \times H_0^1(\Omega)} - a(u_h(t),\pi_h v;t) \\
&\le \big( \|f_h(t)\|_{H^{-1}(\Omega)} + C \|u_h(t)\|_{H^1(\Omega)} \big) \|\pi_h v \|_{H^1(\Omega)}. \end{aligned}$$ Since $\pi_h$ is bounded on $H_0^1(\Omega)$, we further have $\|\pi_h v\|_{H^1(\Omega)} \le C \|v\|_{H^1(\Omega)}$. Using these estimates in the expression for the norm of the time derivative finally yields $$\begin{aligned}
\|u_h(t)\|_{H^{-1}(\Omega)} \le C' \big( \|f_h(t)\|_{H^{-1}(\Omega)} + \|u_h(t)\|_{H^1(\Omega)} \big).\end{aligned}$$ The result then follows by integration over time and the energy estimate derived before.
As a direct consequence of the definition and the similarity of the continuous and the semi-discrete variational problems, we obtain
$ $ \[lem:galerkin\_orthogonality\]\
Let $u$ and $u_h$ denote the solutions of Problems \[prob:weak\] and \[prob:semi\], respectively. Then $$\begin{aligned}
\langle u(t) - u_h(t), v_h\rangle_{H^{-1}(\Omega) \times H_0^1(\Omega)}
+ a(u(t)-u_h(t), v_h;t) = 0
% \qquad \text{for almost every $t \in (0,T)$}.\end{aligned}$$ for almost every $t \in (0,T)$ and all $v_h \in V_h$. Moreover, $\pi_h (u(0)-u_h(0))=0$.
We can now turn to the error analysis of the semi-discretization. To this end, we divide the error into an approximation error and a discrete error as $$\begin{aligned}
u(t) - u_h(t) = (u(t) - \pi_h u(t)) + (\pi_h u(t) - u_h(t)) = (i) + (ii) . \end{aligned}$$ Due to the approximation properties of $\pi_h$, the first term can be bounded readily by
\[lem:semi\_approximation\_error\] Let (A3)-(A4) hold. Then $$\begin{aligned}
\|u - \pi_h u\|_{W(0,T)} \le C h \big( \|u\|_{L^2(0,T;H^2(\Omega)} + \|u'\|_{L^2(0,T;L^2(\Omega))}\big),\end{aligned}$$ and $C$ only depends on the domain and the shape-regularity constants of the mesh.
Using Galerkin orthogonality and the discrete stability of the method, the discrete error component can then be bounded as usual by the approximation error as well.
\[lem:semi\_discrete\_error\] Let (A1)–(A4) hold. Then $$\begin{aligned}
\|\pi_h u - u_h\|_{W(0,T)} \le C \|u - \pi_h u\|_{W(0,T)}.\end{aligned}$$ Moreover, the constant $C$ only depends on the domain, the bounds for the coefficients, and the shape-regularity of the mesh.
The discrete error $e_h = \pi_h u(t) - u_h(t)$ satisfies $$\begin{aligned}
(e_h(t), v_h)_{L^2(\Omega)} + a(e_h(t),v_h; t)
&=(\pi_h u'(t) - u'(t), v_h)_{L^2(\Omega)} + a(\pi_h u(t) - u(t), v_h;t)\\
&=a(\pi_h u(t) - u(t), v_h;t) =: \langle \tilde f(t), v_h\rangle_{H^{-1}(\Omega) \times H^1_0(\Omega)}, \end{aligned}$$ and also $e_h(0) = \pi_h u(t) - u_h(t) = 0$. Using the continuity of the bilinear form, we have $$\begin{aligned}
\|\tilde f(t)\|_{H^{-1}(\Omega)} \le C_a \|u(t) - \pi_h u(t)\|_{H^1(\Omega)}.\end{aligned}$$ By application of Lemma \[lem:discrete\_apriori\] to the equation for the discrete error, we thus obtain $$\begin{aligned}
\|e_h\|_{W(0,T)} \le C \|\tilde f\|_{L^2(0,T;H^{-1}(\Omega))} \le C' \|u - \pi_h u\|_{W(0,T)},\end{aligned}$$ and this already proves the assertion of the lemma.
By combining the previous two lemmas, we readily obtain our first main result.
$ $ \[lem:thm1\]\
Let (A1)-(A4) hold, and let $u$ and $u_h$ denote the solutions of Problems \[prob:weak\] and \[prob:semi\], respectively. Then $$\begin{aligned}
\|u - u_h\|_{W(0,T)} \le C h \big( \|u\|_{L^2(0,T;H^2(\Omega)} + \|u'\|_{L^2(0,T;L^2(\Omega))}\big),\end{aligned}$$ and the constant $C$ only depends on the domain, the bounds for the parameters, and the shape regularity of the mesh.
Let us emphasize that the regularity requirements for the solution are natural. In view of approximation properties of finite elements, they are also almost necessary to guarantee the approximation order one in the energy-norm. Also note that no additional regularity of the domain or the coefficients was required for our proofs.
Time stepping {#sec:time}
=============
Let us now turn to the time discretization of the semi-discrete Problem \[prob:semi\] by the backward Euler method. For ease of presentation, we only consider here uniform time steps of size $\tau=T/N$, and therefore set $t^{n} = n \tau$. To allow for evaluation of the coefficients and the data at individual points in time, we further assume that
- $a,b,c \in W^{1,\infty}(0,T;L^\infty(\Omega))$, and
- $f \in H^1(0,T;H^{-1}(\Omega))$.
The condition (A5) could be relaxed by using a Discontinuous-Galerkin method for the time discretization. In addition, we require that the time step is sufficiently small, i.e.,
- $\tau < 1/\eta$,
where $\eta$ is the constant from the G[å]{}rding inequality . This condition could be avoided by treating the lower order terms in the bilinear form in an explicit manner. To facilitate the notation, we will use in the sequel $$\partial_\tau u_h^{n+1} = \frac{1}{\tau} (u_h^{n+1} - u_h^n)$$ to denote the discrete time derivatives at $t^{n+1}$. Applying the backward Euler scheme for the time discretization of Problem \[prob:semi\] then leads to
$ $ \[prob:full\]\
Set $u_h^0 := \pi_h u_0$ and find $u_h^n \in V_h$ for $1 \le n \le N$, such that $$\begin{aligned}
\label{eq:full}
(\partial_\tau u_h^{n+1}, v_h)_{L^2(\Omega)} + a(u_h^{n+1},v_h;t^{n+1}) = \langle f(t^{n+1}), v_h \rangle_{H^{-1}(\Omega) \times H_0^1(\Omega)}
% \qquad \text{for all } v_h \in V_h.\end{aligned}$$ holds for all test functions $v_h \in V_h$.
The equation amounts to an implicit time-stepping scheme and that the snapshots $u_h^n$ can be computed recursively. The assumption (A7) guarantees that the elliptic problem for each time-step is uniquely solvable. More precisely, we have
$ $ \[lem:full\_apriori\]\
Let (A1)-(A7) hold. Then Problem \[prob:full\] has a unique solution $\{u^n_h\}_{0 \le n \le N}$, and $$\begin{aligned}
\sum\nolimits_{n=1}^N \tau \big( \|\partial_\tau u_h^{n}\|_{H^{-1}(\Omega)}^2 + \|u_h^n\|_{H^1(\Omega)}^2 \big)
\le C \big( \|u_0\|_{L^2(\Omega)} + \|f\|_{H^1(0,T;H^{-1}(\Omega))}\big). \end{aligned}$$
The norm on the left hand side is a discrete version of the energy-norm $\|\cdot\|_{W(0,T)}$.
Testing with $u_h^{n+1}$ and proceeding as in the proof of Lemma \[lem:discrete\_apriori\] yields $$\begin{aligned}
&\frac{1}{2\tau} \|u_h^{n+1}\|_{L^2(\Omega)}^2 + \frac{\alpha}{2} \|u_h^{n+1}\|_{H^1(\Omega)}^2 \\
&\qquad \qquad
\le \frac{1}{2\tau} \|u_h^{n}\|_{L^2(\Omega)}^2 + \eta \|u_h^{n+1}\|_{L^2(\Omega)}^2
+ \frac{1}{2\alpha} \|f(t^{n+1})\|_{H^{-1}(\Omega)}^2.\end{aligned}$$ Via a Gronwall argument, we then obtain the discrete energy estimate $$\begin{aligned}
\max\nolimits_{1 \le n \le N} \|u_h^n\|^2_{L^2(\Omega)} + \sum\nolimits_n \tau \|u_h^n\|^2_{H^1(\Omega)}
\le C \big(\|u_0\|_{H^1(\Omega)}^2 + \sum\nolimits_n \tau \|f(t^n)\|_{H^{-1}(\Omega))}^2\big)\end{aligned}$$ Due to condition (A5), the last term can be estimated via Talyor expansion by $$\begin{aligned}
% \label{eq:trick}
\sum\nolimits_n \tau \|f(t^{n+1})\|_{H^{-1}(\Omega)}^2
\le C \big( \|f\|_{L^2(0,T;H^{-1}(\Omega))}^2 + \tau^2 \|f'\|^2_{L^2(0,T;H^{-1}(\Omega))} \big).\end{aligned}$$ In order to derive the estimate for $\partial_\tau u_h^n$, recall that $$\begin{aligned}
\|\partial_\tau u_h^{n+1}\|_{H^{-1}(\Omega)}
% = \sup_{v \in H_0^1(\Omega)} (\partial_\tau u_h^{n+1}, v)_{L^2(\Omega)} / \|v_h\|_{H^1(\Omega)}
= \sup_{v \in H_0^1(\Omega)} (\partial_\tau u_h^{n+1}, \pi_h v)_{L^2(\Omega)}/\|v\|_{H^1(\Omega)}.\end{aligned}$$ Using $v_h = \pi_h v$ as a test function in the discrete scheme , we further obtain $$\begin{aligned}
(\partial_\tau u_h^{n+1}, \pi_h v)_{L^2(\Omega)}
&= \langle f(t^{n+1}), \pi_h v \rangle_{H^{-1}(\Omega) \times H_0^1(\Omega)} - a(u_h^{n+1},\pi_h v;t^{n+1}) \\
&\le C (\|f(t^{n+1})\|_{H^{-1}(\Omega)} + C \|u_h^{n+1}\|_{H^1(\Omega)} \big) \|\pi_h v\|_{H^1(\Omega)}.\end{aligned}$$ Using the stability of the projection $\pi_h$, this allows to estimate the discrete time derivative by known terms. The assertion then follows from the previous estimates.
Let us now turn to the derivation of error estimates for the full discretization defined in Problem \[prob:full\]. Similar as in the previous section, we use an error decomposition into an approximation error and a discrete error by $$\begin{aligned}
u(t^n) - u_h^n = (u(t^n) - \pi_h u(t^n)) + (\pi_h u(t^n) - u_h^n) = (i) + (ii).\end{aligned}$$ The first component can be bounded by the approximation estimates for $\pi_h$ as follows.
\[lem:full\_approx\_error\] Let (A3)-(A4) hold. Then $$\begin{aligned}
&\sum\nolimits_{n=1}^N \tau \big( \|\partial_\tau u(t^n) - \partial_\tau \pi_h u(t^n)\|_{H^{-1}(\Omega)}^2
+ \|u(t^n) - \pi_h u(t^n)\|_{H^1(\Omega)}^2\big) \\
&\qquad \qquad \le
C \big(
h^2 \| u\|^2_{L^2(0,T;H^2(\Omega))} +
h^2 \| u'\|^2_{L^2(0,T;L^2(\Omega))} +
\tau^2 \|u'\|^2_{L^2(0,T;H^1(\Omega))}
\big).\end{aligned}$$
Using Taylor expansion in time and the properties of the projection, we obtain $$\begin{aligned}
\sum\nolimits_n \tau &\|\partial_\tau u(t^n) - \partial_\tau \pi_h u(t^n)\|^2_{H^1(\Omega)} \\
&\le C \|u' - \pi_h u'\|_{L^2(0,T;H^{-1}(\Omega))}^2
\le C' h^2 \|u'\|_{L^2(0,T;L^2(\Omega))}^2.\end{aligned}$$ In a similar manner, we obtain for the second term $$\begin{aligned}
\sum\nolimits_n \tau &\|u(t^n) - \pi_h u(t^n)\|^2_{H^1(\Omega)} \\
&\le C \big( \|u - \pi_h u\|_{L^2(0,T;H^1(\Omega))}^2 + \tau^2 \|u' - \pi_h u'\|_{L^2(0,T;H^1(\Omega))}^2 \big) \\
&\le C' \big( h^2 \|u\|_{L^2(0,T;H^2(\Omega))} + \tau^2 \|u'\|_{L^2(0,T;H^1(\Omega))}^2 \big).\end{aligned}$$ The assertion then follows by summing up the two contributions.
The discrete stability of the scheme allows us to bound the discrete error as follows.
\[lem:full\_discrete\_error\] Let (A1)-(A7) hold. Then $$\begin{aligned}
&\sum\nolimits_{n=1}^N \tau \big( \|\partial_\tau \pi_h u(t^n) - \partial_\tau u_h^n \|_{H^{-1}(\Omega)}^2
+ \|\pi_h u(t^n) - u_h^n\|_{H^1(\Omega)}^2\big) \\
&\qquad \qquad \le
C \big(
h^2 \| u\|^2_{L^2(0,T;H^2(\Omega))} +
\tau^2 \|u'\|^2_{L^2(0,T;H^1(\Omega))} +
\tau^2 \|u''\|^2_{L^2(0,T;H^{-1}(\Omega))}
\big).
% C \big( \tau^2 \|u'\|^2_{L^2(0,T;H^1(\Omega))} + \tau^2 \|u''\|^2_{L^2(0,T;H^{-1}(\Omega))} + h^2 \| u\|^2_{L^2(0,T;H^2(\Omega))}\big).\end{aligned}$$
Let $e_h^n = \partial_\tau \pi_h u(t^{n}) - u_h^n$ denote the discrete error. Then $$\begin{aligned}
&\partial_\tau (e_h^{n+1},v_h)_{L^2(\Omega)} + a(e_h^{n+1},v_h;t^{n+1}) \\
% &= (\partial_\tau \pi_h u(t^{n+1}) - \partial_\tau u(t^{n+1}),v_h)_{L^2(\Omega)} \\
& \qquad = (\partial_\tau u(t^{n+1}) - u'(t^{n+1}),v_h)_{L^2(\Omega)}
+ a(\pi_h u(t^{n+1})-u(t^{n+1}),v_h;t^{n+1}). \end{aligned}$$ With similar arguments as in the proof of Lemma \[lem:full\_apriori\], we then obtain $$\begin{aligned}
&\max_{1 \le n \le N} \|e_h^n\|_{L^2(\Omega)}^2 + \sum\nolimits_n \tau \|e_h^n\|_{H^1(\Omega)}^2 \\
&\qquad \le C \sum\nolimits_n \tau \big( \|\partial_\tau u(t^n) - u'(t^n)\|_{H^{-1}(\Omega)}^2
+ \|\pi_h u(t^n) - u(t^n)\|_{H^1(\Omega)}^2\big).\end{aligned}$$ Using Taylor expansion, the first term on the right hand side can be further bounded by $$\begin{aligned}
\sum\nolimits_n \tau \|\partial_\tau u(t^n) - u'(t^n)\|_{H^{-1}(\Omega)}^2
\le \tfrac{\tau^2}{2} \|u''\|_{L^2(0,T;H^{-1}(\Omega))}^2, \end{aligned}$$ In a similar way, we obtain for the second term $$\begin{aligned}
&\sum\nolimits_n \tau \|\pi_h u(t^{n+1}) - u(t^{n+1})\|_{H^1(\Omega)}^2\\
&\qquad \le C \big(\|\pi_h u - u\|^2_{L^2(0,T;H^1(\Omega))} + \tau^2 \|\pi_h u' - u'\|^2_{L^2(0,T;H^1(\Omega))} \big) \\
&\qquad \le C' \big(h^2 \| u\|^2_{L^2(t^n,t^{n+1};H^2(\Omega))} + \tau^2 \|u'\|^2_{L^2(t^n,t^{n+1};H^1(\Omega))}\big).\end{aligned}$$ Here we employed Taylor expansions and the approximation and stability properties of the projection $\pi_h$. By combination of the two estimates we obtain $$\begin{aligned}
&\max_{1 \le n \le N} \|\pi_h u(t^n) - u_h^n\|_{L^2(\Omega)}^2 + \sum\nolimits_n \tau \|\pi_h u(t^n) - u_h^n\|_{H^1(\Omega)}^2 \\
&\qquad \qquad
\le C \big(
h^2 \| u\|^2_{L^2(0,T;H^2(\Omega))} +
\tau^2 \|u'\|^2_{L^2(0,T;H^1(\Omega))} +
\tau^2 \|u''\|^2_{L^2(0,T;H^{-1}(\Omega))}
\big).\end{aligned}$$ Using the characterization of the $H^{-1}$-norm for finite element functions, the definition of the discrete scheme , Galerkin-orthogonality, and some basic manipulations, we can further bound the discrete time derivative terms by $$\begin{aligned}
&\sum\nolimits_n \tau \|\partial_\tau \pi_h u(t^n) - u_h^n\|^2_{H^{-1}(\Omega)}
\\ &\qquad
\le C \sum\nolimits_n
\tau \big(
\|\partial_\tau u(t^{n}) - u'(t^n)\|_{H^{-1}(\Omega)}^2
\\ &\qquad \qquad \qquad \quad
+ \|u(t^n) - \pi_h u(t^n)\|_{H^1(\Omega)}^2
+\|u_h^n - \pi_h u(t^n)\|_{H^1(\Omega)}
\big). \end{aligned}$$ The first and second term on the right hand side can be estimated as above, and the third term is an approximation error which has already been bounded.
Summing up the two estimates for approximation error and the discrete error yields
\[lem:full\_discrete\_energy\] Let (A1)-(A7) hold. Then $$\begin{aligned}
&\sum\nolimits_{n=1}^N \tau \big( \|\partial_\tau u(t^{n+1}) - \partial_\tau u_h^{n+1} \|_{H^{-1}(\Omega)}^2 + \|u(t^{n+1}) - u_h^n\|_{H^1(\Omega)}^2\big) \\
& \qquad \qquad \le
C \big( h^2 \| u\|^2_{L^2(0,T;H^2(\Omega))} + h^2 \|u'\|_{L^2(0,T;L^2(\Omega))}
\\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad
+ \tau^2 \|u'\|^2_{L^2(0,T;H^1(\Omega))} + \tau^2 \|u''\|^2_{L^2(0,T;H^{-1}(\Omega))}\big).\end{aligned}$$
To finally obtain an estimate in the energy-norm $\|\cdot\|_{W(0,T)}$, let us denote by $$\begin{aligned}
\label{eq:uhtilde}
\tilde u_h(t) = \frac{t^{n+1}-t}{t^{n+1}-t^n} u_h^{n} + \frac{t-t^n}{t^{n+1}-t^n} u_h^{n+1}, \qquad t^n \le t \le t^{n+1}, \end{aligned}$$ the linear interpolant of the fully discrete approximations $u_h^n$ in time. Using the definition of $\tilde u_h$ and simple manipulations, we now obtain the second main result of this paper.
$ $ \[lem:thm2\]\
Let (A1)-(A7) hold and $u$ be sufficiently smooth. Then $$\begin{aligned}
\|u - \tilde u_h\|_{W(0,T)}
&\le
C \big( h^2 \| u\|^2_{L^2(0,T;H^2(\Omega))} + h^2 \|u'\|_{L^2(0,T;L^2(\Omega))} \\
& \qquad \qquad \qquad \qquad + \tau^2 \|u'\|^2_{L^2(0,T;H^1(\Omega))} + \tau^2 \|u''\|^2_{L^2(0,T;H^{-1}(\Omega))}\big).\end{aligned}$$ Like in the previous estimates, the constant $C$ here only depends on the domain, on the bounds for the coefficients, and on the shape regularity of the mesh.
Let us emphasize that in view of the bounds and , the regularity requirements for the solution are reasonable and that the rates are optimal with respect to the approximation properties of the discretization. Also note, that no additional smoothness assumptions for the domain or on the spatial regularity of the parameters were required.
Numerical tests {#sec:num}
===============
For illustration of the benefits of our results, let us shortly present two irregular test problems, for which the standard error estimates and cannot be applied directly, while our estiamtes and still provide order optimal error estimates.
Lack of smoothness
------------------
Let us consider the one-dimensional heat equation $$\begin{aligned}
\partial_t u(x,t) &= u_{xx}(x,t) + f(x,t), \qquad 0<x<1, \ 0<t<1, \end{aligned}$$ with homogeneous initial and boundary conditions. Here $\partial_t u = u'$ is used for the time derivative synonymously. We assume that the exact solution has the form $$\begin{aligned}
u(x,t) = \sum\nolimits_{n \ge 1} u_n \sin(n \pi x) \sin(n^2 \pi t)
\qquad \text{with} \quad u_n = (1+n^2)^{-5/4-\eps}.\end{aligned}$$ The parameter $\eps>0$ is assumed to be small. The special form of the solution allows to compute various norms of $u$ analytically and, in particular, to show that $$\begin{aligned}
u \in L^2(0,T;H^2(\Omega))
\qquad \text{and} \qquad
u' \in L^2(0,T;L^2(\Omega)).\end{aligned}$$ At the same time, one can easily check that $$\begin{aligned}
u \notin L^\infty(0,T;H^{1+3\eps}(\Omega))
\qquad \text{and} \qquad
u' \notin L^1(0,T;H^{3\eps}(\Omega)).\end{aligned}$$ Due to lack of regularity, the standard a-priori estimate for the semi-discretization therefore cannot be applied, in contrast to our energy-norm estimate , which yields $$\begin{aligned}
\|u - u_h\|_{L^\infty(0,T;L^2(0,\pi))}
\le C \|u - u_h\|_{W(0,T)} \le C' h.\end{aligned}$$ Note that, since $u \notin L^\infty(0,T;H^{1+3\eps}(\Omega))$, the rate of convergence for the error in the norm of $L^\infty(0,T;L^2(\Omega))$ cannot be improved substantially here. For a solution of the form $$\begin{aligned}
u(x,t) = \sum\nolimits_{n \ge 1} u_n \sin(n \pi x) \sin(n^{3/2} \pi t)
\qquad \text{with} \quad u_n = (1+n^2)^{-5/4-\eps},\end{aligned}$$ one can see in a similar manner that $$\begin{aligned}
u \in L^2(0,T;H^2(\Omega)),
\qquad
u' \in L^2(0,T;H^1(\Omega)),
\quad \text{and} \quad
u'' \in L^2(0,T;H^{-1}(\Omega)).\end{aligned}$$ However, $u'' \notin L^2(0,T;L^2(\Omega))$, and therefore the standard estimate for the full discretization cannot be applied due to lack of regularity. On the other hand, our error estimate for the full discretization still allows to guarantee $$\begin{aligned}
\|u(t^n) - u_h(t^n)\|_{L^2(\Omega)}
\le \|u - \tilde u_h\|_{L^\infty(L^2(\Omega))}
\le C \|u - \tilde u_h\|_{W(0,T)} \le C(h+\tau).\end{aligned}$$ Note that by interpolation one has at least $u \in L^\infty(0,T;H^{3/2})$ here, so the rate of convergence for the error $\|u(t) - u_h(t)\|_{L^2(\Omega)}$ in terms of the meshsize may possibly be improved. For sufficiently smooth solutions, the estimates of [@Varga; @Wheeler73; @Thomee] would in fact predict the optimal rate $\|u(t) - u_h(t)\|_{L^2(\Omega)} \le C (h^2+\tau)$ here.
Discontinuous parameters
------------------------
As a second test case, we consider a thermal diffusion problem on a square covered by an inhomogeneous medium. The governing system reads $$\begin{aligned}
\partial_t u(x,t) &= \div (a(x) \nabla u(x,t)) + f(x,t), \qquad x \in (-1,1)^2, \ 0<t<1/2.\end{aligned}$$ As before, we presecribe homogeneous initial and boundary conditions. Moreover, we assume that the diffusion parameter has the form $$\begin{aligned}
a(x) &= \begin{cases} 1, & x_1 \cdot x_2 > 0, \\ \eps, & x_1 \cdot x_2 < 0, \end{cases}\end{aligned}$$ where $\eps$ is some small positive constant. It is well-known [@SaendigNicaise94], that the associated elliptic operator $L u = - \div (a \nabla u)$ for such a problem is rather irregular. More precisely: for every $\beta>0$, one can choose an $\eps>0$, such that $L$ is not an isomorphism from $H^{1+\beta}_0(\Omega)$ to $H^{-1+\beta}(\Omega)$. In particular, the maximal value of $\beta$, such that $$\begin{aligned}
\label{eq:ritzb}
\|R_h u - u\|_{L^2(\Omega)} \le C h^{\beta} \|u\|_{H^1(\Omega)} \end{aligned}$$ holds for arbitrary $u \in H^1_0(\Omega)$ can be made arbitrarily small. Note that the standard estimate does not apply directly here, since only holds instead of . A generalization of the error estimate to non-smooth problems however still allows to guarantee $$\begin{aligned}
\label{eq:res1b}
\|u - u_h\|_{L^\infty(0,T;L^2(\Omega))} \le C h^{2\beta}, \end{aligned}$$ provided that the solution $u$ is sufficiently smooth; see [@Thomee Sec 19] for details. Since $\beta$ can be aribtrarily small in general, this estimate is highly unsatisfactory. Our energy-norm estimate for the semi-discrete approximation however still applies and yields $$\begin{aligned}
\|u-u_h\|_{L^\infty(0,T;L^2(\Omega))} \le C \|u-u_h\|_{W(0,T)} \le C' h, \end{aligned}$$ provided that the solution has the required smoothness. This results hold regardless of the spatial regularity of the diffusion parameter $a(\cdot)$. For illustration of the validity of this theoretical result, we also provide some results of numerical tests. To verify the convergence rate for the semi-discretization, we integrated the semi-discrete problem numerically in time with a very accurate time stepping scheme. The resulting errors obtained on a sequence of uniformly refined meshes are summarized in Table \[tab:1\].
$h$ 0.50000 0.25000 0.12500 0.06250 0.03125 rate
------- --------- --------- --------- --------- --------- ------
$e_1$ 1.65010 0.82062 0.39475 0.19318 0.09710 1.03
$e_2$ 0.37632 0.11211 0.02928 0.00741 0.00186 1.93
: Errors $e_1=\|u-u_h\|_{W(0,T)}$ and $e_2=\|u-u_h\|_{L^\infty(0,T;L^2(\Omega))}$ obtained on a sequence of uniformly refined meshes with meshsize $h$. \[tab:1\]
1em
As predicted by the theory, we observe convergence of the energy-norm error with first order. The numerical tests actually yield a better convergence rate for the error in the norm of $L^\infty(0,T;L^2(\Omega))$, which in fact is the optimal one from an approximation point of view. We however cannot give a full explanation for this observation yet.
Summary {#sec:sum}
=======
Various results concerning the numerical analysis of Galerkin approximations for parabolic problems are available in the literature; see [@Thomee] for a comprehensive overview and further references. This paper contributes to this active field with providing error estimates in the energy-norm $\|u\|_{W(0,T)} =\|u\|_{L^2(0,T;H^1)}+\| u'\|_{L^2(0,T;H^{-1})}$, which seems to be a natural choice from an analytic point of view, but which has not been studied intensively in previous works. In this manuscript, we considered only low order discretizations of a simple model problem. The general approach is however applicable to much wider class of problems and discretization schemes. Morover, our arguments may also be fruiteful for the derivation of a-priori error estimates in other norms [@DouglasDupont77; @Thomee80; @ThomeeWahlbin83; @Wheeler73b; @Wheeler75] and for the derivation of a-posteriori error estimates [@ErikssonJohnson91; @ErikssonJohnson95; @GeorgoulisLakkisVirtanen11; @MakridakisNochetto03]. Our main motivation to consider the energy-norm, was to overcome a sub-optimality of the standard estimates and concerning the regularity requirements for the solution, for the domain, and for the parameters. This sub-optimality is partly due to a loose handling of time derivatives in the estimates, and, on the other hand, stems from the use of the Ritz projector in the error decomposition, which requires duality arguments and regularity of the underlying elliptic problem. We therefore utilize here the $L^2$-projection in our error splitting and carefully estimate time derivatives in the $H^{-1}$-norm.
In our presentation, we focused on a-priori error estimates in the energy-norm $\|\cdot\|_{W(0,T)}$, and we could establish optimal convergence rates under minimal regularity assumptions. By continuous embedding, we could also obtain estimates pointwise in time with the same convergence rates. In our numerical experiments, we observed a better convergence of the error in the norm of $L^\infty(0,T;L^2)$ for a particular problem, which does not follow directly from our results. For sufficiently regular problems, this better convergence is well explained by the standard results [@Thomee; @Wheeler73; @Varga]. A justification for the case of certain irregular problems is however missing.
Acknowledgements {#acknowledgements .unnumbered}
================
The author would like to gratefully acknowledge support by the German Research Foundation (DFG) via grants GSC 233, IRTG 1529, and TRR 154.
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abstract: 'We report on longitudinal and transverse [magnetization]{} measurements performed on single crystal samples of [$\rm Gd_2Ti_2O_7$]{} for a magnetic field applied along the \[100\] direction. The measurements reveal the presence of previously unreported phases in fields below 10 kOe in an addition to the higher-field-induced phases that are also seen for [$H \! \parallel \! [111]$]{}, \[110\], and \[112\]. The proposed $H$-$T$ phase diagram for the $[100]$ direction looks distinctly different from all the other directions studied previously.'
author:
- 'O.A. Petrenko'
- 'M.R. Lees'
- 'G. Balakrishnan'
- 'V.N. Glazkov'
- 'S.S. Sosin'
title: 'Novel magnetic phases in a Gd$_2$Ti$_2$O$_7$ pyrochlore for a field applied along the $[100]$ axis'
---
The model of a Heisenberg antiferromagnet on a pyrochlore lattice has been the focus of theoretical attention for a number of years. The highly degenerate ground state manifold for a system of spins on corner-shared tetrahedra interacting through nearest-neighbour exchange prevents a magnetic ordering both in the quantum and the classical limits.[@Heis_theory1] Further interactions (e.g. dipolar) can remove this degeneracy and stabilize a particular ordered structure.[@Heis_theory3] To date, only two experimental realizations of a Heisenberg pyrochlore lattice antiferromagnet are known: [$\rm Gd_2Ti_2O_7$]{} (GTO) and [$\rm Gd_2Sn_2O_7$]{} (GSO). The spin-orbit coupling is strongly reduced for a Gd$^{3+}$ magnetic ion since its electronic ground state is $^8S_{7/2}$ with $L=0$. On cooling below 1 K, GTO and GSO develop different types of magnetic order[@Raju_PRB_1999; @Champion_PRB_2001; @Bertin_EPJB_2002; @Bonville_JPCM_2003] which have been intensively studied over the last decade (for a recent review see Ref. ).
The application of neutron scattering techniques to the determination of the magnetic structure in the ordered phases of GTO and GSO is hindered by the high neutron absorption cross section of naturally occurring Gd. The zero-field $k=(\frac{1}{2}\frac{1}{2}\frac{1}{2})$ magnetic structure of [$\rm Gd_2Ti_2O_7$]{} is generally believed to be a multi-$k$ structure suggested by Stewart [*et al.,*]{}[@Stewart_JPCM_2004] however, this was recently challenged by Brammal [*et al.*]{}[@Brammal_PRB_2011; @Stewart_PRB_comment_2012]. While the zero-field structure in GSO appears to be simpler,[@Wills_JPCM_2006] its behaviour in field remains largely unexplored because of a lack of single crystal samples.
In this Rapid Communication we revisit the unusual $H$-$T$ phase diagram of GTO. Initial interest in the field-induced behavior of GTO was driven by the heat capacity and susceptibility data obtained on a polycrystalline sample.[@Ramirez_PRL_2002] Later work on single crystals showed that despite a nominal spin-only state, the magnetic properties of GTO are anisotropic and that the $H$-$T$ phase diagram contains three different ordered phases. [@Petrenko_PRB_2004] The sequence of phase transitions in GTO in an applied magnetic field has been studied further using several techniques including muon spin relaxation,[@Dunsiger_PRB_2006] transverse [magnetization]{} measurements [@Glazkov_JPCM] and magnetic resonance.[@Sosin_PRB_2006] Nevertheless, to date, there is no full theoretical description of the observed magnetic phases.
All previous studies of the magnetic phase transitions in GTO were limited to fields applied along the \[111\], \[110\], or \[112\] directions, while no data have been published for [$H \! \parallel \! [100]$]{}. Here we report on longitudinal and transverse [magnetization]{} measurements performed on single crystals samples of [$\rm Gd_2Ti_2O_7$]{} for a magnetic field applied along the \[100\] direction. These measurements reveal the presence of previously unreported field-induced phase transitions that occur in magnetic fields below $\sim10$ kOe, in addition to the transitions at higher applied fields that are also seen for [$H \! \parallel \! [111]$]{}, \[110\], and \[112\].
Single crystal samples were prepared as described previously.[@Balakrishnan_JPCM_1998] The principal axes of the samples were determined using the X-ray diffraction Laue technique; the crystals were aligned to within an accuracy of 2 degrees. Two samples grown and aligned independently have shown no appreciable differences in their [magnetization]{} behavior.
Longitudinal [magnetization]{}, ([$M_\parallel$]{}), measurements were made down to 0.5 K in applied magnetic fields of up to 70 kOe using a Quantum Design Magnetic Properties Measurement System SQuID magnetometer along with an i-Quantum $^3$He insert.[@Shirakawa_JMMM_2004] The [magnetization]{} was measured both as a function of temperature in a constant magnetic field and as a function of applied field at constant temperature. Because of the relatively large magnetic moments observed, de[magnetization]{} effects had to be taken into consideration.
Transverse [magnetization]{}, ([$M_\perp$]{}), measurements were performed using a homemade capacitance torquemeter mounted in a $^3$He cryostat with a base temperature of 0.4 K. The sensor element was a flat capacitor formed by a rigid base and a flexible bronze cantilever with the sample attached. The cantilever was aligned parallel to the external magnetic field. The torque $\mathbf{T} = \mathbf{M_\perp} \times\mathbf{H}$, caused by the [magnetization]{} component normal to the capacitor plates, leads to a bending of the cantilever and to a change in the sensor capacitance. This technique has already been successfully applied to the study of the magnetic ordering in GTO.[@Glazkov_JPCM] The sample for the torque measurements was cut in the shape of a thin plate of approximately $0.15\times 1\times 1$ mm$^3$, with its plane coinciding with a (110)-plane of the crystal. It was glued on to the cantilever with a \[001\] axis parallel to the field and a \[110\] axis normal to the cantilever plane. The field was applied in the sample plane, thus any de[magnetization]{} effects were negligible.
![\[Fig1\_M\] (Colour online) Temperature dependence of the magnetic susceptibility of [$\rm Gd_2Ti_2O_7$]{} measured with a magnetic field applied along the \[100\] direction. The left-hand panel shows the data collected in 100 Oe. The arrows indicate the two transition temperatures at $T_{N1}=1.02$ K and $T_{N2}=0.74$ K reported from the specific heat measurements in zero field.[@Petrenko_PRB_2004] The right-hand panel shows the temperature dependence of the susceptibility measured on warming in various higher fields. These curves are consecutively offset by 0.05 emu/mol for clarity.](Fig1.eps){width="0.95\columnwidth"}
The temperature dependence of the magnetic susceptibility in a magnetic field of 100 Oe applied along \[100\] is shown in the left-hand panel of Fig. \[Fig1\_M\]. Both the upper ([$T_{N1}$]{}) and lower ([$T_{N2}$]{}) critical temperatures are clearly visible in the data. In agreement with the previously reported data for [$H \! \parallel \! [111]$]{},[@Petrenko_JPCM_2011] the upper phase transition is observed at 1.05 K, a temperature marginally higher than $T_{N1}=1.02$ K found from the specific heat measurements,[@Petrenko_PRB_2004] while the lower transition temperature, 0.74 K, is identical for all the measurements. The evolution of the temperature dependence of the magnetic susceptibility with applied field is shown in the right-hand panel of Fig. \[Fig1\_M\].
![\[Fig2\_M\] (Colour online) The upper panel shows the field dependence of [$M_\parallel$]{} of [$\rm Gd_2Ti_2O_7$]{} at 0.5 K with [$H \! \parallel \! [100]$]{}. The data taken for increasing and decreasing fields are almost indistinguishable. Linear fits to the data in the interval 0 to 10 kOe and 10 to 20 kOe reveal a 10% change in the [$M_\parallel (H)$]{} slope at $H\approx 10$ kOe. The lower panel shows the field dependence of the $dM_\parallel/dH$ curves, obtained from the [$M_\parallel (H)$]{} data at different temperatures and then offset by the specified values.](Fig2.eps){width="0.95\columnwidth"}
The field dependence of the longitudinal [magnetization]{} is shown in Fig. \[Fig2\_M\] together with the $dM_\parallel/dH$ curves obtained from the [$M_\parallel (H)$]{} curves measured at different temperatures. No appreciable hysteresis is observed in the [$M_\parallel (H)$]{} data. In the highest applied field of 70 kOe, which translates into 65.5 kOe after taking into account the de[magnetization]{} field, [$M_\parallel$]{} is still growing at a considerable rate. The maximum measured magnetic moment is 6.8$\mu_B$ per Gd ion, close to the value of 7$\mu_B$ expected for a state with $S=7/2$ and $L=0$. The saturation process in [$\rm Gd_2Ti_2O_7$]{} is not trivial. At $T=0.5$ K, instead of a gradual decrease in the gradient of the $M_\parallel(H)$ curve on approaching the saturation field $H_{sat}$, which would be typical for an ordinary antiferromagnet, the gradient increases from a lower field value of $\approx 0.10$ to $\approx 0.13 \mu_B/$kOe per Gd ion at $H_{sat}$. Although an additional transition at 10 kOe can be seen as a $\sim 10$% decrease in the slope of the [$M_\parallel (H)$]{} curve measured at 0.5 K, it becomes more obvious after differentiation. Fig. \[Fig2\_M\] suggests that (a) this transition is not particularly temperature dependent up to 0.8 K, and that (b) its influence on the $dM_\parallel/dH(H)$ curves is much more pronounced than the transition at a half of the saturation field.
The experimentally observed field dependence of the torquemeter capacitances is shown in Fig. \[C(H)\]. The smooth continuous variation of the capacitance with an applied field, as measured at temperatures above the magnetic ordering, is due to a field gradient at the sample position, inhomogeneity in the sample [magnetization]{}, or a slight sample misalignment causing the de[magnetization]{} field to deviate from the direction of the external field. All of these effects are proportional to the longitudinal [magnetization]{}. Since the variation of [$M_\parallel$]{} with temperature amounts to only a few percent over the entire range of temperatures explored, one can use the high-temperature ($T>T_{N1}$) response curves as a background for the low-temperature measurements.
On cooling below [$T_{N1}$]{} the torquemeter response curves change drastically. At the base temperature of the $^3$He cryostat of 0.4 K, the curve has a weak kink around 5 kOe and well defined, abrupt changes around 10, 25, and 55 kOe, presumably corresponding to magnetic field-induced phase transitions. The hysteresis of the torquemeter response curves at the high-field transition is related to a strong magnetocaloric effect.[@Sosin_PRB_2005] Overall the thermodynamics of GTO is quite complicated near the saturation field, but for temperatures below the magnetic ordering the magnetocaloric effect heats the sample if the field is decreasing and cools the sample if the field is increasing for $H<H_{sat}$. Since the sample is thermalised only through the 100 $\mu$m-thick bronze cantilever, this may lead to a deviation of the sample temperature from the sensor temperature. The temperature difference estimated from the spread of the transition points is 25 mK at 0.4 K and up to 100 mK at around 0.7 K. At low fields (15 kOe and below) any hysteresis in the torquemeter response is not sensitive to the field sweep rate, suggesting that in this field regime the hysteresis is not a thermalization issue.
![\[C(H)\] (Colour online) Field dependence of the torquemeter capacitance measured at 0.4 and 1.3 K. Dashed vertical arrows mark the transition fields. Solid arrows indicate increasing and decreasing field sweeps.](Fig3.eps){width="0.95\columnwidth"}
The transverse [magnetization]{} can be recovered from the experimental capacitance curves as follows. The magnetic torque is compensated by an elastic force, which is proportional to the change of the capacitor spacing $\Delta d$. Magnetic torque includes a transverse [magnetization]{} effect $a M_\perp H$ and a background response $b M_{||}(T>T_{N1}) H$. For a flat capacitor $d\propto \frac{1}{C}$, hence: $$\begin{aligned}
\Delta\frac{1}{C} & \propto &(a M_\perp +{b} M_{||}(T>T_{N1}))H \nonumber \\
M_\perp(H,T) & \propto &
\frac{1}{H}\left(\frac{1}{C(H,T)}-\frac{1}{C(H,T>T_{N1})}\right) \nonumber\end{aligned}$$
The field dependence of the extracted [$M_\perp$]{} is shown in Fig. \[Mtrans(H)\]. These curves demonstrate well defined features at the phase transitions. Firstly, as expected in the paramagnetic (saturated) phase, [$M_\perp$]{} is absent above the saturation field of $55\pm1.5$ kOe (here we follow the 0.4 K curve). [$M_\perp$]{} appears below this field and on lowering the field further it vanishes again in a field of $25\pm2$ kOe. This transition field is close to the $\approx 30$ kOe transition field observed for other studies in which the magnetic field was applied along \[111\], \[110\], or \[112\].[@Petrenko_PRB_2004; @Glazkov_JPCM] The transverse moment reappears in lower fields and smoothly increases as the field decreases to $H^{\downarrow}=9.9\pm 0.2$ kOe. At this field, corresponding to a small change in the slope of the [$M_\parallel (H)$]{} curve (see Fig. \[Fig2\_M\]), a sharp step-like increase of the [magnetization]{} component normal to the cantilever plane is observed. On further decrease of the field the transverse [magnetization]{} decreases almost linearly with the field showing a weak kink below 3 kOe.
![\[Mtrans(H)\] (Colour online) Field dependence of the transverse [magnetization]{} at different temperatures for increasing (solid lines) and decreasing (dashed lines) magnetic fields. For clarity, the curves are consecutively offset by -0.2 units of $M_\perp$.](Fig4.eps){width="0.95\columnwidth"}
On sweeping the field back up at the same temperature of 0.4 K, this kink is transformed into a more extended transition region which terminates with a sharp feature at about 6 kOe. Presumably this region, marked by a considerable hysteresis, is associated with the reduction of the system into a single magnetic domain state possessing the largest [magnetization]{} component perpendicular to the sample plane. As the magnetic field is increased further this component decreases sharply by about a factor of 10 at a field $H^{\uparrow}=10.4\pm0.1$ kOe. The hysteresis effects at this critical field remained largely unchanged when the field sweep rate was reduced by a factor of 3, pointing to the first-order nature of the transition. This transition is almost temperature independent, while the kink at 5 kOe evolves rapidly with increasing temperature. At a temperature of 0.55 K, [$M_\perp$]{} is zero in low fields, then it appears in a step-like fashion, increases linearly with field up to 10 kOe and then decreases. At temperatures above 0.6 K, [$M_\perp$]{} remains zero in the entire field range below 10 kOe.
The data presented above, which were collected for a magnetic field applied along the \[100\] direction, allow one to construct an $H$-$T$ phase diagram which looks distinctly different from those observed for other field orientations (see Fig. \[phasediag\]). We associate phase I in this diagram with the zero-field structure identified in neutron diffraction experiments[@Stewart_JPCM_2004] as a partially disordered multi-$k$ structure. At low-temperature, e.g. $T=0.4$ K, phase I is transformed by weak magnetic fields into phase I$^{\prime}$ in the way that resembles the reduction of a magnetic structure into a single domain state. This process is accompanied by a rapid step-like increase in the transverse component of the [magnetization]{} forbidden by the cubic symmetry of the magnetic structure in phase I. Observed for [$H \! \parallel \! [100]$]{} only, this lower symmetry state in which the transverse [magnetization]{} grows linearly in field on the phase diagram is rather unstable. Heating from within phase I$^{\prime}$ to above 0.5-0.6 K restores the state I in which the transverse magnetic moment remains [*exactly*]{} zero for fields below 10 kOe. Phases I and I$^{\prime}$ are separated from phase II by a first-order transition which was not observed in this field range for any other field directions. The phase boundary is practically temperature independent and according to our preliminary electron spin resonance measurements[@ESR_Sosin_2012] is accompanied by an abrupt change of the gaps in the spin-wave spectrum. The transverse moment in phase II (see Fig. \[phasediag\]) is either much smaller, but still nonzero in value, or it is rotated in the plane perpendicular to the applied magnetic field. This phase extending to $\simeq 25$-27 kOe, is also non-cubic in symmetry, and is likely to be equivalent to the low-field states observed in GTO for other field directions.[@Petrenko_PRB_2004; @Glazkov_JPCM] The field of 25-27 kOe is similar in value to the transition fields for other directions, but the situation for [$H \! \parallel \! [100]$]{} is different in that the transition produces no impact on the excitation spectrum,[@ESR_Sosin_2012] and does not result in a significant change in the magnetic susceptibility. This transition from II to III is therefore second-order, unlike the transitions observed for other directions of applied field e.g. [$H \! \parallel \! [111]$]{}.[@Sosin_PRB_2006; @Petrenko_JPCM_2011] The high-field part of the phase diagram (phase III) as well as the higher-temperature phase $\rm I_{HT}$ at $T_{N2}<T<T_{N1}$ are common for all field directions.
![\[phasediag\] (Colour online) Magnetic phase diagram of [$\rm Gd_2Ti_2O_7$]{} for a field applied along the \[100\] axis. The phase transition points detected by measuring the field or temperature dependence of the longitudinal [magnetization]{} are marked with circles. The open and solid squares were obtained from the transverse [magnetization]{} measurements for decreasing and increasing magnetic fields respectively. The lines are guides to the eyes, colour/white areas on the diagram correspond to the phases with/without transverse [magnetization]{}.](Fig5.eps){width="0.95\columnwidth"}
In conclusion, the magnetic phase diagram of [$\rm Gd_2Ti_2O_7$]{} for [$H \! \parallel \! [100]$]{} is established from longitudinal and transverse [magnetization]{} measurements. At low temperatures and fields, it contains an additional phase with a lower than cubic symmetry, which separates a zero-field magnetic structure from higher-field-induced states. At higher temperatures this phase disappears and breaking of the cubic symmetry occurs via a first-order transition at a field of about 10 kOe. A direct identification of the magnetic phases as well as a clarification of the role of the magnetic field in their formation remains a challenge for neutron diffraction experiments.
The Kapitza Institute group acknowledge support from the Russian Foundation for Basic Research (RFBR Grant No.10-02-01105) and the Russian President Program for the Support of the Leading Scientific Schools (Grant No.4889.2012.2). The Warwick group acknowledges financial support from the EPSRC, UK. Some of the equipment used at the University of Warwick was obtained through the Science City Advanced Materials project “Creating and Characterising Next Generation Advanced Material” with support from Advantage West Midlands and part funded by the European Regional Development Fund.
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abstract: 'Based on a perturbative theory of quantum chromodynamics and non-relativistic quark model, associated $J/\psi$ plus open charm photoproduction on charm quarks in a proton via partonic subprocess $\gamma c\to J/\psi c$ is discussed. It is shown that the value and energy dependence of the cross section for such process remarkably depends on the choice of charm distribution function in a proton. In the region of small $z=E_J/E_{\gamma}<0.2$ the contribution of the $\gamma c\to J/\psi c$ subprocess in the inelastic $J/\psi$ photoproduction spectra is larger than the contribution of the photon-gluon fusion subprocess. At the energy range of HERA collider intrinsic charm contribution in the total inclusive $J/\psi$ photoproduction cross section may be equal to 4% of the dominant contribution of photon-gluon fusion mechanism.'
---
2[t]{} 2[u]{} 2[s]{}
[**Intrinsic Charm in Proton and $J/\psi$ Photoproduction\
at High Energies**]{}
[**V.A. Saleev\
Samara State University, 443011 Samara, Russia**]{}
[Introduction]{} The study of the charm quark distribution function in a proton taiks a remarkable interest from the point of view of a investigation of the non-perturbative proton wave function [@1] as well as from the point of view of a calculation of the intrinsic charm quark contribution to the processes of charm particle and quarkonium production in photon-hadron, hadron-hadron and hadron-nucleus interactions at high energies [@2; @3]. The existing data on charm production processes gives us the idea [@1; @4] of a small, but finite (0.3%–0.5%), non-perturbative $c\bar c$-component in proton wave function. It helps to explain some effects in charm production processes which are difficult for understanding in the assumption that charm quarks were produced only in hard partonic subprocesses [@1].
The $J/\psi$ photoproduction process is the best source of information about gluon structure function in a proton [@5], especially in the region of very small $x$[@5b]. In the deep inelastic domain $J/\psi$ photoproduction process may be described using so-called colour singlet model [@6; @7], in which $J/\psi$ is produced in $\gamma g
\to J/\psi g$ partonic subprocess. Obviously, in the proton fragmentation region the contribution of the intrinsic charm in proton via $\gamma c\to J/\psi c$ subprocess is too comparably large [@8]. The partonic subprocess $\gamma c\to J/\psi c$ gives the main contribution to associated $J/\psi$ plus open charm photoproduction and may be very clean test for charm quark sea in a proton.
In Sec.2 we shall obtain amplitude, differential and total cross section for partonic subprocess $\gamma c\to J/\psi c$ using perturbative theory of QCD and non-relativistic quark model. The connection of the partonic and measurable cross sections for processes $\gamma p\to J/\psi c X$ and $\gamma p\to J/\psi X$ is discussed in Sec.3. In that part we also present charm quark distribution functions which are used in calculations. In Sec.4 we shall calculate $z$-spectra and total cross sections for associated $J/\psi$ plus open charm photoproduction process as well as the $\gamma c\to J/\psi c$ subprocess contribution to the inclusive $J/\psi$ photoproduction cross section at high energies.
Partonic subprocess $\gamma c\to J/\psi c$
==========================================
In the lowest order of QCD perturbative theory associated $J/\psi$ plus open charm photoproduction in $\gamma p$-interactions corresponds to partonic subprocess $\gamma c\to J/\psi c$. It is described by Feynman diagrams which are shown in Fig.1. The quarkonium is represented as non-relativistic quark-untiquark bound system in singlet colour state with specified mass $M=2m$ (m is c-quark mass) and spin-parity $J^p=1^-$.
The amplitude of subprocess $\gamma c\to J/\psi c$ can be expressed in the form: $$M=M_1+M_2+M_3+M_4,$$ where $$M_1=e_qeg^2\bar U(q')\hat\varepsilon_{\gamma}\frac{\hat q'-\hat k+m}
{(k-q')^2-m^2}T^a\gamma_{\mu}\frac{\delta^{ab}g_{\mu\nu}}{(p-q)^2}
T^b\hat P\gamma_{\nu}U(q),$$ $$M_2=e_qeg^2\bar U(q')\gamma_{\mu}T^b\hat
P\frac{\delta^{ab}g_{\mu\nu}}{(p+q')^2}
\gamma_{\nu}T^a\frac{\hat k+\hat q+m}
{(k+q)^2-m^2}
\hat\varepsilon_{\gamma}U(q),$$ $$M_3=e_qeg^2\bar U(q')\gamma_{\mu}T^a\hat
P\frac{\delta^{ab}g_{\mu\nu}}{(p+q')^2}
\hat\varepsilon_{\gamma}\frac{\hat p-\hat k+m}
{(p-k)^2-m^2}
\gamma_{\nu}T^bU(q),$$ $$M_4=e_qeg^2\bar U(q')\gamma_{\mu}T^a
\frac{\hat k-\hat p+m} {(p-k)^2-m^2} \hat\varepsilon_{\gamma}\hat P
\frac{\delta^{ab}g_{\mu\nu}}{(p-q)^2}\gamma_{\nu}U(q).$$ In these formula: $$\hat P=\frac{F_c}{\sqrt 2}A\hat\varepsilon_J(\hat p+m),$$ $A=\Psi(0)/\sqrt m$, $F_c=\delta^{kr}/\sqrt{3}$, k and r are colour indexes of charm quarks, $T^a=\lambda^a/2$, $e_q$ is electrical charge of $Ó$-quark in units of $e$. It is well known that $\Psi(0)$, which is equal to $J/\psi$ wave function at zero point, can be extracted in the lowest order of perturbative QCD from the leptonic decay width of the $J/\psi$: $$\Gamma_{ee}=4\pi e_q^2\alpha^2\frac{|\Psi(0)|^2}{m^2}.$$ We shall put in our calculation $\Gamma_{ee}=5.4$ KeV [@9].
If we average and sum over spins and colours of initial and final particles, we obtain the expression for square of matrix element: $$|\bar M|^2=\frac{B_{\gamma c}}{m^2}\sum_{j\geq i=1}^{4}K_{ij}
(\tilde s,\tilde t,\tilde u),$$ where $$B_{\gamma c}=\frac{32 \pi^2\alpha_s^2\Gamma_{ee}m}
{9\alpha},$$ $$\tilde s=\hat s/m^2,\qquad \tilde t=\hat t/m^2,\qquad \tilde u=\hat
u/m^2,$$ $\hat s, \hat t, \hat u$ are usual Mandelstam variables and $\hat s+\hat t+\hat u=6m^2$. The explicit analytical formula for functions $K_{ij}$ have the following forms:
$$\begin{aligned}
K_{11}&=&-(2\s2\t2-2\s2+\t2^2\u2-4\t2^2-8\t2\u2+14
\t2+7\u2-106)\nonumber\\
&&/(4*(\t2^4-4\t2^3+6\t2^2-4\t2+1))\end{aligned}$$
$$\begin{aligned}
K_{12}&=&(\s2^3-\s2^2\t2-6\s2^2-\s2\t2^2-2\s2\t2\u2+16
\s2\t2-\s2\u2^2+8\s2\u2\nonumber\\
&& -28\s2+\t2^3-6\t2^2-\t2\u2
^2+8\t2\u2-28\t2+4\u2^2-126\u2+276)/\nonumber\\
&& (4(\s2^2\t2^2-2\s2^2\t2+\s2^2-2\s2\t2^2+4\s2\t2-
2\s2+\t2^2-2\t2+1))\end{aligned}$$
$$\begin{aligned}
K_{13}&=&(\s2^2\t2+\s2^2+2\s2\t2^2+\s2\t2\u2-22\s2\t2+7
\s2\u2+16\s2+\nonumber\\
&& 2\t2^2\u2-24\t2^2-11\t2\u2+217
\t2-2\u2^2+41\u2-511)/(2*(\s2\t2^2\u2-4\s2\nonumber\\
&& \t2^2-2\s2\t2\u2+8\s2\t2+\s2\u2-4\s2-\t2^2\u2+4
\t2^2+2\t2\u2-8\t2-\u2+4))\end{aligned}$$
$$\begin{aligned}
K_{14}&=&-(\s2^3+\s2^2\t2-18\s2^2+\s2\t2^2+\s2\t2\u2-16
\s2\t2-\s2\u2^2-3\s2\u2+\nonumber\\
&&166\s2-11\t2^2-5\t2\u2+
115\t2+13\u2^2-65\u2-335)/\nonumber\\
&&(2*(\t2^3\u2-4
\t2^3-3\t2^2\u2+12\t2^2+3\t2\u2-12\t2-\u2+4))\end{aligned}$$
$$\begin{aligned}
K_{22}&=&-(\s2^2\u2-4\s2^2+2\s2\t2-8\s2\u2+14\s2-2
\t2+7\u2-106)/\nonumber\\
&&(4*(\s2^4-4\s2^3+6\s2^2-4\s2+1))\end{aligned}$$
$$\begin{aligned}
K_{23}&=&-(\s2^2\t2-11\s2^2+\s2\t2^2+\s2\t2\u2-16\s2\t2-
5\s2\u2+115\s2+\nonumber\\
&&\t2^3-18\t2^2-\t2\u2^2-3\t2\u2+
166\t2+13\u2^2-65\u2-335)/\nonumber\\
&&(4(\s2^3\u2-4
\s2^3-3\s2^2\u2+12\s2^2+3\s2\u2-12\s2-\u2+4))\end{aligned}$$
$$\begin{aligned}
K_{24}&=&(2\s2^2\t2+2\s2^2\u2-24\s2^2+\s2\t2^2+\s2\t2
\u2-22\s2\t2-11\s2\u2+\nonumber\\
&& 217\s2+\t2^2+7\t2\u2+16
\t2-2\u2^2+41\u2-511)/(2*(\s2^2\t2\u2-4\s2^2\t2\nonumber\\
&& -\s2^2\u2+4\s2^2-2\s2\t2\u2+8\s2\t2+2\s2
\u2-8\s2+\t2\u2-4\t2-\u2+4))\end{aligned}$$
$$\begin{aligned}
K_{33}&=&-(2\s2^2+2\s2\t2+6\s2\u2-46\s2+\t2^2\u2-10
\t2^2-\nonumber\\
&&4\t2\u2+74\t2+2\u2^3-22\u2^2+53\u2-118)/
(\s2^2\u2^2-8\s2^2\u2+\nonumber\\
&& 16\s2^2-2\s2\u2
^2+16\s2\u2-32\s2+\u2^2-8\u2+16)\end{aligned}$$
$$\begin{aligned}
K_{34}&=&-(2*(\s2^2+\s2\t2-11\s2+\t2^2+2\t2\u2-19\t2+\u2
^3-9\u2^2+28\u2+11))/\nonumber\\
&&(\s2\t2\u2^2-8\s2
\t2\u2+16\s2\t2-\s2\u2^2+8\s2\u2-16\s2\nonumber\\
&&-\t2\u2^2+8
\t2\u2-16\t2+\u2^2-8\u2+16)\end{aligned}$$
$$\begin{aligned}
K_{44}&=&-(\s2^2\u2-10\s2^2+2\s2\t2-4\s2\u2+74\s2+2
\t2^2+6\t2\u2-\nonumber\\
&&46\t2+2\u2^3-22\u2^2+53\u2-118
)/ (\t2^2\u2^2-8\t2^2\u2+16\t2^2-2\t2\u2\nonumber\\
&& ^2+16\t2\u2-32\t2+\u2^2-8\u2+16)\end{aligned}$$
The differential cross section for subprocess $\gamma c\to J/\psi c$ can be written as follows: $$\frac{d\hat\sigma}{d\hat t}=\frac{1}{16\pi(\hat s-m^2)^2}|\bar M|^2$$ The total cross section will be obtained after integration over $\hat
t$ in limits: $$\hat t_{max\atop{min}}=
m^2-\frac{\hat s-m^2}{2\hat s}[\hat s-3m^2\pm
\sqrt{(\hat s-9m^2)(\hat s-m^2)}].$$ This procedure can be made analytically, we find that: $$\hat\sigma(\gamma c\to J/\psi c)=\frac{B_{\gamma c}}
{16\pi(\hat s-m^2)^2}\sum_{j\geq i=1}^{4}[H_{ij}(\tilde s,\tilde t
_{max})-H_{ij}(\tilde s,\tilde t_{min})],$$ where $\tilde t_{max}=\hat t_{max}/m^2$ and $\tilde t_{min}=\hat
t_{min}/m^2$. The explicit expressions for functions $H_{ij}(\tilde s,\tilde t)$ are more unwieldy than for functions $K_{ij}$ and the FORTRAN expression for $H_{ij}$ can be obtained by E-mail from the author on request [^1].
We shall also calculate the dominant contribution in total and differential inclusive $J/\psi$ photoproduction cross sections from $\gamma g\to J/\psi g$ partonic subprocess. Here we present main formula for this one without discussion (see, for example [@6]).
First the partonic differential cross section is given by: $$\frac{d\hat\sigma}{d\hat t}(\gamma g\to J/\psi g)=
B_{\gamma g}M_J^4F(\hat s,\hat t),$$ where $$B_{\gamma g}=\frac{8\pi\alpha_s^2\Gamma_{ee}}{3\alpha M_J},$$ $$F(\hat s,\hat t)=\frac{1}{\hat s^2}
\left[ \frac{ \hat s^2(\hat s-M_J^2)^2+\hat t^2(\hat t-M_J^2)^2
+\hat u^2(\hat u-M_J^2)^2}
{(\hat s-M_J^2)^2(\hat t-M_J^2)^2(\hat u-M_J^2)^2}\right].$$ Here: $\hat s+\hat t+\hat u=M_J^2$, $\hat t_{max}=0$ É $\hat t_{min}=
-\hat s+M_J^2.$
The total partonic cross section reads: $$\hat\sigma(\gamma g\to J/\psi g)=B_{\gamma g}M_J^4\Phi(\hat s),$$ where $$\Phi(\hat s)=\frac{2}{(\hat s+M_J^2)^2}\left[\frac{\hat s-M_J^2}
{\hat s M_J^2}-\frac{2\ln(\hat s/M_J^2)}{\hat s+M_J^2}
\right]+$$ $$\frac{2(\hat s+M_J^2)}{\hat s^2M_J^2(\hat s-M_J^2)}-
\frac{4\ln(\hat s/M_J^2)}{\hat s(\hat s-M_J^2)^2}.$$
Associated $J/\psi$ plus charm photoproduction on proton
========================================================
Let us consider the kinematic for process $\gamma p\to J/\psi+c+X$ in the rest frame of the proton (the lab. frame). Variables of partonic subprocess $\gamma c\to J/\psi c$ and variables describing measurable process are connected as follows: $$\hat s=xs+m^2\mbox{, }\hat t=5m^2-xzs\mbox{, }\hat u=-xs(1-z),$$ where (in lab. frame) $s=2m_pE_{\gamma}$, $z=E_J/E_{\gamma}$, $x$ is a momentum fraction of charm quark in the proton. In the general factorization approach of QCD the measurable cross section $\sigma$ and partonic cross section $\hat\sigma$ are connected by the following expressions: $$\sigma(\gamma p\to J/\psi c X)=
\int_{x_{min}}^{1}dxC_p(x,Q^2)\hat\sigma(\gamma c\to J/\psi c),$$ where $x_{min}=8m^2/s$, $Q^2=M_J^2$ and $$\frac{d\sigma}{dz}(\gamma p\to J/\psi cX)=
-s\int dx xC_p(x,Q^2)\frac{d\hat\sigma}{d\hat t}(\gamma c\to J/\psi c),$$ where the region of integration over $x$ is defined by the condition: $\hat t_{min}<5m^2-xzs<\hat t_{max}$.
At present the direct experimental information about charm quark distribution function in a proton in wide region of $x$ is practically absent. Existing parameterizations of $C_p(x,Q^2)$ are very different. For comparison we shall use in our calculation “hard” scaling parameterization [@1]: $$C_p(x,Q^2)=C_p(x)=18x^2[(1-x)(1+10x+x^2)/3+2x(1+x)\ln x]$$ and “soft”, based on perturbative QCD, parameterization [@10] at the scale $Q^2=M_J^2$: $$xC_p(x,Q^2)=(s-s_c)^a(1+Bx)(1-x)^D\exp\left (-E+\sqrt{E's^b\ln(1/x)}\right ),$$ where $$a=1.01,\quad b=0.37,\quad s_c=0.888,\quad B=4.24-0.804s,
\quad D=3.46+1.076s,$$ $$E=4.61+1.49s,\quad E'=2.555+1.961s,\quad \mu^2=0.25\quad GeV^2,
\quad \Lambda=0.232\quad GeV,$$ $$s=\ln\left(\frac{\ln(Q^2/\Lambda^2)}{\ln(\mu^2/\Lambda^2)}\right).$$
Note that mean value of the proton momentum, which is carried by charm quarks, is equal to 0.3% in the case of parameterization [@1] and approximately 0.5% in the case of parameterization [@10], which does not contradict data from EMC Collaboration on $F_2^c(x,Q^2)$ [@11].
These parameterizations for $C_p(x,Q^2)$ have very different physical interpretation. The scaling parameterization [@1], so-called “intrinsic”, was obtained assuming existence of a small, but finite, non-perturbative charm component in proton wave function. It is necessary for description of open charm and $J/\psi$ production total cross section and $x_F$ spectra at $x_F\to 1$ in hadron-hadron and hadron-nucleus collisions. On the contrary, the parameterization [@10], so-called “extrinsic”, strongly depends on choice of scale $Q^2$, because it was obtained in perturbative QCD approach assuming that at $Q^2<Q^2_{min}$ charm quarks in the proton are absent and they are generated in QCD cascade only at large $Q^2$.
In Fig.2 the $x$ dependence of the charm distribution function is shown at $Q^2=M_J^2$. As it will be discussed later different $x$- dependences of the charm distribution functions give us very different predictions for total $J/\psi$ photoproduction cross section as a function of energy.
Results and discussion
======================
Result of calculation of total cross section as function of photon energy $E_{\gamma}$ for associated $J/\psi$ plus open charm photoproduction in approach discussed above is shown in Fig.3. “Intrinsic” parameterization [@1] predicts largest value of cross section at energies $E_{\gamma}\le 60$ GeV, but this cross section rapidly decrease as energy growth at $E_{\gamma}\ge 100$ GeV. On the contrary, the cross section, which was calculated using “extrinsic” parameterization [@10], monotonously increase from $5\dot 10^{-3}$ nbn at the energy $E_{\gamma}=50$ GeV to 0.4 nbn at the energy $E_{\gamma}=5$ TeV and at the energy range of $ep$-collider HERA ($\sqrt s_{\gamma
p}=200$ GeV) it is approximately equal to 0.8 nbn (Fig.4). Such a way we have sufficiently large measurable cross section of the $J/\psi$ plus open charm production for direct experimental investigation of the $x$-dependence of charm quark distribution function in a proton.
The study of the $J/\psi$ photoproduction in $\gamma p$- interactions concentrated on inelastic $J/\psi$ production as a tool to obtain information on the gluon distribution function in a proton [@5]. That is why we have calculated the contribution of the $\gamma c\to
J/\psi c$ partonic subprocess in the total inclusive $J/\psi$ photoproduction cross section. Fig.4 shows result of calculation this contribution using “extrinsic” parameterization [@10] as well as contribution of the dominant photon-gluon fusion subprocess as functions of $\sqrt s_{\gamma p}$. It is known that $\gamma g\to
J/\psi g$ partonic subprocess gives contribution in the total inclusive $J/\psi$ photoproduction cross section approximately equal to 50% at the energy $\sqrt s_{\gamma p}=200$ GeV [@5]. Our calculation shows that total contribution of the subprocesses $\gamma
c\to J/\psi c$ and $\gamma\bar c\to J/\psi\bar c$ at the energy $\sqrt
s_{\gamma p}=200$ GeV is equal to 4% of the photon-gluon fusion subprocess contribution or 2% of the total contribution of all mechanisms. The contribution of the charm quarks in $J/\psi$ photoproduction cross section may be significant and approximately equal to contributions of elastic (5%) or diffractive (2%) mechanisms [@5]. But at smaller energies $\sqrt s_{\gamma p}=10-20$ GeV charm quark and untiquark contribution is 15% of photon-gluon fusion contribution in the case of parameterization [@1] for charm quarks in the proton.
The contribution of the proton intrinsic charm in $J/\psi$ photoproduction must be very large in the region of proton fragmentation that is at small $z=E_J/E_{\gamma}$ in the proton rest frame. Figs. 5 and 6 show results of calculation for contributions $\gamma c\to J/\psi c$ and $\gamma g\to J/\psi g$ partonic subprocesses in $z-$spectra of the $J/\psi$ photoproduction at energies $\sqrt s_{\gamma p}=14.7$ GeV and $\sqrt s_{\gamma p}=200$ GeV, correspondingly. The charm quark contribution is larger than photon-gluon fusion subprocess contribution at $z<0.2$ for both energies. At the relatively small energy ($\sqrt s_{\gamma p}=14.7$ GeV) the parameterization [@1] gives more large contribution. At the energy $\sqrt s_{\gamma p}=200$ GeV the contribution of parameterization [@1] is largest only for very small $z< 10^{-3}$, in the region $10^{-3}<z<0.2$ the contribution of parameterization [@10] is dominant.
In conclusion we note that the comparison of our results with data didn’t made because data usually have been obtained in fixed kinematical region of variables $z$ and $p_T$[^2]. In contrary in our approach total kinematical region (It is bounded by conservation laws) of these variables have been took into account. But all calculations were made in the same kinematical approximation and we can to compare relative contributions of the different mechanisms with data. The calculations for total cross section, $z$ and $p_T$ spectra of the $J/\psi$ photoproduction via partonic subprocess $\gamma c\to J/\psi c$ in fixed kinematical region (HERA collider, for example) will be present in future publications.
I would like to thank Prof. N.Zotov and Dr. A.Martynenko for useful discussions and particularly Prof. A.Likhoded for remarkes. The work was supported in part by the Russia Foundation for Basic Research under Grant 93-02-3545.
[99]{} Brodsky S.J. et al. Phys.Lett. 1980, V.93B, P.451; Brodsky S.J., Peterson C. Phys.Rev.1981, V.D23, P.2745. Barger V., Halzen F., Keung W.Y. Phys.Rev.1982, V.D25, P.112. Odorico R. Nucl.Phys.1982, V.B209, P.77. Vogt R., Brodsky S.J., Hoyer P. Preprint SLAC-Pub-5827, May, 1992. Jung H., Schuller G.A., Terron J. Int.J.Mod.Phys.1992, V.A32, P.7955; Preprint DESY-92-028, 1992. Saleev V.A., Zotov N.P. Mod.Phys.Lett.A, 1994, in publ.; In Proc. “Physics at HERA”, V.1, P.637, Hamburg, 1991. Berger E.L., Jones D. Phys.Rev.1981, V.D23, P.1521. Baier R., Ruckl R. Nul.Phys.1983, V.B218, P.289; 1982, V.B20, P.1. Saleev V.A. In Proc. of Conf. “Non-commutative structures in mathematical physics”, Togliatti, p.100-107, October, 1993. Rewiew of Particle Properties. Phys.Rev.1992, V.D45. Gluck M., Reya E., Vogt A. Phys.Rev. 1992, V.D45, P.3986; 1992, V.D46, P.1973. Aubert J.J. et al. Phys.Lett.1982, V.110B, P.73; Nucl.Phys.1983, V.B213, P.31.
[**Figure captions**]{}
1. Diagrams used to describe the partonic subprocess $\gamma c\to J/\psi c$.
2. The charm distribution function in a proton versus $x$ at $Q^2=M_J^2$. The curver 1 corresponds to parameterization [@1], the curve 2 - [@10].
3. The cross section of the process $\gamma p\to J/\psi Ó X$ as a function of photon energy $E_{\gamma}$. Curves as the same Fig.2.
4. The cross section of the process $\gamma p\to J/\psi X$ as a function of $\sqrt s_{\gamma p}$. The curver 1 corresponds to subprocess $\gamma c\to
J/\psi c$ (parameterization for $C_p(x,Q^2)$ is taken from [@10]), the curver 2 - $\gamma g\to J/\psi g$ (parameterization for $G_p(x,Q^2)$ is taken from [@10] too).
5. The $z$- spectrum of $J/\psi$ in $\gamma p$- interaction at $\sqrt s_{\gamma p}=14.7$ çÜ÷. Curvers 1 and 2 - contributions of $\gamma c\to J/\psi c$ subprocess used parameterizations [@1] and [@10], correspondingly. The curver 3 - contribution of photon-gluon fusion mechanism.
6. As the same Fig. 5 at $\sqrt s_{\gamma p}=200$ GeV.
[**Abstract**]{}
Based on a perturbative theory of quantum chromodynamics and non-relativistic quark model, associated $J/\psi$ plus open charm photoproduction on charm quarks in a proton via partonic subprocess $\gamma c\to J/\psi c$ is discussed. It is shown that the value and energy dependence of the cross section for such process remarkably depends on the choice of charm distribution function in a proton. In the region of small $z=E_J/E_{\gamma}<0.2$ the contribution of the $\gamma c\to J/\psi c$ subprocess in the inelastic $J/\psi$ photoproduction spectra is larger than the contribution of the photon-gluon fusion subprocess. At the energy range of HERA collider charm quarks contribution in the total inclusive $J/\psi$ photoproduction cross section may be equal to 4% of the dominant contribution of photon-gluon fusion mechanism.
\#1\#2\#3\#4\#5\#6
(\#1,\#2)
(\#4,\#5)
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(125.00,94.20) (50.00,69.04)[(2.00,9.89)\[\]]{} (32.00,91.19)[(0,0)\[cc\][k]{}]{} (32.00,54.20)[(0,0)\[cc\][q]{}]{} (56.00,86.89)[(0,0)\[cc\][$q'$]{}]{} (44.00,77.86)[(0,0)\[cc\][p]{}]{} (44.00,58.93)[(0,0)\[cc\][p]{}]{} (72.00,68.82)[(0,0)\[cc\][$p_J=2p$]{}]{} (114.00,81.94)[(10.00,2.15)\[\]]{} (50.00,22.58)[(2.00,9.89)\[\]]{} (112.00,27.75)[(2.00,9.89)\[\]]{}
Fig. 1
[^1]: saleev@univer.samara.su
[^2]: The introduction of the normalization K-faktor is needed too
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bibliography:
- 'references.bib'
title: 'Mapping Navigation Instructions to Continuous Control Actions with Position-Visitation Prediction'
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abstract: |
Let $k$ be a field, $G$ be an abelian group and $r\in {\mathbb N}.$ Let $L$ be an infinite dimensional $k$-vector space. For any $m\in {\operatorname{End}}_k(L)$ we denote by $r(m)\in [0,\infty ]$ the rank of $m$. We define by $R(G,r,k)\in [0,\infty ]$ the minimal $R$ such that for any map $A:G \to {\operatorname{End}}_k(L)$ with $r(A(g'+g'')-A(g')-A(g''))\leq r$, $g',g''\in G$ there exists a homomorphism $\chi :G\to {\operatorname{End}}_k(L)$ such that $r(A(g)-\chi (g))\leq R(G, r, k)$ for all $g\in G$.
We show the finiteness of $R(G,r,k)$ for the case when $k$ is a finite field, $G=V$ is a $k$-vector space $V$ of countable dimension. We actually prove a generalization of this result.
In addition we introduce a notion of [*Approximate Cohomology*]{} groups $H^k_{\mathcal F}(V,M)$ (which is a purely algebraic analogue of the notion of ${\epsilon}$-representation ([@ep])) and interperate our result as a computation of the group $H^1_{\mathcal F}(V,M)$ for some $V$-modules $M$.
address:
- 'Einstein Institute of Mathematics, Edmond J. Safra Campus, Givaat Ram The Hebrew University of Jerusalem, Jerusalem, 91904, Israel'
- 'Einstein Institute of Mathematics, Edmond J. Safra Campus, Givaat Ram The Hebrew University of Jerusalem, Jerusalem, 91904, Israel'
author:
- David Kazhdan
- Tamar Ziegler
title: Approximate cohomology
---
[^1]
Introduction
============
Let $k$ be a field, $G$ be an abelian group and $r\in {\mathbb N}.$ Let $L$ be an infinite dimensional $k$-vector space, $End(L)=End_k(L)$ and $M\subset {\operatorname{End}}(L)$ be the subspace of operators of finite rank. For any $m\in {\operatorname{End}}(L)$ we denote by $r(m)\in [0,\infty ]$ the rank of $m$. We define by $R(G,r,k)\in [0,\infty ]$(correspondingly $R^f(G,r,k))$ the minimal number $R$ such that for any map $A:G\to {\operatorname{End}}(L)$(correspondingly a map $A:G\to M)$, $g\to m_g$ with $r(A(g'+g'')-A(g')-A(g''))\leq r$, $g',g''\in G$ there exists a homomorphism $\chi :G\to {\operatorname{End}}(L)$ such that $r(A(g)-\chi (g))\leq R(G,r,k)$ for all $g\in G$.
It is easy to see that in the case when $G={{\mathbb Z}}$ and $\operatorname{char} (k)\neq 2$ we have $R({{\mathbb Z}},1, {\mathbb F})\leq 2$. We sketch the proof: one studies the rank $\le 1$ operators $r_{m,n}=A(m+n)-A(m)-A(n)$. Since $r_{0,0}$ of rank $\le 1$, we replace $r_{m,n}$ by $r_{n,m}-r_{0,0}$ and can then assume that $r_{0,0}=0$. Under this condition one must show that $r_{n,m}$ is a coboundary of rank one operators. The operators $r_{n,m}$ satisfy the equation $r_{a,-a}=r_{a+c, -a}+r_{a,c}$. From this one can deduce that either there is a subspace of codimension $1$ in the kernel of all three operators, or a subspace of dimension $1$ containing the image of all three. One shows inductively that this property holds for all operators $r_{m,n}$. Unfortunately we don’t know whether $R({{\mathbb Z}},2,{{\mathbb C}})<\infty$.
In this paper we first show that $R^f(V,r,k)<\infty$ in the case when $k={\mathbb F}_p$ and $G$ is a $k$- vector space $V$ of countable dimension and then show that $R(V,r,k)=R^f(V,r,k)$.
Actually we prove the analogous bound in a more general case when $M$ is replaced by the space of tensors $L^1\otimes L^2\otimes ...\otimes L^n$. To simplify the exposition we assume that $L_i=L^\vee$ and that $Im(A)$ is contained in the subset $Sym^d(L^\vee)\subset {L^\vee}^{\otimes n}$ of symmetric tensors. In other words we consider map $A:V\to M^d$ where $M^d$ is isomorphic to the space of homogeneous polynomials of degree $d$ on $L$. We denote by $N^d\subset M^d$ the subspace of multilinear polynomials.
Now some formal definitions.
Let $M$ be an abelian group. A [*filtration*]{} ${\mathcal F}$ on $M$ is an increasing sequence of subsets $M_i, 1\leq i\leq \infty$ of $M$ such that for each $i,j$ there exists $c(i,j)$ such that $M_i+M_j\subset M_{c(i,j)}$ and $M=\cup _iM_i$. Two filtrations $M_i,M_i'$ on $M$ are equivalent if there exist functions $a,b: \mathbb Z _+\to \mathbb Z _+$ such that $M_i\subset M_{a(i)}'$ and $M'_i\subset M_{b(i)}$.
\[Algebraic rank filtration\] Let $k$ be a field. Fix $d\geq 2$ and consider the $k$-vector space $ M=M^d$ of homogeneous polynomials $P$ of degree $d$ in variables $x_j$, $j\geq 1$. For a non-zero homogeneous polynomial $P$ on a $k$-vector space $W$, $P\in k[W^\vee ]$ of degree $d\geq 2$ we define the [*rank*]{} $r(P)$ of $P$ as $r$ ,where $r$ is the minimal number $r$ such that it is possible to write $P$ in the form $$P = \sum ^ r_{i=1} l_iR_i,$$ where $l_i, R_i\in \bar k[W^\vee ]$ are homogeneous polynomials of positive degrees (in [@schmidt] this is called the $h$-invariant). We denote by ${\mathcal A}_d$ the filtration on $M$ such that $M_n$ is the subset of polynomials $P$ with $r(P) < n$. For the $k$-space $\mathcal P^d$ of non-homogeneous polynomials of degree $\le d$ define the rank similarly, and denote by ${\mathcal B}_d$ the corresponding filtration.\
Let $V$ be a countable vector space over $k$. We say that a linear map $P:V\to M^d$ is of [*finite rank*]{} if we can write $P$ as a sum $\sum _{k=1}^{d-1}P_k$ where each $P_k$ is a finite sum $P_k=\sum _jQ_jR_j$ where $R_j\in M^k$ and $Q_j$ is a linear map from $V$ to $M^{d-k}$. Denote ${\operatorname{Hom}}_f(V,M^d)$ the subspace of ${\operatorname{Hom}}(V,M^d) $ of finite rank maps.
Now we can formulate our main result.
\[Main\] For any finite field $k={\mathbb F}_p ,d<p-1, r\geq 0$ there exists $R=R(r,k,d)$ such that for any map $$A:V\to M=M^d, \quad r(A(v'+v'')-A(v')-A(v''))\leq r, \quad v',v''\in V$$ there exists a homomorphism $\chi_A : V\to M$ such that $r(A(v)-\chi_A (v))\leq R$. Moreover the homomorphism $\chi _A$ is unique up to an addition of a homomorphism of a finite rank.
$a)$ Is there a bound on $R$ independent of $k$ ? Moreover, does there exist $c(d)$ such that $R(V,r,k,d)\leq c(d)r$?
$b)$ Could we drop the condition $d<p-1$ if $Im(A)\subset N^d$?
Theorem \[Main\] does not hold for $p = d= 2$, see [@tao-example] for a function from ${\mathbb F}_2^n$ to the space of quadratic forms over ${\mathbb F}_2$ such that $f(u+v)-f(u)-f(v)$ is of rank $\le 3$ but for $n$ sufficiently large $f$ does not differ from a linear function by a function taking values in bounded rank quadratics. In the low characteristic case the same proof shows that obstructions come from [*non-classical polynomials*]{} see Remark \[nonclassical\]. In the case when $G$ is a finite cyclic group, $k$ any field, $d=2$, one can show that $C(1,k)\le 2$.
We can reformulate Theorem \[Main\] as an example of a computation of [*approximate cohomology* ]{} groups.
1. Let $M$ be an abelian group, and let ${\mathcal F}=\{M_i\}$ be a filtration on $M$.
2. Let $G$ be a discrete group acting on $M$ preserving the subsets $M_i$. A cochain $r:G^n\to M$ is an [*approximate $n$-cocycle*]{} if $Im
(\partial r)\subset M_i$ for some $i\in {{\mathbb Z}}_+$. It is clear that the set $Z^n_{\mathcal F}$ of approximate $n$-cocycles is a subgroup of the group $C^n$ of $n$-chains which depends only on the equivalence class of a filtration ${\mathcal F}$.
3. A cochain $r:G^n\to M$ is an [*approximate $n$-coboundary*]{} if there exists an $n-1$-cochain $t\in C^{n-1}$ such that $Im (r-\partial t)\subset M_i$ for some $ i\in {{\mathbb Z}}_+$. It is clear that the set $B^n_{\mathcal F}$ of approximate $n$-coboundaries is a subgroup of $ \tilde Z ^n_{\mathcal F}$.
4. We define $H^n_{\mathcal F}=Z^n_{\mathcal F}/B^n_{\mathcal F}$.
5. Since any cocycle is an approximate cocycle and any coboundary is an approximate coboundary we have a morphism $a^n_{\mathcal F}:H^n(G,M)\to H^n_{\mathcal F}(G,M)$.
In this paper we consider the case when the group $V$ acts trivially on $M$. So the group $Z^1(V,M)$ of $1$-cocycles coincides with the group ${\operatorname{Hom}}(V,M)$ of linear maps, the subgroup of coboundaries $B^1(V,M)\subset Z^1(V,M)$ is equal to $\{ 0\}$ and therefore $H^1(V,M)={\operatorname{Hom}}(V,M)$. In this case we can reformulate the Theorem \[Main\] in terms of a computation of the map $a^1_{\mathcal F}$.
\[trivial-action-V\] Let $k$ be a prime finite field of characteristic $p$, $V$ be a countable vector space over $k$ acting trivially on $(M, {\mathcal F})=(M^d,{\mathcal A}_d)$ and assume that $p>d+1$. Then the map $a^1: H^1(V,M)={\operatorname{Hom}}(V,M)\to \tilde H_{\mathcal F}^1(V,M)$ is surjective, and ${\operatorname{Ker}}(a^1) = {\operatorname{Hom}}_f(V,M_d)$.
How to describe $H_{\mathcal F}^n(V,M)$ for $n>1$ ?
\[translation-action-V\] Let $k$ be a prime finite field of characteristic $p$, and let $V$ be a countable vector space over $k$. Consider the filtration $(\mathcal P^d, {\mathcal B}_d)$ with $W=V$ and with $V$-acting by translations $(v .P)(x) = P(x+v)$ and assume that $p>d+1$. Then the map $a^1_{\mathcal B}$ is surjective.
Let $P: V \to \mathcal P^d$ where $\mathcal P^d$ is the space of polynomial of degree $\le d$. . We assume $$\partial P(v,v')(x)= P(v+v')(x)-P(v)(x+v')-P(v')(x)$$ is of rank $\le i$ for any $v,v' \in V$. Let $Q(v)$ be the homogeneous degree $d$ term of $P(v)$. Then since $P(v)(x+v')-P(v)(x)$ is of degree $<d$ we have $$Q(v+v')(x)-Q(v)(x)-Q(v')(x)$$ is of rank $\le i+1$.
The proof of Theorem \[Main\] uses the inverse theorem for the Gowers norms [@btz; @tz-inverse]. One can use also prove the reverse implication modifying the arguments in [@sam], and thus an independent proof of Theorem \[Main\] could lead to a new proof of the inverse conjecture for the Gowers norms.
Proof of Theorem \[Main\]
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For a function $f$ on a finite set $X$ we define $${{\mathbb E}}_{x \in X}f(x) = \frac{1}{|X|}\sum_{x \in X}f(x).$$ We use $X \ll_L Y$ to denote the estimate $|X| \le C(L) |Y|$, where the constant $C$ depends only on $L$. We fix a prime finite field $k$ of order $p$ and degree $d<p-1$ and suppress the dependence of all bounds on $k,d$. We also fix a non-trivial additive character $\psi$ on $k$.\
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For a function $f:G \to H$ a function between abelian groups we denote $\Delta_h f(x) = f(x+h) - f(x)$. if $f:G_1 \times G_2 \to H$ then for $g_1 \in G_1$ we write $\Delta_{g_1} f$ shorthand for $\Delta_{(g_1,0)}f$.\
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Let $V$ be a finite vector space over $k$. Let $F: V \to k$. The $m$-th [*Gowers norm*]{} of $F$ is defined by $$\|\psi(F)\|_{U_m}^{2^m} = {{\mathbb E}}_{v, v_1, \ldots, v_m \in V} \psi(\Delta_{v_m} \ldots \Delta_{v_1} F(v)).$$ These were introduced by Gowers in [@gowers], and were shown to be norms for $m>1$.\
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For a homogeneous polynomial $P$ on $V$ of degree $d$ we define $$\label{multilinear}
\tilde P(x_1, \ldots, x_d) = \Delta_{x_d}\ldots \Delta_{x_1}P(x).$$ This is a multilinear homogeneous form in $x_1, \ldots, x_d \in V$ such that $$P(x) = \frac{1}{d!}\tilde P(x,\ldots, x).$$
\[main-finite\] There exists a function $R(L)$ such that for any finite dimensional $k$-vector space $V$ and a map $P : V \to M= M_d$ such that $r(P(v+v') -P(v) -P(v')) \le L$ for $v,v' \in V$ there exists a linear map $Q : V \to M$ such that $r(P( v) - Q(v))\le R(L)$.
The proof is based on an argument of [@gt-equivalence], [@lovett]. Our aim is to show that if $\partial P(v,v')(x)$ is of rank $\le L$ for all $v,v' \in V$ then there exists a homogeneous polynomial $Q(v)(x)$ of degree $\le d$ such that $\partial P(v,v')=0$ and $P(v)-Q(v) \ll_{L} 1$.
Let $P:V^d \to k$ be a multilinear homogeneous polynomial of degree $d \ge 2$ and rank $L$. Then ${{\mathbb E}}_{\bar x \in V} \psi(P(x))\ge C_{L,d}$ for some positive constant $C_{L,d}$ depending only on $L,d$.
We prove this by induction on $d$. For quadratics: $P(x_1,x_2)= \sum_{i=1}^L l^1_i(x_1)l^2_i(x_2)$. If $x_1 \in \bigcap_i {\operatorname{Ker}}(l_i^1)$ then $ \psi(P({\bar x})) \equiv 1$, thus on a subspace $W$ of codimension at most $L$ we have $ \psi(P({\bar x})) \equiv 1$, so that $$\sum_{x_1 \in W} {{\mathbb E}}_{x_2 \in V }\psi(P({\bar x}))=|W|.$$ If $x_1$ is outside $W$ then the inner sum is nonnegative.\
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Suppose now that $d>2$. Let $P(x_1,\ldots, x_d)$ be multilinear homogeneous polynomial of degree $d$ and rank $L$. Write $$P(x_1, \ldots, x_d) = \sum_{j \le d} \sum_{i\le L_j} l_i^j(x_j)Q_i^j(x_1, \ldots, \hat x_j, \ldots, x_d) + \sum_{k\le M} T_k({\bar x})R_k({\bar x})$$ with ${\bar x} = (x_1, \ldots, x_d)$, $l_i^j$ are linear and $T_k({\bar x})R_k({\bar x})$ is homogenous multilinear in $x_1, \ldots x_d$ such that the degrees of $T_k({\bar x})$ and $R_k({\bar x})$ are $\ge 2$, and $\sum_j L_j+M=L$.\
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If for all $j$ we have $L_j=0$, then $P({\bar x})= \sum_{k\le L} T_k({\bar x})R_k({\bar x})$ with the degree of $T_k({\bar x}),R_k({\bar x}) \ge 2$. For $x_1 \in V$ write $P_{x_1}(x_2, \ldots, x_d) = P(x_1,x_2, \ldots, x_d)$. Then $P_{x_1}$ is of rank $L$ and degree $d-1$ for all $x_1$ and we obtain the claim by induction.\
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Otherwise there is a $j$, such that $L_j>0$, without loss of generality $j=1$. Let $W = \bigcap_{i} {\operatorname{Ker}}(l_i^1)$, then $W$ is of codimension at most $L_1$. For $x_1 \in W$ let $P_{x_1}(x_2, \ldots, x_d) = P(x_1,x_2, \ldots, x_d)$. Consider the sum $${{\mathbb E}}_{x_2, \ldots, x_d \in V} \psi(P_{x_1}(x_2, \ldots, x_d)).$$ For $x_1 \in W$ we have $P_{x_1}$ is of rank $\le L-L_1$ and homogeneous of degree $d-1$. By the induction hypothesis the above sum is $\ge C_{L, d-1}$, so that $$\sum_{x_1 \in W} {{\mathbb E}}_{x_2, \ldots, x_d \in V} \psi(P_{x_1}(x_2, \ldots, x_d)) \ge C_{L, d-1} |W|.$$ For any $x_1 \notin W$, $P_{x_1}$ is of degree $d-1$, and of rank $<\infty$ thus by the induction hypothesis, $${{\mathbb E}}_{x_2, \ldots, x_d \in V} \psi(P_{x_1}(x_2, \ldots, x_d)) \ge 0.$$ Thus $${{\mathbb E}}_{x_1 \in V} {{\mathbb E}}_{x_2, \ldots, x_d \in V} \psi(P_{x_1}(x_2, \ldots, x_d)) \ge C_{L, d-1} |W|/|V|,$$ and we obtain the claim.
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Let $P:V\to M_d$ be a map such that for all $u,v$: $$rk(P(v)+P(u) - P(v+u))<L.$$ We define a function on $V\times W^d$ by $$f(v,x_1, \ldots, x_d) = \psi(\tilde P(v)(x_1, \ldots, x_d)) = \psi(\tilde P(v)({\bar x})).$$
$\|f\|_{U^{d+2}} \ge c_{L}$.
We expand $$\Delta_{(v_{d+2},{\bar h_{d+2}})} \ldots \Delta_{(v_1,{\bar h_1})}\tilde P(v)({\bar x}) = \sum_{k=0}^{d+2} \Delta_{{\bar h_{d+2}}} \ldots \Delta_{{\bar h_{k+1}}}( \Delta_{v_k} \ldots \Delta_{v_1}(\tilde P(v)))({\bar x}+{\bar h_1 + \ldots + h_k})$$ with $v_i \in V$ and ${\bar h_i} \in V^d$. Since $P_v$ is of degree $d$ the above is equal $$\sum_{k=2}^{d+2} \Delta_{{\bar h_{d+2}}} \ldots \Delta_{{\bar h_{k+1}}}( \Delta_{v_k} \ldots \Delta_{v_1} \tilde P(v))({\bar x}+{\bar h_1 + \ldots +\bar h_k})$$ Since $rk(P(v)+P(u) - P(v+u))<L$, for any $v_1+u_1=v_2+u_2$ we have $$rk(P(v_1)+P(u_1) - P(v_2)-P(u_2) )< 2L.$$ that for $k\ge 2$ we have $\Delta_{v_k} \ldots \Delta_{v_1}\tilde P(v)$ is of rank $\ll_L 1$. For fixed $v, v_1, \ldots, v_k$, the above polynomial can be expresses as a multilinear homogeneous polynomial of degree $d$ in $y_1, \ldots, y_d$ with $$y_j= (x_j,h_1^j, \ldots, h_{d+2}^j).$$ which is of rank that is bounded in terms of $L, d$. Now apply previous lemma.
By the inverse theorem for the Gowers norm [@btz; @tz-inverse] there is a polynomial $Q$ on $V^{d+1}$ of degree $d+1$ on with $Q:V \times V^d \to k$ s.t. such that $$|{{\mathbb E}}_{v,x_1, \ldots, x_d }\psi(\tilde P(v)(x_1, \ldots, x_d)-Q(v, x_1, \ldots,x_d)) | \gg_L 1.$$ By an application of the triangle and Cauchy-Schwarz inequalities we obtain $$\begin{aligned}
&|{{\mathbb E}}_{v,x_1, \ldots, x_d }\psi(\tilde P(v)(x_1, \ldots, x_d)-Q(v, x_1, \ldots,x_d)) |^2 \\
& \le \left|{{\mathbb E}}_{v,x_1, \ldots,x_{d-1}}\left| {{\mathbb E}}_{x_d }\psi(\tilde P(v)(x_1, \ldots, x_d)-Q(v, x_1, \ldots,x_d))\right| \right|^2\\
&\le {{\mathbb E}}_{v,x_1, \ldots,x_{d-1}}\left| {{\mathbb E}}_{x_d }\psi(\tilde P(v)(x_1, \ldots, x_d)-Q(v, x_1, \ldots,x_d)) \right|^2\\
&= {{\mathbb E}}_{v,x_1, \ldots,x_{d-1}} {{\mathbb E}}_{x_d,x'_d }\psi(\tilde P(v)(x_1, \ldots, x_d+x'_d) -\tilde P(v)(x_1, \ldots, x_d)-Q(v, x_1, \ldots,x_d+x_d')+Q(v, x_1, \ldots,x_d)) \\
&= {{\mathbb E}}_{v,x_1, \ldots,x_{d-1}} {{\mathbb E}}_{x_d,x'_d }\psi(\tilde P(v)(x_1, \ldots, x_{d-1},x'_d)-\Delta_{x_d'}Q(v, x_1, \ldots,x_d)).
\end{aligned}$$ Where the last equality follows from the fact that $\tilde P$ is homogeneous multilinear form and thus $$\tilde P(v)(x_1, \ldots, x_d+x'_d) -\tilde P(v)(x_1, \ldots, x_d)= \tilde P(v)(x_1, \ldots, x_{d-1},x'_d).$$ Applying Cauchy-Schwarz $d-1$ more times we obtain $${{\mathbb E}}_{v,x_1, x_1', \ldots, x_d,x'_d }\psi(\tilde P_v(x'_1, \ldots, x'_d)-\Delta_{x_1'}\ldots \Delta_{x_d'}Q(v, x_1, \ldots,x_d)) \gg_{L} 1$$ One more application of Cauchy-Schwarz gives $${{\mathbb E}}_{v,v',x_1, x_1', \ldots x_d,x'_d }\psi((\tilde P(v+v')-\tilde P(v))(x'_1, \ldots, x'_d)-\Delta_{v'}\Delta_{x_1'}\ldots \Delta_{x_d'}Q(v, x_1, \ldots,x_d)) \gg_{L} 1.$$ Since $Q$ is a polynomial of degree $d+1$, $\Delta_{v'}\Delta_{x_1'}\ldots \Delta_{x_d'}Q$ is independent of $v,x_1, \ldots, x_d$ so we obtain $$|{{\mathbb E}}_{v,v',x'_1, \ldots, x'_d }\psi((\tilde P(v+v')-\tilde P(v))(x'_1, \ldots, x'_d)-\tilde Q(v', x'_1, \ldots,x'_d)) | \gg_{L} 1$$ with $\tilde Q$ a multilinear homogeneous form in $v', x'_1, \ldots,x'_d$. Denote by $\tilde Q(v)$ the function on $W^d$ given by $\tilde Q(v, \bar x)=Q(v,\bar x)$. Then $${{\mathbb E}}_{v,v'}|{{\mathbb E}}_{x'_1, \ldots, x'_d }\psi((\tilde P(v+v')-\tilde P(v)- \tilde Q(v'))(x'_1, \ldots, x'_d) | \gg_{L} 1.$$ By [@gt-polynomial] (Proposition 6.1) and [@BL] (Lemma 4.17) it follows that for at least $\gg_{L} |V|^2$ values of $v,v'$ we have $ P(v+v')-P(v)- Q'(v')$ is of rank $\ll_{L} 1$, where $Q'(v)(x) = \tilde Q(v)(x, \ldots, x)/d!$ where $Q'(v)$ is linear in $v$. Recall now that $P(v+v')-P(v) - P(v')$ is of rank $\le L$, so that we get a set $E$ of size $\gg_{L} |V|$ of $v$ for which $$P(v) = Q'(v) + R(v)$$ with $R(v)$ of rank $\le L$. Since $E \gg_{L} |V|$, by the Bogolyubov lemma (see e.g. [@wolf])) $2E$ contains a subspace $E'$ of codimension $K \ll_{L} 1$ in $V$. For $v \in E'$, define $$P'(v)=Q'(v).$$ Let $P':V\to M_d$ be any extension of $P'$ linear in $v\in V$. Then $P'(v) - P(v)$ is of rank $\ll_{L} 1$, and $P'(v)$ is a cocycle.
\[nonclassical\] In the case where $p\le d+2$, by the inverse theorem for the Gowers norms over finite fields the polynomial $Q$ in the above argument on $V \times V^d$ would be replaced by a [*nonclassical polynomial*]{} see [@tz-low], and the same argument would give that the approximate cohomology obstructions lie in the nonclassical degree $d$ polynomials - these are functions $P: V \to {\mathbb T}$ satisfying $\Delta_{h_{d+1}} \ldots \Delta_{h_1}P \equiv 0$.
[*Proof of Theorem \[Main\]*]{}. Let $k={\mathbb F}_q$, and let $V$ be an countable vector space over $k$. Denote $V_n=k^n$, then $V=\cup V_n$. Let $M\subset k[x_1,...x_n,...]$ be the subspace of homogeneous polynomials of degree $d$. For any $l\geq 1$ we denote by $N_l\subset M$ the subset of polynomials of in $x_1,...,x_l$ and denote by $p_l:M\to N_l$ the projection defined by $x_i\to 0$ for $ i> l$. Observe that $p_n$ does not increase the rank. By Proposition \[main-finite\] there is a constant $C$ depending only on $L,d$ (and $k$) such that for any $n$ there exists a linear map $\phi _n:V_n\to M$ such that rank $(P(v)-\phi _n (v))\le C$, for $v\in V_n$.
We now show that the existence of such linear maps $\phi _n$ implies the existence of a linear map $\psi :V\to M$ such that $rank (R(v)-\psi (v))\leq C$, for $v\in V$.
\[compatible\] Let $X_n$, $ n\geq 0$ be finite not empty sets and $f_n:X_{n+1}\to X_n$ be maps. Then one can find $x_n\in X_n$ such that $f_n(x_{n+1})=x_n$.
This result is standard, but for the convenience of a reader we provide a proof. The claim is obviously true if the maps $f_n$ are surjective. For any $m>n$ we define the subset $X_{m,n}\subset X_n$ as the image of $$f_n\circ \ldots \circ f_{m-1}:X_m\to X_n$$ It is clear that for a fixed $n$ we have $$X_n\supset X_{n+1,n}\supset \ldots \supset X_{m,n}\supset \ldots$$ \
We define $Y_n$ as the intersection $\cap _{m>n}X_{m,n}$. Since the set $X_n$ is finite, the sets $X_{m,n}$ stabilize as m grows and hence $Y_n$ is not empty. Let ${\tilde}f_n$ be the restriction of $f_n$ on $Y_{n+1}$. Now the maps ${\tilde}f_n:Y_{n+1}\to Y_n$ are surjective, thus the lemma follows.
\[lift\] Let $P:V\to M$ be a map such that for any $n$ there exists a linear map $\phi _n:V_n\to M$ such that rank $(P(v)-\phi _n (v))\leq C$, $v\in V_n$. Then there exists a linear map $\psi :V\to M$ such that $rank (P(v)-\psi (v))\leq C$, for all $v\in V$.
Let $l(n)$ be such that $P(V_n)\subset N_{l(n)}$ and $\psi _n=p_{l(n)}\circ \phi _n$. Since $p_n$ does not increase the rank, and since $(p_{l(n)}\circ P)(V_n)= R(V_n)$ we have $$(\star _n) \qquad rank(P(v)-\psi _n (v))\leq C, \quad v\in V_n.$$ We apply Lemma \[compatible\] to the case when $X_n$ is the set of linear maps $\psi _n:V_n\to N_{l(n)}$ satisfying $(\star _n)$ and $f_n$ are the restriction from $V_{n+1}$ onto $V_n$ we find the existence of linear maps $\psi _n:V_n\to N_{l(n)}$ satisfying the condition $(\star _n)$ and such that the restriction of $\psi _{n+1}$ onto $V_n$ is equal to $\psi _n.$ The system $\{ \psi _n\}$ defines a linear map $\psi :V\to M$.
We now prove the result stated in the abstract by proving the equality $R(V,r,k)=R^f(V,r,k)$. Let $\Gamma$ be an abelian group, $\Gamma =\cup \Gamma _n$ where $\Gamma_n$ are finitely generated groups. Let $k$ be a finite field, and let $V,W$ be $k$-vector spaces with bases $v_j, w_j$. For $n\geq 1$ let $V_n, W_n$ be the spans of $v_j,w_j,1\leq j\leq n$. We denote by $i_n:V_n\to V_{n+1}$ the natural imbedding and by $\beta _n:W_{n+1}\to W_n$ the natural projection. Denote ${\operatorname{Hom}}^f(V,W)$ the finite rank homomorphisms from $V \to W$.
Suppose there exists $C=C(c)$ such that for any map $a^f:\Gamma \to {\operatorname{Hom}}^f(V,W)$ such that $$r(a^f({\gamma}'+{\gamma}'')-a^f({\gamma}')-a^f({\gamma}''))\leq c$$ there exists a homomorphism $\chi^f:{\Gamma}\to {\operatorname{Hom}}^f(V,W)$ such that $r(a^f({\gamma})-\chi ^f({\gamma}))\leq C$. Then for any map $a:{\Gamma}\to {\operatorname{Hom}}(V,W)$ such that $$r(a({\gamma}'+{\gamma}'')-a({\gamma}')-a({\gamma}''))\leq c$$ there exists a homomorphism $\chi :\Gamma \to {\operatorname{Hom}}(V,W)$ such that $r(a({\gamma})-\chi ({\gamma}))\leq C$.
We will use the following fact that is an immediate consequence of König’s lemma : Let $X$ be a locally finite tree, $x\in X$. If for any $N$ there exists a branch starting at $x$ of length $N$ then there exists an infinite branch starting at $x$.\
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Let $a:\Gamma \to {\operatorname{Hom}}(V,W)$ be a map such that $$r(a({\gamma}'+{\gamma}'')-a({\gamma}')-a({\gamma}''))\leq c$$ We define $F_n={\operatorname{Hom}}(V_n,W_n)$. Let $Y_n={\operatorname{Hom}}(\Gamma _n, F_n)$ and $q_n :Y_{n+1}\to Y_n$ be given by $$q_n (\chi _{n+1})=\beta _n\circ \chi '_{n+1}\circ i_n$$ where $\chi '_{n+1}$ is the restriction of $\chi _{n+1}$ on ${\Gamma}_n$.\
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We denote by $X_n\subset Y_n$ the subset of homomorphisms $\chi _n$ of $\Gamma _n$ such that $$r(\beta _n\circ a({\gamma})\circ i_n -\beta _n\circ \chi _n({\gamma})\circ i_n )\leq C.$$ Let $X$ be the disjoint union of $X_n$ and we connect $\chi _n\in X_n$ with $\chi _{n+1}\in X_{n+1}$ if $\chi _n=q_n(\chi _{n+1})$.\
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By the assumption $X_n$ are finite not empty sets and for any $n$ there exists a branch from $X_0$ to $X_n$ (any $\chi _n\in X_n$ defines such a branch). Now the Lemma \[compatible\] implies the existence a character $\chi :\Gamma \to {\operatorname{Hom}}(V,W)$ such that $r(a({\gamma})-\chi ({\gamma}))\leq C$.
To conclude the proof of Theorem \[Main\] we calculate the kennel of the map $a^1$:
\[kernel\] The kernel of $a^1$ consists of maps $P$ of finite rank.
Suppose $V, W$ are of dimension $n_1, n_2$ respectively. All the bounds below are independent of $n_1, n_2$. Suppose $P:V \to M_d$ is a linear map with $r(P) \le L$. Let $\tilde P$ be the multilinear version of $P$ as in . Let $f(v,\bar x) = \psi(\tilde P(v)(\bar x))$. Now $\tilde P(v)(\bar x)$ is a multilinear polynomial on $V\times W^d$ of degree $d+1$.\
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For any fixed $v$ we have $$\mathbb E_{\bar x \in W^d} \psi(\tilde P(v)(\bar x)) \ge C(L),$$ and thus $$\mathbb E_{v\in V} \mathbb E_{\bar x \in W^d} \psi(\tilde P(v)(\bar x)) \ge C(L).$$ It follows that $\tilde P(v)(\bar x)$ is of bounded rank $\ll_L 1$ and thus of the form $$\tilde P(v)(\bar x) = \sum_{j=1}^K \tilde Q_j(v,\bar x) \tilde R_j(v, \bar x)$$ with $\tilde Q_j, \tilde R_j$ of degree $\ge 1$ ,for any fixed $v$ also $\tilde Q_j, \tilde R_j$ are of degree $\ge 1$, and $K \ll_L 1$. For any fixed $x$, $\tilde P(V)$ is linear and thus either $\tilde Q_j$ or $\tilde R_j$ are constant as a function of $v$. Recall that $P(v)(x) = \frac{1}{d!}\tilde P(v)(x,\ldots, x)$, and let $Q_j(v,x) = \frac{1}{d!}\tilde Q_j(v,x,\ldots, x)$, similarly $R_j$. Let $J$ be the set of $j$ in the sum $P=\sum_j Q_jR_j$ such that $Q_j$ is linear in $v$ and does not depend on $x$. Let $V'=\bigcap _j {\operatorname{Ker}}Q_j$. The restriction of $P$ to $V'$ has finite (that is by a constant which does not depend on $n_1,n_2$ ) rank. Since $codim (V')\leq |J|$ we see that $P$ has rank that is bounded by a constant which does not depend on $n_1$ and $n_2$.\
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Now let $V, W$ be infinite. Let $V_n, l(n), p_{l(n)}$ be as in Lemma \[lift\]. Len $ P_n(V_n)= (p_{l(n)}\circ P)(V_n)$. Now apply Lemma \[compatible\] for $X_n$ the collection of finite rank maps from $V_n \to N_{l(n)}$, and $f_n$ the restriction as before. This finishes a proof of Theorem \[Main\].
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[^1]: The second author is supported by ERC grant ErgComNum 682150
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abstract: 'Global oil price is an important factor in determining many economic variables in the world’s economy. It is generally modeled as a stochastic process and have been studied through different techniques by comparing the historic time series of demand, supply and the price itself. However, there are many historic events where the demand or supply changes are not sufficient in explaining the price changes. In such cases, it is the expectations on the future changes of demand or supply that causes heavy and quick influences on the price. There are many parameters and variables that shape these expectations, and are usually neglected in traditional models. In this paper, we have proposed a model based on System Dynamics approach that takes into account these non-traditional factors. The validity of the proposed model is then evaluated using real and potential scenarios in which the proposed model follows the trend of the real data.'
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bibliography:
- 'MohammadRaf.bib'
title: 'A Data-Driven Approach for Modeling Stochasticity in Oil Market '
---
Introduction {#sec:intro}
============
It is widely accepted that the price of a commodity is determined by the interacting and opposing forces of the demand and supply in any free market [@abel2008macroeconomics]. Like any other market this statement is correct for the oil market, as well. However, in the oil market, the total demand and supply are not the only determinants of the price [@chevillon2009physical; @kilian2014oil; @sadorsky2006modeling; @rafieisakhaei2016analysis]. There are several examples in the history of oil market where the oil price has experienced sharp jumps in such a short period of time that the existing models have failed to capture the causes and reasons [@chiroma2015evolutionary; @shin2013prediction; @he2012crude; @fan2008generalized].
It is important to note that the types of oil production resources changes a lot both geographically and in time [@nehring1978giant; @rafieisakhaei2017effects]. Particularly, the advances of researches in the oil-related industries not only introduce new resources in the market, but they also make the costs of the production in the conventional oil wells much lower [@16_macrae_2011]. This makes the old models which aggregate the total production as merely one type to be less effective in capturing the sharp changes of the price [@sterman1988modeling; @fan2008generalized]. Moreover, some of the other models like [@he2012crude; @salisu2013modelling; @musaddiq2012modeling; @narayan2007modelling; @yang2002analysis] model the oil price as a stochastic process, and are successful in characterizing the oil price features including its volatility. However, these models usually do not consider the underlying factors and variables in determining the oil price. For instance, parts of added demand on the market has been responded through the growth of the production in the shale industry [@bartis2005oil; @schmidt2003new]. Particularly, US oil drillers has increased their production by 70$ \% $ since 2008, reducing the US oil imports from the Organization of Petroleum Exporting Countries (OPEC) by 50$ \% $ [@2_krauss_2015]. Likewise, production of oil from the Tar sands of Canada [@demaison1977tar] holds another reason for the increased supply in the oil market. As reported by the International Energy Agency (IEA) and OPEC, the average production of OPEC has only increased from 29.81 mb/d to 30.52 mb/d since the last quarter of 2014 till the first quarter of 2015, whereas the global oil supply has increased from 91.89 mb/d to 94.34 mb/d, which is much more significant [@23_generator_2015].
Similar to the carbon market [@rafieisakhaei2017efficacy], the nature of the supply resources are different in the oil market. Henceforth, the risk factors and dependencies of the variables that are involved in the oil production process are different for different resources. For instance, the cost of setting up a conventional oil well is usually lower in the Middle East [@4_the_economist_2014] whereas it is much higher in the northern parts of Russia. Moreover, the cost of setting up a shale oil well is much less than that of the conventional wells, both time and money-wise [@4_the_economist_2014]. However, the amount of oil that is produced from the latter is much more while the shale oil wells’ production usually decreases sharply by 60-70$ \% $ after the first year [@5_the_economist_2014]. Therefore, the shale oil industry needs a constant rate of investment as the new oil wells need to be created with much higher rates. These show that the variables involving the supply chain on the different types of resources need to be modeled with different mathematical formulas. For instance, researchers say that the current break-even price for the American projects is around \$65-70 [@5_the_economist_2014]. However, the producing wells are extremely profitable. On the other hand, the conventional oil well producers can tolerate a much lower price. However, many of the OPEC countries’ and Russia’s budget is heavily dependent on oil and this is one of the factors that reduces the OPEC countries’ tolerance of the very low prices [@6_kvue_2015].
To address this issue, the proposed model considers the effect that expectation whether on the demand side or supply side could have of the oil price. Details of the model is described in the following section.
Model, Loops and Formula {#sec:Model, Loops and Formula}
========================
In this section, we provide the proposed model, its important loops and some of the main formula that was used in the model.
Utilized Numerical Methods for Integration
------------------------------------------
In this subsection, we provide the mathematical basis that has been used in the model. We have used Euler method and Runge-Kutta methods to numerically solve the differential equations that result from modeling the stock variables. Let us suppose that we are given a differential equation of the form $$\begin{aligned}
\frac{dy}{dt} &= f(t, y(t)),\label{eq:differential equation}\\
\nonumber y &= y_0;\end{aligned}$$ where $ f $ is a Lipschitz continuous function, i.e. there exists a real number $ C>0 $ such that $$\begin{aligned}
|\frac{f(t_k, y(t_k))-f(t_k,y_k)}{y(t_k)-y_k}|<C,\end{aligned}$$ where $ y(t_k) $ is the real value of the stock variable $ y $, and $ y_k $ is the numerical approximation of this value at time step $ t_k $. This equation can be modeled in Vensim [@VensimP] as figure \[fig:stock\_flow\].
*Euler method:* In this method, the first order Taylor series expansion is used to approximate the value of the stock variable. In particular, at time step $ t_k $, $ y $ is approximated as: $$\begin{aligned}
y_{k+1} = y_k+\delta tf(t_k,y_k), ~1\le k\le K\end{aligned}$$ Therefore, the new value $ y_{k+1} $ is computed using the previous value $ y_{k} $ succeeded by moving forward using the slope given by . It can be shown that by using this method, both global and local error decay linearly with $ \delta t $. Therefore, it is important to choose small value for $ \delta t $. One of the disadvantages of this method is that it requires explicit form of derivatives of $ f $. However, we will obtain an approximation of the derivative using the real data, which we discuss later.
*Forth order Runge-Kutta method:* This method is still a one step method but utilizes the value of the variable at different points. The value $ y_k $ at time step $ t_k $ is approximated using the following set of equations: $$\begin{aligned}
\nonumber r_1 & = \delta t f(t_k, y_k)\\
\nonumber r_2 & = \delta t f(t_k + 0.5\delta t, y_k+0.5 r_1)\\
\nonumber r_3 & = \delta t f(t_k + 0.5\delta t, y_k+0.5 r_2)\\
\nonumber r_2 & = \delta t f(t_k + \delta t, y_k+ r_3)\\
y_{k+1} &= y_k +(r_1+r_2+r_3+r_4)/6.\end{aligned}$$ The advantage in using this method is that, it does not need explicit form of the derivative at time $ t_k $.
Now that we have provided the mathematical basis for our analysis, we proceed to provide the regression results based on this analysis, in addition to the main loops of the model.
The Main Loop
-------------
In this subsection, we provide the core structure of the model that involves the main variables in determining the oil price. Figure \[fig:Main Loop\] depicts the main loop along with the sign of causal effects that the corresponding variables have on each other. The rest of this subsection explains the various parameters in this loop.
*Components of the Main Loop:* Similar to [@rafieisakhaei2016modeling] the Main Loop consists of ‘Expected Total Oil Supply ($ TOS $)’, ‘Expected Total Oil Demand ($ TOD $)’, ‘Supply to Demand Ratio ($ TOS/TOD $)’, ‘Expectation on Oil Supply ($ ExS $)’ and ‘Expectation on Oil Demand ($ ExD $)’ which interact with each other to determine the ‘Oil Price ($ P $)’. In addition, there are two variables named as ‘External Variables on Demand’ and ‘External variables on Supply’ which summarize external factors that may not directly influence the oil price (reflected in Fig. \[fig:Main Loop\] as the External Variables on Supply and Demand, respectively). These factors can also depend on various variables that are determined in the economy as a whole (including the energy sector). We will elaborate them in the later sections.
Total Demand (Mb/Quarter) Total Supply (Mb/Quarter) Supply to Demand Ratio Price (USD) Price Change Rate
---------- --------------------------- --------------------------- ------------------------ ------------- -------------------
Min 86.8 87.03 0.99 76.06 -12.53
Max 92.8 91.74 1.02 105.83 11.78
Mean 89.82 89.64 1 92.65 1.45
Median 89.7 89.71 1 94.04 1.19
STD 1.69 1.45 0.01 8.84 7.23
Kurtosis -0.47 -1.11 0.28 -0.47 -0.74
Skewness 0.03 -0.21 0.61 -0.51 -0.46
*Supply to Demand Ratio and Oil Price:* As mentioned before, oil price like any other commodity in the economics, is mainly determined by the ratio of supply to demand [@fattouh2007drivers]. It is known that the price inversely affects the demand. Therefore, the higher the demand, the lower the supply to demand ratio and the higher the oil price. Moreover, increased oil price usually lowers the growth of the demand rate. On the other hand, oil price directly affects supply, which means that the higher the price, the higher the supply rate and the higher the supply to demand ratio. Thus, because of the inverse relation of the supply to demand ratio with the oil price, these loops stabilize and the price gets determined based on the ratio of the total supply and the total demand. We have used a first order regression using the West Texas Intermediate (WTI) index data in the normal states of the oil price to mathematically determine this negative feedback relationship, and this is what we discuss next.
*Regression analysis for Price Change Rate (PCR) with the assumption of Euler method:* Now, given a time series real data of $ P(t),~ TOS(t) $, and $ TOD(t) $ at equidistant time steps $ t_k, 1\le k\le K $, we obtain the time series $ PCR(t_k) $ as follows: $$\begin{aligned}
PCR(t_k) = \frac{P(t_{k+1})-P(t_{k})}{\delta t},\end{aligned}$$ where $ \delta t := t_{k+1}-t_{k} $ is a constant period of time which we choose for our analysis and in fact represents the resolution of the data that we have used. Note that, $ \delta t $ in this analysis can be different than $ \Delta t = TIME ~STEP = 0.0625~ (Day) $ that we use in our simulations, however, $ \delta t/\Delta t\in \mathbb{N} $ where $ \mathbb{N} $ is the set of Natural numbers. Next, we do a linear regression analysis in “R-Project” to find the relation between $ PCR $’ and $ TOS/TOD $ as follows: $$\begin{aligned}
\label{eq:Supply to Demand Ratio and the Oil Price}
PCR &= \alpha_1 TOS/TOD +\beta_1,\end{aligned}$$ where $ \alpha_1 $ and $ \beta_1 $ are the results of the regression analysis on the quarterly data of supply, demand and price over the first quarter of 2010 to the second quarter of 2014 [@1_eia_gov_2015; @1_company_2013; @23_generator_2015] whose statistics is summarized in table \[table:PCR-Ratio\]. The Stock and Flow model that is used for this part of the model is shown in Fig \[subfig:Price\_model\] which will be more clarified in the next parts.
Note that in obtaining equation , we searched for a period of time during which the price volatility was not too high in the long-term analysis, and more importantly the period of time was recent enough to fit our model. The nature of the oil price imposes some limitations in doing the regression analysis. For instance, one cannot simply choose any arbitrary period of time and impose the regression results to fit for all the time. Because, in some periods, it is seen that the price does not follow a negative relation with the supply to demand ratio. This analysis confirms our previous conjecture that in the oil market, the oil price does not only depend on the supply to demand ratio. This is because, if this was the case, we should have seen a merely non-increasing relation between the ratio $ TOS/TOD $ and price change rate, which is not the case. Therefore, the period of time is important, as well. Moreover, as it is seen in the above equations, only a naive regression analysis is not enough to model the price and supply to demand ratio’s relations. Since, we noticed that the residual remains high even with higher order polynomials. This shows that when we are considering long period of time, the relation is much more complex than some low-order polynomial. Moreover, we cannot just over-fit a very high order polynomial to the data, since, that would be over-simplification of the problem and would not be realistic or practical. Therefore, once again, this confirms that in the oil market, there are more hidden variables which heavily contribute to the price changes and should be taken into consideration in the models. As mentioned before, we call these variables the expectational variables and provide their related causal loops in the following sections.
It is worth mentioning that given two known values of $ P(t_k) $ and $ P(t_{k+1}) $, since in the forth-order Runge-Kutta method four values are unknown, it is not possible to do the above regression method based on that method, unless some other heuristics and approximations are used. Therefore, we only use Euler’s method for the regression analysis. However, in the integration, we use the Runge-Kutta method, as well.
*Expectation on Oil Demand and Supply:* As mentioned before, unlike many other commodities, oil market is heavily influenced by the factors other than the total demand and supply [@chevillon2009physical]. Some of these variables like ‘Policies on Energy Consumption’ that include policies on transportation are affected by the oil price whereas some of the others may not be directly influenced by the oil price. For instance, we have defined a variable named as ‘Economic Depression’ which tries to model the effects of the economic depression that is still felt in the EU zone on the oil price. The ‘External Variables on Demand’ and ‘External Variables on Supply’ in Fig. \[fig:Main Loop\] are dummy variables that reflect these parameters on the main loop.
*Effects of the expectational variables on the Oil Price:* In our model, the only variable that directly affects the oil price, is Supply to Demand Ratio. The novelty of our work is in the way that we model the effects of the other factors on the oil price only through this ratio. As shown in Fig. \[fig:Main Loop\], the expectational variables act as a booster or suppressor of the total demand or supply. In other words, they only act as a coefficient that amplify the effects of the supply or demand on the price. Their corresponding values are determined internally in the model in their corresponding loops which will be explained later in the paper. The reason behind this modeling is that whenever there is a change in the $ ExD $, if it is a positive expectation, then the oil price increases. In such a case, it is as if the demand has slightly increased. On the other hand, if there is a negative expectation on the growth of the demand, the oil price faces a decrease. It is similar to the condition that the demand has decreased slightly. Even though the actual demand or supply might not change for some while, these expectations change the price much faster than the actual changes. As an instance, on the mid-August 2015, it was expected that China’s demand will reduce, and this caused a very sharp decay in the oil price. In particular, the oil price went down from \$45.51 on 13th of Aug. to 4-year low record of \$38.18 on 25th of Aug. [@1_oilpricecom_2015]. It should be noted that, these expectations changes affect the price with the same sign that the demand affects the price. Therefore, we model all these variables as the expectational variables and aggregate their interacting effects on the ‘Expectation on the Oil Demand’ variable, whose value is multiplied in the demand to amplify or suppress the effects of the actual demand on the oil price. The same line of reasoning applies for the supply side, as well. In a word, the expectational parameters model the expectational changes of the demand and supply, and therefore, have the same effect on the price that the demand and supply have. Hence, we have modeled them as if the total demand or supply have changed by a factor that is determined by them.
Simulations and Scenarios {#sec:simulations}
=========================
In this section, we simulate several scenarios using Vensim PLE 6.3 [@VensimP] and bring their corresponding results to support our model. In each of the scenarios, some specific circumstance that can happen in the oil market have been investigated and the corresponding oil price changes have been reflected.
![WTI oil price history.[]{data-label="subfig:WTI_history"}](WTI_history_3.png){width="3.5in" height="6cm"}
![Parts of the WTI oil price history that we have used in our simulations, shown with rectangles.[]{data-label="subfig:WTI_history_marked"}](WTI_history_marked.png){width="3.5in" height="6cm"}
Scenario A: Neutral and Usual Case
----------------------------------
In this subsection, we simulate the model in a neutral condition, where there is no specific (or unexpected) event that can change the price significantly. The time horizon is considered to be a month and the initial oil price is supposed to be \$100.
*Results for the neutral case:* Figure \[subfig:Oil\_Price\_scenario\_A\_2\] depicts the results corresponding to this case. As it is seen, the global oil price, is roughly constant over a time horizon of about one month, where the demand is predicted to be almost constant (since there is no significant event) and the supply is predicted to be slightly growing from US side. In our simulations, due to the slight amount of surplus in the supply, the price faces a very slow decay, reaching to \$99.16 within 43 days. The real data sequence that is used for this scenario involves the daily WTI price from 30th of Dec. 2011 to 10th of Jan. 2012.
![The Oil Price changes in Scenario A, outputs shown from Vensim and real data.[]{data-label="subfig:Oil_Price_scenario_A_2"}](Price_Scenario_A.png){width="3.5in" height="6cm"}
![The Oil Price changes in Scenario B, outputs shown from Vensim and real data.[]{data-label="subfig:Scenario_B_vensim"}](Price_Scenario_B.png){width="3.5in" height="6cm"}
![The Oil Price changes in Scenario B in a case that the OPEC Decision is set to zero (meaning that they allow the reduction of their production levels), outputs shown from Vensim and real data.[]{data-label="subfig:Scenario_E_vensim"}](Price_Scenario_B_OPEC.png){width="3.5in" height="6cm"}
Scenario B: Presence of Upset
-----------------------------
In this subsection, we consider a case where there is a Geopolitical Upset in the OPEC countries that results in the reduction of the OPEC’s production. The results of this scenario is reflected in the Fig. \[subfig:Scenario\_B\_vensim\].
*Effect of an upset on the Price:* In such a case, the immediate result is an increase on oil price due to the shortage in the supply which is boosted by the speculations on the future amount of supply. These speculations are reflected in the price by the factors Expectations on Supply. We additionally consider that the OPEC countries set-up a meeting with some delay in order to respond to the supply shortage. In such a case, two possible outcomes are possible. The first outcome is that they decide to keep the supply as it is, and let the prices go up. In the second scenario, they decide to use the spare supply resources to level up the supply to the previous levels. However, the resulting effect is that the prices remain high, although its change rate becomes nearly zero. This is due to the fact that the spare supply of the OPEC countries usually can be fed into the market with a delay and can remain at least for 90 days, but the speculations on the future of the oil productions still remain high due to the conflicts and the fact that the OPEC countries might not want to use their spare resources for a long period of time. In such a case, if the upsets are still existent, the price once again keeps climbing. In other word, despite the fact that the oil supply might change back to its normal amount, the expectations on oil supply still remain speculative.
Conclusion and Future Work {#sec:conclusion}
==========================
In this paper, we introduced a system dynamic model that explains the relations among some of the main variables in determining the global oil price. Through historic examples and analysis we showed that unlike many other commodities, oil price cannot be predicted only based on the aggregated supply and demand. Rather, it is through the ‘expected’ demand and supply that the oil price gets determined. In our model we introduced the Expected Total Oil Demand and Expected Total Oil Supply to mathematically elaborate the effects of the expectations on the futures changes of supply and demand on the price. Particularly, we could still keep the traditional model of the commodity prices in which the price is obtained from the supply to demand ratio; however, instead of the actual values, we used the expected values of supply and demand in determining the ratio.
|
---
abstract: 'We study the limiting occupation density process for a large number of critical and driftless branching random walks. We show that the rescaled occupation densities of $\lfloor sN\rfloor$ branching random walks, viewed as a function-valued, increasing process $\{g_{s}^{N}\}_{s\ge 0}$, converges weakly to a pure jump process in the Skorohod space $\mathbb D([0, +\infty), \mathcal C_{0}({\mathbb{R}}))$, as $N\to\infty$. Moreover, the jumps of the limiting process consist of i.i.d. copies of an Integrated super-Brownian Excursion (ISE) density, rescaled and weighted by the jump sizes in a real-valued stable-1/2 subordinator.'
address:
- |
Steven P. Lalley\
Department of Statistics\
University of Chicago\
5747 S. Ellis Avenue\
Chicago, IL 60637, USA
- |
Si Tang\
Department of Mathematics\
Lehigh University\
17 Memorial Drive East\
Bethlehem, PA 18015, USA
author:
- 'Steven P. Lalley and Si Tang'
title: Occupation densities of Ensembles of Branching Random Walks
---
Introduction {#sec:intro}
============
In a branching random walk on the integers, individuals live for one generation, reproduce as in a Galton-Watson process, giving rise to offspring which then independently jump according to the law of a random walk. A branching random walk is said to be *critical* if the offspring distribution $\nu$ has mean $1$, and *driftless* if the jump distribution $F$ has mean $0$ and finite variance. We will assume throughout that (i) the offspring distribution $\nu$ has mean one (so that the Galton-Watson process is critical) and finite, positive variance $\sigma^{2}_{\nu} $; and (ii) the step distribution $F$ for the random walk has span one, mean zero and finite, positive variance $\sigma^{2}_{F}$. (Thus, the spatial locations of individuals will always be points of the integers $\mathbb{Z}$.)
To any branching random walk can be associated a randomly labeled Galton-Watson tree $\mathcal{T}$, where the Galton-Watson tree describes the lineage of the individuals and the label of each vertex marks the spatial location of the corresponding individual. This labeled tree $$\mathcal T= (T, \{l(v)\}_{ v\in T})$$ is generated as follows.
(i) Let $T$ be the genealogical tree of a Galton-Watson process with a single ancestral individual and offspring distribution $\nu$, with the root node $\rho$ representing this ancestral individual. Since $\nu$ has mean $1$, the tree $T$ is finite with probability one.
(ii) Assign the label $l(\rho)=0$ to the root.
(iii) Conditional on $T$, let ${
\left\{ \xi_{e} \right\}}_{e \in \mathcal{E}(T)}$ be a collection of i.i.d. random variables, with common distribution $F$ (the “step distribution”) indexed by the (directed) edges $e=(u,v)$ of the tree $T$, where $u$ is the parent vertex of $v$. For any such directed edge $e=(u,v)$, define $$l(v)=l(u)+\xi_{e}.$$
Given the labeled tree $\mathcal T$ associated with the branching random walk, the occupation measures can be recovered as follows. For any time $n \in \mathbb{Z}_{+}$ and any site $x\in \mathbb{Z}$, the number $Z_{n}(x)$ of individuals at location $x$ at time $n$ is the number of vertices $v \in T$ at height $n$ (i.e., at distance $n$ from the root) with label $l(v)=x$. The *vertical profile*, or the *occupation measure*, of $\mathcal T$ (see [@ISE2006-2]) is the random counting measure on $\mathbb Z$ defined by $$X(x; \mathcal{T})=\sum _{n=0}^{\infty}Z_{n}(x).$$ In this paper, we study the limiting behavior of the occupation measure and its connection to super-Brownian motions. Before stating our main result, we review a few results about the occupation measure of random labeled trees.
The study of such occupation measures dates back to Aldous [@AldousISE1993], who introduced an object called the *integrated super-Brownian excursion* (ISE), denoted by $\mu_{\textup{ISE}}$, a (probability) measure-valued random variable that arises as the scaling limit of the occupation measure of certain labeled random planar trees and tree embeddings. In [@marckert2004rotation], Marckert proved that the rescaled occupation measure of a random binary tree of $n$ vertices converges weakly to $\mu_{\text{ISE}}$ as $n\to
\infty$.
Bousquet-Mélou and Janson [@ISE2006-2] later proved a local version of Marckert’s result: they showed that the *density* of the rescaled occupation measure of random binary trees, random complete binary trees, or random plane trees on $n$ vertices converges to the density of $\mu_{\textup{ISE}}$, denoted by ${f_{\textup{ISE}}}$, which is known to be a random Hölder$(\alpha)$-continuous function for every $\alpha <1$ with compact support. This local convergence was later extended in [@condGW-profile Theorem 1.1] to general branching random walks conditioned to have exactly $n$ vertices, as long as $\nu$ and $F$ satisfy the assumptions above.
In this paper, we consider an ensemble of critical, driftless branching random walks, all with the same offspring and step distributions $\nu$ and $F$, and study the limiting behavior of the total occupation density. Our first result shows that the total occupation density converges in the Skorohod space $\mathbb D([0, +\infty), \mathcal C_{0}(\mathbb R)\,)$, which can be characterized by a super-Brownian motion.
Let $\mathcal T^{1}, \mathcal{T}^{2},\ldots$ be the random labeled trees associated with an infinite sequence $(Z_{1,n})_{n\ge 0}$, $(Z_{2,n})_{n\ge 0}, \cdots $ of independent copies of the branching random walk. For each integer $m \geq 1$ define $$\label{eq:occupation-measure}
X^{m}(j):= \sum_{i=1}^{m} X(j;\mathcal{T}^{i})$$ to be the total number of vertices in the first $m$ trees with label $j\in \mathbb Z$, and define $\bar X^{m}(x) $ to be the linear interpolation to $x \in \mathbb{R}$. Observe that $X^{m}$ can be viewed as the occupation measure of the branching random walk initiated by the $m$ ancestral particles that engender the branching random walks $Z_{1},Z_{2}, \cdots Z_{m}$. Clearly, the function $\bar X^{m}(x) $ is an element of $ \mathcal {C}_{0}(\mathbb
R)$. Finally, define the $\mathcal C_{0}(\mathbb R)$-valued process $\{g_{s}^{N}\}_{s\ge 0}$ by $$\label{eq:occupation-measure-rescaled}
g_{s}^{N}(x) :=N^{-3/2} \bar X^{\lfloor sN\rfloor}(\sqrt{N} x).$$
\[thm1.1\] As $N \to \infty$, the rescaled density processes $\{ g_{s}^{N}\}_{s\ge 0} $ converge weakly in the Skorohod space $\mathbb D:=\mathbb D([0, +\infty), \mathcal C_{0}(\mathbb R))$ to a process $\{g_{s}\}_{s\ge 0}$. Moreover, the limiting process satisfies $$\begin{aligned}
\label{eqn:defgs}
g_{s}(x) &{\overset{\mathcal D}{=\joinrel=}}\int_{0}^{\infty}Y^{s}\left(t,
x\right) dt
$$ where $\{Y^{s}(t, x), x\in {\mathbb{R}}\}_{t\ge
0}$ is the density process for a super-Brownian motion $\{Y^{s}_{t}\}_{t\ge 0}$ with variance parameters $(\sigma_{\nu}^{2}, \sigma_{F}^{2})$, started from the initial measure $Y^{s}_{0}=s \delta_{0}$.
Super-Brownian motion $\{Y_{t}\}_{t\ge 0}$ is, by definition (see for instance [@etheridge2000introduction], ch. 1) a measure-valued stochastic process that can be constructed as a weak limit of rescaled counting measures associated with branching random walks. In one dimension, for each $t>0$, the random measure $Y_{t}$ is absolutely continuous relative to the Lebesgue measure, and the Radon-Nikodym derivative $Y(t, x)$ is jointly continuous in $(t,x)$ [@KonnoShiga1988]. Super-Brownian motion is singular in higher dimensions and thus the representation does not exist in higher dimensions. When the dependence on the variance parameters $\sigma_{\nu}^{2}$ and $\sigma_{F}^{2}$ must be emphasized, we do so by adding them as extra superscripts, i.e., $$Y^{s,\sigma_{\nu}^{2},\sigma_{F}^{2}}(t,x).$$ When $\sigma_{\nu}^{2}=\sigma^{2}_{F}=1$, the measure-valued process associated with $Y^{s,1,1}(t,x)$ is a *standard* super-Brownian motion. The density processes for different variance parameters obey a simple scaling relation: $$Y^{s, \sigma_{\nu}^{2}, \sigma_{F}^{2}}\left(t, x\right) {\overset{\mathcal D}{=\joinrel=}}\sigma_{\nu}\sigma_{F}^{-1}Y^{s,1,1}\left(\sigma_{\nu}^{2}t,
\sigma_{\nu}\sigma_{F}^{-1}x\right),\ \ \quad \textrm{for all} \, t>0,
\, x \in \mathbb{R}.$$ Thus, we can rewrite in terms of the density function of standard super-Brownian motion as follows: $$g_{s}(x) {\overset{\mathcal D}{=\joinrel=}}\sigma_{\nu}\sigma_{F}^{-1}
\int_{0}^{\infty}Y^{s,1,1}\left(\sigma_{\nu}^{2}t,
\sigma_{\nu}\sigma_{F}^{-1}x\right)dt$$
\[rem:indef-integral\] For any fixed $s>0$ and each integer $N\geq 1$, the random function $g^{N}_{s} (\cdot)$ is the (rescaled) occupation density of the branching random walk gotten by amalgamating the branching random walks generated by the first $\lfloor sN\rfloor$ initial particles. Because this sequence of branching random walks is governed by the fundamental convergence theorem of Watanabe [@Watanabe1968] and its extension to densities by Lalley [@lalley1d], the limiting random function $g_{s} (\cdot)$ must (after the appropriate scaling) be the integrated occupation density of the super-Brownian motion with initial measure $s\delta_{0}$. This explains relation . But even for fixed $s>0$ the weak convergence $g^{N}_{s}(x) \Longrightarrow\int_{0}^{\infty}Y^{s,
\sigma_{\nu}^{2}, \sigma_{F}^{2}}\left(t, x\right) dt$ does not follow directly from the local convergence of the density process proved in [@lalley1d Theorem 2], for two reasons. First, the local convergence result in [@lalley1d] requires that the initial densities must, after Feller-Watanabe rescaling, converge to a density function $Y^{s}(0, \cdot)\in \mathcal C_{0}(\mathbb R)$. In Theorem \[thm1.1\], however, the limiting initial density $Y^{s}_{0}=s \delta_{0}$ is not absolutely continuous with respect to the Lebesgue measure. Second, even if the local convergence could be shown to remain valid under the initial condition $Y^{s}_{0}=s\delta_{0}$, the indefinite integral operator on $\mathbb C([0, +\infty), \mathcal C_{0}(\mathbb R))$ is not bounded, and so it would not follow, at least without further argument, that the integral of the discrete densities would converge to that of the super-Brownian motion density over the time interval $[0, +\infty)$.
![A simulation of $g_{s}^{N}$, for $N=1000$ and $s=0.1, 0.2, \ldots, 1$. The offspring and step distributions are $\nu = \text{Poi}(1)$ and $F = (\delta_{-1}+\delta_{0}+\delta_{1})/3$.[]{data-label="Fig:simuG"}](simuG-5.pdf){width="80.00000%"}
For each $N\geq 1$, the process $\{g_{s}^{N}\}_{s\ge 0}$ is nondecreasing[^1] in $s$ (relative to the natural partial ordering on $\mathcal C_{0} ({\mathbb{R}})$) and has stationary, independent increments. Therefore, the limiting process $\{g_{s}\}_{s\ge0}$ must also be nondecreasing, with stationary, independent increments. We prove the following properties of the limiting process.
\[thm:property-gs\] The limiting function-valued process $\{g_{s}\}_{s\ge 0}$ has the following properties.
1. It obeys the scaling relation $g_{s}(x)
\overset{\mathcal D}{=\joinrel=} s^{3/2}g_{1}(x/\sqrt{s})$.
2. The real-valued process $\{I_{s}\}_{s\ge 0}$, where $I_{s} := g_{s}(0)$ is the occupation density at zero, is a stable subordinator with exponent $\alpha = 2/3$.
3. The real-valued process $\{\theta_{s}\}_{s\ge 0}$, where $\theta_{s} := \int_{-\infty}^{\infty}g_{s}(x)dx$ is the total rescaled occupation density, is a stable subordinator with exponent $\alpha = 1/2$.
4. $\{g_{s}\}_{s\ge 0}$ is a pure-jump subordinator in the Banach lattice $\mathcal C_{0}(\mathbb
R)$ (see Definition \[def:banachlattice\]).
As we will show, the limiting process $g_{s}$ in Theorem \[thm1.1\] has jump discontinuities, that is, there are times $t>0$ such that the function $g_{t}-g_{t-}$ is non-zero (and hence positive) over some interval. The jumps that occur before time $t=1$ can be ordered by total area $\int (g_{t}-g_{t-})
(x)\,dx$, i.e., the jump size in the stable-1/2 process $\{\theta_{s}\}_{s\ge 0}$. Denote these jumps (viewed as elements of $\mathcal C_{0} ({\mathbb{R}})$) by $$J_{1} (x), J_{2} (x),\dots \quad \text{where} \quad \int
J_{1}>\int J_{2}>\int J_{3}>\dots .$$ (In Section \[sec:Levyjump\], we will see that no two jump sizes can be the same.) For each $N$, the Galton-Watson trees $\mathcal{T}_{i}$ with $i\leq N$ can also be ordered by their size (i.e., the number of vertices). The corresponding jumps in the (rescaled) occupation density $g^{N}_{s}$ will be denoted by $$J^{N}_{1} (x), J^{N}_{2} (x), J^{N}_{3} (x),\dots .$$ (Thus, if the $j$th largest tree among the first $N$ trees is $\mathcal T_{\lfloor s_{j}N\rfloor} $, then $J^{N}_{j}=
g^{N}_{s_{j}}-g^{N}_{\frac{\lfloor s_{j}N\rfloor-1}{N}}$.)
\[cor:order-stats\] For each $m\geq 1$, $$\label{eq:jointConvergence}
(J^{N}_{1},J^{N}_{2},\dots ,J^{N}_{m})\Longrightarrow
(J_{1},J_{2},\dots ,J_{m}),$$ where the weak convergence is relative to the $m$-fold product topology on $\mathcal C_{0} ({\mathbb{R}})$.
This is an immediate consequence of Theorem \[thm1.1\], because weak convergence in the Skorohod topology on ${\mathbb{D}}$ implies weak convergence of the ordered jump discontinuities.
Theorem \[thm1.1\] can be regarded as an unconditional version of the local convergence in [@ISE2006-2 Theorem 3.1] and [@condGW-profile Theorem 1.1]. The connection between Theorem \[thm1.1\] and the results of Bousquet-Mélou/Janson and Aldous leads to a reasonably complete description of the Lévy-Khintchine representation of the pure-jump process $\{g_{s}\}_{s\ge 0}$.
\[thm:levy-measure\] The point process of jumps of $\{g_{s}\}_{s\ge 0}$ is a Poisson point process $\{\mathcal N(B)\}_{B \in \mathcal B}$ on the space $(\mathbb R_{+}\times
\mathbb R_{+}\times \mathcal C_{0}(\mathbb R), \mathcal B:= \mathcal B_{\mathbb
R_{+}\times \mathbb R_{+}\times \mathcal C_{0}(\mathbb R)})$ with intensity (Lévy-Khintchine) measure $$\label{eqn:gs-intensity}
\chi(dt, dl, dh) := dt\cdot \frac{dl}{\sqrt{2\pi l^{3}}}\cdot f_{\textup{ISE}}(dh).$$ Consequently, the process $g_{s}$ can be written as $$\label{eqn:pointprocess}
g_{s}(\cdot) {\overset{\mathcal D}{=\joinrel=}}\frac{1}{\sigma_{\nu}\sigma_{F}}\iiint \mathbf 1_{[0,
s]}(t) l^{3/4}h\left(l^{-1/4}\,\frac{ \sigma_{\nu}}{\sigma_{F}} \cdot\,
\right) \mathcal N(dt, dl, dh).$$
The remainder of the paper is organized as follows. Section \[sec:proof1.1\] is devoted to the proof of Theorem \[thm1.1\], where we make use of Aldous’ stopping time criterion [@Aldous-tightness] to show the tightness of the sequence of processes $\{g^{N}_{s}\}_{s\ge 0}$. In Section \[sec:Levyjump\], we prove the properties of the limiting process $g_{s}$ enumerated in Theorem \[thm:property-gs\] and the Lévy-Khintchine representation .
Proof of Theorem \[thm1.1\]\[sec:proof1.1\]
===========================================
Preliminaries on the Skorohod space $\mathbb D$
-----------------------------------------------
Let $(\mathcal S, d)$ be a separable and complete metric space, and let $\mathbb D (\mathcal S){{\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =\,}}\mathbb D ([0, +\infty), (\mathcal S,
d))$ be the spaces of all $\mathcal S$-valued *càdlàg* functions $f$ with domain $[0, +\infty)$, i.e., $f\in \mathbb D(\mathcal S)$ is right-continuous and has left limits. The space $\mathbb D(\mathcal S)$ is metrizable, and under the usual *Skorohod metric*, the space $\mathbb D(\mathcal S)$ is complete and separable. We refer to [@Billingsley1999 Section 13] for details on the Skorohod topology. Here, we quote the following theorem, which gives a sufficient condition for the weak convergence in $\mathbb D(\mathcal S)$.
\[thm:convD\] Let $\{X_{t}^{N}\}_{t \ge 0}$ and $\{X_{t}\}_{t \ge 0} \in \mathbb D(\mathcal S)$ be $\mathcal S$-valued processes. Let $\Delta$ be some dense subset of $[0, +\infty)$. If the sequence $\{X^{N}\}_{t\ge 0}$ is tight (relative to the Skorohod metric) and if $(X^{N}_{t_{1}}, X^{N}_{{t_{2}}}, \ldots, X_{t_{m}}^{N})
\Longrightarrow (X_{t_{1}}, X_{{t_{2}}}, \ldots, X_{t_{m}})$ as $N \to \infty$ for all $t_{1}, \ldots, t_{m} \in \Delta$, then $\{X^{N}_{t}\}_{t\ge 0 }\Longrightarrow \{X_{t}\}_{t \ge 0}$ as $N \to \infty$.
For the particular case where $(\mathcal S, d)$ is the real line with the Euclidean metric, Aldous [@Aldous-tightness Theorem 1] gave a useful sufficient condition for the tightness of a sequence $\{X^{N}_{t}\}_{t \in [0,1]}$ in the space $\mathbb D([0, 1], (\mathbb R, \,\lvert\cdot\rvert))$. He also pointed out [@aldous1981weak Theorem 4.4] that, with a slight modification, the criterion can be generalized to $\mathcal S$-valued stochastic processes over the half line $[0, +\infty)$ as long as $(\mathcal S, d)$ is a complete and separable metric space. We state Aldous’ criterion in this form below.
\[thm:tightness\] Let $(\mathcal S, d)$ be a complete and separable metric space and let $\{X^{N}_{s}\}_{s\ge 0} \in \mathbb D ([0, +\infty), (\mathcal
S, d))$ be a sequence of $\mathcal S$-valued stochastic processes. A sufficient condition for tightness of the sequence ${
\left\{ X^{N}_{s} \right\}}_{s \geq 0}$ is that the following two conditions hold:
. For each $s$, the sequence ${
\left\{ X_{s}^{N} \right\}}_{N\in \mathbb N}$ is tight in $(\mathcal S, d)$, and\
. For any $L > 0$, any sequence of constant $\delta_{N} \downarrow 0$, and any sequence of stopping times $\tau_{N} $ for $\{X_{s}^{N}\}_{s\ge 0}$ that are all upper bounded by $L$, $$\label{eq:condition2}
d(X_{\tau_{N}+\delta_{N}}^{N}, X_{\tau_{N}}^{N}) \stackrel{P}{\longrightarrow} 0.$$
In the case when $(\mathcal S, d) = (\mathcal C_{0}({\mathbb{R}}), \lVert\cdot\lVert_{\infty})$, is equivalent to $$\label{eq:condition2-1}
\lVert X_{\tau_{N}+\delta_{N}}^{N} - X_{\tau_{N}}^{N} \rVert_{\infty}\stackrel{P}{\longrightarrow} 0.$$
Proof of Tightness
------------------
In this section, we prove that the sequence $\{g^{N}_{s}\}_{N\in \mathbb N}$ is tight in $\mathbb D (\mathcal C_{0}(\mathbb R))$ by verifying Condition 1$^{\circ}$ and Condition $2^{\circ}$.
To verify
Condition $1^{\circ}$, we will show that for any $s\ge 0$ fixed and any $\epsilon > 0$, we can find a compact subset $\mathsf K \subset \mathcal C_{0}(\mathbb R)$ such that ${\mathbb P}(g_{s}^{N}\in \mathsf K) > 1-\epsilon$ for all $N$ large. Let $\zeta_{sN}$ be the extinction time of the branching random walk gotten by amalgamating the branching random walks $(Z_{1,n})_{n\ge 0}, (Z_{2,n})_{n\ge 0}, \cdots
,(Z_{\lfloor sN\rfloor, n})_{n\ge 0}$, that is, $\zeta_{sN}$ is the maximum of the extinction times of the branching random walks initiated by the first $\lfloor sN\rfloor$ ancestral particles. By a fundamental theorem of Kolmogorov, $$P {
\left\{ \zeta_{1}>n \right\}} \sim \frac{2}{n \sigma_{\nu}^{2}} \quad
\textrm {as} \; n \rightarrow\infty;$$ consequently, for every $\epsilon > 0$, there exists $H=H_{s, \epsilon}>0$ such that $$\label{eq:kolmogorov-tail}
{\mathbb P}(\zeta_{sN} > NH) < \epsilon /2.$$ Therefore, it suffices to prove that there is a compact set $\mathsf K \subset \mathcal
C_{0}(\mathbb R)$ such that for all $N$ large, $$\label{eq:tightness1}
{\mathbb P}\left(\{g_{s}^{N}\in \mathsf K\} \cap G_{sN}\right) \ge
1-\frac{\epsilon}{2}, \quad \textrm {where} \quad
G_{sN}:= {
\left\{ \zeta_{sN}\le NH \right\}}.$$
To establish inequality we will use Kolmogorov-Čentsov criterion (see, e.g., [@GTM-BM1991 Chapter 2, Problem 4.11]). It suffices to prove that $$\begin{aligned}
\label{eqn:momentbd0}
\tag{2.5a}
\sup_{N}{\ensuremath{\mathbb{E}}}[g_{s}^{N}(0)\mathbf 1_{G_{sN}}] < \infty,\end{aligned}$$ and that for some $m\ge 3$, there exists $C = C(s, m, H)>0$ such that for all $x, y \in\mathbb R$ and for all $N$ sufficiently large, $$\begin{aligned}
\label{eqn:momentbd}
\tag{2.5b}
\mathbb E \left [g_{s}^{N}(x) - g_{s}^{N}(y)\right ]^{2m}
\mathbf 1_{G_{sN}}\le C |x-y|^{\frac{2m}{5}}. \end{aligned}$$ Note that the requirement $m \geq 3$ in ensures that the exponent $2m/5$ is larger than $1$, as is needed for the Kolmogorov-Čentsov criterion. We will rely on the following estimates of [@lalley1d] to compute these bounds.
\[prop:lalley71\] [@lalley1d Proposition 5] Let $Z_{n}(x)$ be the number of particles at location $x\in \mathbb Z$ and time $n\in \mathbb Z_{+}$ in a branching random walk, started from a single particle at $0\in\mathbb Z$, with offspring distribution $\nu$ and step distribution $F$. For each $m\in \mathbb N$, there is constant $C_{m}$ such that for all $x, y\in \mathbb Z$ and all $ n\ge 1$, $$\begin{aligned}
\setcounter{equation}{5}
{\ensuremath{\mathbb{E}}}Z_{n}(x)^{m}&\le C_{m}n^{m/2-1} \label{eqn:lalley70} \\
|{\ensuremath{\mathbb{E}}}(Z_{n}(x) - Z_{n}(y))^{m}| &\le C_{m}n^{2m/5-1}|x-y|^{m/5}
\label{eqn:lalley71}\end{aligned}$$
\[prop:claim\] Let $Z_{n}(x)$ be as in Proposition \[prop:lalley71\]. Then, for any $H > 0$ and $m \ge 1$, there exists $C=C_{m, H}$ such that for all $N\ge 1$ and $x, y \in \mathbb Z/\sqrt{N}$, $$\begin{aligned}
\label{eqn:claim}
\sum_{n_{1}, \ldots, n_{m}=1}^{\lfloor NH \rfloor} \left | \mathbb E
\prod_{l=1}^{m} \left( Z_{n_{l}} (\sqrt{N}x)-Z_{n_{l}}
(\sqrt{N}y)\right) \right | \le C |x-y|^{\frac{m}{5}}N^{\frac{3m}{2}-1}. \end{aligned}$$
Suppose first $m$ is even. Then by a trivial extension of Hölder’s inequality (see, e.g.,[@Pons-Ineq2013 pp. 4]) and Proposition \[prop:lalley71\], $$\begin{aligned}
& \ \ \ \ \ \sum_{n_{1}, \ldots, n_{m}=1}^{\lfloor NH \rfloor} \left
| \mathbb E \prod_{l=1}^{m} \left( Z_{n_{l}}
(\sqrt{N}x)-Z_{n_{l}} (\sqrt{N}y)\right) \right |
\\
& \le \sum_{n_{1}, \ldots, n_{m}=1}^{\lfloor NH \rfloor} \prod_{l=1}^{m} \left | \mathbb E\left( Z_{n_{l}} (\sqrt{N}x)-Z_{n_{l}} (\sqrt{N}y) \right)^{m} \right|^{\frac{1}{m}}\\
& = \left \{ \sum_{n_{1}=1}^{\lfloor NH \rfloor} \left | \mathbb E\left( Z_{n_{1}} (\sqrt{N}x)-Z_{n_{1}} (\sqrt{N}y) \right)^{m} \right|^{\frac{1}{m}} \right \}^{m}\\
&\le \left \{ \sum_{n_{1}=1}^{\lfloor NH \rfloor} \left [ C_{m}n_{1}^{\frac{2m}{5}-1}|x-y|^{\frac{m}{5}}N^{\frac{m}{10}}\right]^{\frac{1}{m}} \right \}^{m}\\
&\le C_{m, H}|x-y|^{\frac{m}{5}}N^{\frac{3m}{2}-1}.\end{aligned}$$ The case when $m$ is odd is similar.
The bound in is easy to check using with $m=1$ and $x=0$ and the linearity of expectation. In particular, $$\begin{aligned}
{\ensuremath{\mathbb{E}}}[g_{s}^{N}(0)\mathbf 1_{G_{sN}}] & \le \frac{sN}{N^{3/2}} \left(1+\sum_{n=1}^{NH} C_{1}n^{1/2-1}\right) = \frac{s}{\sqrt{N}}\left(1+O(\sqrt{N})\right) < \infty.\end{aligned}$$ For , first of all, by triangle inequality and the assumption that $\bar X^{N} (x)$ is defined by linear interpolation, we need only consider $x, y \in \mathbb Z/\sqrt{N}$ in .
Let $Z_{i, n}(x)$ be the number of particles at site $x \in \mathbb Z$ in generation $n$ of the $i$-th ancestral particle. The left side of is clearly bounded by $$\begin{aligned}
N^{-3m}\cdot \mathbb E\left \{ \sum_{i=1}^{\lfloor
sN\rfloor}\sum_{n=0}^{\lfloor NH\rfloor} \left[ Z_{i,
n}(\sqrt{N}x)-Z_{i, n}(\sqrt{N}y)\right] \right \}^{2m}. \end{aligned}$$ We expand the product under the expectation sign and write it as a sum of expectations: $$\begin{aligned}
\label{eqn:oneterm}
N^{-3m}\cdot \sum_{i_{1}, \ldots, i_{2m}=1}^{\lfloor sN \rfloor} \sum_{n_{1}, \ldots, n_{2m}=0}^{\lfloor NH \rfloor} \mathbb E\prod_{j=1}^{2m}\left[ Z_{i_{j}, n_{j}}^{\lfloor sN\rfloor}(\sqrt{N}x)-Z_{i_{j}, n_{j}}^{\lfloor sN\rfloor}(\sqrt{N}y)\right].\end{aligned}$$ When the product inside the expectations is expanded, each term is a product of $2m$ differences of occupation counts in one of the branching random walks $Z_{i_{j}}$ in some generation $n_{j}$. Observe that repetitions of the indices $i_{j}$ and $n_{j}$ are allowed.
Note that Proposition \[prop:lalley71\] applies only for generation $n\ge 1$, whereas $n_{1}, \ldots, n_{2m}$ in run from generation $0$. However, since originally all particles are placed at the origin, we lose nothing by summing from $1$ to $\lfloor NH\rfloor$ as long as $xy\ne 0$. The case when $xy=0$ will be treated separately at the end.
Suppose $x\ne 0$, $y\ne 0$. If $i_{j_{1}} \ne i_{j_{2}}$ are indices of two distinct ancestral individuals, then the differences $\left[Z_{i_{j_{1}}, n_{j_{1}}}(\sqrt{N}x)-Z_{i_{j_{1}},
n_{j_{1}}}(\sqrt{N}y)\right]$ and $\left[Z_{i_{j_{2}}, n_{j_{2}}}(\sqrt{N}x)-Z_{i_{j_{2}},
n_{j_{2}}}(\sqrt{N}y)\right]$ are independent. Let $r$ be the number of distinct $i_{j}$’s inside the expectation in ; then can be written as $$\begin{aligned}
\label{eqn:factors}
N^{-3m}\sum_{r=1}^{2m}\sum_{\substack{1\le i_{1}<\cdots < i_{r}\le \lfloor sN\rfloor\\\sum_{j}m_{j}=2m}}
\prod_{j=1}^{r}\left[ \sum_{n^{j}_{1}\ldots
n^{j}_{m_{j}}=1}^{\lfloor NH \rfloor} \mathbb E \prod_{l=1}^{m_{j}}
\left( Z_{i_{j}, n^{j}_{l}}
(\sqrt{N}x)-Z_{i_{j}, n^{j}_{l}}
(\sqrt{N}y)\right)\right]. \end{aligned}$$ For a particular term with $r$ distinct ancestors $i_{1}, \ldots, i_{r}$ in which $i_{j}$ occurs $m_{j}$ times ($j=1,2,\ldots, r$), the expectation can be factored as a product of $r$ expectations, where each expectation is an expectation of the differences involving the offspring of only one ancestor at time 0. Thus, we always have $\sum_{j=1}^{r}m_{j}=2m$. For each bracketed factor in , for each ancestor $i_{j}$, the summation is over all possible choices of the generations $n_{1}^{j}, \ldots, n_{m_{j}}^{j}$; this can be bounded using Corollary \[prop:claim\] above. It follows that $$\begin{aligned}
&\ \ N^{-3m} \sum_{r=1}^{2m}\sum_{\substack{1\le i_{1}<\cdots < i_{r}\le \lfloor sN\rfloor\\\sum_{j}m_{j}=2m}} \prod_{j=1}^{r}C(m_{j},
H)|x-y|^{\frac{m_{j}}{5}}N^{\frac{3m_{j}}{2}-1}\\
&\le\ C_{1}(m, s, H) \,N^{-3m} \sum_{r=1}^{2m}N^{r} \cdot
N^{\frac{3}{2}\cdot
2m-r}|x-y|^{\frac{2m}{5}}\\
&\le \ C(m, s, H)|x-y|^{\frac{2m}{5}},\end{aligned}$$ and is proved.
Finally, we must deal with the case when $xy=0$. If $x=y=0$, then both sides of are zero. If $x=0$ and $y\ne 0$, then, because all initial $\lfloor sN \rfloor$ particles are placed at zero, we can write the left side of as $$\mathbb E\left \{ \left(\sum_{i=1}^{\lfloor sN\rfloor}\sum_{n=1}^{NH}
\frac{\left[ Z_{i, n}(0)-Z_{i, n}(\sqrt{N}y)\right]}{ N^{3/2}}\right) +\frac{ \lfloor sN \rfloor}{N^{3/2}}\right \}^{2m}.$$ It is not difficult to see that for large $N$, the first term dominates, because this term can be handled exactly as in the case $xy \not =
0$. This proves that Condition 1$^{\circ}$ holds.
For each $N$ the process $g^{N}_{s}$ is piecewise constant in $s$, with jumps only at times $s$ that are integer multiples of $1/N$. Consequently, in verifying Condition 2 we may restrict attention to stopping times $\tau_{N}$ such that $N \tau_{N}$ is an integer between $0$ and $NL$. It is obvious from its definition that the discrete-time process $g^{N}_{s}$, with $s=0,1/N, 2/N, \cdots
$, is non-decreasing and has stationary, independent increments; therefore, for any stopping time $\tau^{N}$ and any constant $\delta
>0$, the increment $g^{N}_{\tau_{N}+\delta}-g^{N}_{\tau_{N}}$ has the same distribution as $g^{N}_{\delta}$. Therefore, to prove Condition 2$^{\circ}$ it is enough to show that for any $\epsilon
>0$ there exists $\delta>0$ such that for all $N$ sufficiently large, $${\mathbb P}\left (\sup_{x\in \mathbb R}g_{\delta}^{N}(x) \ge \epsilon
\right) \le \epsilon,$$ equivalently, $$\label{eq:tightness2}
{\mathbb P}\left (\sup_{x\in \mathbb R}g_{1}^{\lfloor \delta N \rfloor}(x) \ge \epsilon \delta^{-3/2}
\right) \le \epsilon.$$ But we have already proved, in Condition 1$^{\circ}$, that the sequence of $\mathcal{C}_{0}(\mathbb{R})-$valued processes $g^{N}_{1}$ is tight, so there exists $K=K_{\epsilon}<\infty$ so large that for all $m \in \mathbb{N}$, $${\mathbb P}\left (\sup_{x\in \mathbb R}g_{1}^{m}(x) \ge K
\right) \le \epsilon.$$ By choosing $\delta >0$ so small that $\epsilon/\delta^{3/2}>K$, we obtain for $N$ sufficiently large.
Uniqueness of the Limit Process
-------------------------------
Since $\{g_{s}^{N}\}_{s\ge 0}$ has stationary and independent increments, any weak limit will also have these properties. Therefore, to prove the uniqueness of the limit process it suffices to show that for any fixed time $s>0$ there is only one possible limit for the sequence $\{g^{N}_{s}\}_{N \in\mathbb{N}}$.
For any $N\in\mathbb{N}$, the random function $g^{N}_{s}(\cdot )$ is defined by rescaling the occupation measure $X^{\lfloor sN \rfloor}$ of the branching random walk initiated by the first $\lfloor sN \rfloor$ ancestral individuals (cf. equation ). The occupation measure $X^{\lfloor sN \rfloor}$ is defined by , which can be rewritten as $$X^{m}(j)=\sum_{i=1}^{m} \sum_{n=0}^{\infty} Z_{i,n}(j),$$ where $Z_{i,n}(j)$ is the occupation counts at location $j$ of individuals in the $n$-th generation of the $i$-th labeled tree. As discussed in Remark \[rem:indef-integral\], to avoid invoking an indefinite integral operator in the weak limit, we consider the truncated occupation counts and the associated occupation density up to the $\lfloor NH\rfloor$-th generation for some $H>0$ fixed. Define $$\begin{aligned}
X^{m,H}(j)&=\sum_{i=1}^{m} \sum_{n=0}^{\lfloor NH\rfloor} Z_{i,n}(j) \quad
\textrm{and} \\
g^{N}_{s,H}(x)&=N^{-3/2}\bar{X}^{\lfloor sN \rfloor,H} (\sqrt{N}x)\end{aligned}$$ where, as earlier, the bar denotes the function obtained by linear interpolation. The same calculations as in the proof of Condition 1$^{\circ }$ show that for any fixed $H>0$ and $s>0$ the sequence $g^{N}_{s,H}$ is tight.
Watanabe’s convergence theorem states that for any $s>0$, the rescaled measure-valued process converges weakly to the super-Brownian motion, i.e., $$\left\{Y_{t}^{s, N}\right\}_{0\le t\le H} :=\left\{\frac{1}{N}\sum_{j\in \mathbb Z}\sum_{i=1}^{\lfloor Ns\rfloor}Z_{i, \lfloor tN \rfloor}(j)\delta_{j/\sqrt{N}}\right\} \Longrightarrow \left\{Y_{t}^{s}\right\}_{0\le t\le H},$$ Viewing the measure-valued process $Y_{t}^{s,N}$ as nonnegative continuous functions over $\mathbb R$, we define $\bar Y_{t}^{s, N}\in \mathcal C_{0}(\mathbb R)$ by setting $$\bar Y_{t}^{s, N}(x) := \frac{1}{N}\sum_{i=1}^{\lfloor Ns\rfloor}Z_{i, \lfloor tN \rfloor}(\sqrt{N}x), \quad \text{if } x\in \mathbb Z/\sqrt{N}.$$ and then doing a linear interpolation. The above convergence implies that the weak convergence of the rescaled total occupation measure of the first $\lfloor NH\rfloor$ generations: $$\frac{1}{N}\sum_{t=0}^{\lfloor NH\rfloor} Y^{s, N}_{\frac{t}{N}}(x) \Longrightarrow \int_{0}^{H} Y_{t}^{s}(x) dt.$$ Notice that the left side is indeed $\frac{X^{\lfloor sN\rfloor, H}(\sqrt{N}x)}{N^{2}}$, which has densities $g^{N}_{s,H}$. Consequently, any possible weak subsequential limit of $\{g^{N}_{s,H}\}_{N\ge 1}$ in the function space $\mathcal{C}_{0}(\mathbb{R})$ must be a density for the occupation measure of the super-Brownian motion, that is, as $N\to\infty$ $$g^{N}_{s,H} \Longrightarrow g_{s,H} \quad \textrm {where} \quad g_{s,
H}(x){{\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =\,}}\int_{0}^{H}Y^{s}(t, x)dt .$$ But by inequality , for any $\epsilon>0$ there exists $H=H_{\epsilon}<\infty$ so large that for any $N$, $${\mathbb P}{
\left\{ g^{N}_{s}\not = g^{N}_{s,H} \right\}}\leq \epsilon.$$ Consequently, the sequence $\{g^{N}_{s}\}_{N\in \mathbb N}$ must converge weakly to $\int_{0}^{\infty}Y^{s}(t, x)dt$.
\[sec:Levyjump\]Properties of the Limiting Process
==================================================
In this section, we prove properties of the limiting process $\{g_{s}\}_{s\ge 0}$ (Theorem \[thm:property-gs\]) and characterize it using a Poisson point process (Theorem \[thm:levy-measure\]). In order to make sense of the notion of a “subordinator” on the function space $\mathcal C_{0} ({\mathbb{R}})$, we first briefly review the definition of a *Banach lattice*.
\[def:banachlattice\] A [Banach lattice]{} is a triple $(E, \lVert \cdot\rVert, \le)$ such that
1. $(E, \lVert \cdot \rVert)$ is a Banach space with norm $\lVert\cdot\rVert$;
2. $(E, \le)$ is an ordered vector space with the partial ordering $\le$;
3. under $\le$, any pair $x, y \in E$ has a least upper bound denoted by $x \vee y$ and a greatest lower bound denoted by $x \wedge
y$ (this is the “lattice” property); and
4. Set $|x| {{\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =\,}}x \vee (-x)$. Then $|x|\le |y|$ implies $\lVert x \rVert \le \lVert y\rVert$, $\forall x, y \in E$ (i.e., $\lVert \cdot \rVert$ is “a lattice norm”).
The Banach space $(\mathcal C_{0}(\mathbb R), \lVert \cdot
\rVert_{\infty})$ has a natural partial ordering, defined by $$f \le g \textup{ if and only if } g(x)-f(x) \ge 0 \quad \text{for all}
\quad x\in \mathbb R.$$ The triple $(\mathcal C_{0}(\mathbb R), \lVert \cdot \rVert_{\infty},
\le)$ clearly satisfies (a) and (b) in Definition \[def:banachlattice\]. The least upper bound and the greatest lower bound are defined pointwise: $$(f\vee g) (x) = f(x) \vee g(x), \qquad (f\wedge g) (x) = f(x) \wedge g(x).$$ Condition (d) can be verified easily.
Let $(E, \lVert \cdot\rVert, \le)$ be a Banach lattice. An $E$-valued stochastic process $\{X_{t}\}_{t\ge 0}$ is a *subordinator* if $\{X_{t}\}_{t \ge 0}$ is a Lévy process (that is, $\{X_{t}\}_{t \ge 0}$ has stationary, independent increments) and with probability one, for all $t \ge s \ge 0$, $$X_{t}-X_{s} \ge 0.$$ A subordinator $\{X_{t}\}_{t\ge 0}$ is a *pure jump* process if for every $t$, $$X_{t} = \sum_{s\le t}(X_{s}-X_{s-}).$$
For (i), we have for each $N\ge 1$, $$g_{s}^{N}(x) = \frac{\bar X^{\lfloor sN \rfloor}(\sqrt N x)}{N^{3/2}} = \frac{\lfloor sN\rfloor ^{3/2}}{N^{3/2}}\cdot \frac{\bar X^{1\cdot \lfloor sN \rfloor}\left (\sqrt{N} x\right)}{\lfloor sN\rfloor ^{3/2}} = \frac{\lfloor sN\rfloor ^{3/2}}{N^{3/2}} g_{1}^{\lfloor sN\rfloor}\left (\sqrt{\frac{N}{\lfloor sN \rfloor}}\, x \right).$$ Taking $N\to\infty$ gives (i). The claim that $\{I_{s}\}_{s\ge 0}$ is a stable-2/3 subordinator follows from monotonicity of $g_{s}$ and the scaling relation above at $x=0$, which yields $$I_{s} = s^{3/2} I_{1}.$$ For (iii), recall that a version of the stable–$\frac{1}{2}$ subordinator on ${\mathbb{R}}$ is the inverse local-time process of a standard Brownian motion $\{B_{t}\}_{t\ge 0}$ $$\tilde{\tau}_{s}=\inf \{t\geq 0\,:\, L^{0}_{t}>s\},$$ where $L^{0}_{t}$ is the Brownian local time at location $0$ up to time $t$. The jumps of the process $\{\tilde{\tau}_{s} \}_{s\geq 0}$ are the lengths of the excursions of the Brownian path.
Now consider a sequence of independent critical Galton-Watson trees $\mathcal{T}_{i}$ with offspring distribution $\nu$, initiated by particles $i=1,2,3,\dots$. Let $|\mathcal{T}_{i}|$ be the size (number of vertices) of the $i$-th tree, and set $A_{N}{{\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =\,}}\sum_{i=1}^{N}|\mathcal{T}_{i}|$, the total number of vertices in the first $k$ trees. Then by a theorem of Le Gall [@LeGall2005], as $N\to\infty$, $$\{ A_{\lfloor sN\rfloor}/N^{2}\}_{s\geq 0} \Longrightarrow \{\tilde{\tau}_{s/\sigma_{\nu}}
\}_{s\geq 0} {\overset{\mathcal D}{=\joinrel=}}\{\sigma_{\nu}^{-2}\tilde{\tau}_{s}
\}_{s\geq 0},$$ where the last equality follows from the scaling rule of a stable–$\frac{1}{2}$ process. Next, suppose that branching random walks are built on the Galton-Watson trees $\mathcal{T}_{i}$ by labelling the vertices, as described earlier. Then clearly $$A_{\lfloor sN\rfloor } = \sum_{j\in {\mathbb{Z}}/\sqrt{N}}X^{\lfloor sN\rfloor} (\sqrt{N}j).$$ By Theorem \[thm1.1\], $\{g_{s}^{N}\}_{s\ge 0} \Rightarrow \{g_{s}\}_{s\ge 0}$ in $\mathbb D([0, +\infty), \mathcal C_{0}(\mathbb R))$. Considering the space-truncated occupation density $g_{s}^{N}(x)\mathbf 1_{[-B, B]}(x)$ (i.e., truncated in space) for sufficiently large $B>0$ and following the same strategy as when proving the uniqueness of the limiting process $\{g_{s}\}_{s\ge 0}$, one would obtain $$\left\{\frac{1}{\sqrt{N}}\sum_{x\in {\mathbb{Z}}/\sqrt{N}}g_{s}^{N}(x)\right\}_{s\geq 0} \Longrightarrow \left\{ \int_{{\mathbb{R}}} g_{s}(x)dx\right\}_{s\ge 0} := \{\theta_s\}_{s\ge 0},$$ where the left side is indeed $$\frac{1}{\sqrt{N}}\sum_{x\in {\mathbb{Z}}/\sqrt{N}}g_{s}^{N}(x) = \frac{\sum_{x\in {\mathbb{Z}}/\sqrt{N}}X^{\lfloor sN\rfloor}(\sqrt{N}x)}{N^{2}} = \frac{A_{\lfloor sN\rfloor}}{N^{2}}.$$
Consequently, the processes $\{\theta_{s}\}_{s\ge 0}$ and $\{\sigma_{\nu}^{-2}\tilde{\tau}_{s} \}_{s\geq 0}$ have the same law, and so $\{\theta_{s}\}_{s\ge 0}$ is a stable-1/2 subordinator.
For (iv), we have already observed that $g_{s}$ has stationary, independent increments and increasing sample paths relative to the natural partial order $\leq$ on $\mathcal C_{0} ({\mathbb{R}})$. To show that $\{g_{s}\}_{s\ge 0}$ has pure jumps, we make use of the fact that the total area process $\{\theta_{s}\}_{s\ge 0}$ is a stable–$\frac{1}{2}$ subordinator and thus has pure jumps. Let $\mathcal J$ be the set of jump times of the process $\{\theta_{s}\}_{s\geq 0}$, that is, the set of all $t\geq 0$ for which $\theta
(t)-\theta (t-)>0$. Define $$\tilde{g}_{s}=\sum_{t\in\mathcal J\cap [0,s]} (g_{t}-g_{t-}),$$ a process that collects the changes in $\{g_{s}\}_{s\ge 0}$ at those times when the limiting total area process $\{\theta_{s}\}_{s\ge 0}$ makes jumps. Clearly, the process $\tilde{g}_{s}$ is an increasing process in $\mathcal C_{0} ({\mathbb{R}})$, and since $\tilde g_{s}$ only gathers the jumps of $g_{s}$, we have $$\tilde{g}_{s}\leq g_{s} \quad \text{for every} \;\; s\geq 0.$$ But since the area process $\theta_{s}$ is pure jump, $g_{s}$ and $\tilde{g}_{s}$ bound the same total area for every $s$, that is, $$\int_{{\mathbb{R}}} g_{s} (x)\,dx = \theta_{s}= \int_{{\mathbb{R}}} \tilde{g}_{s} (x)\,dx.$$ By continuity of both $g_{s}$ and $\tilde g_{s}$, we have $g_{s}=\tilde{g}_{s}$ for every $s$, and thus the process $g_{s}$ is a pure jump process in $\mathcal C_{0} ({\mathbb{R}})$.
\[Proof of Theorem \[thm:levy-measure\]\] We have already proved in Theorem \[thm:property-gs\] that the process $g_{s}$ consists of pure jumps. It remains to show that the point process of jumps is a Poisson point process with intensity given by and then the representation would follow automatically. Consider the point process of jumps of $g_{s}$ for $s\leq 1$ (the case $s\leq s_{*}$, for arbitrary $s_{*}>0$, can be handled in analogous fashion). Let $J_{1}, J_{2},\dots$ be the jumps ordered by size from largest to smallest, as in Corollary \[cor:order-stats\]. Since by Theorem \[thm:property-gs\], the limiting process $\{g_{s}\}_{s\ge 0}$ is a pure jump subordinator, we have $$\begin{aligned}
\label{eqn:repg1}
g_{1}=\sum_{i=1}^{\infty}J_{i}.\end{aligned}$$ Theorem \[thm:property-gs\] also implies that the jump sizes $|J_{i}|=\int J_{i} (x)\,dx$ are distributed (jointly) as the ordered excursion lengths of a standard Brownian motion run up to the first time $t$ that $L^{0}_{t}=1$, rescaled by $\sigma_{\nu}^{-2}$. By Corollary \[cor:order-stats\], for any $m\geq 1$, as $k \rightarrow \infty$, $$\begin{aligned}
\label{eqn:jump-cov}
(J^{N}_{1},J^{N}_{2},\dots ,J^{N}_{m})\Longrightarrow
(J_{1},J_{2},\dots ,J_{m}),\end{aligned}$$ where $J^{N}_{1},J^{N}_{2},\dots$ are the ordered jumps in the (rescaled) occupation density processes $g^{N}_{s}$ for $s\leq 1$ for the branching random walk obtained by amalgamating the first $N$ trees. Consequently, the joint distribution of the random variables $$|J^{N}_{i}|=|\mathcal{T}^{N}_{(i)}|/N^{2}$$ (where $\mathcal{T}^{N}_{(i)}$ is the $i$-th largest tree among the first $N$ trees) converges to the joint distribution of the sizes $|J_{i}|$. In particular, the largest, second largest, etc., trees among the first $N$ trees have sizes of order $N^{2}$ — and so as $N
\rightarrow \infty$, these will be large.
To identify the limiting distribution of the rescaled jumps, we now make use of Theorem 1.1 in [@condGW-profile], which states that the occupation density of a conditioned branching random walk scaled by the size of the tree converges to that of the ISE density $f_{\textup{ISE}}$, as the size of the tree becomes large. This implies, for each $i=1,2,\dots $, as $N \rightarrow \infty$, $$\begin{aligned}
\label{eqn:jump-char}
\frac{J^{N}_{i} (|J_{i}^{N}|^{1/4}\ \cdot\, )}{|J_{i}^{N}|^{3/4}}
\Longrightarrow \gamma f^{(i)}_{\textup{ISE}}(\gamma \ \cdot\,),\ \
\text{where } \gamma = \sigma_{F}^{-1}\sigma_{\nu}^{1/2},\end{aligned}$$ and the limiting ISE densities, $ f^{(1)}_{\textup{ISE}}$, $ f^{(2)}_{\textup{ISE}}$, $\ldots$ are i.i.d copies of $f_{\textup{ISE}}$. By and , we can describe the joint distribution of $J_{1},J_{2},\dots$ as follows: (a) let $\varepsilon_{1}>\varepsilon_{2}>\varepsilon_{3}>\dots $ be the ordered excursion lengths of a standard Brownian motion run until the first time $t$ such that $L^{0}_{t}=1$; (b) let $f_{1},f_{2},f_{3},\dots$ be i.i.d. copies of the ISE density $f_{\textup{ISE}}$ which are independent of the $\varepsilon_{i}$’s; and (c) set $$\begin{aligned}
J_{i} (\cdot) &{{\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =\,}}(\sigma_{\nu}^{-2}\varepsilon_{i})^{3/4}\gamma f_{i} (\gamma (\sigma_{\nu}^{-2}\varepsilon_{i})^{-1/4}\ \cdot\,)\\
&= \frac{1}{\sigma_{\nu}\sigma_{F}}\, \varepsilon_{i}^{3/4} f_{i} (\sigma_{\nu}\sigma_{F}^{-1}\varepsilon_{i}^{-1/4}\ \cdot\,).\end{aligned}$$ Since the ordered excursion lengths $\varepsilon_{1}>\varepsilon_{2}>\varepsilon_{3}>\dots $ have the distribution of the ordered points in a Poisson point process on ${\mathbb{R}}_{+}$ with intensity measure $
\frac{dy}{\sqrt{2\pi y^{3}}},
$ the representation follows from .
[**Acknowledgment.**]{} The authors are grateful to the anonymous referee for valuable comments.
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[^1]: By “nondecreasing”, we mean that for all $t\ge s\ge 0$, $g^{N}_{t}-g^{N}_{s} \ge 0$. A simulation is shown in Figure \[Fig:simuG\].
|
---
abstract: 'We consider a multilingual weakly supervised learning scenario where knowledge from annotated corpora in a resource-rich language is transferred via bitext to guide the learning in other languages. Past approaches project labels across bitext and use them as features or gold labels for training. We propose a new method that projects model expectations rather than labels, which facilities transfer of model uncertainty across language boundaries. We encode expectations as constraints and train a discriminative CRF model using Generalized Expectation Criteria [@Mann:2010:JMLR]. Evaluated on standard Chinese-English and German-English NER datasets, our method demonstrates F$_1$ scores of 64% and 60% when no labeled data is used. Attaining the same accuracy with supervised CRFs requires 12k and 1.5k labeled sentences. Furthermore, when combined with labeled examples, our method yields significant improvements over state-of-the-art supervised methods, achieving best reported numbers to date on Chinese OntoNotes and German CoNLL-03 datasets.'
author:
- Mengqiu Wang
- |
Christopher D. Manning\
Computer Science Department\
Stanford University\
Stanford, CA 94305 USA\
[{mengqiu,manning}@cs.stanford.edu ]{}
bibliography:
- 'CLiPPER.bib'
nocite: '[@Collins:1999:EMNLP; @Klein:2005:Thesis; @Liang:2005:Thesis; @Smith:2006:Thesis; @Goldberg:2010:Thesis]'
title: |
Cross-lingual Pseudo-Projected Expectation Regularization for\
Weakly Supervised Learning
---
[UTF8]{}[gbsn]{}
Introduction {#sec:intro}
============
Supervised statistical learning methods have enjoyed great popularity in Natural Language Processing (NLP) over the past decade. The success of supervised methods depends heavily upon the availability of large amounts of annotated training data. Manual curation of annotated corpora is a costly and time consuming process. To date, most annotated resources resides within the English language, which hinders the adoption of supervised learning methods in many multilingual environments.
To minimize the need for annotation, significant progress has been made in developing unsupervised and semi-supervised approaches to NLP (Collins and Singer 1999; Klein 2005; Liang 2005; Smith 2006; Goldberg 2010; *inter alia*) . More recent paradigms for semi-supervised learning allow modelers to directly encode knowledge about the task and the domain as constraints to guide learning [@Chang:2007:ACL; @Mann:2010:JMLR; @Ganchev:2010:JMLR]. However, in a multilingual setting, coming up with effective constraints require extensive knowledge of the foreign[^1] language.
Bilingual parallel text (bitext) lends itself as a medium to transfer knowledge from a resource-rich language to a foreign languages. project labels produced by an English tagger to the foreign side of bitext, then use the projected labels to learn a HMM model. More recent work applied the projection-based approach to more language-pairs, and further improved performance through the use of type-level constraints from tag dictionary and feature-rich generative or discriminative models [@Das:2011:ACL; @Tackstrom:2013:ACL].
In our work, we propose a new project-based method that differs in two important ways. First, we never explicitly project the labels. Instead, we project expectations over the labels. This pseudo-projection acts as a soft constraint over the labels, which allows us to transfer more information and uncertainty across language boundaries. Secondly, we encode the expectations as constraints and train a model by minimizing divergence between model expectations and projected expectations in a Generalized Expectation (GE) Criteria [@Mann:2010:JMLR] framework.
We evaluate our approach on Named Entity Recognition (NER) tasks for English-Chinese and English-German language pairs on standard public datasets. We report results in two settings: a weakly supervised setting where no labeled data or a small amount of labeled data is available, and a semi-supervised settings where labeled data is available, but we can gain predictive power by learning from unlabeled bitext.
Related Work {#sec:related_work}
============
Most semi-supervised learning approaches embody the principle of learning from constraints. There are two broad categories of constraints: multi-view constraints, and external knowledge constraints.
Examples of methods that explore multi-view constraints include self-training [@Yarowsky:1995:ACL; @McClosky:2006:NAACL],[^2] co-training [@Blum:1998:COLT; @Sindhwani:2005:ICML], multi-view learning [@Ando:2005:ACL; @Carlson:2010:WSDM], and discriminative and generative model combination [@Suzuki:2008:ACL; @Druck:2010:ICML].
An early example of using knowledge as constraints in weakly-supervised learning is the work by . They showed that the addition of a small set of “seed” rules greatly improve a co-training style unsupervised tagger. proposed a constraint-driven learning (CODL) framework where constraints are used to guide the selection of best self-labeled examples to be included as additional training data in an iterative EM-style procedure. The kind of constraints used in applications such as NER are the ones like “the words CA, Australia, NY are <span style="font-variant:small-caps;">Location</span>” [@Chang:2007:ACL]. Notice the similarity of this particular constraint to the kinds of features one would expect to see in a discriminative model such as MaxEnt. The difference is that instead of learning the validity (or weight) of this feature from labeled examples — since we do not have them — we can constrain the model using our knowledge of the domain. also demonstrated that in an active learning setting where annotation budget is limited, it is more efficient to label features than examples. Other sources of knowledge include lexicons and gazetteers [@Druck:2007:NIPS; @Chang:2007:ACL].
While it is straight-forward to see how resources such as a list of city names can give a lot of mileage in recognizing locations, we are also exposed to the danger of over-committing to hard constraints. For example, it becomes problematic with city names that are ambiguous, such as Augusta, Georgia.[^3] To soften these constraints, proposed the Generalized Expectation (GE) Criteria framework, which encodes constraints as a regularization term over some score function that measures the divergence between the model’s expectation and the target expectation. The connection between GE and CODL is analogous to the relationship between hard (Viterbi) EM and soft EM, as illustrated by .
Another closely related work is the Posterior Regularization (PR) framework by . In fact, as have shown, in a discriminative model these two methods optimize exactly the same objective.[^4] The two differ in optimization details: PR uses a EM algorithm to approximate the gradients which avoids the expensive computation of a covariance matrix between features and constraints, whereas GE directly calculates the gradient. However, later results [@Druck:2011:Thesis] have shown that using the Expectation Semiring techniques of , one can compute the exact gradients of GE in a Conditional Random Fields (CRF) [@Lafferty:2001:ICML] at costs no greater than computing the gradients of ordinary CRF. And empirically, GE tends to perform more accurately than PR [@Bellare:2009:UAI; @Druck:2011:Thesis].
Obtaining appropriate knowledge resources for constructing constraints remain as a bottleneck in applying GE and PR to new languages. However, a number of past work recognizes parallel bitext as a rich source of linguistic constraints, naturally captured in the translations. As a result, bitext has been effectively utilized for unsupervised multilingual grammar induction [@Alshawi:2000:MT; @Snyder:2009:ACL], parsing [@Burkett:2008:EMNLP], and sequence labeling [@Naseem:2009:JMLR].
A number of recent work also explored bilingual constraints in the context of simultaneous bilingual tagging, and showed that enforcing agreements between language pairs give superior results than monolingual tagging [@Burkett:2010:CONLL; @Che:2013:NAACL; @Wang:2013:AAAI]. They also demonstrated a *uptraining* [@Petrov:2010:EMNLP] setting where tag-induced bitext can be used as additional monolingual training data to improve monolingual taggers. A major drawback of this approach is that it requires a readily-trained tagging models in each languages, which makes a weakly supervised setting infeasible. Another intricacy of this approach is that it only works when the two models have comparable strength, since mutual agreements are enforced between them.
Projection-based methods can be very effective in weakly-supervised scenarios, as demonstrated by , and . One problem with projected labels is that they are often too noisy to be directly used as training signals. To mitigate this problem, designed a label propagation method to automatically induce a tag lexicon for the foreign language to smooth the projected labels. filter out projection noise by combining projections from from multiple source languages. However, this approach is not always viable since it relies on having parallel bitext from multiple source languages. proposed the use of crowd-sourced Wiktionary as additional resources for inducing tag lexicons. More recently, combined token-level and type-level constraints to constrain legitimate label sequences and and recalibrate the probability distribution in a CRF. The tag dictionary used for POS tagging are analogous to the gazetteers and name lexicons used for NER by .
Our work is also closely related to . They used a two-step projection method similar to for dependency parsing. Instead of using the projected linguistic structures as ground truth [@Yarowsky:2001:NAACL], or as features in a generative model [@Das:2011:ACL], they used them as constraints in a PR framework. Our work differs by projecting expectations rather than Viterbi one-best labels. We also choose the GE framework over PR. Experiments in and suggest that in a discriminative model (like ours), GE is more accurate than PR.
{width=".95\textwidth"}
Approach
========
Given bitext between English and a foreign language, our goal is to learn a CRF model in the foreign language from little or no labeled data. Our method performs [**C**]{}ross-[**Li**]{}ngual [**P**]{}seudo-[**P**]{}rojection [**E**]{}xpectation [**R**]{}egularization ([**CLiPPER**]{}).
Figure \[fig:process\] illustrates the high-level workflow. For every aligned sentence pair in the bitext, we first compute the posterior marginal at each word position on the English side using a pre-trained English CRF tagger; then for each aligned English word, we project its posterior marginal as expectations to the aligned word position on the foreign side.
We would like to learn a CRF model in the foreign language that has similar expectations as the projected expectations from English. To this end, we adopt the Generalized Expectation (GE) Criteria framework introduced by . In the remainder of this section, we follow the notation used in [@Druck:2011:Thesis] to explain our approach.
CLiPPER
-------
The general idea of GE is that we can express our preferences over models through constraint functions. A desired model should satisfy the imposed constraints by matching the expectations on these constraint functions with some target expectations (attained by external knowledge like lexicons or in our case transferred knowledge from English). We define a constraint function $\bphi_{i, l_j}$ for each word position $i$ and output label assignment $l_j$ as a label identity indicator: $$\phi_{i, l_j}(\by) = \left\{ \begin{array}{rl}
1 &\mbox{ if $l_j = y_i$ and $A_i \neq \emptyset$} \\
0 &\mbox{ otherwise}
\end{array} \right.$$ The set $\{l_1, \cdots, l_m\}$ denotes all possible label assignment for each $y_i$, and $m$ is number of label values. $A_i$ is the set of English words aligned to Chinese word $i$. The condition $A_i \neq \emptyset$ specifies that the constraint function applies only to Chinese word positions that have at least one aligned English word. Each $\phi_{i, l_j}(\by)$ can be treated as a Bernoulli random variable, and we concatenate the set of all $\phi_{i, l_j}$ into a *random vector* $\bphi(\by)$, where $\bphi_k = \phi_{i, l_j}$ if $k = i * m + j$. We drop the $(\by)$ in $\bphi$ for simplicity.
The target expectation over $\phi_{i, l_j}$, denoted as $\tilde\phi_{i, l_j}$, is the expectation of assigning label $l_j$ to English word $A_i$[^5] under the English conditional probability model.
The expectation over $\bphi$ under a conditional probability model $P(\by|\bx;{\btheta})$ is denoted as $E_{P(\by|\bx;{\btheta})}[\bphi]$, and simplified as $E_{\btheta}[\bphi]$ whenever it is unambiguous.
The conditional probability model $P(\by|\bx;{\btheta})$ in our case is defined as a standard linear-chain CRF:[^6] $$\begin{aligned}
P(\by|\bx;{\btheta}) = \frac{1}{Z(\bx;\btheta)}{\exp\left( \sum\limits_{i}^{n}{\btheta \mathbf{f}({\bx, y_i, y_{i-1}}) } \right)}\end{aligned}$$ where $\mathbf{f}$ is a set of feature functions; $\btheta$ are the matching parameters to learn; $n = |\bx|$.
The objective function to maximize in a standard CRF is the log probability over a collection of labeled documents: $$\begin{aligned}
L_{CRF}(\btheta) = \sum\limits_{a=1}^{a'}\log P(\by_{a}^{*}|\bx_{a};{\btheta})
\label{eqn:obj_crf}\end{aligned}$$ $a'$ is the number of labeled sentences. $\by^*$ is an observed label sequence.
The objective function to maximize in GE is defined as the sum over all unlabeled examples (foreign side of bitext), over some cost function $S$ between between the model expectation ($E_{\btheta}[\bphi] $) and the target expectation ($\tilde\bphi$) over $\bphi$.
We choose $S$ to be the negative $L_2^2$ squared error,[^7] defined as: $$\begin{aligned}
L_{GE}(\btheta) &= \sum\limits_{b=1}^{n'} S\left(E_{P(\by|\bx_b;\btheta)}[\bphi(\by_b)], \tilde\bphi(\by_b\right) \nonumber \\
& = \sum\limits_{b=1}^{b'} -\| \tilde\bphi(\by_b) - E_{\btheta}[\bphi(\by_b)] \|_2^2
\label{eqn:obj_ge}\end{aligned}$$ $b'$ is the total number of unlabeled bitext sentence pairs.
When both labeled and bitext training data are available, the joint objective is the sum of Eqn. \[eqn:obj\_crf\] and \[eqn:obj\_ge\]. Each is computed over the labeled training data and foreign half in the bitext, respectively.
We can optimize this joint objective by computing the gradients and use a gradient-based optimization method such as L-BFGS. Gradients of $L_{CRF}$ decomposes down to the gradients over each labeled training example $(\bx, \by^{*})$, computed as: $$\begin{aligned}
\vspace{-3cm}
\frac{\partial}{\partial \btheta}(\log P(\by^{*}_{a}|\bx_{a};{\btheta}) = \tilde{E}[\btheta] - E[\btheta] \\
\vspace{-3cm}\end{aligned}$$ where $\tilde E[\btheta]$ and $E[\btheta]$ are the empirical and expected feature counts, respectively.
Computing the gradient of $L_{GE}$ decomposes down to the gradients of $S(E_{P(\by|\bx_b;\btheta}[\bphi])$ for each unlabeled foreign sentence $\bx$ and the constraints over this example $\bphi$ . The gradients can be calculated as: $$\begin{aligned}
\frac{\partial}{\partial \btheta}S(E_{\btheta}[\bphi] ) & = -\frac{\partial}{\partial \btheta}\left( \tilde\bphi - E_{\btheta}[\bphi] \right)^{T} \left( \tilde\bphi - E_{\btheta}[\bphi] \right) \\
& = 2\left( \tilde\bphi - E_{\btheta}[\bphi] \right)^{T} \left( \frac{\partial}{\partial \btheta} E_{\btheta}[\bphi] \right)\end{aligned}$$ We redefine the penalty vector $\mathbf{u} = 2\left( \tilde\bphi - E_{\btheta}[\bphi] \right)$ to be $u$. $\frac{\partial}{\partial \btheta} E_{\btheta}[\bphi]$ is a matrix where each column contains the gradients for a particular model feature $\theta$ with respect to all constraint functions $\bphi$. It can be computed as: $$\begin{aligned}
&\frac{\partial}{\partial \btheta} E_{\btheta}[\bphi] = \sum\limits_{\by} \bphi(\by) \frac{\partial}{\partial \btheta} P(\by|\bx;\btheta)\nonumber \\
=& \sum\limits_{\by} \bphi(\by) \frac{\partial}{\partial \btheta} \left( \frac{1}{Z(\bx;\btheta)} {\exp({\btheta^{T}\mathbf{f}({\bx,\by}))}} \right) \nonumber \\
=& \sum\limits_{\by} \bphi(\by) \Bigg( \frac{1}{Z(\bx;\btheta)} \left( \frac{\partial}{\partial\btheta} {\exp({\btheta^{T}\mathbf{f}({\bx,\by}))}} \right) \nonumber \\
& + {\exp({\btheta^{T}\mathbf{f}({\bx,\by}))}} \left( \frac{\partial}{\partial\btheta} \frac{1}{Z(\bx;\btheta)} \right) \Bigg) \nonumber\end{aligned}$$ $$\begin{aligned}
= & \sum\limits_{\by} \bphi(\by) \bigg( P(\by | \bx; \btheta) \mathbf{f}({\bx,\by})^{T} \nonumber\\
& - P(\by | \bx; \btheta)\sum_{\by'} P(\by' | \bx; \btheta) \mathbf{f}({\bx,\by'})^{T} \bigg) \nonumber \\
=& \sum\limits_{\by} P(\by | \bx; \btheta)\sum_{\by} \bphi(\by) \mathbf{f}({\bx,\by})^{T} \nonumber \\
&- \big( \sum\limits_{\by} P(\by | \bx; \btheta) \bphi(\by) \big) \big( \sum_{\by} P(\by | \bx; \btheta) \mathbf{f}({\bx,\by})^{T} \big) \nonumber \\
= & \; \; \mathrm{COV}_{P(\by|\bx;\btheta)}\left( \bphi(\by), \mathbf{f}(\bx,\by) \right) \label{eqn:cov}\\
= & \; \; E_{\btheta}[\bphi\mathbf{f}^{T}] - E_{\btheta}[\bphi]E_{\btheta}[\mathbf{f}^{T}] \label{eqn:before}\end{aligned}$$ Eqn. \[eqn:cov\] gives the intuition of how optimization works in GE. In each iteration of L-BFGS, the model parameters are updated according to their covariance with the constraint features, scaled by the difference between current expectation and target expectation. The term $E_{\btheta}[\bphi\mathbf{f}^{T}]$ in Eqn. \[eqn:before\] can be computed using a dynamic programming (DP) algorithm, but solving it directly requires us to store a matrix of the same dimension as $\mathbf{f}^{T}$ in each step of the DP. We can reduce the complexity by using the following trick: $$\begin{aligned}
& \frac{\partial}{\partial \btheta}S(E_{\btheta}[\bphi] ) = u^{T} \left( \frac{\partial}{\partial \btheta} E_{\btheta}[\bphi] \right) \nonumber \\
= & \mathbf{u}^{T} \left( E_{\btheta}[\bphi\mathbf{f}^{T}] - E_{\btheta}[\bphi]E_{\btheta}[\mathbf{f}^{T}] \right) \nonumber \\
= & E_{\btheta}[\mathbf{u}^{T} \bphi\mathbf{f}^{T}] - E_{\btheta}[\mathbf{u}^{T} \bphi]E_{\btheta}[\mathbf{f}^{T}] \nonumber \\
= & E_{\btheta}[\bphi'\mathbf{f}^{T}] - E_{\btheta}[\bphi']E_{\btheta}[\mathbf{f}^{T}] \label{eqn:after} \\
& \bphi' = \mathbf{u}^{T} \bphi \nonumber\end{aligned}$$ Now in Eqn. \[eqn:after\], $E_{\btheta}[\bphi']$ becomes a scalar value; and to compute the term $E_{\btheta}[\bphi'\mathbf{f}^{T}] $, we only need to store a vector in each step of the following DP algorithm [@Druck:2011:Thesis 93]: $$\begin{aligned}
E_{\btheta}[\bphi'\mathbf{f}^{T}] = & \sum\limits_{i=1}^{n}\sum\limits_{y_i}\sum\limits_{y_{i+1}}\bigg\{ \Big[ \sum\limits_{j=1}^{n} \sum\limits_{y_j} P(y_i, y_{i+1}, y_j | \bx) \\
\cdot & \; \bphi(y_j) \Big] \cdot \mathbf{f}(y_i, y_{i+1}, \bx)^{T} \bigg\}\end{aligned}$$ The bracketed term can be broken down to two parts: $$\begin{aligned}
&\sum\limits_{j=1}^{n} \sum\limits_{y_j} P(y_i, y_{i+1}, y_j | \bx) \bphi(y_j) \\
= &\sum\limits_{j=1}^{i} \sum\limits_{y_j} P(y_i, y_{i+1}, y_j | \bx) \bphi(y_j) \\
+ & \sum\limits_{j=i+1}^{n} \sum\limits_{y_j} P(y_i, y_{i+1}, y_j | \bx) \bphi(y_j) \\
= & \; \alpha(y_i, y_{i+1}, i) + \beta(y_i, y_{i+1}, i)\end{aligned}$$ $$\begin{aligned}
& \alpha(y_0, y_1, 0) \equiv P(y_0, y_1 | \bx) \bphi(y_0) \\
& \alpha(y_i, y_{i+1}, i) \equiv P(y_i, y_{i+1} | \bx) \bphi(y_i) + \\
& P(y_{i+1} | y_i, \bx) \sum\limits_{y_{i-1}} \alpha(y_{i-1}, y_i, i-1) \\
& \beta(y_{n-1}, y_n, n-1) \equiv P(y_{n-1}, y_n | \bx) \bphi(y_n) \\
& \beta(y_{i}, y_{i+1}, i) \equiv P(y_{i}, y_{i+1} | \bx) \bphi(y_{i+1}) + \\
& P(y_{i} | y_{i+1}, \bx) \sum\limits_{y_{i+2}} \beta(y_{i+1}, y_{i+2}, i+1)\end{aligned}$$
The resulting algorithm has complexity $O(nm^2)$, which is the same as the standard forward-backward inference algorithm for CRF.
Hard vs. Soft Projection {#sec:hard}
------------------------
Projecting expectations instead of one-best label assignments from English to foreign language can be thought of as a soft version of the method described in [@Das:2011:ACL] and [@Ganchev:2009:ICML]. Soft projection has its advantage: when the English model is not certain about its predictions, we do not have to commit to the current best prediction. The foreign model has more freedom to form its own belief since any marginal distribution it produces would deviates from a flat distribution by just about the same amount. In general, preserving uncertainties till later is a strategy that has benefited many NLP tasks [@Finkel:2006:EMNLP]. Hard projection can also be treated as a special case in our framework. We can simply recalibrate posterior marginal of English by assigning probability mass $1$ to the most likely outcome, and zero everything else out, effectively taking the $\argmax$ of the marginal at each word position. We refer to this version of expectation as the “hard” expectation. In the hard projection setting, GE training resembles a “project-then-train” style semi-supervised CRF training scheme [@Yarowsky:2001:NAACL; @Tackstrom:2013:ACL]. In such a training scheme, we project the one-best predictions of English CRF to the foreign side through word alignments, then include the newly “tagged” foreign data as additional training data to a standard CRF in the foreign language. The difference between GE training and this scheme is that they optimize different objectives: CRF optimizes maximum conditional likelihood of the observed label sequence, whereas GE minimizes squared error between model’s expectation and “hard” expectation based on the observed label sequence. We compare the hard and soft variants of GE with the project-then-train style CRF training in our experiments and report results in Section \[sec:min-results\].
Experiments
===========
We conduct experiments on Chinese and German NER. We evaluate CLiPPER in two learning settings: weakly supervised and semi-supervised. In the weakly supervised setting, we simulate the condition of having no labeled training data, and evaluate the model learned from bitext alone. We then vary the amount of labeled data available to the model, and examine the model’s learning curve. In the semi-supervised setting, we assume our model has access to the full labeled data; our goal is to improve performance of the supervised method by learning from additional bitext.
Dataset and Setup
-----------------
We used the latest version of Stanford NER Toolkit[^8] as our base CRF model in all experiments. Features for English, Chinese and German CRFs are documented extensively in [@Che:2013:NAACL] and [@Faruqui:2010:KONVENS] and omitted here for brevity. It it worth noting that the current Stanford NER models include recent improvements from semi-supervise learning approaches that induces distributional similarity features from large word clusters. These models represent the current state-of-the-art in supervised methods, and serve as a very strong baseline.
For Chinese NER experiments, we follow the same setup as to evaluate on the latest OntoNotes (v4.0) corpus [@Hovy:2006:NAACL].[^9] A total of 8,249 sentences from the parallel Chinese and English Penn Treebank portion [^10] are reserved for evaluation. Odd-numbered documents are used as development set, and even-numbered documents are held out as blind test set. The rest of OntoNotes annotated with NER tags are used to train the English and Chinese CRF base taggers. There are about 16k and 39k labeled sentences for Chinese and English training, respectively. The English CRF tagger trained on this training corpus gives F$_1$ score of 81.68% on the OntoNotes test set. Four entities types (<span style="font-variant:small-caps;">Person</span>, <span style="font-variant:small-caps;">Location</span>, <span style="font-variant:small-caps;">Organization</span> and <span style="font-variant:small-caps;">GPE</span>) are used with a BO tagging scheme. The English-Chinese bitext comes from the Foreign Broadcast Information Service corpus (FBIS).[^11] It is first sentence aligned using the Champollion Tool Kit,[^12] then word aligned with the BerkeleyAligner.[^13]
For German NER experiments, we evaluate using the standard CoNLL-03 NER corpus [@Sang:2003:CoNLL]. The labeled training set has 12k and 15k sentences. We used the de-en portion of the *News Commentary*[^14] data from WMT13 as bitext. The English CRF tagger trained on CoNLL-03 English training corpus gives F$_1$ score of 90.4% on the CoNLL-03 test set.
We report standard entity-level precision (P), recall (R) and F$_1$ score given by <span style="font-variant:small-caps;">ConllEval</span> script on both the development and test sets. Statistical significance tests are done using a paired bootstrap resampling method with 1000 iterations, averaged over 5 runs. We compare against three recently approaches that were introduced in Section \[sec:related\_work\]. They are: semi-supervised learning method using factored bilingual models with Gibbs sampling [@Wang:2013:AAAI]; bilingual NER using Integer Linear Programming (ILP) with bilingual constraints, by [@Che:2013:NAACL]; and constraint-driven bilingual-reranking approach [@Burkett:2010:CONLL]. The code from [@Che:2013:NAACL] and [@Wang:2013:AAAI] are publicly available,[^15]. Code from [@Burkett:2010:CONLL] is obtained through personal communications.[^16]
Since the objective function in Eqn. \[eqn:obj\_ge\] is non-convex, we adopted the early stopping training scheme from [@Turian:2010:ACL] as the following: after each iteration in L-BFGS training, the model is evaluated against the development set; the training procedure is terminated if no improvements have been made in 20 iterations.
Weakly Supervised Results {#sec:min-results}
-------------------------
The top four figures in Figure \[fig:learning\_curve\] show results of weakly supervised learning experiments. Quite remarkably, on Chinese test set, our proposed method (CLiPPER) achieves a F$_1$ score of 64.4% with 80k bitext, when no labeled training data is used. In contrast, the supervised CRF baseline would require as much as 12k labeled sentences to attain the same accuracy. Results on the German test set is less striking. With no labeled data and 40k of bitext, CLiPPER performs at F$_1$ of 60.0%, the equivalent of using 1.5k labeled examples in the supervised setting. When combined with 1k labeled examples, performance of CLiPPER reaches 69%, a gain of over 5% absolute over supervised CRF. We also notice that supervised CRF model learns much faster in German than Chinese. This result is not too surprising, since it is well recognized that Chinese NER is more challenging than German or English due to the lack of orthographical features, such as word capitalization. Chinese NER relies more on lexicalized features, and therefore needs more labeled data to achieve good coverage. The results also suggest that CLiPPER seems to be very effective at transferring lexical knowledge from English to Chinese.
The bottom two figures in Figure \[fig:learning\_curve\] compares soft GE projection with hard GE projection and the “project-then-train” style CRF training scheme (*cf.* Section \[sec:hard\]). We observe that both soft and hard GE projection significantly outperform the “project-then-train” style training scheme. The difference is especially pronounced on the Chinese results when fewer labeled examples are available. Soft projection gives better accuracy than hard projection when no labeled data is available, and also has a faster learning rate.
Semi-supervised Results
-----------------------
-- ------- ------------------------------------------------------------------------------- ------- ------- ------- ------- -------
P R F$_1$ P R F$_1$
79.87 63.62 70.83 88.05 73.03 79.84
10k 81.36 65.16 72.36 85.23 77.79 81.34
20k **[81.79]{} & 64.80 & 72.31 & 88.11 & 75.93 & 81.57\
& 40k & 79.24 & **[66.08]{} & 72.06 & **[88.25]{} & 76.52 & **[81.97]{}\
& 80k & 80.26 & 65.92 & **[72.38]{} & 87.80 & **[76.82]{} & 81.94************
-- ------- ------------------------------------------------------------------------------- ------- ------- ------- ------- -------
: Chinese and German NER results on the development set using CLiPPER with varying amounts of unlabeled bitext (10k, 20k, etc.). Best number of each column is highlighted in bold. The F$_1$ score improvements over CRF baseline in all cases are statistically significant at 99.9% confidence level. []{data-label="tbl:dev_results"}
In the semi-supervised experiments, we let the CRF model use the full set of labeled examples in addition to the unlabeled bitext. Table \[tbl:dev\_results\] shows results on the development dataset for Chinese and German using 10-80k bitext. We see that with merely 10k additional bitext, CLiPPER is able to improve significantly over state-of-the-art CRF baselines by as much as 1.5% F$_1$ on both Chinese and German. With more unlabeled data, we notice a tradeoff between precision and recall on Chinese. The final F$_1$ score on Chinese at 80k level is only marginally better than 10k. On the other hand, we observe a modest but steady improvement on German as we add more unlabeled bitext, up until 40k sentences. We select the best configurations on development set (80k for Chinese and 40k for German) to evaluate on test set.
--------------- --------------- ----------------------------------------------------------------------------------------------------------------------- ------------------- ----------- --------------- -----------
CRF 79.09 63.56 70.48 86.77 71.30 78.28
CRF$_{ptt}$ [**84.01**]{} 45.29 58.85 81.50 [**75.56**]{} 78.41
WCD13 [80.31]{} [65.78]{} [72.33]{} [85.98]{} [72.37]{} [78.59]{}
CWD13 81.31 [65.50]{} [72.55]{} [85.99]{} [ 72.98]{} [78.95]{}
BPBK10 79.25 65.67 71.83 - - -
CLiPPER$_{h}$ [83.67]{} 64.80 73.04$^{\S\ddag}$ 86.52 72.02 78.61
CLiPPER$_{s}$ 82.57 **[65.99]{} & **[73.35]{}$_{\diamond\ast}^{\S\dag\star}$ & **[87.11]{} & 72.56 & **[79.17]{}$^{\S\ddag\star}$********
--------------- --------------- ----------------------------------------------------------------------------------------------------------------------- ------------------- ----------- --------------- -----------
: Chinese and German NER results on the test set. Best number of each column is highlighted in bold. <span style="font-variant:small-caps;">CRF</span> is the supervised baseline. <span style="font-variant:small-caps;">CRF$_{ptt}$</span> is the “project-then-train” semi-supervised scheme for CRF. <span style="font-variant:small-caps;">WCD13</span> is [@Wang:2013:AAAI], <span style="font-variant:small-caps;">CWD13</span> is [@Che:2013:NAACL], and <span style="font-variant:small-caps;">BPBK10</span> is [@Burkett:2010:CONLL]. <span style="font-variant:small-caps;">CLiPPER$_s$</span> and <span style="font-variant:small-caps;">CLiPPER$_h$</span> are the soft and hard projections. $\S$ indicates F$_1$ scores that are statistically significantly better than CRF baseline at 99.5% confidence level; $\star$ marks significance over <span style="font-variant:small-caps;">CRF$_{ptt}$</span> with 99.5% confidence; $\dagger$ and $\ddagger$ marks significance over <span style="font-variant:small-caps;">WCD13</span> with 99.9% and 94% confidence; and $\diamond$ marks significance over <span style="font-variant:small-caps;">CWD13</span> with 99.7% confidence; $\ast$ marks significance over <span style="font-variant:small-caps;">BPBK10</span> with 99.9% confidence. []{data-label="tbl:test_results"}
Results on the test set are shown in Table \[tbl:test\_results\]. All semi-supervised baselines are tested with the same number of unlabeled bitext as CLiPPER in each language. The “project-then-train” semi-supervised training scheme severely hurts performance on Chinese, but gives a small improvement on German. Moreover, on Chinese it learns to achieve high precision but at a significant loss in recall. On German its behavior is the opposite. Such drastic and erratic imbalance suggest that this method is not robust or reliable. The other three semi-supervised baselines (row 3-5) all show improvements over the CRF baseline, consistent with their reported results. <span style="font-variant:small-caps;">CLiPPER$_s$</span> gives the best results on both Chinese and German, yielding statistically significant improvements over all baselines except for <span style="font-variant:small-caps;">CWD13</span> on German. The hard projection version of CLiPPER also gives sizable gain over CRF. However, in comparison, <span style="font-variant:small-caps;">CLiPPER$_s$</span> is superior.
The improvements of <span style="font-variant:small-caps;">CLiPPER$_s$</span> over CRF on Chinese test set is over 2.8% in absolute F$_1$. The improvement over CRF on German is almost a percent. To our knowledge, these are the best reported numbers on the OntoNotes Chinese and CoNLL-03 German datasets.
Efficiency
----------
Another advantage of our proposed approach is efficiency. Because we eliminated the previous multi-stage “project-then-train” paradigm, but instead integrating the semi-supervised and supervised objective into one joint objective, we are able to attain significant speed improvements. Table \[tbl:time\] shows the training time required to produce models that give results in Table \[tbl:test\_results\].
--------------- -------- --------
CRF 19m30s 7m15s
CRF$_{ptt}$ 34m2s 12m45s
WCD13 3h17m 1h1m
CWD13 16h42m 4h49m
BPBK10 6h16m -
CLiPPER$_{h}$ 1h28m 16m30s
CLiPPER$_{s}$ 1h40m 18m51s
--------------- -------- --------
: Timing stats during model training. []{data-label="tbl:time"}
Error Analysis and Discussion
=============================
Figure \[fig:example\] gives two examples of CLiPPER in action. Both examples have a named entity that immediately proceeds the word “纪念碑” (monument) in the Chinese sentence. In Figure \[fig:example\_PER\], the word “高岗” has literal meaning of *a hillock located at a high position*, which also happens to be the name of a former vice president of China. Without having previously observed this word as a person name in the labeled training data, the CRF model does not have enough evidence to believe that this is a <span style="font-variant:small-caps;">Person</span>, instead of <span style="font-variant:small-caps;">Location</span>. But the aligned words in English (“Gao Gang”) are clearly part of a person name as they were preceded by a title (“Vice President”). The English model has high expectation that the aligned Chinese word of “Gao Gang” is also a <span style="font-variant:small-caps;">Person</span>. Therefore, projecting the English expectations to Chinese provides a strong clue to help disambiguating this word. Figure \[fig:example\_LOC\] gives another example: the word “黄河”(Huang He, the Yellow River of China) can be confused with a person name since “黄”(Huang or Hwang) is also a common Chinese last name.[^17]. Again, knowing the translation in English, which has the indicative word “River” in it, helps disambiguation.
Conclusion {#sec:conc}
==========
We introduced a domain and language independent semi-supervised method for training discriminative models by projecting expectations across bitext. Experiments on Chinese and German NER show that our method, learned over bitext alone, can rival performance of supervised models trained with thousands of labeled examples. Furthermore, applying our method in a setting where all labeled examples are available also shows improvements over state-of-the-art supervised methods. Our experiments also showed that soft expectation projection is more favorable to hard projection. This technique can be generalized to all sequence labeling tasks, and can be extended to include more complex constraints. For future work, we plan to apply this method to more language pairs and examine the formal properties of the model.
[^1]: For experimental purposes, we designate English as the resource-rich language, and other languages of interest as “foreign”. In our experiments, we simulate the resource-poor scenario using Chinese and German, even though in reality these two languages are quite rich in resources.
[^2]: A multi-view interpretation of self-training is that the self-tagged additional data offers new views to learners trained on existing labeled data.
[^3]: This is a city in the state of Georgia in USA, famous for its golf courses. It is ambiguous since both Augusta and Georgia can also be used as person names.
[^4]: The different terminology employed by GE and PR may be confusing to discerning readers, but the “expectation” in the context of GE means the same thing as “marginal posterior” as in PR.
[^5]: An English word aligned to foreign word at position $i$. When multiple English words are aligned to the same foreign word, we average the expectations.
[^6]: We simplify notation by dropping the $L_2$ regularizer in the CRF definition, but apply it in our experiments.
[^7]: In general, other loss functions such as KL-divergence can also be used for $S$. We found $L_2^2$ to work well in practice.
[^8]: [<http://www-nlp.stanford.edu/ner>]{}
[^9]: LDC catalogue No.: LDC2011T03
[^10]: File numbers: chtb\_0001-0325, ectb\_1001-1078
[^11]: LDC catalogue No.: LDC2003E14
[^12]: [[champollion.sourceforge.net](champollion.sourceforge.net)]{}
[^13]: [[code.google.com/p/berkeleyaligner](code.google.com/p/berkeleyaligner)]{}
[^14]: [<http://www.statmt.org/wmt13/training-parallel-nc-v8.tgz>]{}
[^15]: [<https://github.com/stanfordnlp/CoreNLP>]{}
[^16]: Due to technical difficulties, we are unable to replicate experiments on German NER, therefore only Chinese results are reported.
[^17]: In fact, a people search of the name 黄河 on the Chinese equivalent of Facebook (www.renren.com) returns over 13,000 matches.
|
---
abstract: 'We study the causal structure of the minimal surface of the four-gluon scattering, and find a world-sheet wormhole parametrized by Mandelstam variables, thereby demonstrate the EPR = ER relation for gluon scattering. We also propose that scattering amplitude is the change of the entanglement entropy by generalizing the holographic entanglement entropy of Ryu-Takayanagi to the case where two regions are divided in space-time.'
author:
- Shigenori Seki
- 'Sang-Jin Sin'
date: '3 April 2014, Revised: 12 June 2014'
title: EPR = ER and Scattering Amplitude as Entanglement Entropy Change
---
Introduction
============
Quantum entanglement is one of the most subtle and intriguing property of the nature in the entire physics history. When two pair-created particles fly away from each other, their states are entangled even after their separation is beyond causal contact. That is the Einstein-Podolsky-Rosen (EPR) pair [@EPR]. On the other hand, Einstein-Rosen bridge [@ER] connects far separated regions by short wormholes. Both of them shares the common nature such that causally disconnected objects or region are tied although no information can be transmitted through them. Recently Maldacena and Susskind [@MS] conjectured that any EPR pair might be connected through a wormhole of some kind. It was dubbed as ‘EPR = ER’. If true, it would be a fascinating connection between quantum mechanics and space-time geometry giving an enlightenment on this long standing mystery of modern physics.
Soon after this suggestion, Jensen and Karch [@JK] and Sonner [@So] discussed the entanglement of a pair of accelerating quark and antiquark in the context of the AdS/CFT correspondence, using the corresponding minimal surface obtained by Ref. [@Xi]. It allows one to consider only a classical world-sheet configuration where causal structure makes sense. It was shown that the trajectories of quark and antiquark are connected by a line that has to pass through the world-sheet wormhole zone, thereby supporting the EPR = ER with the space-time wormhole replaced by the world-sheet one. Ref. [@CGP] suggested that the gluonic radiation between the quark and antiquark induces their entanglement. It is very interesting to see what happens to other exactly known world-sheet configuration [@JO; @GP; @Ni; @HM].
In this paper we shall consider the four-gluon scattering, whose minimal surface was well studied by Alday and Maldacena [@AM]. We shall study its causal structure in its T-dual space-time picture and conclude that EPR = ER is also supported in this case.
Another related question is how to quantify the degree of entanglement. Notice that without interaction, unentangled state can not be entangled and vice versa. For example, in a scattering process of two particles starting with unentangled initial state, the final state is entangled if and only if there is an interaction, because the time evolution operator $U=\exp[-it (H_1+H_2+H_{\rm int})]$ factorizes iff $H_{\rm int}=0$. The entanglement entropy (EE) of the final state is the [*change*]{} of EE, $\Delta S_E$, created by the interaction during the scattering process. So [*the change of EE must be related to the interaction*]{}, hence we expect that the EE change is related to the scattering amplitude itself. In the AdS/CFT correspondence, the scattering amplitude can be related to the area of the minimal surface of the Wilson loop of trajectories of scattering particles [@RSZ; @JP], one way is to extend the EE derived from the minimal surface by Ryu and Takayanagi [@RT]. The relations between the EE and Wilson loop have been pointed out [@KP; @LW; @LM] for simple shape of the Wilson loop. We assume that the relation hold to more general cases. The gluon scattering amplitude was given from a polygonal Wilson loop in Ref. [@AM]. Using all such data, we shall write down how these are connected.
Minimal surface for gluon scattering
====================================
Alday and Maldacena have considered the $AdS_5$ of momentum space, of which metric is denoted by $$ds^2 = {R^2 \over r^2} \left(\eta_{\mu\nu}dy^\mu dy^\nu + dr^2 \right) \,, \quad
\eta_{\mu\nu} = {\rm diag}(-1,1,1,1) \,, \label{adsmom}$$ and have found the minimal surface solution corresponding to the gluon scattering [@AM], $$\begin{aligned}
&&r = {\alpha \over \ch u_1 \ch u_2 + \beta \sh u_1 \sh u_2} \,, \nonumber \\
&&y_0 = r \sqrt{1+\beta^2} \sh u_1 \sh u_2 \,, \quad
y_3 = 0 \,, \nonumber \\
&&y_1 = r \sh u_1 \ch u_2 \,, \quad
y_2 = r \ch u_1 \sh u_2 \,, \label{AMsolmom} \end{aligned}$$ where $\sh \equiv \sinh$ and $\ch \equiv \cosh$. $u_1$ and $u_2$ are the world-sheet coordinates. The boundary of this surface is a closed sequence of four light-like segments due to momentum conservation of gluons. $\alpha$ and $\beta$ are associated with Mandelstam variables [^1] as $$-s\,(2\pi)^2 = {8\alpha^2 \over (1-\beta)^2} \,, \quad
-t\,(2\pi)^2 = {8\alpha^2 \over (1+\beta)^2} \,, \label{sttoab}$$ ($0 \leq \beta \leq 1$). In this paper we assume $s,t < 0$, that is to say, the $u$-channel. $\beta \to 1$ corresponds to the Regge limit, namely, $-s \to \infty$ with $-t$ fixed. Note that changing the sign of $\beta$ ([*i.e.*]{}, $-1 \leq \beta \leq 0$) is equivalent to exchanging $s$ and $t$.
We calculate the world-sheet induced metric on the surface (\[AMsolmom\]), $$ds_{\rm ws}^2 = R^2 \left( du_1^2 + du_2^2 \right) \,, \label{momwsmet}$$ and this induced metric is flat and Euclidean.
In order to obtain the surface for gluon scattering in the position space $(x^\mu, z)$, we use the “T-dual” transformation [@KT] (Fig. \[figtdual\]), $$\partial_m y^\mu = {R^2 \over z^2} \epsilon_{mn}\partial_n x^\mu \,, \quad
z = {R^2 \over r} \,, \label{tdualtrf}$$ so that the metric (\[adsmom\]) is interpreted as an anti-de Sitter space again, $ds^2 = (R^2/z^2) (\eta_{\mu\nu}dx^\mu dx^\nu + dz^2 )$.
![\[figtdual\] The minimal surfaces in momentum space (left) and in position space (right).](tdual.eps)
The transformation leads the solution (\[AMsolmom\]) to $$\begin{aligned}
z &=& {R^2 \over 2\alpha}\left[ (1+\beta)\ch u_+ +(1-\beta)\ch u_-
\right] \,, \nonumber \\
x_0 &=& -{R^2 \over 2\alpha}\sqrt{1+\beta^2}\sh u_+ \sh u_- \,, \quad
x_3 = 0 \,, \nonumber\\
x_+ &=& -{R^2 \over 2\sqrt{2}\alpha} \left[ (1+\beta)u_- +(1-\beta) \ch u_+
\sh u_- \right] \,, \nonumber\\
x_- &=& {R^2 \over 2\sqrt{2}\alpha} \left[ (1-\beta)u_+ +(1+\beta) \sh u_+
\ch u_- \right] \,, \label{solLC}\end{aligned}$$ where we employed the space-time coordinates $x_\pm \equiv (x_1 \pm x_2)/\sqrt{2}$ and the world-sheet coordinates $u_\pm \equiv u_1 \pm u_2$ for convenience of calculation [@GPS]. Note that $x_\pm$ and $u_\pm$ are not light-cone coordinates and that $dx_1^2 + dx_2^2$ is equal to $dx_+^2 +dx_-^2$.
Causal structure on world-sheet and entanglement
================================================
The induced metric on the world-sheet (\[solLC\]) in the position space is written down as $$ds_{\rm ws}^2 = R^2 \left(
g_{++}du_+^2 +2g_{+-}du_+ du_- +g_{--}du_-^2
\right) \,, \label{wsmetpm}$$ with $$\begin{aligned}
g_{\pm\pm} &&= {2 \over \left[(1 +\beta)\ch u_+ +(1-\beta)\ch u_-\right]^2}\bigl[ (1\pm\beta)^2\sh^2 u_\pm \nonumber \\
&&+(1+\beta^2) -4^{-1}( (1\pm\beta)\ch u_\pm -(1\mp\beta)\ch u_\mp )^2 \bigr] \,, \nonumber \\
g_{+-} &&= {2(1-\beta^2)\sh u_+ \sh u_- \over \left[(1+\beta)\ch u_+ +(1-\beta)\ch u_-\right]^2} \,. \label{wsmetpmcomp} \end{aligned}$$ On this world-sheet there are two kinds of horizons: one is given by $g_{--} = 0$, [*i.e.*]{}, $$\begin{aligned}
&&(1-\beta)\ch u_- +2\sqrt{(1-\beta)^2\sh^2 u_- +1 +\beta^2} \nonumber \\
&=& (1+\beta)\ch u_+ \,, \label{horminus}\end{aligned}$$ and the other is given by $g_{++} = 0$, [*i.e.*]{}, $$\begin{aligned}
&& (1+\beta)\ch u_+ +2\sqrt{(1+\beta)^2\sh^2 u_+ +1 +\beta^2} \nonumber \\
&=& (1-\beta)\ch u_- \,. \label{horplus}\end{aligned}$$ Note that the causal structure is induced in the world-sheet in position space by the “T-dual” transformation (\[tdualtrf\]), although the world-sheet in momentum space (\[momwsmet\]) is Euclidean. We introduce the rescaled coordinates, $X_\mu \equiv (\alpha/R^2) x_\mu$ ($\mu = 0,+,-,3$), and $Z \equiv (\alpha/R^2) z$. Furthermore, in order to explicitly visualize the structure around infinity of $X_\pm$, we also use the coordinates, ${\hat X}_\pm \equiv (2 / \pi)\arctan X_\pm \in [-1,1]$.
![\[figsingularity\] (a) The causal structure on the minimal surface in position space ($\beta = 1/2$). (b) The blue lines are the singularity.](singularity.eps)
![\[figcausaldiag\]The causal structure on world-sheet. (a) $\beta = 0$, (b) $\beta = 1/2$, (c) $\beta = 1$.](causaldiag.eps)
We depict the projection of minimal surfaces (\[solLC\]) onto the $(X_+,X_-)$-plane in Fig. \[figsingularity\](a) and the $({\hat X}_+,{\hat X}_-)$-plane in Fig. \[figcausaldiag\].
Firstly we consider the case $0 \leq \beta < 1$. Especially $\beta = 0$ implies that the scattering is symmetric with respect to $s$ and $t$ (see Eqs. (\[sttoab\])). The causal structure on world-sheet is drawn in Fig. \[figcausaldiag\](a,b). The red solid lines are the horizons by $g_{--}=0$, [*i.e.*]{}, Eq. (\[horminus\]), and the red dashed lines are the horizons by $g_{++}=0$, [*i.e.*]{}, Eq. (\[horplus\]). In the red shaded regions, both of $g_{++}$ and $g_{--}$ are positive. In every figure, $g_{++} >0$ and $g_{--} <0$ in the upper and lower white regions, while $g_{++} <0$ and $g_{--} >0$ in the left and right white regions. Therefore these white regions are Lorentzian, and are separated by the (red) Euclidean region, that is, a wormhole.
Note that $g_{--}$ is negative in the upper and lower Lorentzian regions, while $g_{++}$ is negative in the left and right Lorentzian regions, and that $g_{++}$ is equal to $g_{--}$ on the blue dotted lines given by $(1+\beta)\sh u_+ = \pm (1-\beta)\sh u_-$. It means that we can define world-sheet time as an appropriate coordinate depending on the region. Since the vertex operators can be inserted anywhere on the boundary of disk, this is completely natural. Consider a static gauge, $(\tau, \sigma) = (X_0, Z)$. The time $\tau\, (=X_0)$ begins at the upper-left and lower-right corners and ends up at the upper-right and lower-left ones. The thin blue (red) lines are negative (positive) constant $\tau$ lines. On the axes, $X_\pm =0$, $\tau$ is equal to zero. The thin green lines are constant $Z$ lines. The surface (\[solLC\]) implies $Z \geq 1$. $Z$ has a minimum, $Z=1$, at the origin in Fig. \[figtdual\](b) and Fig. \[figcausaldiag\](a,b). $Z$ becomes infinity on the square bounding boxes, which are the AdS boundary [^2], in Fig. \[figcausaldiag\](a,b). Therefore we can recognize the thin blue and red lines as the time evolution of open strings whose endpoints are located on the AdS boundary.
The horizons (\[horminus\]) and (\[horplus\]) are at least the stationary limit curves but might be different from a horizon of usual black hole. So let us check whether there is a singularity. The Kretschmann scalar on the world-sheet (\[wsmetpm\]), $R_{ijkl}R^{ijkl}$ ($i,j,k,l = \pm$), diverges on $(1-\beta)\ch u_- -(1+\beta)\ch u_+ = \pm 2\sqrt{1+\beta^2}$, in other words, these curves are singularity. From Fig. \[figsingularity\](b), we can see that the singularity is in the interior of horizons, hence the horizons themselves are not singularity.
Next we focus on the case $\beta =1$. It is so-called the Regge limit, namely, $-s \to \infty$ with $-t$ fixed. The world-sheet metric (\[wsmetpm\]) is reduced to $$R^{-2} ds_{\rm ws}^2 = \left({3 \over 2} -{1 \over \ch^2 u_+}\right)du_+^2
-\left({1 \over 2} -{1 \over \ch^2 u_+}\right) du_-^2 \,. \label{wsmetreg}$$ While $g_{++}$ is positive definite, $g_{--}$ is negative when $\ch u_+ > \sqrt{2}$, [*i.e.*]{}, $|u_+| > \log (\sqrt{2} +1)$. Therefore the world-sheet horizons appear at $u_+ = \pm \log (\sqrt{2} +1)$. The causal structure on world-sheet is depicted in Fig. \[figcausaldiag\](c), in which the red thick lines are the horizons given by $g_{--} = 0$. In this case different from those in $0 \leq \beta < 1$, two Lorentzian regions, where $g_{++} >0$ and $g_{--}<0$, exist, and are separated by a Euclidean wormhole (red shaded).
Since a gluon is described by an open string itself, we can see two kinds of entanglement: one is the entanglement of string endpoints in a gluon, and the other is the entanglement of gluons.
![\[figentangle\]The gluon scattering world-sheet projected onto $({\hat X}_+,{\hat X}_-)$. The boundary is denoted by the green box. The red region is a wormhole.](entangle.eps)
In Fig. \[figentangle\], $A_{L,R}$ and $B_{L,R}$ denote the endpoints of open strings describing gluons on the boundary. Since the upper-left and lower-right corners are at $X_0 = -\infty$ and the lower-left and upper-right corners are at $X_0 = \infty$, in the static gauge we can regard $g_1$ and $g_2$ as the incoming gluons and $g_3$ and $g_4$ as the outgoing gluons. The gluons, $g_1$ and $g_2$ at $X_0 = t_1\, (<0)$ and $g_3$ and $g_4$ at $X_0 = t_2\, (>0)$, can be described as the entangled states of open string endpoints, namely, $$\begin{aligned}
|g_1(t_1) \rangle\!\rangle &= \sum_{i,j} c^{(1)}_{ij}|A_{Li}(t_1)\rangle \otimes |A_{Rj}(t_1)\rangle \,, \nonumber \\
|g_2(t_1) \rangle\!\rangle &= \sum_{i,j} c^{(2)}_{ij}|B_{Li}(t_1)\rangle \otimes |B_{Rj}(t_1)\rangle \,, \nonumber \\
|g_3(t_2) \rangle\!\rangle &= \sum_{i,j} c^{(3)}_{ij}|A_{Li}(t_2)\rangle \otimes |B_{Rj}(t_2)\rangle \,, \nonumber \\
|g_4(t_2) \rangle\!\rangle &= \sum_{i,j} c^{(4)}_{ij}|B_{Li}(t_2)\rangle \otimes |A_{Rj}(t_2)\rangle \,. \label{epentangle}\end{aligned}$$ Each entanglement in the states (\[epentangle\]) is interpreted to the fact that each open string crosses over the wormhole (see Fig. \[figentangle\]). Let us focus on the vicinities of the corners of (green) bounding box. The causal structure on the world-sheet which describes the entanglement of string endpoints in each gluon ([*e.g.*]{} $A_R \to B_L$) is similar to that of accelerating quark and antiquark in Ref. [@JK] .
At $X_0 = 0$ the open strings, $g_1$ and $g_2$, join and split to $g_3$ and $g_4$, in other words, the color exchange of gluonic interaction happens at the mid-point $M$ of open strings (Fig. \[figentangle\]). Therefore $X_0 = 0$ is the moment that the entanglement between gluons is gained. Even if the initial state of gluons is not entangled, the final state of gluons is entangled due to the interaction. From a geometric viewpoint, any paths connecting the open string gluons ([*e.g.*]{} $A_R(t_2)B_L(t_2)$ and $A_L(t_2)B_R(t_2)$) must cross the wormhole region (see the blue ribbon in Fig. \[figentangle\]).
Entanglement entropy and scattering amplitude
=============================================
How can we quantify entanglement of two interacting particles? In Refs. [@KP; @LW; @LM], the EE is associated with a Wilson loop by $S_E = (1-c\lambda \partial_\lambda)\log \langle W \rangle$.[^3] Note that the EE itself is associated with a quantum state at a time while the Wilson loop $\langle W \rangle$ depends on the entire time dependent process. Therefore we should consider the left hand side of above mentioned equation as the [*change*]{} of the EE, $\Delta S_E$. So the gluon scattering amplitude [@AM] is related to the change of the EE in leading order of large $\lambda$ by $$\begin{aligned}
\Delta S_E
&\sim& {(1-{1\over 2}c)\sqrt{\lambda} \over 8\pi}\left(\log{s \over t}\right)^2 \nonumber \\
&=& {(1-{1 \over 2}c)\sqrt{\lambda} \over 2\pi}\left(\log{1+\beta \over 1-\beta}\right)^2 \,, \label{EEscat}\end{aligned}$$ where we neglected the IR divergent pieces.
We introduce another characteristic quantity concerning about the entanglement of gluons. Let us consider the proper lengths of lines, $A_R(0)B_R(0)$ and $A_L(0)B_L(0)$, at the contacting instance $X_0=0$; $$\ell_\pm(\beta) = R\int_{-u_{\pm\infty}}^{+u_{\pm\infty}} du_\pm \sqrt{g_{\pm\pm}}\big|_{u_\mp = 0} \,, \label{gluentlength}$$ where we introduced the cutoff, $z_\infty\,(\gg 1)$, such that $(2\alpha / R^2) z_\infty = (1\pm\beta)\ch u_{\pm\infty}+1\mp\beta$.
![\[figgluent\]The open string world-sheets and Feynman-like diagrams in the two channels of gluon interaction.](gluent.eps)
$\ell_+$ and $\ell_-$ correspond to the two channels of gluon interaction. In one channel (Fig. \[figgluent\](a)), the gluons $g_1$ and $g_2$ flow to $g_3$ and $g_4$ respectively. Then, the region $\Sigma$ on the boundary corresponding to the gluon $g_1 \to g_3$ is drawn by the thick green line segments and the region ${\overline \Sigma}$ corresponding to the gluon $g_2 \to g_4$ is drawn by the dotted green line segments. Since the blue line $A_R(0)B_R(0)$ in the bulk connects the boundary $\partial \Sigma$, $\ell_+$ is related with the entanglement of gluons in the sense of Ref. [@RT]. In the same way, we can consider the other channel, [*i.e.*]{}, $g_1 \to g_4$ and $g_2 \to g_3$, in which $\ell_-$ characterizes a part of the entanglement of gluons (Fig. \[figgluent\](b)).
Eq. (\[gluentlength\]) is computed as $\ell_\pm(\beta) = -\sqrt{6}R\log(1\pm \beta)$, where we subtracted the divergent piece, $\sqrt{6}R \log (2\alpha z_\infty/R^2)$, for $z_\infty \to \infty$. Then the EE change (\[EEscat\]) is also described as $$\Delta S_E \sim {1-{1 \over 2}c \over 4\pi^{3/2}}\biggl({\ell_+ - \ell_- \over \ell_s} \biggr)^2 \,, \label{entfinite}$$ where we used $R^2 = \sqrt{4\pi \lambda} \ell_s^2$.
We comment on the Regge limit, $\beta = \pm 1$. Since the finite part of $\Delta S_E$ becomes minimum at $\beta = 0$ and diverges at $\beta = \pm 1$, the Regge limit is the case with maximal $\Delta S_E$. Actually Fig. \[figcausaldiag\](c) shows that, at $\beta =1$, one of the endpoints of $g_1$ ($g_2$) always coincides with that of $g_4$ ($g_3$), and $\ell_-(1)$ diverges.
Can we generalize above result to more general scattering particles? We believe this is the case. To show this we give a construction by which $\Delta S_E$ can be identified as the scattering amplitude. First we can extend the Ryu-Takayanagi formulation of EE by allowing the subspace A and B to be the space-time regions (rather than spatial regions) whose minimal surface in AdS generates the change in the EE. In case of world-line of scattering quark-antiquark pair, it is nothing but the minimal surface calculating the Wilson lines. That is, for any two scattering particles A and B, there is an infinite line $l(t)$ connecting them at each time $t$.
 The scattering surface in the boundary theory. (b) The minimal surface of Wilson lines.](scatsurface.eps)
As time evolves, $l(t)$ generates a two-dimensional surface in the entire space-time of boundary field theory, which we call a scattering surface (Fig. \[figscatsurface\](a)). Then the world-lines of the two particles will divide the scattering surface into two, $\Sigma$ and ${\overline \Sigma}$. Considering the constantly accelerating particles whose minimal surfaces is found in Ref. [@Xi], we can exemplify these idea. The trajectory of two particles forms a circle in Eudlideanized space-time. The minimal surface of circle is well studied, and its area is given by $-\sqrt{\lambda}/(2\pi)$ independent of the acceleration [@DGO].
This construction shows a way to identify the scattering amplitude as a change of EE. Notice that the change of EE between initial and final states is a function of the whole scattering process. Therefore this change should be related to S-matrix.
The term, $\lambda \partial_\lambda \log \langle W \rangle$, in $\Delta S_E$ comes from the replica trick in the derivation of EE. On the other hand, in the language of scattering, that term corresponds to Bremsstralung of radiative correction. Actually in the case of accelerating quark-antiquark, $\lambda \partial_\lambda \log \langle W \rangle$ is proportional to the Bremsstralung function [@CHMS]. Therefore the change of EE, $\Delta S_E$, is related with S-matrix, which gives a scattering amplitude in principle including a radiative correction.
Conclusion
==========
We studied the causal structure on the open string world-sheet of gluon scattering minimal surface in position space. On this world-sheet there exists the wormhole which separates the Lorentzian regions including the boundary. Gluons are given by the open strings. We have shown that any paths connecting such two open string gluons at any time slices pass through the wormhole. Therefore a wormhole can always be associated with the entanglement of interacting gluons. This result supports the EPR = ER conjecture.
Below, we discuss a few points which needs clarification:
- One may ask why entanglement should be related to the interaction, because entanglement is property of the state not the hamiltonian. Consider scattering of two particles which are initially (at $t=-\infty$) far separated and un-entangled. We can construct the basis of in- and out- states by tensor product of free particle states. Let the initial state $|i\rangle$ to be a tensor product state $|i\rangle =|i_1\rangle \otimes |i_2\rangle $ They approach each other and interact and then go force-free region after long time $ t=+\infty$. Such time evolution is given by the evolution operator $U=\exp[-iT (H_1+H_2+H_{\rm int})]$ or S-matrix: $$|\psi \rangle = \lim_{T\to \infty} U|i\rangle
= \sum _f |f\rangle\langle f|S|i \rangle
=\sum_f |f\rangle S_{fi}$$ which is entangled in general unless interaction $H_{\rm int}=0$ so that $U$ is factorized. So the final state of two free particles are entangled and its EE can be identified as the ‘change’ of EE of the two particle system. Our question is that how to relate the latter to the S-matrix itself, which seems to be non-trivial task in field theory setting.
- If the final state involves sum over all possible quantum states, why one can consider only one world-sheet? In classical mechanics, final configuration is completely determined if initial one is given. Now in the AdS/CFT, due to the large $N$ nature, classical discussion can be made. That is, when we consider a minimal surface whose boundary is the trajectories of two scattering particles, we implicitly assumed that such classical picture is valid in describing the gluon-gluon or heavy quark-antiquark scatterings. Therefore we do not sum over trajectories and hence not sum over the world-sheets. This is the reason why we can consider the causal structure of a single world sheet of the gluon scattering instead of summing over such world-sheets. The same philosophy was implicitly assumed in the discussions of causal structure of world-sheet in recent literature. With these understanding, we observed the EE change (\[EEscat\]), following the holographic calculations by Ref. [@LM]. This EE change becomes minimum at $\beta = 0$ and diverges at the Regge limit $\beta =\pm 1$. The relation shows that the change of EE is a function of dynamical process, which is natural. Here it was shown by holographic argument and mostly likely it is true only in the holographic context where semi-classical nature holds. It would be interesting to see how this relation in the general quantum field theory can be written.
- Another point that should be discussed further in the future is the conjecture we used: the EE of Wilson loop can be calculated by the minimal surface associated with the Wilson loop expectation value. Which was proven only simplest cases. Even providing more examples will be interesting.
This work was supported by Mid-career Researcher Program through the National Research Foundation of Korea (NRF) grant No. NRF-2013R1A2A2A05004846. SS was also supported in part by Basic Science Research Program through NRF grant No. NRF-2013R1A1A2059434. SS is grateful to Institut des Hautes [' E]{}tudes Scientifiques (IH[' E]{}S) for their hospitality and to Thibault Damour and Robi Peschanski for helpful comments. SJS wants to thank Tadashi Takayanagi for illuminating discussions on EE at ICTP.
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[^1]: the Mandelstam variables are defined by $-s = (k_1 + k_2)^2 = 2k_{1\mu} k_2{}^\mu$, $-t = (k_1 + k_4)^2 = 2k_{1\mu} k_4{}^\mu$ and $-u = (k_1 + k_3)^2 = 2k_{1\mu} k_3{}^\mu = s+t$.
[^2]: The minimal surface (\[solLC\]) at $Z=\infty$ is laid on the AdS boundary, because simultaneously $X_\pm$ also goes to infinity (see Appendix A in Ref. [@AM]).
[^3]: The undetermined constant $c$ depends on the shape of scattering Wilson loop and is not relevant to our purpose here. ([*cf.*]{} $c=4/3$ for a circular Wilson loop Ref. [@LM].)
|
---
abstract: |
[^1] [^2]
A new proof is given of the existence of the solution to electromagnetic (EM) wave scattering problem for an impedance body of an arbitrary shape. The proof is based on the elliptic systems theory and elliptic estimates for the solutions of such systems.
author:
- |
Alexander G. Ramm\
Department of Mathematics, Kansas State University,\
Manhattan, KS 66506, USA\
ramm@math.ksu.edu\
Martin Schechter\
Department of Mathematics, University of California,\
Irvine, CA 92697-3875, U.S.A.\
mschecht@math.uci.edu
title: Existence of the solution to electromagnetic wave scattering problem for an impedance body of an arbitrary shape
---
Introduction {#S:1}
============
Let $D\subset \mathbb{R}^3$ be a bounded domain with a connected smooth boundary $S$, $D':= \mathbb{R}^3\setminus D$, $k^2=const>0$, $\omega>0$ is frequency, $\zeta=const$, Re$\zeta\ge 0$, be the boundary impedance, $\epsilon>0$ and $\mu>0$ are dielectric and magnetic constants, $\epsilon'=\epsilon +i\frac{\sigma}{\omega}$, $\sigma=const\ge 0$, $x\in D'$, $r=|x|$, $N$ is the unit normal to $S$ pointing into $D'$.
Consider the problem \[e1\] e=ih,h=-i’ h D’, \[e2\] r(e\_r-i k e)=o(1), r, \[e3\] \[N,\[e,N\]\]-\[N,curl e\]=-f.Here $f$ is a given smooth tangential field to $S$, $[A,B]=A\times B$ is the cross product of two vectors, $A\cdot B$ is their scalar product. Problem (1)-(3), which we call [*problem I*]{}, is the scattering problem for electromagnetic (EM) waves for an impedance body $D$ of an arbitrary shape. This problem has been discussed in many papers and books. Uniqueness of its solution has been proved (see, e.g., [@R635], pp.81-83). Existence of its solution was discussed much less (see [@CK], pp.254-256). Explicit formula for the plane EM wave scattered by a small impedance body ($ka\ll 1$, $a$ is the characteristic size of this body) of an arbitrary shape is derived in [@R635]. There one can also find a solution to many-body scattering problem in the case of small impedance bodies of an arbitrary shape.
The aim of this paper is to outline a method for proving the existence of the solution to [*problem I*]{} based on elliptic theory and on a result from [@R643]. It is clear that [*problem I*]{} is equivalent to [*problem II*]{}, which consists of solving the equation \[e4\] (+k\^2)e=0 D’,assuming that $e$ satisfies conditions , and \[e5\] Div e=0 S.Conditions , , and equation imply $\nabla\cdot e=0$ in $D'$. If [*problem II*]{} has a solution $e$, then the pair $\{e,h\}$ solves [*problem I*]{}, provided that $h=curl\, e/(i\omega \mu)$. The solution to [*problem II*]{}, if it exists, is unique, because [*problem I*]{} has at most one solution and is equivalent to [*problem II*]{}. This solution satisfies the following estimate: \[e6\] e\^2:=e\_0:=\_[D’]{}|e(x)|\^2w(x)dxc, w(x):=(1+|x|)\^[-d]{},d=const>1.We denote by $H^m(D',w)$ the weighted Sobolev space with the weight $w$, by $\|e\|_m$ the norm in $H^m(D',w)$, and by $|e|_m$ the norm in $H^m(S)$, where $H^m(S)$ is the usual Sobolev space of the functions on $S$ and $m$ need not be an integer.
Let us outline the ideas of our proof.
[*Step 1.*]{} One checks that problem , , and equation is an elliptic problem, i.e., equation is elliptic (this is obvious) and the boundary conditions , , satisfy the Lopatinsky-Shapiro (LS) condition (see, e.g., [@A] for the definition of LS condition which is also called ellipticity condition for the operator in and the boundary conditions , , or the complementary condition, see also [@S977] ).
[*Step 2.*]{} Reduction of [*problem II*]{} to the form from which it is clear that [*problem II*]{} is of Fredholm type and its index is zero.
[*Step 3.*]{} Derivation of the estimate: \[e7\] |e|\_[m+1]{}c |f|\_[m]{},m>1/2, where Re$\zeta>0$, $c=const>0$ does not depend on $e$ or $f$.
Let us formulate our result.
\[T:1\] For any tangential to $S$ field $f\in H^m(S)$ [*problem II*]{} has a (unique) solution $e \in H^{m+(3/2)}(D',w) $, $e|_S\in H^{m+1}(S)$, and estimates and hold.
In [Section \[S:2\]]{} we prove Theorem 1.
Proof of Theorem 1 {#S:2}
===================
[*Step 1.*]{} The principal symbol of the operator in is $\xi^2 \delta_{pq}$, $\delta_{pq}$ is the Kronecker delta, $\xi^2=\sum_{j=1}^3 \xi_j^2$, so system is elliptic. Let us rewrite and boundary conditions and as follows: \[e4’\] P(D)e=(D\_1\^2 +D\_2\^2 +D\_3\^2 -k\^2)e=0 D’,\[e3’\] B(D)e:={\[N,curl e\] -\[N,\[e,N\]\]=f,\_[p=1]{}\^3 D\_pe\_p=0} S,where $D_j= -i\partial /\partial x_j$ and $D=(D_1,D_2,D_3).$ The principal part of , which defines its principal symbol, is \[e4”\] P’(D)=D\_1\^2 +D\_2\^2 +D\_3\^2,where the prime in $P'(D)$ denotes the principal part of . If we take the local coordinate system in which $N=(0,0,1),$ then the principal part of the boundary operator is the matrix \[8’\] B’(D):= (
[clcr]{} -D\_3 &0 &D\_1\
0 &-D\_3 &D\_2\
D\_1 &D\_2 &D\_3\
)and its symbol is \[e8"\]B’()=(
[clcr]{} -i\_3 &0 &i\_1\
0 &-i\_3 &i\_2\
i\_1 &i\_2 &i\_3\
) The operator $\frac {\partial}{\partial x_p}$ is mapped onto $i\xi_p$. The principal symbol of the operators in the boundary conditions , is calculated in the local coordinates in which $x_3$-axis is directed along $N$. The third row in matrix corresponds to condition . The first two rows correspond to the expression $[N, curl\, e]:= ( curl\,e)_\tau$, which is responsible for the principal symbol corresponding to boundary condition .
To check if the LS condition is satisfied, we must show that the only rapidly (exponentially) decreasing solution of the problem \[e4”’\] P’(\_1,\_2, D\_t) u(,t)=0, t>0,\[e3”’\] B’(\_1,\_2, D\_t) u(,0)=0,is the zero solution. Here $D_t= -i\partial /\partial t$.
The set of rapidly decreasing solutions to the equation, corresponding to the principal symbol of , is $\{v_me^{-t\rho}\}$, where $\rho:=(\xi_1^2+\xi_2^2)^{1/2}$ and vectors $v_m$ are linearly independent. Thus, if $\xi':=\{\xi_1, \xi_2\}$ and $u(\xi',t) $ is a rapidly decreasing solution of , then $D_tu(\xi',t) = i\rho u(\xi',t).$ Therefore, \[e8\]B’(’,D\_t)u(’,t)= (
[clcr]{} -D\_t &0 &i\_1\
0 &-D\_t &i\_2\
i\_1 &i\_2 &D\_t\
) u(’,t)= (
[clcr]{} -i&0 &i\_1\
0 &-i&i\_2\
i\_1 &i\_2 &i\
)u(’,t). The LS condition holds if the matrix
\[e9\](
[clcr]{} -i&0 &i\_1\
0 &-i&i\_2\
i\_1 &i\_2 &i\
) is non-singular for all $\xi'\neq 0$. The determinant of this matrix is $-2i\rho^3\neq 0$ for $\rho>0$. Therefore, the LS condition holds. [$\Box$]{}
[*Step 2 and Step 3.*]{} To check that $\kappa=0$, where $\kappa$ is the index of [*problem II*]{}, let us transform this problem using the result in [@R643], where it is proved that for $\zeta=0$ the solution $e$ to [*problem II*]{} exists, $e$ is uniquely determined by $f$, and $e$ has the same smoothness as $f$. This follows from the results in [@R643] under the assumption that the domain $D$ is small, which implies that $k^2$ is not an eigenvalue of the Dirichlet Laplacean in $D$. If $D$ is not small then this result follows from the fact that the relation $curl \int_{D'}g(x,t)J(t)dt=0$ in $D'$ implies $J=0$ on $S$ if $J$ is a tangential to $S$ field. In proving this one assumes that $k^2$ is not an eigenvalue of the Dirichlet Laplacean in $D$. This is not an essential restriction: see [@R190], p.20, Section 1.3. The map $V:f\to e_\tau$, where $e_\tau$ is the tangential to $S$ component of $e$, acts from $ H^m(S)$ onto $H^{m}(S)$, and $V$ is an isomorphism of $ H^m(S)$ onto itself.
Rewrite equation as \[e10\] e\_=-Vf+V( (curle)\_).The operator $V( ( curl\,e)_\tau)$ preserves the smoothness of $( curl\,e)_\tau$, and so, if Re$\zeta> 0$, it acts from $H^{m}(S)$ into $H^{m+1}(S)$ due to equation . Indeed, if $e_\tau$ belongs to $H^{m+1}(S)$ then $( curl\,e)_\tau$ belongs to $H^{m}(S)$. On the other hand, equation implies that the smoothness of $V( ( curl\,e)_\tau)$ is not less than the smoothness of $e_\tau$, which is $H^{m+1}(S)$. Also, equation implies that the smoothness of $e_\tau$ is not less than the smoothness of $Vf$, which is $H^{m+1}(S)$ if $f\in H^{m+1}(S)$. Therefore, if Re$\zeta> 0$ then $V$ acts from $H^{m}(S)$ into $H^{m+1}(S)$, so it is is compact in $H^{m}(S)$, and equation is of Fredholm type with index zero. Knowing $e_\tau$ on $S$ one can uniquely recover $e$ in $D'$.
Since problem - is an elliptic system obeying the LS condition, we have the elliptic estimate (see, e.g., [@S977]): \[e100\] e\_m C(P(D)e\_[m-2]{} + |B(D)e|\_[m-(3/2)]{} + e\_0), e H\^m(D’,w),where $m>3/2 $ and $\psi \in C^\infty_0(\R^3).$ Recall that $P(D)e=0$. Hence, a solution of , satisfies \[e101\] e\_m C(|f|\_[m-(3/2)]{} + e\_0). Equation has at most one solution if Re$\zeta\ge 0$ because [*problem II*]{} has at most one solution. Therefore, by the Fredholm alternative, equation has a solution, this solution is unique, and estimate holds due to ellipticity of the [*problem II*]{}.
Estimate holds because $e=O(1/r)$ when $r\to \infty$.
We now want to prove that $e$ belongs to $H^m(D',w)$ where $m$ is determined by the smoothness of $f$.
[**Lemma 1.**]{} [*The following estimate holds for a solution to problem II with Re$\zeta>0$*]{}: \[e11\] e\_mc|f|\_[m-(3/2)]{}. Here and below $c>0$ stand for various estimation constants.
[*Proof.*]{} If is false, then there is a sequence of $f_n$ such that \[e12\] e\_n\_mn|f\_n|\_[m-(3/2)]{}, e\_n\_m=1. Thus, there is a subsequence denoted again $e_n$ and an $e \in H^m(D',w)$ such that $e_n \to e$ weakly in $ H^m(D',w),$ strongly in $H^{m'}(D',w)$, $m'<m$, and almost everywhere in $D'$. Since $\psi \in C^\infty_0(\R^3),$ we have $\psi e_n \to \psi e$ strongly in $L^2(D').$ Hence by , e\_j-e\_k\_m C(|f\_j-f\_k|\_[m-(3/2)]{} +e\_j-e\_k\_0 ) 0as $j,k\to \infty.$ Thus, $e_n \to e$ in $H^m(D',w)$, so that $\|e\|_m=1$ while $f=0.$ Consequently, $e$ solves [*problem II*]{} with $f=0$, so $e=0$. This contradicts the fact that $\|e\|_m=1$.
Let us give an alternative proof of the convergence of $e_n$ to $e$ in $H^m(D',w)$.
One has, by Green’s formula, \[e13\] e\_n(x)=\_S( e\_n(t)g\_N(x,t)-g(x,t)(e\_n)\_N(t))dt. Pass to the limit $n\to \infty$ in this formula, use convergence $|e_n-e|_{m-(3/2)}\to 0$ and $|(e_n)_N-e_N|_{m-(3/2)}\to 0 $ and get \[e14\] e(x)=\_S( e(t)g\_N(x,t)-g(x,t)e\_N(t))dt. This equation implies that $e$ solves equation and satisfies the radiation condition . Furthermore, it satisfies equation $\nabla \cdot e=0$ in $D'$ because one can pass to the limit $n\to \infty$ in equation for $e_n$, and if $Div e=0$ on $S$ then $\nabla \cdot e=0$ in $D'$. Indeed, $\nabla \cdot e$ satisfies equation and the radiation condition , so if it vanishes on $S$ then it vanishes in $D'$. Moreover, $e$ satisfies equation with $f=0$ because $f_n\to 0$ as $n\to \infty$. By the uniqueness theorem, $e=0$ in $D'$.
Let us check that $\|e_n-e\|_m\to 0$. Locally this convergence is already checked, so one has to check convergence in the weighted norm near infinity. Estimate implies \[e15\] |e\_n(x)|c|x|\^[-1]{}, where the constant $c>0$ does not depend on $n$ because of the convergence $|e_n-e|_{m-0.5}\to 0$ and $|(e_n)_N-e_N|_{m-1.5}\to 0$. Estimate implies the desired convergence near infinity in the weighted norm because of the assumption $d>1$. Therefore, we have a contradiction: $\|e_n\|_m=1$ and $\|e_n-e\|_m=\|e_n\|_m\to 0$. This contradiction proves Lemma 1.[$\Box$]{}
This completes the outline of the proof of [Theorem \[T:1\]]{}.
[1000]{} S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, [*Comm. pure appl. math.*]{}, 17, 35-92, (1964).
D. Colton, R. Kress, [*Inverse acoustic and electromagnetic scattering theory*]{}, Springer-Verlag, Berlin, 1992.
A. G. Ramm, [*Scattering of Acoustic and Electromagnetic Waves by Small Bodies of Arbitrary Shapes. Applications to Creating New Engineered Materials*]{}, Momentum Press, New York, 2013
A. G. Ramm, Electromagnetic wave scattering by small perfectly conducting particles and applications, [*J. Math. Phys.*]{}, 55, 083505, (2014).
A. G. Ramm, [*Scattering by obstacles*]{}, D.Reidel, Dordrecht, 1986.
M. Schechter, [*Modern Methods in Partial Differential Equations*]{}, McGraw-Hill, New York, 1977; Dover, 2013.
[^1]: MSC: 78A45; 78A25
[^2]: key words: electromagnetic wave scattering; small impedance body; scatterer of an arbitrary shape; complementary condition; elliptic systems
|
---
abstract: 'Vision-and-Language Navigation (VLN) is a task where agents must decide how to move through a 3D environment to reach a goal by grounding natural language instructions to the visual surroundings. One of the problems of the VLN task is data scarcity since it is difficult to collect enough navigation paths with human-annotated instructions for interactive environments. In this paper, we explore the use of counterfactual thinking as a human-inspired data augmentation method that results in robust models. Counterfactual thinking is a concept that describes the human propensity to create possible alternatives to life events that have already occurred. We propose an adversarial-driven counterfactual reasoning model that can consider effective conditions instead of low-quality augmented data. In particular, we present a model-agnostic adversarial path sampler (APS) that learns to sample challenging paths that force the navigator to improve based on the navigation performance. APS also serves to do pre-exploration of unseen environments to strengthen the model’s ability to generalize. We evaluate the influence of APS on the performance of different VLN baseline models using the room-to-room dataset (R2R). The results show that the adversarial training process with our proposed APS benefits VLN models under both seen and unseen environments. And the pre-exploration process can further gain additional improvements under unseen environments.'
author:
- 'Tsu-Jui Fu'
- Xin Wang
- Matthew Peterson
- Scott Grafton
- Miguel Eckstein
- |
William Yang Wang\
University of California, Santa Barbara, CA, USA\
[tsu-juifu@ucsb.edu, {xwang, william}@cs.ucsb.edu]{}\
[{peterson, scott.grafton, miguel.eckstein}@psych.ucsb.edu]{}
bibliography:
- 'egbib.bib'
title: 'Counterfactual Vision-and-Language Navigation via Adversarial Path Sampling'
---
Introduction
============
Vision-and-language navigation (VLN) [@anderson2018r2r; @chen2019touchdown] is a complex task that requires an agent to understand natural language, encode visual information from the surrounding environment, and associate critical visual features of the scene and appropriate actions with the instructions to achieve a specified goal (usually to move through a 3D environment to a target destination).
![The comparison between randomly-sampled (rand) and APS-sampled (aps) under validation-seen set for Seq2Seq over different ratios of augmented path used.[]{data-label="fig:pre-vs-aps"}](imgs/fig_pre-vs-aps.png){width="0.95\linewidth"}
To accomplish the VLN task, the agent learns to align linguistic semantics and visual understanding and also make sense of dynamic changes in vision-and-language interactions. One of the primary challenges of the VLN task for artificial agents is data scarcity; for instance, while there are more than 200K possible paths in the Room-to-Room (R2R) dataset [@anderson2018r2r], the R2R training data comprises only 14K sampled paths. This scarcity of data makes learning the optimal match between vision and language within the interactive environments quite challenging. Meanwhile, humans often lack extensive experience with joint access to visual experience and accompanying language instructions for navigating novel or unfamiliar environments, yet the human mind can navigate environments despite this data scarcity by incorporating mechanisms such as counterfactual reasoning [@roese1997counterfactual] and self-recovered missing information. For example, if a human follows an instruction to “turn right" and they see a door in front of them, they can also consider what they may have encountered had they turned left instead. Or, if we stop in front of the dining table instead of walking away from it, what should the instruction be? The premise, then, is that counterfactual reasoning can improve performance in a VLN task through exploration and consideration of alternative actions that the agent did not actually make. This may allow the agent to operate in data-scarce scenarios by bootstrapping familiarity of environments and the links between instructions and multiple action policy options.
Counterfactual thinking has been used to increase the robustness of models for various tasks [@kusner2017cf-fair; @garg2019cf-text]. However, no explicitly counterfactual models have been applied to the VLN task specifically. Speaker-Follower [@fried2018sf], which applies a back-translated speaker model to reconstruct the instructions for randomly-sampled paths as augmented training examples, is probably the VLN model that comes closest to instantiating counterfactual thinking. While the use of augmented training examples by the Speaker-Follower agent resembles a counterfactual process, the random sampling method is too arbitrary. Fig. \[fig:pre-vs-aps\] reports the performance of the model trained with randomly-sampled augmented data (the line in a light color) over different ratios of the augmented path used. It shows that the success rate stops increasing once augmented paths account for 60% or more of the training data [@huang2019multi-dis-model]. Since those paths are all randomly sampled, it can limit the benefit of counterfactual thinking to data augmentation.
In this paper, we propose the use of adversarial-driven counterfactual thinking where the model learns to consider effective counterfactual conditions instead of sampling ample but uninformative data. We introduce a model-agnostic adversarial path sampler (APS) that learns how to generate augmented paths for training examples that are challenging, and thus effective, for the target navigation model. During the adversarial training process, the navigator is trying to accomplish augmented paths from APS and thus optimized for a better navigation policy, while the APS aims at producing increasingly challenging paths, which are therefore more effective than randomly-sampled paths. Moreover, empowered by APS, the model can adapt to unseen environments in a practical setting—environment-based pre-exploration, where when deployed to a new environment, the robot can first pre-explore and get familiar with it, and then perform natural language guided tasks within this environment.
Experimental results on the R2R dataset show that the proposed APS can be integrated into a diverse collection of VLN models, improving their performance under both seen and unseen environments. In summary, our contributions are four-fold:
- We integrate counterfactual thinking into the vision-and-language navigation task, and propose the adversarial path sampler (APS) to progressively sample challenging and effective paths to improve the navigation policy.
- The proposed APS method is model-agnostic and can be easily integrated into various navigation models.
- Extensive experiments on the R2R dataset validate that the augmented paths generated by APS are not only useful in seen environments but also capable of generalizing the navigation policy better in unseen environments.
- We demonstrate that APS can also be used to adapt the navigation policy to unseen environments under environment-based pre-exploration.
Related Work
============
**Vision-and-Language Navigation** Navigation in 3D environments based on natural language instruction has recently been investigated by many studies [@anderson2018r2r; @chen2019touchdown; @jain2019stay-path; @ke2019tactical; @ma2019regretful; @huang2019vln-trans; @wang2018rpa; @ma2019self-m; @fried2018sf; @wang2019rcm; @tan2019envdrop; @hemachandra2015unseen]. For vision-and-language navigation (VLN), fine-grained human-written instructions are provided as guidance to navigate a robot in indoor environments. But data scarcity is a critical issue in VLN due to the high cost of data collection.
In order to augment more data for training, the Speaker-Follower model [@fried2018sf] applies a back-translated speaker model to generate instructions for randomly-sampled paths. In spite of obtaining some improvements from those extra paths, a recent study [@huang2019multi-dis-model] shows that only a limited number of those augmented paths are useful and after using 60% of the augmented data, the improvement diminishes with additional augmented data. In this paper, we present a model-agnostic adversarial path sampler that progressively produces more challenging paths via an adversarial learning process with the navigator, therefore forcing the navigation policy to be improved as the augmented data grows. **Counterfactual Thinking** Counterfactual thinking is a concept that describes the human propensity to create possible alternatives to life events that have already occurred. Humans routinely ask questions such as: “What if ...?" or “If there is only ..." to consider the outcomes of different scenarios and apply inferential reasoning to the process. In the field of data science, counterfactual thinking has been used to make trained models explainable and more robust [@kusner2017cf-fair; @garg2019cf-text; @goyal2019cf-vis]. Furthermore, counterfactual thinking is also applied to augment training targets [@zmigrod2019cfd-text; @chen2019cfc-sg; @ashual2019cf-isg]. Although previous studies have shown some improvements over different tasks, they all implement counterfactual thinking arbitrarily without a selection process to sample counterfactual data that might optimize learning. This can limit the effectiveness of counterfactual thinking. In this paper, we combine the adversarial training with counterfactual conditions to guide models that might lead to robust learning. In this way, we can maximize the benefit of counterfactual thinking.
**Adversarial Training** Adversarial training refers to the process by which two models try to detrimentally influence each other’s performance and as a result, both models improve by competing against each other. Adversarial training has been successfully used to guide the target during model training [@goodfellow2014gan; @wu2017adv-re; @chou2018adv-pe; @miyato2017adv-text; @hong2019self-adv; @agmon2017adv-robot]. Apart from leading the training target, adversarial training is also applied to data augmentation [@antoniou2017gan-da; @zhang2018meta-gan]. While previous studies just generate large amounts of augmented examples using a fixed pre-trained generator. In this paper, the generator is updated along with the target model and serves as a path sampler which samples challenging paths for effective data augmentation.
**Pre-Exploration under Unseen Environments** Pre-exploration under unseen environments is a popular method to bridge the gap between seen and unseen environments. Speaker-Follower [@fried2018sf] adopts a state-factored beam search for several candidate paths and then selects the best one. RCM [@wang2019rcm] introduces self-imitation learning (SIL) that actively optimized the navigation model to maximize the cross-matching score between the generated path and the original instruction. Nevertheless, beam search requires multiple runs for each inference, and SIL utilizes the original instructions in the unseen environments for optimization. EnvDrop [@tan2019envdrop] conducts pre-exploration by sampling shortest paths from unseen environments and augments them with back-translation, which however utilizes the meta-information of unseen environments (, the shortest path planner that the robot is not supposed to use). In addition, EnvDrop puts the augmented paths from all the unseen environments together to optimize the policy, which is also not practical in real life as the robot deployed to a house can hardly access to other houses. In contrast, we propose to use the adversarial path sampler for *environment-based pre-exploration*, where the agent pre-explore an environment only for the tasks within the same environment with no meta-information of it.
Methodology
===========
Background
----------
#### Visual-and-Language Navigation (VLN)
At each time step $t$, the environment presents the image scene $s_t$. After stepping an action $a_t$, the environment will transfer to next image scene $s_{t+1}$: $$s_{t+1} = \text{Environment}(s_t, a_t).$$ To carry out a VLN task, the navigation model steps a serious of actions $\{a_t\}_{t=1}^{T}$ to achieve the final goal described in the instruction. Though previous studies propose different architectures of navigation model (NAV), in general, NAV is a recurrent action selector based on the visual feature of the image scene, navigation instruction, and previous history: $$\begin{split}
f_t &= \text{VisualFeature}(s_t), \\
a_t &= \text{softmax}(\text{NAV}(f_t, I, h_t)),
\end{split}$$ where $f_t$ is the visual feature of the image scene $s_t$ at time step $t$, $I$ is the navigation instruction, $h_t$ represents the previous history of image scenes, and $a_t$ is the probability of each action to step at time step $t$. With $a_t$, we can decide which action to step based on greedy decoding (step the action with the highest probability).
In this work, we experiment under navigator with 3 different architectures, Seq2Seq [@anderson2018r2r], Speaker-Follower [@fried2018sf], and RCM [@wang2019rcm].
#### Back-Translated Speaker Model
Introduced in Speaker-Follower [@fried2018sf], the back-translated speaker model (Speaker) generates the instruction of a navigation path: $$I = \text{Speaker}(\{(f_1, a_1), (f_2, a_2), ..., (f_L, a_L)\}),$$ where $f_t$ is the visual feature of the image scene, $a_t$ is the action taken at time step $t$, $L$ represents the length of the navigation path, and $I$ is the generated instruction. Speaker is trained with pairs of navigation paths and human-annotated instructions in the training data. With Speaker, we can sample various paths in the environments and augment their instructions.
![The learning framework of our adversarial path sampler (APS), where Speaker denotes the back-translated speaker model and NAV denotes the navigation model.[]{data-label="fig:overview"}](imgs/fig_overview.png){width="\linewidth"}
Overview
--------
The overall learning framework of our model-agnostic adversarial path sampler (APS) is illustrated in Fig. \[fig:overview\]. At first, APS samples batch of paths $P$ and we adopt the Speaker [@fried2018sf] to obtain the reconstructed instructions $I$. With the pairs of ($P$, $I$), we obtain the navigation loss $\mathcal{L}_{\text{NAV}}$. NAV minimizes $\mathcal{L}_{\text{NAV}}$ to improve navigation performance. While APS learns to sample paths that NAV can not perform so well by maximizing $\mathcal{L}_{\text{NAV}}$. Hence, there is an adversarial situation for $\mathcal{L}_{\text{NAV}}$ between NAV and APS, where APS aims at sampling challenging paths and NAV tries to solve the navigation tasks from APS.
By the above adversarial training process, we collect all of the ($P$, $I$) sampled from APS to compose the adversarial augmented data which can be more helpful to NAV than randomly-sampled one. Both Speaker and NAV are pre-trained using the original training set and Speaker keeps fixed during adversarial training. We collect all ($P$, $I$) sampled from APS as APS-sampled augmented path and further train NAV. More detail can be seen in Sec. \[sec:train-aps\].
![The architecture of the adversarial path sampler (APS).[]{data-label="fig:aps"}](imgs/fig_aps.png){width="\linewidth"}
Architecture of APS
-------------------
As shown in Fig. \[fig:aps\], the proposed APS is a recurrent action sampler $\pi_\text{APS}$ which samples series of actions $\{a_t\}_{t=1}^{T}$ (with the scene images $\{f_t\}_{t=1}^{T}$ presented from the environment) and combines as the path output, where $f_t$ means the visual feature (e.g., extracted from the convolutional neural networks). For the panoramic image scene, $f_{t, j}$ represents the visual feature of the image patch at viewpoint $j$ at time step $t$. At each time step $t$, the history of previous visual feature and $a_{t-1}$ is encoded as $h_{t}$ by a long short-term memory (LSTM) [@hochreiter1997lstm] encoder: $$h_t = \text{LSTM}([v_t, a_{t-1}], h_{t-1}),$$ where $a_{t-1}$ is the action taken at previous step and $v_t$ is the weighted sum of visual feature of each image path for the panoramic image scene. $v_t$ is calculated using the attention [@bahdanau2015att] between the history $h_{t-1}$ and the image patches $\{f_{t,j}\}_{j=1}^m$: $$\begin{split}
v_t &= \text{Attention}(h_{t-1}, \{f_{t, j}\}_{j=1}^{m}) \\
&=\sum_{j} \text{softmax}(h_{t-1}W_{h}(f_{t, j}W_{f})^{T})f_{t, j},
\end{split}$$ where $W_h$ and $W_f$ are learnable projection matrics. The above equation of $v_t$ is for panoramic scene with $m$ viewpoints. APS also supports the navigator which uses visuomotor view as input (e.g., Seq2Seq [@anderson2018r2r]) and the single visual feature $f_t$ is seen as $v_t$ directly. Finally, APS decides which action to step based on the history $h_t$ and action embedding $u$: $$a_t = \text{softmax}(h_{t}W_{c}(u_{k}W_{u}^{T})),$$ where $u_{k}$ is the action embedding of the $k$-th navigable direction. $W_c$ and $W_u$ are learnable projection matrics.
NAV: the target navigation model Speaker: the back-translated instruction model APS: the adversarial path sampler $\text{aug}_{\text{aps}}$: collected APS-sampled augmented data\
Pre-train NAV with original training set Pre-train Speaker with original navigation path Initialize APS $\text{aug}_{\text{aps}}$ $\leftarrow$ $\varnothing$\
$P = \{(f_1, a_1), (f_2, a_2), ..., (f_L, a_L)\}$ $\leftarrow$ APS samples $I$ $\leftarrow$ back-translated by Speaker with $P$ $\mathcal{L}_\text{NAV}$ $\leftarrow$ student-forcing loss of NAV using ($P$, $I$)\
Update NAV by minimizing $\mathcal{L}_\text{NAV}$ Update APS by maximizing $\mathcal{L}_\text{NAV}$ using Policy Gradient $\text{aug}_{\text{aps}}$ $\leftarrow$ $\text{aug}_{\text{aps}}$ $\cup$ ($P$, $I$)\
Train NAV with $\text{aug}_\text{aps}$ Fine-tune NAV with original training set
Adversarial Training of APS {#sec:train-aps}
---------------------------
After each unrolling of APS, we comprise the navigation history $\{a_t\}_{t=1}^{T}$ and $\{f_{t, j}\}_{j=1}^{m}$ to obtain the path $P$. To be consistent with the original training data whose navigation paths are all shortest paths [@anderson2018r2r], we transform the sampled paths by APS into shortest paths[^1] (same start and end nodes as in the sampled paths). Then we employ the Speaker model [@fried2018sf] to produce one instruction $I$ for each sampled path $P$, and eventually obtain a set of new augmented pairs ($P$, $I$). We train the navigation model (NAV) with ($P$, $I$) using student-forcing [@anderson2018r2r]. The training loss ($\mathcal{L}_{\text{NAV}}$) can be seen as an indicator of NAV’s performance under ($P$, $I$): the higher $\mathcal{L}_{\text{NAV}}$ is, the worse NAV performs. Hence, in order to create increasingly challenging paths to improve the navigation policy, we define the loss function $\mathcal{L}_\text{APS}$ of APS as: $$\mathcal{L}_\text{APS} = - \mathbb{E}_{p(\text{P};\pi_\text{APS})}\mathcal{L}_{\text{NAV}}.$$
Since the path sampling process is not differentiable, we adopt policy gradient [@sutton2000pg] and view $\mathcal{L}_{\text{NAV}}$ as the reward $R$ to optimize the APS objective. According to the REINFORCE algorithm [@williams1992reinforce], the gradient is computed as following: $$\resizebox{\linewidth}{!}{$
\begin{aligned}
\nabla_{\pi_\text{APS}} \mathcal{L}_\text{APS} &\approx -\sum_{t=1}^{T} [\nabla_{\pi_\text{APS}} \log p(a_t | a_{1:t-1} ; \pi_\text{APS}) R] \\
&\approx -\sum_{t=1}^{T} [\nabla_{\pi_\text{APS}} \log p(a_t | a_{1:t-1} ; \pi_\text{APS}) (R-b)],
\end{aligned}
$}$$ where $b$ is the baseline estimation to reduce the variance and we treat $b$ as the mean of all previous losses. Note that APS is model-agnostic and can be easily integrated into different navigation models, since it only considers the training loss from a navigation model regardless of its model architecture.
Algorithm \[algo:aps\] illustrates the training process of APS. APS aims at maximizing the navigation loss $\mathcal{L}_{\text{NAV}}$ of NAV to create more challenging paths, while NAV tries to minimize $\mathcal{L}_{\text{NAV}}$ to do better navigation: $$\begin{split}
\min_{\text{NAV}} & \max_{\text{APS}} \mathcal{L}_{\text{NAV}}.
\end{split}$$ After collecting the challenging paths augmented by APS, we train NAV on them and finally fine-tune NAV with the original training set. The detailed analysis of APS-sampled augmented data is shown in Sec. \[sec:ayz-aps\].
![The optimization flow of environment-based pre-exploration under unseen environments. APS samples paths from the unseen environment to optimize NAV and make it more adaptive. Then, NAV runs each instruction in a single turn.[]{data-label="fig:pre-exp"}](imgs/fig_pre-exp.png){width="\linewidth"}
Environment-based Pre-Exploration
---------------------------------
Pre-exploration is a technique that adapts the navigation model to unseen environments. The navigator can explore the unfamiliar environment first and increase the chance to carry out the navigation instructions under unseen environments. For previous pre-exploration methods like beam search [@fried2018sf] or self-imitation learning (SIL) [@wang2019rcm], they are instruction-based which optimizes for each instruction. This will make the navigation path excessive long since it first runs many different paths and then selects the possible best one. In the real world, when we deploy a robot into a new environment, it might pre-explore and get familiar with the environment, and then efficiently execute the tasks following natural language instructions within this environment. So unlike previous approaches [@wang2019rcm; @tan2019envdrop] that either optimize the given instructions or assume access to all the unseen environments at once, we propose to use our APS method to do the environment-based pre-exploration where the agent pre-explore an environment only for the tasks within the same environment with no prior knowledge of it. Under an unseen environment, we adopt APS to sample multiple paths ($P'$) and generate the instructions ($I'$) of the sampled paths[^2] with the Speaker model [@fried2018sf]. We then use ($P'$, $I'$) to optimize NAV to adapt to the unseen environment as illustrated in Fig. \[fig:pre-exp\]. Note that during pre-exploration, we only optimize NAV and let APS fixed[^3]. We also present a detailed analysis of our proposed environment-based pre-exploration method in Sec. \[sec:ayz-pre-exp\].
-------------------------------------------- ----------------- ---------------- --------------- ---------------- ---------- ----------------- ---------------- --------------- ---------------- ---------- ----------------- ---------------- --------------- ----------------
Model NE $\downarrow$ OSR $\uparrow$ SR $\uparrow$ SPL $\uparrow$ NE $\downarrow$ OSR $\uparrow$ SR $\uparrow$ SPL $\uparrow$ NE $\downarrow$ OSR $\uparrow$ SR $\uparrow$ SPL $\uparrow$
Seq2Seq [@anderson2018r2r] 6.0 51.7 39.4 33.8 7.8 27.7 22.1 19.1 7.9 26.6 20.4 18.0
+ $\text{aug}_\text{rand}$ 5.3 58.1 43.7 37.2 7.7 28.9 22.6 19.9 7.8 26.2 21.0 18.8
+ $\text{aug}_\text{aps}$ **5.0** **60.8** **48.2** **40.1** **7.1** **32.7** **24.2** **20.4** **7.5** **30.1** **22.5** **19.3**
+ $\text{aug}_\text{aps}$+pre-exploration **6.6** **37.8** **27.0** **24.6** **6.7** 29.4 **23.2** **20.8**
Speaker-Follower [@fried2018sf] 5.0 61.6 51.7 44.4 6.9 40.7 29.9 21.0 7.0 41.2 30.9 24.0
+ $\text{aug}_\text{rand}$ 3.7 74.2 66.4 59.8 6.6 46.6 36.1 28.8 6.6 43.4 34.8 29.2
+ $\text{aug}_\text{aps}$ **3.3** **74.9** **68.2** **62.5** **6.1** **46.7** **38.8** **32.1** **6.5** **44.2** **36.1** 28.8
+ $\text{aug}_\text{aps}$+pre-exploration **5.2** **49.1** **42.0** **35.7** **5.9** **46.4** **37.6** **32.4**
RCM [@wang2019rcm] 5.7 53.8 47.0 44.3 6.8 43.0 35.0 31.4 6.7 43.5 35.9 33.1
+ $\text{aug}_\text{rand}$ 4.1 66.9 61.9 58.6 5.7 52.4 45.6 41.8 5.9 52.4 44.5 40.8
+ $\text{aug}_\text{aps}$ **3.9** **69.3** **63.2** **59.5** **5.4** **56.6** **47.7** **42.8** **5.8** **53.9** **45.1** **40.9**
+ $\text{aug}_\text{aps}$+pre-exploration **5.3** 56.2 **48.0** **42.8** **5.5** **55.6** **45.9** **40.9**
-------------------------------------------- ----------------- ---------------- --------------- ---------------- ---------- ----------------- ---------------- --------------- ---------------- ---------- ----------------- ---------------- --------------- ----------------
Experiments
===========
Experimental Setup
------------------
**R2R Dataset** We evaluate the proposed method on the Room-to-Room (R2R) dataset [@anderson2018r2r] for vision-and-language navigation. R2R is built upon the Matterport3D [@angel2017mp3d], which contains 90 different environments that are split into 61 for training and validation-seen, 11 for validation-unseen, and 18 for testing sets. There are 7,189 paths and each path has 3 human-written instructions. The validation-seen set shares the same environments with the training set. In contrast, both the validation-unseen and the testing sets contain distinct environments that do not appear during training.
**Evaluation Metrics** To compare with the existing methods, we report the same used evaluation metrics: Navigation Error (NE), Oracle Success Rate (OSR), Success Rate (SR), and Success Rate weighted by Path Length (SPL). NE is the distance between the agent’s final position and goal location. OSR is the success rate at the closest point to the goal that the agent has visited. SR is calculated as the percentage of the final position within 3m from the goal location. SPL, defined in [@anderson2018spl], is the success rate weighted by path length which considers both effectiveness and efficiency.
**Baselines** We experiment with the effectiveness of the model-agnostic APS on 3 kinds of baselines:
- *Seq2Seq* [@anderson2018r2r], the attention-based seq2seq model that is trained with student forcing (or imitation learning) under the visuomotor view and action space;
- *Speaker-Follower* [@fried2018sf], the compositional model that is trained with student forcing under the panoramic view and action space;
- *RCM* [@wang2019rcm], the model that integrates cross-modal matching loss, and is trained using reinforcement learning under the panoramic view and action space.
In the following sections, we use the notations as:
- $\text{aug}_\text{rand}$: the randomly-sampled augmented path;
- $\text{aug}_\text{aps}$: the APS-sampled augmented path;
- $\text{model}_\text{rand}$: the model trained with $\text{aug}_\text{rand}$;
- $\text{model}_\text{aps}$: the model trained with $\text{aug}_\text{aps}$.
For example, $\text{Speaker-Follower}_\text{aps}$ is the Speaker-Follower model trained with the APS-sampled augmented path.
For each baseline, we report the results of the model trained without any augmented data, trained with $\text{aug}_\text{rand}$, and trained with $\text{aug}_\text{aps}$. For the unseen environments, we also report the results under the pre-exploration.
**Implementation Details** To follow the previous studies [@anderson2018r2r; @fried2018sf; @wang2019rcm], we adopt ResNet-152 [@he2016res-net] to extract visual features (2048d) for all scene images without fine-tuning; for the navigation instructions, the pre-trained GloVe embeddings [@pennington2014glove] are used for initialization and then fine-tuned with the model training. For baseline models, we apply the same batch size 100, LSTM with 512 hidden units, learning rate 1e-4, RL learning rate 1e-5, and dropout rate 0.5. For our proposed APS, the hidden unit of LSTM is also 512, the action embedding size is 128, and the learning rate is 3e-5. We adopt the learning rate 1e-5 under the pre-exploration for the unseen environments. All models are optimized via Adam optimizer [@kingma2015adam] with weight decay 5e-4.
For $\text{aug}_\text{rand}$, we use the same 17K paths as Speaker-Follower [@fried2018sf]. To compare fairly, APS also adversarially samples the same amounts of paths for data augmentation. The navigation models are first trained using augmented data for 50K iterations and then fine-tuned with original human-written instructions for 20K iterations.
{width="\linewidth"}
Quantitative Results
--------------------
Table. \[table:result\] presents the R2R results for Seq2Seq [@anderson2018r2r], Speaker-Follower [@fried2018sf], and RCM [@wang2019rcm] under validation-seen, validation-unseen, and testing sets. All models are trained without augmented data, with $\text{aug}_\text{rand}$, and with $\text{aug}_\text{aps}$. First, we can observe that under validation-seen set, $\text{Seq2Seq}_\text{aps}$ outperforms $\text{Seq2Seq}_\text{rand}$ on all evaluation metrics, , 4.5% absolute improvement on Sucess Rate and 2.9% on SPL. Similar trends can be found for Speaker-Follower and RCM where models trained with APS-sampled paths comprehensively surpass models trained with randomly-sampled paths. Since APS can sample increasingly challenging and custom-made paths for the navigator, APS-sampled paths are more effective than randomly-sample paths and bring in larger improvements on all metrics for all navigation models. For the unseen environments, all models trained with APS consistently outperform $\text{model}_\text{rand}$ with 1.6%-2.7% success rate under validation-unseen set and 0.6%-1.5% under testing set. The improvement shows that APS-sampled paths are not only helpful under the seen environments, but also strengthens the model’s generalizability under the unseen environments. The results under validation-seen, validation-unseen, and testing sets demonstrate that our proposed APS can further improve the baseline models in all terms of visuomotor view, panoramic view, imitation learning, and reinforcement learning.
And under the pre-exploration, all models gain further improvement, especially on SPL for Seq2Seq and Speaker-Follower due to the prior exploration experience which can shorten the navigation path length. For RCM, they adopt reinforcement learning which may increase the path length but still obtain improvement on success rate.
------------------------------ ---------- ---------------------------- ---------------------------
Model train $\text{aug}_{\text{rand}}$ $\text{aug}_{\text{aps}}$
Seq2Seq 71.3 20.3 17.7
$\text{Seq2Seq}_\text{rand}$ **81.4** 26.4 23.8
$\text{Seq2Seq}_\text{aps}$ 78.5 **27.3** **24.8**
------------------------------ ---------- ---------------------------- ---------------------------
: The success rate under training, randomly-sampled augmented ($\text{aug}_{\text{rand}}$), and APS-sampled augmented ($\text{aug}_{\text{aps}}$) sets for Seq2Seq trained without augmented data, with $\text{aug}_{\text{rand}}$ ($\text{Seq2Seq}_{\text{rand}}$), and with $\text{aug}_{\text{aps}}$ ($\text{Seq2Seq}_{\text{aps}}$).[]{data-label="table:diff-pre-aps"}
Ablation Study
--------------
#### Random Path Sampling Adversarial Path Sampling {#sec:ayz-aps}
To investigate the advantage of APS, we perform a detailed comparison between randomly-sampled and APS-sampled data. Fig. \[fig:aps-s2s-sf\] presents the R2R success rate over different ratios of augmented data used for Seq2Seq and Speaker-Follower. The trend line in light color shows that $\text{Seq2Seq}_\text{rand}$ cannot gain additional improvement when using more than 60% augmented data. However, for our proposed APS, the sampled augmented path can keep benefiting the model when more data used and achieve 4.5% and 1.6% improvement under validation-seen and validation-unseen sets, respectively. Since $\text{aug}_\text{rand}$ is sampled in advance, the help to the model is limited. While on the other hand, our proposed APS adversarially learns to sample challenging paths that force the navigator to keep improving. A similar trend can be found for Speaker-Follower where the improvement of $\text{Speaker-Follower}_\text{rand}$ is also stuck but $\text{Speaker-Follower}_\text{aps}$ can lead to even better performance.
**Difficulty and Usefulness of the APS-sampled Paths ** For a more intuitive view of the difficulty and usefulness of the APS-sampled paths, we conduct experiments shown in Table \[table:diff-pre-aps\] to quantitatively compare them with randomly-sample paths. As you can see, the APS-sampled paths seem to be the most challenging as all models perform worst on them. These paths can in turn help train a more robust navigation model ($\text{Seq2Seq}_\text{aps}$) that outperforms the model trained with randomly sampled paths. Moreover, $\text{Seq2Seq}_{\text{aps}}$ even performs better on $\text{aug}_\text{rand}$ than $\text{Seq2Seq}_{\text{rand}}$ which shows that $\text{aug}_\text{aps}$ is not only challenging but also covers useful paths over $\text{aug}_\text{rand}$.
![The success rate under validation-unseen set under different pre-exploration steps for Seq2Seq and Speaker-Follower.[]{data-label="fig:pre-exp-s2s-sf"}](imgs/fig_pre-exp-s2s-sf.png){width="\linewidth"}
{width="0.8\linewidth"}
![The improvement of success rate over the scene feature difference under each validation-unseen environment under the pre-exploration. Each point represents a distinct validation-unseen environment.[]{data-label="fig:pre-exp-each"}](imgs/fig_pre-exp-each.png){width="\linewidth"}
**Pre-Exploration ** \[sec:ayz-pre-exp\] Table. \[table:result\] has shown the improvement brought from the pre-exploration. While, those paths in training, validation, and testing sets are all shortest path but the paths sampled from our APS under unseen environments are not promised to be the shortest. With more pre-exploration steps, the model has more opportunities to explore the unseen environment but at the same time, those too complicated paths sampled from APS may hurt the model. Fig. \[fig:pre-exp-s2s-sf\] presents the success rate under different pre-exploration steps. It shows a trade-off between the model performance and the iterations of the pre-exploration. For Seq2Seq, 15 steps of pre-exploration come out the best result and 40 steps are most suitable for Speaker-Follower.
We also analyze the performance under the pre-exploration under each unseen environments. Fig. \[fig:pre-exp-each\] demonstrates the improvement of the success rate over the scene feature difference. Each point represents a distinct validation-unseen environment. The feature difference under each unseen environment is calculated as the mean of the L2-distance between the visual feature of all scenes from that environment and all scenes in the training environments. In general, most of the unseen environments gain improvement under the pre-exploration. We also find a trend that under the environment which has a larger feature difference, it can improve more under the pre-exploration. It shows that under more different environments, the pre-exploration can be more powerful which makes it practical to be more adaptive and generalized to real-life unseen environments.
**Qualitative Results ** Fig. \[fig:vis\] demonstrates the visualization results of the navigation path without and with pre-exploration for the instruction *“Walk out of the bathroom"*. Under the unseen environment, it is difficult to find out a path to get out of the unfamiliar bathroom, and as is shown in Fig. \[fig:vis\](a), the model without pre-exploration is stuck inside. In contrast, with the knowledge learned during the pre-exploration phase, the model can successfully walk out of the bathroom and eventually achieve the final goal.
Conclusion
==========
In this paper, we integrate counterfactual thinking into the vision-and-language navigation (VLN) task to solve the data scarcity problem. We realize counterfactual thinking via adversarial learning where we introduce an adversarial path sampler (APS) to only consider useful counterfactual conditions. The proposed APS is model-agnostic and proven effective in producing challenging but useful paths to boost the performances of different VLN models. Due to the power of reasoning, counterfactual thinking has gradually received attention in different fields. We believe that our adversarial training method is an effective solution to realize counterfactual thinking in general, which can possibly benefit more tasks.
[^1]: Note that transforming the sampled paths into shortest paths can only be done under seen environments. For pre-exploration under unseen environments, we directly use the sampled paths because the shortest path planner should not be exploited in unseen environments.
[^2]: Note that the shortest-path information is not used during pre-exploration.
[^3]: We have tried to update APS simultaneously with NAV during pre-exploration, but it turns out that under a previous unseen environment without any regularization of human-annotated paths, APS tends to sample too difficult paths to accomplish, , back and forth or cycles. However, those paths will not improve NAV and may even hurt the performance. To avoid this kind of dilemma, we keep APS fixed under the pre-exploration.
|
---
abstract: 'Methods of *deep machine learning* enable to to reuse low/level representations efficiently for generating more abstract high/level representations. Originally, deep learning has been applied passively (e.g., for classification purposes). Recently, it has been extended to estimate the value of actions for autonomous agents within the framework of *reinforcement learning* (RL). Explicit models of the environment can be learned to augment such a value function. Although “flat” connectionist methods have already been used for model/based RL, up to now, only model/free variants of RL have been equipped with methods from deep learning. We propose a variant of *deep model/based RL* that enables an agent to learn arbitrarily abstract hierarchical representations of its environment. In this paper, we present research on how such hierarchical representations can be grounded in sensorimotor interaction between an agent and its environment.'
author:
- |
[[Mark Wernsdorfer]{}, Ute Schmid]{}[^1]\
Cognitive Systems Group\
Faculty of Information Systems and Applied Computer Sciences\
Otto-Friedrich-Universität Bamberg\
An der Weberei 5, 96047 Bamberg, Germany\
`{mark.wernsdorfer, ute.schmid}@uni-bamberg.de`
bibliography:
- 'iclr2015.bib'
title: '[Grounding Hierarchical Reinforcement Learning Models for Knowledge Transfer]{}'
---
Introduction {#sec:intro}
============
Machine learning algorithms are derived from models of the problem/to/be/solved. In RL, problems are modeled in the form of *Markov decision processes* (MDPs). A MDP can be defined as the 4-tuple $(S, A, \mathcal{T}, \mathcal{R}) $. The first element $S $ describes the *set of wold states*—in most cases it is interpreted as the agent’s sensory perception. The second element $A $ describes the *set of possible actions.* The *transition function* determines transitions in state space. It has the general form $\mathcal{T}: S \times A \rightarrow S $. A particular action in a particular state causes a particular successor state. Lastly, the *reward function* gives a scalar value which acts as reward. It indicates whether a particular action in a particular state is to be repeated or avoided. It has the general form $\mathcal{R}: S \times A \rightarrow R $ where usually $R = \mathbb{Z} $.
Although $S $ might be chosen in a way that captures the whole state of the world, for realistic applications, it must be restricted to cover only a small subset of the statements that are currently true about the world. Consider the following examples.
1. Observable world state
- The agent perceives that it is in position (2, 13).
- It moves north to enter sensor state (2, 12).
2. Hidden world state
- The agent perceives a wall in front of it.
- It turns around to enter a sensor state that suggests free passing.
To model the second case, the agent’s restricted perception must be taken into account. To cover this, MDPs can be generalized into *partially observable MDPs* (POMDPs). They can be defined as the 6-tuple $(S, A, \mathcal{T}, \mathcal{R}, \Omega, \mathcal{O}) $. In addition to the four elements above, these models also comprise the *set of possible observations* $\Omega $—possibly distinct from the set of world states—and the *observation function* which has the general form $\mathcal{O}: S \times A \rightarrow \Omega $. A particular action in a particular world state causes a particular observation. [@Kaelbling1998]
POMDPs can describe a plethora of tasks for RL agents. If the set of world states is accessible to the agent, it can infer a particular *belief* about what the current world state is. Indications for the current world state can be inferred from previous beliefs and those world states, that recently observed transitions match with. The result is a belief in the form of a probability distribution over all world states.
As a consequence, the resulting *belief MDP* describes transitions between continuous probability distributions instead of discrete states. To generate probability distributions over wold states, however, these world states need to be known beforehand. Providing the set of world states a priori implies two fatal consequences.
Semantic Load of Representations {#ssec:semantic}
--------------------------------
The first problem concerns the *semantic load* of predefined representations. The set of world states, and therefore the beliefs inferred from them, can be regarded as *representations of the environment.* To provide an agent with such a fixed set of representations, however, means to inject considerable amounts of knowledge into the system.
It tacitly implies that these representations are appropriate for solving the POMDP at hand. Having appropriate representations implies knowledge about how those representations are to be used. To know how to use representations, on the other hand, means to already know how to handle the task at hand.
@Bengio2013 mention the necessary trade/off between the “richness” of representations and the effort necessary to process them. @Diuk2008 show that the selection of representations has immense influence on the performance of RL algorithms. More specifically, they show that providing *semantically rich* representations (e.g. representations of objects) can simplify a given task considerably compared to *semantically sparse* representations (e.g. representations of world states).
Transfer of Knowledge
---------------------
The second problem is that *knowledge transfer* with objective inputs requires elaborate methods for joining discrete data. Partial observability is frequently avoided by providing the world state as sensory input. In any (stochastically) determinate world, complete world states enable to learn an interaction policy which optimizes the cumulative reward received over time. The detachment of objective “snapshots” of the whole world, however, masks the actually relevant information in interaction data.
The first example of observable world states is such a case of *objective interaction.* Knowledge transfer with objective data is successfully achieved if the equivalence of seemingly different states is discovered. It might be discovered, for example, that column 2 in a grid world is always devoid of obstacles. Therefore, state transitions to the north in row 2 can be modeled by approximating the transition function $\mathcal{T} $ with $T_2((x, y), a_n) = (x, y-1) $. All states in column 2 can be pooled according to their successor state when the agent moves to the north. The data structure implementing $T_2 $ can be regarded as *a high/level representation* of column 2.
The second example of hidden world states describes a case of *subjective interaction.* The agent has no access to absolute information about its relation to the environment. Assuming the existence of an objective agent position, its perception must therefore be regarded as ambiguous. Every single location in column 2 might be perceived as identical, although, from an objective perspective, they differ quite obviously in their value of $x $, the absolute position of the agent on a vertical axis.[^2]
Depending on the interaction paradigm, the problem of knowledge transfer takes two contrastive forms. Objective interaction makes it necessary to *analyze* states which appear to the agent as if they were different and *pool* them to recognize their *similarity.* In return, the agent does not have ambiguous perceptions from an objective observer’s perspective.
Subjective interaction, on the other hand, makes it necessary to *synthesize* states which appear to the agent as if they were identical and *differentiate* them to recognize their *distinctness.* In return, knowledge transfer comes for free: although the agent might be in objectively different situations, as long as there is no reason for differentiation, knowledge which has been been acquired from identical perceptions and actions is applied effortlessly, in fact even unnoticed by the agent.
[.48]{}
[.48]{}
shows the corridor environment for testing knowledge transfer in RL agents. illustrates the training environment and \[fig:testing\_env\] the environment where the acquired knowledge is tested. In each time step, the agent receives a reward of $-1 $. If it reaches the goal position $G $, the current episode ends, its position is reset to the starting position $S $, and it receives a reward of $+10 $.
*Objective interaction data* consists of the agent’s absolute position as sensor input. Objective motor output determines whether it enters the cell to its north, east, south, or west in the next time step in case it is not occupied by a wall.[^3] The new part of the corridor in the larger testing environment contains qualitatively new objective data for the agent (i.e., coordinates with an unknown horizontal component). Therefore, it cannot benefit from the knowledge acquired in the training environment. It learns a new policy by successively overwriting the old one, as if it were in a completely new environment.
*Subjective interaction data* consists of the four surrounding cells as sensor input. Subjective motor output determines whether the agent turns 90 degrees to its right, to its left, or if it enters the cell in front of it (in case it is not occupied by a wall). Although the new part of the testing corridor is objectively different to the smaller training corridor, they subjectively appear identical to the agent. Subjective interaction data enables the agent to efficiently reapply its knowledge without even knowing so.
When knowledge transfer is the issue, the drawback of objective data becomes obvious. The drawback of subjective data is that any sufficiently complex environment presents behavior optimizing algorithms with POMDPs. To avoid semantically loaded representations, however, the set of world states cannot be provided a priori. POMDPs *without* knowing the complete set of world states cannot be solved by any single policy. As our explicit goal is the reuse of representations in objectively different situations, however, it still seems reasonable to use subjective interaction data and cope with these problems as they occur.
Semantic load and knowledge transfer are essentially two stages of coping with the same problem. The problem of semantic load shows *that* transferring information (i.e., semantics) might increase performance but it impairs autonomy at the same rate. Knowledge transfer, on the other hand, enables to investigate *which* information can be transferred and *how* to transfer it. The difference is, that the former is usually understood as pathological because the transferred knowledge originates *in the designer,* while the latter is understood as beneficial because knowledge is transferred *within the same system.*
In \[sec:intro\] we have differentiated two perspectives on agent interaction that are of utmost importance for knowledge transfer. presents related work from the complementary fields of deep representation learning and hierarchical RL. In \[sec:overview\], we present the general components of our architecture and the type of information they exchange. sketches an instantiation of the architecture and the concrete machine learning methods applied. In \[sec:results\] we present and discuss the results from testing our architecture in typical hierarchical RL gridworld environments. outlines subsequent research.
Related Work {#sec:related}
============
Usually, the contrast between objective and subjective interaction is reduced to the difference between MDPs and POMDPs. To develop a machine learning algorithm from an appropriate model of the problem/to/be/solved, however, we see it fit not to take an observer’s perspective but the agent’s perspective on the problem. From the agent’s perspective, there are no world states to begin with.
First and foremost, beliefs are *identical* to the world perceived and influenced by the agent. To refrain from one’s believes is an accomplishment of highly developed animals and by no means something to take for granted. POMDPs already imply a differentiation between belief and world states in separately modeling a probability distribution (i.e., a belief) and its targets (i.e., world states).
Therefore, POMDPs might be the perfect model for the problem of RL from an observer’s perspective (i.e., for developing algorithms for objective interaction),[^4] to model learning *as it is experienced,* however, requires to refrain from maintaining representations of objective world states. Instead, those states need to be constructed by the agent itself in a process of *deep representation learning.*
Deep Representation Learning
----------------------------
An unbiased learner cannot rely on representations that have been provided by its designer. This insight dates back at least to the philosophical criticism that has been brought forward towards artificial intelligence in the early 80ies in the form of the *Chinese room argument* [@Searle1980], the *epistemological frame problem* [@Dennett1981], and, more explicitly, the *symbol grounding problem* [@Harnad1990]. The semantic implications of representations have been recognized and are investigated in the field of *representation learning.*
The most famous application of representation learning at the time being is in *deep learning.* Deep learning is commonly performed with connectionist methods (e.g. restricted Boltzmann machines, auto/encoders, or artificial neural networks). Its eponymous feature, however, is not its class of methods but how they are employed.
One intuition behind deep learning is that data complexity can be reduced considerably by generating appropriate representations from the dynamics of interaction between a particular system and its environment.[^5] Representation learning methods can be “stacked” by providing the representations generated in one layer to a higher/level instance of a similar representation learner. Therefore, in each layer, data complexity can be reduced more and more by recognizing and exploiting patterns. Data complexity is transformed into structural complexity of representations. As recognized patterns can be applied repeatedly, however, structural complexity only grows logarithmically with input complexity.
Connectionist variations of deep representation learners effectively treat the output of a layer as input for the layer above. The idea, however, can be generalized beyond connectionist implementations. Stacking any method that reduces data complexity by generating representations with less degrees of freedom (e.g. dimensionality reduction, principal component analysis, lossy, or loss/less compression) must eventually lead to a hierarchy of representations, with the most abstract at the top and the most concrete at the bottom.
If representations are generated not only from proto/features, sub/symbolic information, raw input, or however one may call the first data to enter the system, but also from *previously learned representations,* then the system effectively generates layers containing abstract representations. A hierarchical organization of cognition comes naturally if a learning system is able to *reuse* the representations in different layers. Benefiting from the same set of representations in objectively different situations is a case of knowledge transfer.
Hierarchical representations of a *passive data space,* e.g. in image analysis [@Schmidhuber2014] or loss/less sequence compression [@Nevill-Manning1997], have been proposed in various fields of computer science and with various applications. Approaches to learn the structure of hierarchical representations of an *interactive data space* (i.e., a policy) go back at least to the early 90ies ([@Schmidhuber1990]; [@Schmidhuber1991]; [@Feldkamp1998]), but only recently, *RL* methods have been presented that are efficiently able to generate deep policies from scratch ([@Mnih2013]; [@Guo2014]).
Reinforcement Learning
----------------------
RL is concerned with optimizing behavior in unknown environments. Optimality is quantified as reward function that provides an evaluation feedback for the agent, indicating whether an action should be considered as good or bad. To achieve optimal behavior over time, the agent optimizes not the immediate but the *cumulative* reward it expects to receive by following the current policy. This cumulative reward is discounted over future states because immediate rewards might matter more than those far in the future.
In general, such an evaluation can be estimated by remembering which action in which sensor state was followed by which discounted, cumulative reward. The future discounted, cumulative reward $R $ of unknown perception/action tuple $(s, a) \in S \times A $ can be predicted by estimating a value function $V $. $$\begin{aligned}
\label{eq:value}
V: S \times A \rightarrow R\end{aligned}$$ Choosing a perception/action pair that maximizes the value function enables optimal behavior over a certain period of time.
The same action in the same state may not always lead to the same successor state. Non/stationary probability distributions can force the value function to adapt. More importantly, however, the same state/action pair might produce different states *systematically.* To avoid conflating the rewards of those states, the value function can be augmented by past experience about state transitions in the observed environment. *Model/based RL* explicitly models these state transitions.
Optimal models are identical to the transition function of the environment $\mathcal{T} $. Interpolating transition function $T $ in \[eq:trans\] enables predicting the outcome of perception/action pairs that have not been observed before. $$\begin{aligned}
\label{eq:trans}
T: S \times A \rightarrow S\end{aligned}$$
In model/based RL, instead of associating cumulative reward with perception/action pairs, immediate rewards are associated with sensor states. Knowledge about state transitions in the specific environment enables to choose attractive successor states via their estimated immediate reward *and* the probability of reaching them with a particular action. Choosing an action that maximizes the estimated reward of the estimated successor states enables planned behavior.
More importantly, if a reliable model of the environment is available, the value of each perception/action tuple need not be tediously approximated by actually experiencing the future cumulative reward over all of its possible consequences. Instead, state transitions can be “simulated” arbitrarily far into the (estimated) future by exploiting the internal model.
Data structures which implement such a transition model are *representations* of parts of the environment. Value and model function \[eq:value,eq:trans\] determine the current interaction policy $\pi = (V, T) $ for the agent.
*Hierarchical RL* explores hierarchical models of the environment. Most of recent research uses objective interaction data [@Sammut2011]. First and foremost reason is that objective data produces MDPs. If the agent is in a determinate sensor state, and any given action has (stochastically) predictable outcome (i.e., successor state and reward), transition and value functions can be approximated precisely.
Intuitively, transitions of abstract states in a high layer of the hierarchy occur less often than transitions in a low layer. Therefore, *semi MDPs* are frequently used to model such time/sensitivity ([@Thrun1995]; [@McGovern1997]; [@Dietterich1999]; [@McGovern2001]; [@Hengst2002]). This generalization of MDPs extends the model of state transitions by *duration* as proposed by @Sutton1999.
The structure of hierarchical models, however, is mostly *trained*: stochastic information is integrated into a fixed hierarchical structure. It is desirable to enable a RL agent to generate hierarchical representations *autonomously,* on the one hand, to avoid the problem of semantically loaded representations and, on the other hand, to resolve the various ambiguities that result from subjective interaction in POMDPs. Deep representation learning methods can help to achieve this goal.
Overview {#sec:overview}
========
In the following, we will present a rough outline of our approach to the grounding of deep models for RL. We present the general form of the functions we used. We implemented several agents according to this architecture. In this section, however, we present only the general structure such that the individual methods remain exchangeable.
We have shown that subjective interaction requires the disambiguation of seemingly identical states. To differentiate identical observations, additional information is necessary. This information is not directly accessible, it needs to be “uncovered in,” or “generated from,” observable data. This hidden information can be modeled as *latent states.* In belief MDPs, the current world state is such a latent state. To avoid semantic load, however, we have determined that representations (i.e., the set of latent states) cannot be predefined but must be inferred from the dynamics of interaction between agent and environment alone. These dynamics can be stored in *trajectories* or *interaction histories.*[^6]
A history $h $ is a sequence of single *experiences* $e $. In the literature, experiences usually consist of the 4-tuple $(s_t, a_t, s_{t+1}, r_{t+1}) $ with perception and action at time step $t $, and perception and reward at time step $t+1 $ [@Lin1993]. The perceived history $h $ can be recognized as an instance of a latent/state/specific model $M_l $, where $\forall l \in L.\; l \leftrightarrow M_l $ during latent state $l $.
Inspired by Peircean semiotics, our definition of representation consists of shape (i.e., a latent state $l $), content (i.e., a policy $\pi_l = (V_l, M_l) $ of value and model function), and reference (i.e., history $h_l $). A history is considered as the reference of a particular latent state, if the likelihood of the history being generated from the latent state’s model exceeds a particular threshold.[^7] Latent states are the shapes of *abstract representations* for situations in which their transitions hold. These situations are interaction histories.
Learning Query Processes
------------------------
Transitions between abstract representations can be described by the *query function.* The query function is analogue to the transition function $\mathcal{T} $ in MDPs. Notice, however, that MDPs assume a perception/action tuple to determine a single successor perception, whereas in an abstract representation, various queries from abstract representations might succeed and/or fail. $$\begin{aligned}
\label{eq:latent_query}
\mathcal{Q}: L \times L \rightarrow S \qquad S = \{\top, \bot \}\end{aligned}$$
Choosing an abstract representation, and therefore its corresponding model, *is not a MDP.* The reason is that inducing a policy is not as straight/forward as performing an action. Whether the model function of a queried latent state matches the subsequent history is not fully under control of the agent. Among abstract representations, there is no unambiguous distribution of responsibilities between agent and environment as it is among perceptions and actions.
In general, transitions of representations follow *query processes* (QPs). A QP is defined by the 3-tuple $(L, \mathcal{Q}, \mathcal{R}) $. It contains the open set of latent states, the query function in \[eq:latent\_query\], and the *reward function* $\mathcal{R}: L \rightarrow R $, where $R = \mathbb{Z} $. The reward function in QPs is analogue to the reward function in MDPs.
An agent must be able to estimate whether a latent state can be induced from another latent state at all. To perform successful queries, therefore, the query function must be approximated by a *model function.* This model function is analogue to the model function in \[eq:trans\]. Optimal model functions are identical to the query function of the environment $\mathcal{Q} $. $$\begin{aligned}
\label{eq:model}
M: L \times L \rightarrow S \qquad S = \{\top, \bot \}\end{aligned}$$
Abstract representations also need to be chosen according to the expected change in discounted, cumulative reward caused by their policy. This can be estimated by the *abstract value function.* The abstract value function is analogue to the value function in \[eq:value\]. $$\begin{aligned}
\label{eq:context_value}
V_l: L \rightarrow R\end{aligned}$$
determine the policy $\pi = (V_l, M) $ for solving QPs, just like \[eq:value,eq:trans\] determine the policy $\pi = (V, T) $ for solving MDPs. The experiences in history $h $ enable to iteratively approximate value and transition function.[^8]
Abstracting Query Processes
---------------------------
Associating histories with models enables abstract representations of concrete observations in the form of latent states. These latent states model the agent’s abstract perception of the environment: they represent the environment to the agent. A function to differentiate a perceived history according to the latent state whose model matches best has the general form $$\begin{aligned}
\label{eq:abstract}
A: H \rightarrow L\end{aligned}$$ where $H = [e_{t-n}, \ldots, e_t] $, $t $ is the current time/step, and $n $ is the length of the history. This function realizes an abstract representation of a history referenced by a the shape of a latent state via the model it contains. Therefore, we call it *“abstraction function.”*
The environment responds to queries by either accepting or rejecting the queried state, depending on whether the interaction history that has been observed after querying a representation actually matches the according model function. If it does, the query is successful. Such a measure of “matching” must be integrated in the abstraction function, such that only latent states are returned that are associated with models compatible to the observed history. Each abstract representation provides an interaction policy that consists of value and model function. To effectively reuse policies, they must be efficiently retrievable. The general form of a retrieval function must be inverse to $A $: not from observed histories to representations, but from representations to concrete interaction policies. This *application function* has the following form: $$\begin{aligned}
A': L \rightarrow \Pi\end{aligned}$$
Environment and agent determine *together* where the transition from the current representation leads to. Consequentially, the intension to induce a particular successor representation $l_q $ may fail, dependent on whether the current history which has been evoked by the queried policy $h_t^{\pi_q} $ is actually compatible to this policy. Whether a query succeeds or fails is determined in \[eq:query\_success\]. $$\begin{aligned}
h_t^{\pi_q} & = [e_{t-n}, \dots, e_t] \nonumber \\
l_t & = A(h_t^{\pi_q}) \nonumber \\
\mathcal{Q}(l_{t-n-1}, l_q) & \leftrightarrow l_t = l_q \label{eq:query_success}\end{aligned}$$
Stacking Query Processes
------------------------
There are obvious parallels between observable and latent states. Value functions are applicable straight/forwardly in both. In both cases, choices influence the future reward. The forms of the model for observable state transitions in \[eq:trans\] and the model for representation transitions in \[eq:model\], however, differ. MDPs regard *observable states as conditional on a perception/action tuple,* whereas QPs regard *the success of a query as conditional on a tuple of two consecutive latent states.* As we have seen, latent states cannot be differentiated into perception and action. The general form of the model function of MDPs in \[eq:trans\] can therefore not be applied to them.
To enable an unbounded hierarchy, it is desirable to unify the model functions in each layer. QPs can be stacked. Abstract model functions can be generated which describe the transitions of latent states. Latent states, on the other hand, contain low/level model functions themselves. To be grounded in observable states, however, concrete sensorimotor interaction must also be described by QPs.
If we perform some modifications to what we understand as an observable state, the model function of QPs in \[eq:model\] can be applied to observable states. We have shown that observable states in RL are commonly associated with the sensor states of an agent. Its actions, on the other hand, are also observable, but essentially different. Action $a_t $ and perception $s_t $ are separate. To translate MDPs into QPs, action and perception need to be integrated. We introduce *sensorimotor states* by simple concatenation of the last action and the current observation: $x = (a_{t-1}, s_t) $ and $x \in SM $ where $SM = A \times S $.
Sensorimotor states can be queried like latent states. A sensorimotor state, however, does not imply an interaction policy, because sensorimotor states are the lowest layer of interaction. Instead, querying a sensorimotor state implies the execution of the part of the state the agent has control over: the motor component. The agent has only partial influence on the success of a query. The sensor component is controlled by the environment. If the sensorimotor state in the query contains another sensor component than the agent’s perception in the next time step, the query will fail.
Modifying observable states also implies modifying experiences. For QPs we differentiate two types of experiences. One contributing to the value function and one contributing to the model function. *Value/related experiences* consist of a sensorimotor state and the simultaneously received reward $e_v = (x_t, r_t) $. *Model/related experiences* consist of an observed sensorimotor state, a queried sensorimotor state, and whether the query was successful $e_m = (x, q, s) $ where $s \in \{\top, \bot\} $ and $x, q \in SM $.
Implementation {#sec:implementation}
==============
In the following, we describe how we implemented the above functions and related methods from conventional RL. The *value function* estimates the discounted, cumulative reward. The *model function* estimates which sensorimotor states can be queried. The query function $\mathcal{Q} $ in QPs replaces the transition function $\mathcal{T} $ in MDPs. Finally, the *abstraction function* generalizes histories of experiences of sensorimotor states into high/level representations. The *application function* merely remembers which value and model functions are associated with which abstract state. Therefore, we will not cover it in more detail. We will present practical methods to approximate these functions.[^9]
Value Function
--------------
RL literature provides numerous methods for approximating a value function. Two of the most prominent ones are *Q-learning* (originally [@Watkins1989]) and *SARSA-learning* (interestingly, originally in a connectionnist context [@Rummery1994]). The former generates a value function which describes the *optimal interaction policy* (i.e., offline learning), the latter generates a value function which describes the *current interaction policy* (i.e., online learning).
Consider an agent that performs its (arbitrarily chosen) actions only with a certain probability. In some cases it performs a random action instead. The task is to learn to walk the shortest possible path along a straight cliff. Q-agents learn to walk close to the edge, because it is the shortest path with the most cumulative reward. If a random action occurs, however, they might drop, suffering large amounts of negative reward. Q-learning performs offline learning.
Online policy learners, on the other hand, learn from actually performed actions. In the cliff walking example, they learn to follow an arc that keeps a safe distance to the edge. Any autonomously learning agent needs to perform random actions to explore unknown environments. For life/long learning, therefore, online learning seems preferable.
For this reason we adopted SARSA-learning. The original update function for separate sensor and motor states in SARSA-learning is as follows. $$\begin{aligned}
\label{eq:sarsa}
V(s_{t-1}, a_{t-1}) \leftarrow V(s_{t-1}, a_{t-1}) + \alpha [r_t + \gamma V(s_t, a_t) - V(s_{t-1}, a_{t-1})] \qquad \alpha, \gamma \in [0, 1]\end{aligned}$$
This iterative process is parametrized with learning factor $\alpha $ and discount factor $\gamma $. The learning factor determines the readiness to overwrite previous experiences with new ones. The bigger $\alpha $ is, the more weight new value/related experiences have in comparison to older ones. The discount factor $\gamma $ determines the importance of future rewards.
Q- and SARSA-learning use MDPs as a model. Therefore, both separate sensor and motor states. They are, however, easily adaptable to accommodate for an element of the set of all sensorimotor states $x \in SM $. We modified the SARSA/value update in the following way. Notice that in our modification, the simultaneously received reward is used to update the value of a state, instead of the next reward as in \[eq:sarsa\]. $$\begin{aligned}
V_l(x_{t-1}) \leftarrow V_l(x_{t-1}) + \alpha [r_{t-1} + \gamma V_l(x_t) - V_l(x_{t-1})] \qquad \alpha, \gamma \in [0, 1]\end{aligned}$$
If the task is segmented into episodes, the expected cumulative reward has an upper bound (i.e., the cumulative reward to be expected during the remainder of the current episode). If the task is ongoing, however, the value function must be bounded by normalization. $$\begin{aligned}
\label{eq:myvalueupdate}
V_l(x_{t-1}) \leftarrow \frac{V_l(x_{t-1}) + \alpha [r_{t-1} + \gamma V_l(x_t) - V_l(x_{t-1})]}{1 + \alpha} \qquad \alpha, \gamma \in [0, 1]\end{aligned}$$
Q- and SARSA-learning are used in model/free RL. The interaction policy followed by a Q- or SARSA-agent is determined by the value function: the agent performs the one action that maximizes the value function. In model/based RL, however, the reward estimate is only one component responsible for a particular policy.
Model Function
--------------
The environment can be modeled by approximating the transition function for observable states. Transition probabilities are increased for actually observed transitions, see \[eq:obs\_trans\], and decreased for all other transitions, see \[eq:oth\_trans\]. $$\begin{aligned}
T(s_{t-1}, a_{t-1}, s_t) \leftarrow & \; T(s_{t-1}, a_{t-1}, s_t) + \alpha (1 - T(s_{t-1}, a_{t-1}, s_t)) \label{eq:obs_trans} \\
\forall s' \in S \wedge s' \neq s_t. T(s_{t-1}, a_{t-1}, s') \leftarrow & \; T(s_{t-1}, a_{t-1}, s') + \alpha (0 - T(s_{t-1}, a_{t-1}, s')) \label{eq:oth_trans}\end{aligned}$$
Estimates of state transitions enable to plan behavior. Traditionally, transition functions realize models that enable to estimate transition probabilities from one sensor state via action to another sensor state. They are usually not combined with value functions that provide estimates of *cumulative* reward, because the expected cumulative reward can be “simulated” with an appropriate model. shows how model/based approaches rather estimate the *immediate* reward of perception/action pairs. $$\begin{aligned}
R(s_t, a_t) \leftarrow & \; R(s_t, a_t) + \alpha (r_t - R(s_t, a_t)) \label{eq:reward}\end{aligned}$$
Given a comprehensive model of state transitions and expected rewards, model/based reinforcement agents are able to chose the action that maximizes the sum of expected rewards weighted by the probability of expected future states. The probability of future states is estimated by the model function. By recursively retrieving the estimated value $V $ of potential successor states, model/based agents are able to determine values that take states into account which are arbitrarily far in the future. $$\begin{aligned}
V(s, a) = R(s, a) + \gamma \sum_{s'}^S T(s, a, s') \max_{a'}^A V(s', a') \label{eq:model_value}\end{aligned}$$
The recursive call in \[eq:model\_value\] enables to use an existing model for estimating future cumulative reward, instead of visiting each state several times to cover all of its possible consequences like in in \[eq:sarsa\] of model/free RL.
The downside is that a recursion necessitates the definition of a termination condition. Introducing conditions always comes at the risk of corrupting universality. Secondly, the value function has to be evaluated several times in each iteration and, lastly but most importantly, an approximately correct model of the environment needs to be already available to estimate the value of states.
By incorporating the iterative value update in \[eq:myvalueupdate\], we are able to avoid a potentially expensive recursion. Our architecture tries to exploit both: a model of the environment *and* an estimate of the long/term attractiveness of a state. With each query it considers only those sensorimotor states that the model deems possible. Among those states, the value function enables to choose the most promising ones.
Our approximation of the model function can be seen in the following. The value of $s $ indicates whether the environment accepted the last query $q_{t-1} \in SM $ or not (i.e., whether $q_{t-1} $ is identical to the current state $x_t $). It follows that the “inducibility” of query $q $ from sensorimotor state $x $ can be determined by \[eq:induce\]. $$\begin{aligned}
I_l(x_{t-1}, q_{t-1}) \leftarrow I_l(x_{t-1}, q_{t-1}) + \alpha (s - I_l(x_{t-1}, q_{t-1})) \qquad s =
\begin{cases}
1,& \text{if } q_{t-1} = x_t \\
0,& \text{otherwise}
\end{cases} \label{eq:induce}\end{aligned}$$
Action Selection
----------------
If the value function enables to evaluate perception/action tuples, then action selection simply returns the action $a $ from the a tuple that maximizes the estimated value among all perception/action tuples. Drawbacks are costs linear to the sum of the number of perceptions and the number of actions. shows the *decision function* in conventional model/based and model/free RL during perception $s_t $. $$\begin{aligned}
D(s_t) = a \qquad V(s_t, a) = \max_{s_t, a'}^{S \times A} V(s_t, a') \label{eq:dec_action}\end{aligned}$$
Instead of the probability of successor states, query selection must consider the “inducibility” of potential successor states. The approach also performs a simple maximization. Its specific form, however, is quite different from conventional action selection in RL. The decision function realizes query selection determined by the value function and the approximation of the model function. $$\begin{aligned}
D(x_t) = x \qquad V_l(x) = \max_{x'}^{X_i} V_l(x'), \quad \forall x_i \subset X_i.\; I(x_t, x_i) \geq c \quad c \in [0, 1] \label{eq:query}\end{aligned}$$
enables to select sensorimotor *and* latent states alike. The state $x $ can either be a sensorimotor or a latent state. For each query, first, the set of most inducible states with a certainty above $c $ is selected. From this set, a query is selected that maximizes the value function. Both parts of the current policy $\pi $, value function $V $ and model function $M $, are determined by the currently active latent state $l $. This latent state, on the other hand, is subject to simple QPs, just as low/level sensorimotor states are. From the explications above the architecture in \[fig:architecture\] emerges.
\[fig:architecture\]
\[fig:motmem\]
Results {#sec:results}
=======
The more abstract representations get, the more blurry the line between action and perception becomes. In contrast to MDPs, QPs can be applied to abstract state transitions. By continuously performing queries for desirable states, an agent can learn transition probabilities between these states. Once the query function $\mathcal{Q} $ has been sufficiently approximated, the agent can choose those successor states that promise to maximize a normalized variant of expected future cumulative reward. This behavior effectively realizes a value optimizing policy in a particular QP.
Although both, Markovian policy and query policy, converge in the small corridor environment from \[fig:training\_env\] after 14 episodes, QPs take much longer to complete an episode.[^10] The performance of the query policy suffers from condensing sensor and motor states. The decline in performance can be explained by the increased number of sensorimotor states. Whereas a Markovian model function needs to keep track of $|S \times A \times S| $ entries ($432 $ values in the small corridor environment with subjective interaction), the query model function needs to track $2 * |S \times A |^2 $ entries ($2592 $ values, respectively). The value function $V $ in model/based Markovian policies contains $|S| $ values ($12 $ in our example), the value function in query policies contains $|S \times A |$ entries ($36 $, respectively).
Consider, however, that subjective policies do not grow with new objective input. As long as objectively unknown input can be interpreted as subjectively known, the policy size remains unchanged. Of course, this advantage of subjective interaction also holds for Markovian policies. As we have shown, however, the distinction between perception and action effectively prevents any straight/forward application of Markov policies to latent states. Any ambitions towards a hierarchy of representations with MDPs will first have to solve the problem of how to integrate perception and action.
illustrates the labyrinth environment. Reward and interaction conditions are identical to the corridor setting. In contrast to the small corridor environment, however, the labyrinth cannot be solved by a Markovian policy given subjective interaction data. To reach the goal, an agent has to go straight at some crossroads but turn at others. Without further means of differentiation, subjective Markov policies can only perform one of both each time. shows that the labyrinth cannot be solved by MDPs with subjective interaction data.
[.48]{}
[.48]{}
The policy of the objective Markov agent converges at the optimum after roughly 140 episodes. shows that the policy of the subjective query agent converges at sub/optimal performance after only 18 episodes. A query policy is able to differentiate the identical perceptions at crossroads according to the last performed action: it develops a strategy to solve the labyrinth.
The following policy is one of the successful, sub/optimal policies developed by a query agent. The policy has been condensed, such that the underlying search strategy becomes obvious. The conditionals to the left are sensorimotor states, consisting of the last action and the current perception.
- (turn, a wall to the right): turn left
- (turn, at a crossroad): move forward until one of the following holds.
- (forward, at a crossroad): turn left
- (forward, facing a wall): turn left
By storing the last motor activation, query policies with sensorimotor states realize a sort of short/term memory. Subjective single/layer query policies can solve tasks that cannot be solved by subjective single/layer Markovian policies. Such a “graceful decent” of performance is highly desirable if the relevant states of the task are not known in advance. Even if the agent is not able to develop a policy that is optimal *from an observer’s point of view,* it can still produce solutions that take its own limitations into account.
Future Work {#sec:future}
===========
The greatest challenge for extruding a grounded query policy into a hierarchical architecture is to find an appropriate abstraction function $A $ that reliably and repeatedly provides the same models given only linear histories of them. The fact that the dimensionality of the history is always smaller or equal to the dimensionality of the model complicates reliable comparisons. The history might not contain the information relevant for deciding whether is corresponds to one model or another.
Additional information needs to be acquired to resolve ambiguities. It is clear, however, that this information cannot be extracted from the structure of history or model. Representation learning might be able to generalize histories, but any generalization that considers only the structure of history and model is confronted with underdetermined exemplars: it cannot be decided which one of a structurally equally similar set of models a history belongs to. In cognitive science, a similar situation presents a problem in explaining the universality of human object recognition. It is known as *the problem of vanishing intersections* [@Harnad1990].
A hierarchical architecture like in \[fig:architecture\], however, can estimate the model of an observed history from information beyond sheer structure. The transitions of policies, and therefore the transitions of models, that are motivated by observing a particular history are at the same time abstract state transitions in the layer above. To be able to model and predict abstract state transitions (e.g. as QPs) implies the ability to model and predict lower policy transitions.
The top layer effectively provides prior probabilities for policy transitions. Even without any useful structural information (i.e., no significant intersection of history and model), the likelihood for a particular history for having been generated by a particular model can be acquired by previously experienced abstract state transitions. This realizes an *inductive expectation bias.*
QPs provide the formal grounds for augmenting base layer query selection $D^1 $ with an unbounded number of more abstract and less frequent query selections $D^m $ where $m = [1,\infty] $. Layers grow as the acquired knowledge is no longer applicable in every situation. The general case, however, is to assume that every environment can be described in terms of a QP. No sooner than when this assumption is violated, new data structures (i.e., representations) are generated. Until then, available knowledge is transfered onto every objectively new situation *per se.*
[^1]: <http://www.uni-bamberg.de/kogsys/>
[^2]: This concerns not only perception. In relative interaction, actions also depend on hidden world states: moving north, east, south, or west is not part of the agent’s motor capabilities. Instead, its absolute direction of movement depends on its current orientation, which is not directly observable. The objective direction of movement is not only for the agent to decide.
[^3]: The agent does not receive additional punishment for running into walls.
[^4]: Although \[ssec:semantic\] casts doubt on this.
[^5]: Interestingly, this insight corresponds to recent developments in cognitive sciences that have been labeled as *“embodied cognition”* [@Shapiro2011].
[^6]: Although we regard both as equivalent (in contrast, see [@Singh2000]), here, we will stick with the latter term, as it has already been established in connectionist RL [@Lin1993]. The former is more common in applications of *dynamic systems theory* (e.g. [@Guenter2007]).
[^7]: Note that only *linear* histories can be observed. Models, on the other hand contain *nonlinear* information about *possible* transitions, whereas histories only contain *actual* transitions.
[^8]: At this stage, the overall architecture resembles a *hidden Markov model.* As more and more layers are added, however, the binary differentiation between latent and observable layers makes way for an unbounded hierarchy of stacked QPs. Also, notice that transitions between latent states are not determined by incoming information alone but they are also a means to inject information into the environment via actions.
[^9]: In the present research, we evaluate QPs as a means for *grounding* a hierarchical model for RL. Our ongoing research investigates the possibility to effectively generate such a model.
[^10]: In all experiments we set $\alpha = .5 $ and $\gamma = .5 $ with an initial optimistic value of $5 $.
|
---
abstract: 'Aspect based sentiment analysis (ABSA) aims to identify the sentiment polarity towards the given aspect in a sentence, while previous models typically exploit an aspect-independent (weakly associative) encoder for sentence representation generation. In this paper, we propose a novel **A**spect-**G**uided **D**eep **T**ransition model, named AGDT, which utilizes the given aspect to guide the sentence encoding from scratch with the specially-designed deep transition architecture. Furthermore, an aspect-oriented objective is designed to enforce AGDT to reconstruct the given aspect with the generated sentence representation. In doing so, our AGDT can accurately generate aspect-specific sentence representation, and thus conduct more accurate sentiment predictions. Experimental results on multiple SemEval datasets demonstrate the effectiveness of our proposed approach, which significantly outperforms the best reported results with the same setting[^1].'
author:
- |
Yunlong Liang^1^[^2] , Fandong Meng^2^, Jinchao Zhang^2^, Jinan Xu^1^[^3] , Yufeng Chen^1^ and Jie Zhou^2^\
^1^Beijing Jiaotong University, China\
^2^Pattern Recognition Center, WeChat AI, Tencent Inc, China\
`{yunlonliang,jaxu,chenyf}@bjtu.edu.cn`\
`{fandongmeng,dayerzhang,withtomzhou}@tencent.com`\
bibliography:
- 'emnlp-ijcnlp-2019.bib'
title: |
A Novel Aspect-Guided Deep Transition Model\
for Aspect Based Sentiment Analysis
---
Introduction {#sec:introduction}
============
Aspect based sentiment analysis (ABSA) is a fine-grained task in sentiment analysis, which can provide important sentiment information for other natural language processing (NLP) tasks. There are two different subtasks in ABSA, namely, aspect-category sentiment analysis and aspect-term sentiment analysis [@Pontiki:14; @weixueGCAE:18]. Aspect-category sentiment analysis aims at predicting the sentiment polarity towards the given aspect, which is in predefined several categories and it may not appear in the sentence. For instance, in Table \[tbl:testE\], the aspect-category sentiment analysis is going to predict the sentiment polarity towards the aspect “[*food*]{}”, which is not appeared in the sentence. By contrast, the goal of aspect-term sentiment analysis is to predict the sentiment polarity over the aspect term which is a subsequence of the sentence. For instance, the aspect-term sentiment analysis will predict the sentiment polarity towards the aspect term “[*The appetizers*]{}”, which is a subsequence of the sentence. Additionally, the number of categories of the aspect term is more than one thousand in the training corpus.
As shown in Table \[tbl:testE\], sentiment polarity may be different when different aspects are considered. Thus, the given aspect (term) is crucial to ABSA tasks [@jiang-etal-2011-target; @Ma:17; @WangS:18; @DBLP:journals/corr/abs-1905-07719; @liang2019context]. Besides, @li2018transformation show that not all words of a sentence are useful for the sentiment prediction towards a given aspect (term). For instance, when the given aspect is the “[*service*]{}”, the words “[*appetizers*]{}” and “[*ok*]{}” are irrelevant for the sentiment prediction. Therefore, an aspect-independent (weakly associative) encoder may encode such background words (e.g., “[*appetizers*]{}” and “[*ok*]{}”) into the final representation, which may lead to an incorrect prediction.
Numerous existing models [@Tang:16b; @Yi:17; @Fan:18; @weixueGCAE:18] typically utilize an aspect-independent encoder to generate the sentence representation, and then apply the attention mechanism [@D15-1166] or gating mechanism to conduct feature selection and extraction, while feature selection and extraction may base on noised representations. In addition, some models [@Tang:16a; @Wang:16; @Majumder:18] simply concatenate the aspect embedding with each word embedding of the sentence, and then leverage conventional Long Short-Term Memories (LSTMs) [@Hochreiter:1997:LSM:1246443.1246450] to generate the sentence representation. However, it is insufficient to exploit the given aspect and conduct potentially complex feature selection and extraction.
To address this issue, we investigate a novel architecture to enhance the capability of feature selection and extraction with the guidance of the given aspect from scratch. Based on the deep transition Gated Recurrent Unit (GRU) [@Cho:14; @journals/corr/PascanuGCB13; @W17-4710; @Meng:19], an aspect-guided GRU encoder is thus proposed, which utilizes the given aspect to guide the sentence encoding procedure at the very beginning stage. In particular, we specially design an aspect-gate for the deep transition GRU to control the information flow of each token input, with the aim of guiding feature selection and extraction from scratch, i.e. sentence representation generation. Furthermore, we design an aspect-oriented objective to enforce our model to reconstruct the given aspect, with the sentence representation generated by the aspect-guided encoder. We name this **A**spect-**G**uided **D**eep **T**ransition model as AGDT. With all the above contributions, our AGDT can accurately generate an aspect-specific representation for a sentence, and thus conduct more accurate sentiment predictions towards the given aspect.
We evaluate the AGDT on multiple datasets of two subtasks in ABSA. Experimental results demonstrate the effectiveness of our proposed approach. And the AGDT significantly surpasses existing models with the same setting and achieves state-of-the-art performance among the models without using additional features (e.g., BERT [@bert]). Moreover, we also provide empirical and visualization analysis to reveal the advantages of our model. Our contributions can be summarized as follows:
- We propose an aspect-guided encoder, which utilizes the given aspect to guide the encoding of a sentence from scratch, in order to conduct the aspect-specific feature selection and extraction at the very beginning stage.
- We propose an aspect-reconstruction approach to further guarantee that the aspect-specific information has been fully embedded into the sentence representation.
- Our AGDT substantially outperforms previous systems with the same setting, and achieves state-of-the-art results on benchmark datasets compared to those models without leveraging additional features (e.g., BERT).
Model Description
=================
As shown in Figure \[fig:DTSA\_aspect\], the AGDT model mainly consists of three parts: *aspect-guided encoder*, *aspect-reconstruction* and *aspect concatenated embedding*. The aspect-guided encoder is specially designed to guide the encoding of a sentence from scratch for conducting the aspect-specific feature selection and extraction at the very beginning stage. The aspect-reconstruction aims to guarantee that the aspect-specific information has been fully embedded in the sentence representation for more accurate predictions. The aspect concatenated embedding part is used to concatenate the aspect embedding and the generated sentence representation so as to make the final prediction.
Aspect-Guided Encoder
---------------------
The aspect-guided encoder is the core module of AGDT, which consists of two key components: Aspect-guided GRU and Transition GRU [@Cho:14].
**A-GRU:** Aspect-guided GRU (A-GRU) is a specially-designed unit for the ABSA tasks, which is an extension of the L-GRU proposed by @Meng:19. In particular, we design an aspect-gate to select aspect-specific representations through controlling the transformation scale of token embeddings at each time step.
At time step $t$, the hidden state $\mathbf{h}_{t}$ is computed as follows: $$\begin{aligned}
\label{eq:gru_h_l}
\mathbf{h}_{t} &= (1 - \mathbf{z}_{t}) \odot \mathbf{h}_{t-1} + \mathbf{z}_{t} \odot \widetilde{\mathbf{h}}_{t}\end{aligned}$$ where $\odot$ represents element-wise product; $\mathbf{z}_{t}$ is the update gate [@Cho:14]; and $\widetilde{\mathbf{h}}_{t}$ is the candidate activation, which is computed as: $$\begin{split}
\label{eq:l_gru_h_f}
\widetilde{\mathbf{h}}_{t} &= \text{tanh}(\mathbf{g}_{t} \odot (\mathbf{W}_{xh}\mathbf{x}_{t}) + \mathbf{r}_{t} \odot (\mathbf{W}_{hh}\mathbf{h}_{t-1})) \\
&\quad+ \mathbf{l}_{t} \odot \textbf{H}_1(\mathbf{x}_{t}) + \mathbf{g}_{t} \odot \textbf{H}_2(\mathbf{x}_{t})
\end{split}$$ $$\begin{aligned}
\nonumber
\widetilde{\mathbf{h}}_{t} &= \text{tanh}(\mathbf{g}_{t} \odot (\mathbf{W}_{xh}\mathbf{x}_{t}) + \mathbf{r}_{t} \odot (\mathbf{W}_{hh}\mathbf{h}_{t-1})) \\
\label{eq:l_gru_h_f}
&\quad+ \mathbf{l}_{t} \odot \textbf{H}_1(\mathbf{x}_{t}) + \mathbf{g}_{t} \odot \textbf{H}_2(\mathbf{x}_{t}) \end{aligned}$$ where $\mathbf{g}_{t}$ denotes the aspect-gate; $\mathbf{x}_{t}$ represents the input word embedding at time step $t$; $\mathbf{r}_{t}$ is the reset gate [@Cho:14]; $\textbf{H}_1(\mathbf{x}_{t})$ and $\textbf{H}_2(\mathbf{x}_{t})$ are the linear transformation of the input $\mathbf{x}_{t}$, and $\mathbf{l}_{t}$ is the linear transformation gate for $\mathbf{x}_{t}$ [@Meng:19]. $\mathbf{r}_{t}$, $\mathbf{z}_{t}$, $\mathbf{l}_{t}$, $\mathbf{g}_{t}$, $\textbf{H}_{1}(\mathbf{x}_{t})$ and $\textbf{H}_{2}(\mathbf{x}_{t})$ are computed as: $$\begin{aligned}
\label{eq:gru_r}
&\mathbf{r}_{t} = \sigma(\mathbf{W}_{xr}\mathbf{x}_{t} + \mathbf{W}_{hr}\mathbf{h}_{t-1})\\
\label{eq:gru_z}
&\mathbf{z}_{t} = \sigma(\mathbf{W}_{xz}\mathbf{x}_{t} + \mathbf{W}_{hz}\mathbf{h}_{t-1}) \\
\label{eq:gru_l}
&\mathbf{l}_{t} \, = \sigma(\mathbf{W}_{xl}\mathbf{x}_{t} + \mathbf{W}_{hl}\mathbf{h}_{t-1}) \\
\label{eq:aspect_gate}
&\mathbf{g}_{t} = \text{relu}(\mathbf{W}_{a}\mathbf{a} + \mathbf{W}_{hg}\mathbf{h}_{t-1}) \\
\label{eq:H_t1}
&\mathbf{\textbf{H}_1(\mathbf{x}_{t})} = \mathbf{W}_{1}\mathbf{x}_{t} \\
\label{eq:H_t2}
&\mathbf{\textbf{H}_2(\mathbf{x}_{t})} = \mathbf{W}_{2}\mathbf{x}_{t}\end{aligned}$$ where “$\mathbf{a}$" denotes the embedding of the given aspect, which is the same at each time step. The update gate $\mathbf{z}_t$ and reset gate $\mathbf{r}_t$ are the same as them in the conventional GRU.
In Eq. (\[eq:l\_gru\_h\_f\]) $\sim$ (\[eq:H\_t2\]), the aspect-gate $\mathbf{g}_{t}$ controls both nonlinear and linear transformations of the input $\mathbf{x}_{t}$ under the guidance of the given aspect at each time step. Besides, we also exploit a linear transformation gate $\mathbf{l}_{t}$ to control the linear transformation of the input, according to the current input $\mathbf{x}_t$ and previous hidden state $\mathbf{h}_{t-1}$, which has been proved powerful in the deep transition architecture [@Meng:19].
As a consequence, A-GRU can control both non-linear transformation and linear transformation for input $\mathbf{x}_{t}$ at each time step, with the guidance of the given aspect, i.e., A-GRU can guide the encoding of aspect-specific features and block the aspect-irrelevant information at the very beginning stage.
**T-GRU:** Transition GRU (T-GRU) [@journals/corr/PascanuGCB13] is a crucial component of deep transition block, which is a special case of GRU with only “state” as an input, namely its input embedding is zero embedding. As in Figure \[fig:DTSA\_aspect\], a deep transition block consists of an A-GRU followed by several T-GRUs at each time step. For the current time step $t$, the output of one A-GRU/T-GRU is fed into the next T-GRU as the input. The output of the last T-GRU at time step $t$ is fed into A-GRU at the time step $t+1$. For a T-GRU, each hidden state at both time step $t$ and transition depth $i$ is computed as: $$\begin{aligned}
\label{eq:t_gru_h}
\mathbf{h}_{t}^i &= (1 - \mathbf{z}_{t}^i) \odot \mathbf{h}_{t}^{i-1} + \mathbf{z}_{t}^i \odot \widetilde{\mathbf{h}}_{t}^i
\\
\label{eq:t_gru_h_}
\widetilde{\mathbf{h}}_{t}^i &= \text{tanh}(\mathbf{r}_{t}^i \odot (\mathbf{W}_{h}^i\mathbf{h}_{t}^{i-1}))\end{aligned}$$ where the update gate $\mathbf{z}_{t}^i$ and the reset gate $\mathbf{r}_{t}^i$ are computed as: $$\begin{aligned}
\label{eq:t_gru_z}
\mathbf{z}_{t}^i &= \sigma(\mathbf{W}_{z}^i\mathbf{h}_{t}^{i-1}) \\
\label{eq:t_gru_r}
\mathbf{r}_{t}^i &= \sigma(\mathbf{W}_{r}^i\mathbf{h}_{t}^{i-1}) \end{aligned}$$
The AGDT encoder is based on deep transition cells, where each cell is composed of one A-GRU at the bottom, followed by several T-GRUs. Such AGDT model can encode the sentence representation with the guidance of aspect information by utilizing the specially designed architecture.
Aspect-Reconstruction
---------------------
We propose an aspect-reconstruction approach to guarantee the aspect-specific information has been fully embedded in the sentence representation. Particularly, we devise two objectives for two subtasks in ABSA respectively. In terms of aspect-category sentiment analysis datasets, there are only several predefined aspect categories. While in aspect-term sentiment analysis datasets, the number of categories of term is more than one thousand. In a real-life scenario, the number of term is infinite, while the words that make up terms are limited. Thus we design different loss-functions for these two scenarios.
For the aspect-category sentiment analysis task, we aim to reconstruct the aspect according to the aspect-specific representation. It is a multi-class problem. We take the softmax cross-entropy as the loss function: $$\label{eq:aspect_soft_loss}
\begin{split}
\mathcal L_{c} &= min (- \sum_{i=0}^{C1}{y}_{i}^{c} \log({p}_{i}^{c}))
\end{split}$$ where C1 is the number of predefined aspects in the training example; ${y}_{i}^{c}$ is the ground-truth and ${p}_{i}^{c}$ is the estimated probability of a aspect.
For the aspect-term sentiment analysis task, we intend to reconstruct the aspect term (may consist of multiple words) according to the aspect-specific representation. It is a multi-label problem and thus the sigmoid cross-entropy is applied: $$\begin{split}
\label{eq:aspect_sigm_loss}
\mathcal L_{t} &= min\{- \sum_{i=0}^{C2} [{y}_{i}^{t} \log({p}_{i}^{t}) \\
& \quad + (1-{y}_{i}^{t})\log(1-{p}_{i}^{t})]\}
\end{split}$$ where C2 denotes the number of words that constitute all terms in the training example, ${y}_{i}^{t}$ is the ground-truth and ${p}_{i}^{t}$ represents the predicted value of a word.
Our aspect-oriented objective consists of $\mathcal L_{c}$ and $\mathcal L_{t}$, which guarantee that the aspect-specific information has been fully embedded into the sentence representation.
Training Objective
------------------
The final loss function is as follows: $$\label{eq:J_loss}
\begin{split}
\boldsymbol{J}&= min( \underline {- \sum_{i=0}^{C}{y}_{i} \log({p}_{i})} +\boldsymbol{\lambda} \mathcal L)
\end{split}$$ where the underlined part denotes the conventional loss function; C is the number of sentiment labels; ${y}_{i}$ is the ground-truth and ${p}_{i}$ represents the estimated probability of the sentiment label; $\mathcal L$ is the aspect-oriented objective, where Eq. \[eq:aspect\_soft\_loss\] is for the aspect-category sentiment analysis task and Eq. \[eq:aspect\_sigm\_loss\] is for the aspect-term sentiment analysis task. And $\boldsymbol{\lambda}$ is the weight of $\mathcal L$.
As shown in Figure \[fig:DTSA\_aspect\], we employ the aspect reconstruction approach to reconstruct the aspect (term), where “softmax” is for the aspect-category sentiment analysis task and “sigmoid” is for the aspect-term sentiment analysis task. Additionally, we concatenate the aspect embedding on the aspect-guided sentence representation to predict the sentiment polarity. Under that loss function (Eq. \[eq:J\_loss\]), the AGDT can produce aspect-specific sentence representations.
Experiments
===========
Datasets and Metrics {#ssec:ExperSet}
--------------------
#### Data Preparation.
We conduct experiments on two datasets of the aspect-category based task and two datasets of the aspect-term based task. For these four datasets, we name the full dataset as “DS". In each “DS", there are some sentences like the example in Table \[tbl:testE\], containing different sentiment labels, each of which associates with an aspect (term). For instance, Table \[tbl:testE\] shows the customer’s different attitude towards two aspects: “[*food*]{}” (“[*The appetizers*]{}") and “[*service*]{}”. In order to measure whether a model can detect different sentiment polarities in one sentence towards different aspects, we extract a hard dataset from each “DS”, named “HDS”, in which each sentence only has different sentiment labels associated with different aspects. When processing the original sentence *$s$* that has multiple aspects *${a}_{1},{a}_{2},...,{a}_{n}$* and corresponding sentiment labels *${l}_{1},{l}_{2},...,{l}_{n}$* (*$n$* is the number of aspects or terms in a sentence), the sentence will be expanded into (s, ${a}_{1}$, ${l}_{1}$), (s, ${a}_{2}$, ${l}_{2}$), ..., (s, ${a}_{n}$, ${l}_{n}$) in each dataset [@Ruder:16; @ruder-etal-2016-hierarchical; @weixueGCAE:18], i.e, there will be *$n$* duplicated sentences associated with different aspects and labels.
#### Aspect-Category Sentiment Analysis.
For comparison, we follow @weixueGCAE:18 and use the restaurant reviews dataset of SemEval 2014 (“restaurant-14”) Task 4 [@Pontiki:14] to evaluate our AGDT model. The dataset contains five predefined aspects and four sentiment labels. A large dataset (“restaurant-large”) involves restaurant reviews of three years, i.e., 2014 $\sim$ 2016 [@Pontiki:14]. There are eight predefined aspects and three labels in that dataset. When creating the “restaurant-large” dataset, we follow the same procedure as in @weixueGCAE:18. Statistics of datasets are shown in Table \[tbl:aspect-category sentiment analysis\].
#### Aspect-Term Sentiment Analysis.
We use the restaurant and laptop review datasets of SemEval 2014 Task 4 [@Pontiki:14] to evaluate our model. Both datasets contain four sentiment labels. Meanwhile, we also conduct a three-class experiment, in order to compare with some work [@Wang:16; @Ma:17; @li2018transformation] which removed “conflict” labels. Statistics of both datasets are shown in Table \[tbl:aspect-term sentiment analysis\].
#### Metrics.
The evaluation metrics are accuracy. All instances are shown in Table \[tbl:aspect-category sentiment analysis\] and Table \[tbl:aspect-term sentiment analysis\]. Each experiment is repeated five times. The mean and the standard deviation are reported.
Implementation Details
----------------------
We use the pre-trained 300d Glove[^4] embeddings [@glove:14] to initialize word embeddings, which is fixed in all models. For out-of-vocabulary words, we randomly sample their embeddings by the uniform distribution $U(-0.25, 0.25)$. Following @Tang:16b [@Chen:17; @Liu:17], we take the averaged word embedding as the aspect representation for multi-word aspect terms. The transition depth of deep transition model is 4 (see Section \[sec:impactofdepth\]). The hidden size is set to 300. We set the dropout rate [@dropout:14] to 0.5 for input token embeddings and 0.3 for hidden states. All models are optimized using Adam optimizer [@Adam:14] with gradient clipping equals to 5 [@DBLP:journals/corr/abs-1211-5063]. The initial learning rate is set to 0.01 and the batch size is set to 4096 at the token level. The weight of the reconstruction loss $\boldsymbol{\lambda}$ in Eq. \[eq:J\_loss\] is fine-tuned (see Section \[sec:impactofloss\]) and respectively set to 0.4, 0.4, 0.2 and 0.5 for four datasets.
--------------------------------------- -------------------- -------------------- -------------------- --------------------
**ATAE-LSTM**[@Wang:16]\* 78.29$\pm$0.68 45.62$\pm$0.90 83.91$\pm$0.49 66.32$\pm$2.28
**CNN**[@DBLP:journals/corr/Kim14f]\* 79.47$\pm$0.32 44.94$\pm$0.01 84.28$\pm$0.15 50.43$\pm$0.38
**GCAE**[@weixueGCAE:18]\* 79.35$\pm$0.34 50.55$\pm$1.83 85.92$\pm$0.27 70.75$\pm$1.19
**AGDT** **81.78**$\pm$0.31 **62.02**$\pm$1.31 **87.55**$\pm$0.17 **75.73**$\pm$0.50
--------------------------------------- -------------------- -------------------- -------------------- --------------------
---------------------------- -------------------- -------------------- -------------------- --------------------
**TD-LSTM**[@Tang:16a]\* 73.44$\pm$1.17 56.48$\pm$2.46 62.23$\pm$0.92 46.11$\pm$1.89
**ATAE-LSTM**[@Wang:16]\* 73.74$\pm$3.01 50.98$\pm$2.27 64.38$\pm$4.52 40.39$\pm$1.30
**IAN**[@Ma:17]\* 76.34$\pm$0.27 55.16$\pm$1.97 68.49$\pm$0.57 44.51$\pm$0.48
**RAM**[@Chen:17]\* 76.97$\pm$0.64 55.85$\pm$1.60 68.48$\pm$0.85 45.37$\pm$2.03
**GCAE**[@weixueGCAE:18]\* 77.28$\pm$0.32 56.73$\pm$0.56 69.14$\pm$0.32 47.06$\pm$2.45
**AGDT** **78.85**$\pm$0.45 **60.33**$\pm$1.01 **71.50**$\pm$0.85 **51.30**$\pm$1.26
---------------------------- -------------------- -------------------- -------------------- --------------------
Baselines
---------
To comprehensively evaluate our AGDT, we compare the AGDT with several competitive models.
**ATAE-LSTM.** It is an attention-based LSTM model. It appends the given aspect embedding with each word embedding, and then the concatenated embedding is taken as the input of LSTM. The output of LSTM is appended aspect embedding again. Furthermore, attention is applied to extract features for final predictions.
**CNN.** This model focuses on extracting n-gram features to generate sentence representation for the sentiment classification.
**TD-LSTM.** This model uses two LSTMs to capture the left and right context of the term to generate target-dependent representations for the sentiment prediction.
**IAN.** This model employs two LSTMs and interactive attention mechanism to learn representations of the sentence and the aspect, and concatenates them for the sentiment prediction.
**RAM.** This model applies multiple attentions and memory networks to produce the sentence representation.
**GCAE.** It uses CNNs to extract features and then employs two Gated Tanh-Relu units to selectively output the sentiment information flow towards the aspect for predicting sentiment labels.
Main Results and Analysis {#ssec:Res}
-------------------------
### Aspect-Category Sentiment Analysis Task {#aspect-category-sentiment-analysis-task .unnumbered}
We present the overall performance of our model and baseline models in Table \[tbl:result\_aspect-category sentiment analysis\]. Results show that our AGDT outperforms all baseline models on both “restaurant-14” and “restaurant-large” datasets. ATAE-LSTM employs an aspect-weakly associative encoder to generate the aspect-specific sentence representation by simply concatenating the aspect, which is insufficient to exploit the given aspect. Although GCAE incorporates the gating mechanism to control the sentiment information flow according to the given aspect, the information flow is generated by an aspect-independent encoder. Compared with GCAE, our AGDT improves the performance by 2.4% and 1.6% in the “DS” part of the two dataset, respectively. These results demonstrate that our AGDT can sufficiently exploit the given aspect to generate the aspect-guided sentence representation, and thus conduct accurate sentiment prediction. Our model benefits from the following aspects. First, our AGDT utilizes an aspect-guided encoder, which leverages the given aspect to guide the sentence encoding from scratch and generates the aspect-guided representation. Second, the AGDT guarantees that the aspect-specific information has been fully embedded in the sentence representation via reconstructing the given aspect. Third, the given aspect embedding is concatenated on the aspect-guided sentence representation for final predictions.
The “HDS”, which is designed to measure whether a model can detect different sentiment polarities in a sentence, consists of replicated sentences with different sentiments towards multiple aspects. Our AGDT surpasses GCAE by a very large margin (+**11.4%** and +**4.9%** respectively) on both datasets. This indicates that the given aspect information is very pivotal to the accurate sentiment prediction, especially when the sentence has different sentiment labels, which is consistent with existing work [@jiang-etal-2011-target; @Ma:17; @WangS:18]. Those results demonstrate the effectiveness of our model and suggest that our AGDT has better ability to distinguish the different sentiments of multiple aspects compared to GCAE.
### Aspect-Term Sentiment Analysis Task {#aspect-term-sentiment-analysis-task .unnumbered}
As shown in Table \[tbl:result\_aspect-term sentiment analysis\], our AGDT consistently outperforms all compared methods on both domains. In this task, TD-LSTM and ATAE-LSTM use a aspect-weakly associative encoder. IAN, RAM and GCAE employ an aspect-independent encoder. In the “DS” part, our AGDT model surpasses all baseline models, which shows that the inclusion of A-GRU (aspect-guided encoder), aspect-reconstruction and aspect concatenated embedding has an overall positive impact on the classification process.
In the “HDS” part, the AGDT model obtains +3.6% higher accuracy than GCAE on the restaurant domain and +4.2% higher accuracy on the laptop domain, which shows that our AGDT has stronger ability for the multi-sentiment problem against GCAE. These results further demonstrate that our model works well across tasks and datasets.
### Ablation Study {#ablation-study .unnumbered}
We conduct ablation experiments to investigate the impacts of each part in AGDT, where the GRU is stacked with 4 layers. Here “AC” represents aspect concatenated embedding , “AG” stands for A-GRU (Eq. (\[eq:gru\_h\_l\]) $\sim$ (\[eq:H\_t2\])) and “AR” denotes the aspect-reconstruction (Eq. (\[eq:aspect\_soft\_loss\]) $\sim$ (\[eq:J\_loss\])).
From Table \[tbl:aspect-category sentiment analysis\_albated\] and Table \[tbl:aspect-term sentiment analysis\_albated\], we can conclude:
1. Deep Transition (DT) achieves superior performances than GRU, which is consistent with previous work [@W17-4710; @Meng:19] ( vs. ).
2. Utilizing “AG” to guide encoding aspect-related features from scratch has a significant impact for highly competitive results and particularly in the “HDS” part, which demonstrates that it has the stronger ability to identify different sentiment polarities towards different aspects. ( vs. ).
3. Aspect concatenated embedding can promote the accuracy to a degree ( vs. ).
4. The aspect-reconstruction approach (“AR”) substantially improves the performance, especially in the “HDS" part ( vs. ).
5. the results in show that all modules have an overall positive impact on the sentiment classification.
### Impact of Model Depth {#sec:impactofdepth .unnumbered}
We have demonstrated the effectiveness of the AGDT. Here, we investigate the impact of model depth of AGDT, varying the depth from 1 to 6. Table \[tbl:model depth\] shows the change of accuracy on the test sets as depth increases. We find that the best results can be obtained when the depth is equal to 4 at most case, and further depth do not provide considerable performance improvement.
### Effectiveness of Aspect-reconstruction Approach {#sec:impactofaspect-reconstruction .unnumbered}
Here, we investigate how well the AGDT can reconstruct the aspect information. For the aspect-term reconstruction, we count the construction is correct when all words of the term are reconstructed. Table \[tbl:accofaspectreconstruction\] shows all results on four test datasets, which shows the effectiveness of aspect-reconstruction approach again.
### Impact of Loss Weight $\boldsymbol{\lambda}$ {#sec:impactofloss .unnumbered}
We randomly sample a temporary development set from the “HDS" part of the training set to choose the lambda for each dataset. And we investigate the impact of $\boldsymbol{\lambda}$ for aspect-oriented objectives. Specifically, $\boldsymbol{\lambda}$ is increased from 0.1 to 1.0. Figure \[fig:cee\] illustrates all results on four “HDS" datasets, which show that reconstructing the given aspect can enhance aspect-specific sentiment features and thus obtain better performances.
### Comparison on Three-Class for the Aspect-Term Sentiment Analysis Task {#comparison-on-three-class-for-the-aspect-term-sentiment-analysis-task .unnumbered}
We also conduct a three-class experiment to compare our AGDT with previous models, i.e., IARM, TNet, VAE, PBAN, AOA and MGAN, in Table \[tbl:aspect-term sentiment analysis3class\]. These previous models are based on an aspect-independent (weakly associative) encoder to generate sentence representations. Results on all domains suggest that our AGDT substantially outperforms most competitive models, except for the TNet on the laptop dataset. The reason may be TNet incorporates additional features (e.g., position features, local ngrams and word-level features) compared to ours (only word-level features).
Analysis and Discussion {#sec:CSV}
=======================
#### Case Study and Visualization.
To give an intuitive understanding of how the proposed A-GRU works from scratch with different aspects, we take a review sentence as an example. As the example “*the appetizers are ok, but the service is slow.*” shown in Table \[tbl:testE\], it has different sentiment labels towards different aspects. The color depth denotes the semantic relatedness level between the given aspect and each word. More depth means stronger relation to the given aspect.
Figure \[fig:gate\_vis\] shows that the A-GRU can effectively guide encoding the aspect-related features with the given aspect and identify corresponding sentiment. In another case, “*overpriced Japanese food with mediocre service.*”, there are two extremely strong sentiment words. As the above of Figure \[fig:gate\_w\_r\] shows, our A-GRU generates almost the same weight to the word “[*overpriced*]{}” and “[*mediocre*]{}”. The bottom of Figure \[fig:gate\_w\_r\] shows that reconstructing the given aspect can effectively enhance aspect-specific sentiment features and produce correct sentiment predictions.
#### Error Analysis.
We further investigate the errors from AGDT, which can be roughly divided into 3 types. **1)** The decision boundary among the sentiment polarity is unclear, even the annotators can not sure what sentiment orientation over the given aspect in the sentence. **2)** The “conflict/neutral” instances are extremely easily misclassified as “positive” or “negative”, due to the imbalanced label distribution in training corpus[^5]. **3)** The polarity of complex instances is hard to predict, such as the sentence that express subtle emotions, which are hardly effectively captured, or containing negation words (e.g., [*never*]{}, [*less*]{} and [*not*]{}), which easily affect the sentiment polarity.
Related Work
============
#### Sentiment Analysis.
There are kinds of sentiment analysis tasks, such as document-level [@thongtan-phienthrakul-2019-sentiment], sentence-level[^6] [@zhang-zhang-2019-tree; @zhang-etal-2019-latent], aspect-level [@Pontiki:14; @wang-etal-2019-aspect] and multimodal [@8052551; @akhtar2019multi] sentiment analysis. For the aspect-level sentiment analysis, previous work typically apply attention mechanism [@D15-1166] combining with memory network [@Jason:WestonCB14] or gating units to solve this task [@Tang:16b; @he-etal-2018-effective; @Huang:18:PCNN:b; @weixueGCAE:18; @duan-etal-2018-learning-sentence; @Tang:ACL2019; @yang2019aspect; @bao-etal-2019-attention], where an aspect-independent encoder is used to generate the sentence representation. In addition, some work leverage the aspect-weakly associative encoder to generate aspect-specific sentence representation [@Tang:16a; @Wang:16; @Majumder:18]. All of these methods make insufficient use of the given aspect information. There are also some work which jointly extract the aspect term (and opinion term) and predict its sentiment polarity [@schmitt-etal-2018-joint; @DBLP:journals/corr/abs-1811-05082; @ma-etal-2018-joint; @DBLP:journals/corr/abs-1808-08858; @he_acl2019; @Luo2019doer; @hu2019open; @song2019; @Wang:2019:ASA:3308558.3313750]. In this paper, we focus on the latter problem and leave aspect extraction [@DBLP:journals/corr/ShuXL17] to future work. And some work [@bert1; @bert2; @he-etal-2018-exploiting; @Xu:18; @chen-qian-2019-transfer; @he_acl2019] employ the well-known BERT [@bert] or document-level corpora to enhance ABSA tasks, which will be considered in our future work to further improve the performance.
#### Deep Transition.
Deep transition has been proved its superiority in language modeling [@journals/corr/PascanuGCB13] and machine translation [@W17-4710; @Meng:19]. We follow the deep transition architecture in @Meng:19 and extend it by incorporating a novel A-GRU for ABSA tasks.
Conclusions {#sec:conclusions}
============
In this paper, we propose a novel aspect-guided encoder (AGDT) for ABSA tasks, based on a deep transition architecture. Our AGDT can guide the sentence encoding from scratch for the aspect-specific feature selection and extraction. Furthermore, we design an aspect-reconstruction approach to enforce AGDT to reconstruct the given aspect with the generated sentence representation. Empirical studies on four datasets suggest that the AGDT outperforms existing state-of-the-art models substantially on both aspect-category sentiment analysis task and aspect-term sentiment analysis task of ABSA without additional features.
Acknowledgments {#acknowledgments .unnumbered}
===============
We sincerely thank the anonymous reviewers for their thorough reviewing and insightful suggestions. Liang, Xu, and Chen are supported by the National Natural Science Foundation of China (Contract 61370130, 61976015, 61473294 and 61876198), and the Beijing Municipal Natural Science Foundation (Contract 4172047), and the International Science and Technology Cooperation Program of the Ministry of Science and Technology (K11F100010).
[^1]: The code is publicly available at: <https://github.com/XL2248/AGDT>
[^2]: Work was done when Yunlong Liang was an intern at Pattern Recognition Center, WeChat AI, Tencent Inc, China.
[^3]: Jinan Xu is the corresponding author.
[^4]: Pre-trained Glove embeddings can be obtained from <http://nlp.stanford.edu/projects/glove/>
[^5]: More details can be seen in the dataset or see here: <http://alt.qcri.org/semeval2014/>
[^6]: <https://nlp.stanford.edu/sentiment/>
|
---
abstract: 'We present an analysis of the quantum state resulting from the dissociation of diatomic molecules prepared in a condensate vortex state. The many-body state preserves the rotational symmetry of the system in quantum correlated states by having two equally populated components with angular momentum adding to unity. A simple two-mode analysis and a full quantum field analysis is presented for the case of non-interacting atoms and weak depletion of the molecular condensate.'
author:
- 'Uffe V. Poulsen and Klaus Mølmer'
title: Pair correlated atoms with a twist
---
Introduction
============
Since the 1995 experiments with the first production of atomic Bose-Einstein condensates, degenerate quantum gasses have constituted a very active field of research. A wide variety of means exists for detection and control of the properties of these systems, and the early works have been followed by progress on degenerate fermionic systems, on mixtures of different species, and on conversion between atomic and molecular quantum gasses.
The coherence properties of the degenerate quantum states have been verified implicitly by measurements of the response properties of the systems and explicitly by the observation of robust interferences [@andrews97:_obser_inter_between_two_bose_conden] and topological structures [@madison00:_vortex_format_stirr_bose_einst_conden]. Recently, a vortex lattice in a system of fermionic atoms was shown to extend over the BCS-BEC crossover towards formation of bosonic diatomic molecules on the molecular side of a field controlled Feschbach resonance [@zwierlein05:_vortices_fermi].
Many properties of condensates are well described by mean field theories and the Gross-Pitaevskii equation, but in some cases mean field theories may completely fail and totally forbid processes, which occur peacefully according to a full quantum state analysis. Important examples of such processes are the period-doubling observed in a shaken lattice [@gemelke05:_param_amp_matter_waves] and the break-up of a moving condensate in a lattice due to four wave mixing [@campbell06:_param_amp_atom_pairs]. Both processes are driven by collisions of pairs of condensate atoms which emerge in two new momentum states. Within a mean field theory, these processes, like the equivalent parametric amplification process in optical down-conversion and four wave mixing will only get initiated if a non-vanishing mean field with the final state character is seeded to the solution of the Gross-Pitaevskii equation [@hilligsoe05:_four_wave_mixing]. From a formal perspective, the fact that only the sum of the phases of the two final state components and not the individual phases are locked to the initial state wave functions causes the fields to remain in the vacuum state in order not to break the phase symmetry. If the physical problem is simple, e.g., in the few-mode quantum optical problems, one may alternatively have recourse to a full quantum many-body theoretical analysis. In the context of quantum gasses, we have previously [@poulsen01:_quant_states_bose_einst] studied the “degenerate down-conversion” of a molecular condensate by dissociation to a single trapped atomic condensate in such a full quantum treatment and shown that despite the absence of an atomic condensate phase, the atomic component does indeed accumulate in a single preferred quantum state. Dissociation into two atomic beams have smilarly been proposed as a means for generation of Einstein-Podolsky-Rosen correlations [@kheruntsyan05:_epr_dis_mol_bec].
Note that in four-wave mixing processes a mean-field solution not only breaks the phase symmetry of the field, it also breaks the translational invariance of the problem as dramatically observed as period doubling in [@gemelke05:_param_amp_matter_waves], and likewise the different populated momentum components in [@campbell06:_param_amp_atom_pairs] will have spatial interference patterns with periods exceeding the one of the lattice potential used in the experiments. In this paper, we consider a process where the symmetry breaking is similarly spectacular, namely the down conversion by photo dissociation of a molecular condensates which is prepared in a single vortex state. In a cylindrically shaped trap, a vortex state is a topologically stable state of a quantum gas with a vanishing density along the cylinder axis and a phase which changes by $2\pi$ as one follows a closed loop around the axis. The rotational symmetry around the $z$ axis is a fundamental property of the many-body Hamiltonian and both mean field solutions and more elaborate theories must respect this symmetry. Microscopically, the vortex solution is consistent with a Hartree product state description with a product of single particle $M=1$ eigenstates of the azimuthal angular momentum. In this state, every molecule has a unit angular momentum around the condensate axis, which cannot be transferred to a pair of atoms populating the same ($m=1/2$ ?) quantum state. Conversely, any mean field solution for the atomic system will necessarily correspond to an integer angular momentum per atom, and would hence imply an even angular momentum of every pair, i.e, of the molecule.
The molecules must dissociate to at least two different atomic states with azimuthal quantum numbers which add to the molecular value, $m+m'=M$. The easiest situation to deal with is one where only two states with angular momentum $m=0$ and $m'=1$ get populated. We shall address this case in Sec. II. In particular we shall discuss to which extent a strongly number correlated quantum state of this kind state is distinguishable from a state which macroscopically populates an even weight superposition of the same two single particle states. The restriction to only two finally occupied states should be justified by a more elaborate treatment of the many-body Hamiltonian, and in Sec. III, we model the actual process in which the atoms are not put directly into two states extending over vast and partly non-overlapping regions of the atomic single particle quantum states, but where the molecular dissociation enforces a localized common origin of the matter waves. The problem is solved exactly by a Bogoliubov tranformation, and we identify different regimes and the validity of the few-mode Ansatz.
Two-mode analysis {#sec:two-mode_analysis}
=================
For simplicity we ignore interaction between atoms, between molecules, and between atoms and molecules. We also make the assumption that only one quantum state for the molecules and two quantum states for the atoms are relevant. The molecule state should be the macroscopically occupied state of the molecular condensate. The two atomic states are assumed to be selected by a resonance condition: They could for example be the two lowest 2D single particle eigenfunctions in a harmonic trap \[cf. Eq. (\[eq:def\_phi\_nmk\]) below\], $\Phi_{00}(x,y)\propto
\exp(-\rho^2/2{\ensuremath{a_\text{osc}}}^2),\Phi_{01}(x,y)\propto
(x+iy)\exp(-\rho^2/2{\ensuremath{a_\text{osc}}}^2)$, where $\rho^2=x^2+y^2$, and where the oscillator length ${\ensuremath{a_\text{osc}}}=\sqrt{\hbar/{\ensuremath{m_\text{atm}}}\omega}$ is defined in terms of the atomic mass ${\ensuremath{m_\text{atm}}}$ and the trap frequency $\omega$. The lowest state $\Phi_{00}$ has angular momentum $m=0$, while $\Phi_{01}$ has $m'=1$ so that they fulfill the condition for angular momentum conservation when molecules with $M=1$ are dissociated, $m+m'=M$.
If we can drive the system exactly at resonance, i.e., either the atomic and molecular states are degenerate, or in a laser induced dissociation process, the field frequencies involved exactly fulfill the Bohr frequency condition for quantum transitions, our model Hamiltonian becomes $$\label{Hsim}
\hat{H}
=
\beta\, \hat{c} \hat{a}_0^\dagger \hat{a}_1^\dagger
+\beta^*\, \hat{c}^\dagger \hat{a}_0\hat{a}_1
.$$ Here $\beta$ quantifies the strength of the microscopic coupling, the operator $\hat{c}$($\hat{c}^\dagger$) annihilates (creates) a molecule, and the operators $\hat{a}_0,\hat{a}_1$($\hat{a}_0^\dagger,\hat{a}_1^\dagger$) annihilate (create) atoms in the two atomic states. This Hamiltonian is well known in quantum optics where it describes the non-degenerate optical parametric oscillator. In general it leads to a complicated entanglement between the pump beam ($\hat{c}$) and the signal and idler beams $\hat{a}_0,\hat{a}_1$ which can of course be made subject to detailed investigation [@dechoum04:_nondegen_para_oscillator], but in the limit of a strong pump, the depletion can often be ignored and the pump operators can be replaced by c-numbers. We will assume here that the molecular condensate is sufficiently large that depletion can be neglected, and hence we shall consider the simpler Hamiltonian $$\label{Hsimple}
\hat{H}
=
\chi \hat{a}_0^\dagger \hat{a}_1^\dagger
+\chi^* \hat{a}_0\hat{a}_1$$ The coupling strength $\chi$ now includes the c-number describing the molecular field. Note that the quadratic Hamiltonian (\[Hsimple\]) does not conserve the number of atoms, and the final state appears to be a coherent superposition of states with different numbers of atoms. This is, however, only because we omit the meticulous writing of the associated molecular components of the states, which precisely account for the conservation of atom numbers. As we shall only be interested in number conserving observables such as the atomic density distribution, we shall make no errors in applying the symmetry breaking Hamiltonian (\[Hsimple\]).
Starting in a state with no atoms, the atomic vacuum state, in both modes, the time evolution leads to the production of the two-mode state $$\label{squeezed}
|\Psi\rangle
= \sqrt{1-|s|^2}
\sum_n s^n
|n,n\rangle,$$ where $$\label{eq:def_s}
s = -i\frac{\chi}{|\chi|} \tanh(|\chi| t/\hbar).$$ As discussed in detail in the Introduction, this state does not follow from a mean field analysis and, indeed, the mean values of the field operators $\hat{a}_i$ vanish exactly, whereas the state has a mean atom number of $$\label{eq:N_of_t}
\langle \hat{N}_i \rangle
=
\langle \hat{a}_i^\dagger \hat{a}_i \rangle
=
|s|^2/(1-|s|^2)
=
\sinh^2(|\chi|t/\hbar)$$ in each mode. The atom number distributions are exponential (thermal) and thus the fluctuations in the atom number are large, $\mathrm{Var}( \hat{N}_i)=\langle \hat{N}_i\rangle^2+\langle
\hat{N}_i\rangle$.
A number of papers have discussed observational difference between systems populating several single particle states macroscopically and a coherent superposition of the same states. It has been argued [@javanainen96:_quant_phase_bose_einst_conden; @castin97:_relat_bose_einst; @moelmer97:_opt_coh_fiction; @cirac96:_contin_bose] that such differences are small or insignificant for systems with many atoms. The general argument was supplemented by simulations of actual detection records, where the back action due to local measurements on the system turned out precisely to establish a definite relative phase of the two components.
The restriction to only two modes implies that the one-body density matrix is fully characterized by the 2x2 matrix with $\rho_{ij}=\langle a^\dagger_i a_j\rangle$. In particular the spatial density is given by $$\label{dens}
\begin{split}
n(x,y)
&=
\rho_{00}|\Phi_{00}(x,y)|^2
+\rho_{11}|\Phi_{01}(x,y)|^2
\\
&\phantom{=}+2{\mathrm{Re}}[\rho_{01}\Phi_{00}(x,y)\Phi_{01}^*(x,y)].
\end{split}$$ Registration of a particle at position $(x_d,y_d)$, chosen according to this probability distribution, causes the application of the annihilation operator $\Phi_{00}(x_d,y_d)\hat{a}_0+\Phi_{01}(x_d,y_d)\hat{a}_1$ on the state vector expanded in the two mode number components as in (3), followed by a renormalization. We are thus able to compute the up-dated 2x2 density matrix and by repeating these steps to simulate the subsequent detection of a number $k$ of atoms. The present problem differs in two way from the simulations reported in Refs. [@javanainen96:_quant_phase_bose_einst_conden; @moelmer97:_opt_coh_fiction]: the two modes populated are not plane waves but they have different spatial dependencies, implying that some atomic detection events can be ascribed solely to one component (e.g., only the $m=0$ component contributes to the atomic density on the vortex axis), and the inital state does not have a well defined number of atoms, but much larger than Poissonian fluctuations, and hence the rigid mathematical analysis of the emergence of a relative phase in Ref. [@castin97:_relat_bose_einst] does not apply. We have carried out simulations, and despite these two differences, the 2x2 density matrix indeed approaches a pure state projector and we obtain detection patterns that are compatible with the density of a single coherent quantum state.
Multi-mode analysis {#sec:many_mode_analysis}
===================
Let us now analyze the problem without making the two-mode simplification. Retaining the approximation of replacing molecule operators with c-numbers, a full many-mode version of Eq.(\[Hsimple\]) reads $$\label{eq:H_many_modes}
\begin{split}
H =& \int\!d^3\!r\; \hat\Psi^\dagger({\ensuremath{\mathbf{r}}})h({\ensuremath{\mathbf{r}}})\hat\Psi({\ensuremath{\mathbf{r}}})
\\
&+ \left\{ \int\!d^3\!rd^3\!r'\;
\chi({\ensuremath{\mathbf{r}}},{\ensuremath{\mathbf{r}}}')\hat\Psi^\dagger({\ensuremath{\mathbf{r}}})\hat\Psi^\dagger({\ensuremath{\mathbf{r}}}') +
\text{h.c.} \right\} .
\end{split}$$ The single particle part consists of kinetic energy and external trapping potential $$\label{eq:def_h}
h({\ensuremath{\mathbf{r}}})=\frac{{\ensuremath{\mathbf{p}}}^2}{2{\ensuremath{m_\text{atm}}}} + V_\mathrm{ex}({\ensuremath{\mathbf{r}}})-\Delta_\text{bare}
.$$ The energy offset $\Delta_\text{bare}$ is half of the two-atom detuning: the energy of a trapped molecule in the vortex state is the energy of two free atoms plus $2\Delta_\text{bare}$.
Typical traps are well approximated by a harmonic potential $V_\mathrm{ex}({\ensuremath{\mathbf{r}}})={\ensuremath{m_\text{atm}}}(\omega_x^2 x^2+\omega_y^2 y^2 + \omega_z^2
z^2)/2$. We will consider situations with axial symmetry and we therefore let $\omega_x=\omega_y=\omega$. For the purpose of discussing the dissociation of a molecular condensate with a vortex along $z$, it is useful to make one of two simplifying assumptions about the dynamics along $z$:
- A quasi cylindrical situation with $\omega_z \ll
\omega_\perp$ such that the $z$ component of the momentum, $p_z$, becomes approximately conserved.
- A quasi two-dimensional situation where $\omega_z \gg
\omega_\perp$ so that the dynamics along $z$ is effectively frozen to take place in a single quantum state.
We focus on the second case in the following. The modifications needed to treat the first case are briefly discussed in the Appendix.
The Hamiltonian (\[eq:H\_many\_modes\]) can describe many processes where pairs of particles are created: one must choose the correct coupling kernel $\chi({\ensuremath{\mathbf{r}}},{\ensuremath{\mathbf{r}}}')$ for the problem under consideration. When dealing with dissociation of molecules, $\chi({\ensuremath{\mathbf{r}}},{\ensuremath{\mathbf{r}}}')$ depends on the wavefunction of individual molecules (both relative motion and center-of-mass), and on the possible spatial variation of the external fields mediating the dissociation. We will assume the relative motion molecular wavefunction to be very well localized compared to other length scales in the problem. This means that the dependence of $\chi({\ensuremath{\mathbf{r}}},{\ensuremath{\mathbf{r}}}')$ on the separation ${\ensuremath{\mathbf{r}}}-{\ensuremath{\mathbf{r}}}'$ can be approximated by a delta-function. We further assume that the center-of-mass molecular wavefunction and the relevant external fields are rotationally symmetric around the $z$-axis so that $$\label{eq:chi_delta}
\chi({\ensuremath{\mathbf{r}}},{\ensuremath{\mathbf{r}}}')
=
\delta({\ensuremath{\mathbf{r}}}-{\ensuremath{\mathbf{r}}}'){\ensuremath{\chi_\mathrm{cm}}}(\rho,z)e^{iM\phi}
,$$ in usual cylindrical coordinates where $x=\rho\cos\phi$ and $y=\rho\sin\phi$. In the simplest case the dissociating external fields are spatially uniform over the extend of the molecular condensate. Then $M$ is the charge of the molecular vortex state and the remaining spatial variation of ${\ensuremath{\chi_\mathrm{cm}}}(\rho,z)$ is simply given by the norm of the c-number field that describes the molecular condensate (squareroot of the density) [^1]. To specify the overall strength of the coupling a microscopic model of the dissociation must be decided upon. In Ref. [@heinzen00:_super], a two-photon Raman process between a bound molecular state and “free” (except for external trapping) atoms is considered. The dissociation is via an excited molecular state, which should ideally be only negligibly populated since spontaneous decay from it will be an unwanted loss mechanism. The particular example in Ref. [@heinzen00:_super] shows that a peak value of our $\chi$ comparable to the trap level spacing is fully realistic. This is the regime we focus on below.
We emphasize that this physical modelling does not favor any particular pair of atomic modes, and the Hamiltonian (\[eq:H\_many\_modes\]) cannot be brought on the form (\[Hsimple\]). Atoms are coherently prepared from the entire region populated by the molecules, but pairs are initially much more tightly located both radially and azimuthally than the single mode functions suggested in the previous section. In particular, the atoms are not restricted to low values of the angular momentum quantum number. The temporal evolution of the system due to the kinetic energy and external potential, and the role of energy conservation, however, favors the population of only few modes as we shall see below.
Frozen $z$ dynamics {#sec:frozen_z}
-------------------
In the case of tight $z$ confinement for the atoms, we get a simple description. Let the only accessible atomic $z$ mode be $\phi_z$, $\int dz |\phi_z|^2=1$. We can then expand the atomic field operators on the discrete set of mode operators $$\label{eq:Psi_one_z}
\hat\Psi({\ensuremath{\mathbf{r}}})
=
\sum_{nm} \hat{a}_{nm}
\times
\Phi_{nm}(\rho)
\times
\sqrt{\frac{1}{2\pi}} e^{im\phi}
\times
\phi_z(z)$$ where the radial modefunction $\Phi_{nm}$ are $$\label{eq:def_phi_nmk}
\begin{split}
\Phi_{nm}&(\rho) = \sqrt{\frac{n!}{2\left( n+|m|
\right)!{\ensuremath{a_\text{osc}}}^{2}}}
\\
&\times
\exp\left(-\frac{1}{2}\frac{\rho^2}{{\ensuremath{a_\text{osc}}}^2}\right)
\left(\frac{\rho}{{\ensuremath{a_\text{osc}}}}\right)^{|m|}
L^{|m|}_n\left(\frac{\rho^2}{{\ensuremath{a_\text{osc}}}^2}\right) ,
\end{split}$$ with $L^m_n$ the $m$’te associated Laguerre polynomum of order $n$ and ${\ensuremath{a_\text{osc}}}=\sqrt{\hbar/{\ensuremath{m_\text{atm}}}\omega}$ is the oscillator length. The commutation relations of the $\hat{a}_{nm}$ are $$\label{eq:froz_rad_commu}
\begin{split}
\Bigl[\hat{a}_{nm},\hat{a}_{n'm'}\Bigr]
&=
0
\\
\Bigl[\hat{a}_{nm},\hat{a}^\dagger_{n'm'}\Bigr]
&=
\delta_{nn'}\delta_{mm'}
.
\end{split}$$ The Hamiltonian becomes $$\begin{gathered}
\label{eq:H_frozen}
\hat{H} = \sum_{nm} E_{nm} \hat{a}^\dagger_{nm}\hat{a}_{nm}
\\
+ \sum_{m} \sum_{nn'} \left\{ K_{nn'm}
\hat{a}^\dagger_{nm}\hat{a}^\dagger_{n'(M-m)} +\text{h.c.}
\right\} ,\end{gathered}$$ where $K_{nn'm}$ is defined via$$\label{eq:def_K_nnm}
\begin{split}
&K_{nn'm}
=
\\
&\int\! d\!z d\!\rho \rho \;
\chi_\mathrm{cm}(\rho,z) \;
|\phi(z)|^2 \;
\Phi^*_{nm}(\rho) \Phi_{n'M-m}(\rho)
.
\end{split}$$ and $E_{nm}$ is given by $$\label{eq:E_nm}
E_{nm}=\left(2n+|m|+1\right)\hbar\omega-\Delta
.$$ Here $\Delta$ is the effective detuning, adjusted for the energy associated with $\phi_z$: $$\label{eq:def_Delta}
\Delta=\Delta_\text{bare}-\int \!dz\;
\phi_z^*\left(\frac{p_z^2}{2m}+\frac{1}{2}m\omega_z^2z^2\right)\phi_z
.$$
Bogoliubov diagonalization {#sec:bogo_diag}
==========================
The Hamiltonian (\[eq:H\_frozen\]) is a quadratic form of creation and annihilation operators and as such it can in general be decoupled to a collection of independent harmonic oscillators by a *Bogoliubov transformation*. Please note that this transformation is usually applied in connection with the *Bogoliubov approximation* in theoretical studies of Bose-Einstein condensed systems. The Bogoliubov approximation provides a quadratic Hamiltonian by appeal to the macroscopic population of the condensate mode. Here the quadratic form of the Hamiltonian is rather a consequence of (i) that two atoms are created in each fundamental dissociation process and (ii) that we ignore the dynamics (in particular the depletion) of the molecules and describe them by a stationary c-number field. One could in principle extend our approach to also treat the interaction among the created atoms in a Bogoliubov approximation, but since we focus on situations with initially *no* atoms present this would neither be necessary nor indeed well justified. For an example of *time-dependent* quadratic Hamiltonians, see the work by Ziń *et al*. on colliding condensates [@zin05:_q_multimode_elast_scat; @zin06:_elast_scat_bec].
Two-mode Bogoliubov {#sec:two_mode_bogo}
-------------------
Let us first look at the simple case of two modes like in Sec. \[sec:two-mode\_analysis\]. In addition to the pair creation and annihilation terms of Eq. (\[Hsimple\]) we include mode energies $\hbar\omega_0$ and $\hbar\omega_1$: $$\label{eq:Hsimple_delta}
\begin{split}
\hat{H}
&=
\hbar\omega_0 \left( \hat{a}_0^\dagger \hat{a}_0+\frac{1}{2}\right)
+ \hbar\omega_1 \left(\hat{a}_1^\dagger \hat{a}_1+\frac{1}{2}\right)
\\
&\phantom{=}
+\chi \hat{a}_0^\dagger \hat{a}_1^\dagger+\chi^* \hat{a}_0 \hat{a}_1
.
\end{split}$$ A useful and compact notation is $$\label{eq:H_simple_compact}
\hat{H}
=
\frac{1}{2}\hat{A}^\dagger \mathrm{h} \hat{A}$$ with $$\label{eq:def_A}
\hat{A}
=
\begin{bmatrix}
\hat{a}_0 \\
\hat{a}_1 \\
\hat{a}_0^\dagger \\
\hat{a}_1^\dagger
\end{bmatrix}
,
\quad
A^\dagger
=
\begin{bmatrix}
\hat{a}_0^\dagger &
\hat{a}_1^\dagger &
\hat{a}_0 &
\hat{a}_1
\end{bmatrix}
,$$ and $$\label{eq:def_h_rm}
\mathrm{h}
=
\begin{bmatrix}
\hbar\omega_0 & 0 & 0 & \chi \\
0 & \hbar\omega_1 & \chi & 0 \\
0 & \chi^* & \hbar\omega_0 &0 \\
\chi^* & 0 & 0 & \hbar\omega_1
\end{bmatrix}
.$$ We now seek to simplify $\hat{H}$ by a Bogoliubov transformation, i.e. we define new creation and annihilation operators implicitly by $$\label{eq:def_bogo}
\begin{bmatrix}
\hat{a}_0 \\
a_1 \\
\hat{a}_0^\dagger \\
a_1^\dagger
\end{bmatrix}
=
\begin{bmatrix}
U & V^* \\
V & U^*
\end{bmatrix}
\begin{bmatrix}
\hat{b}_0 \\
\hat{b}_1 \\
\hat{b}_0^\dagger \\
\hat{b}_1^\dagger
\end{bmatrix}$$ where the 2$\times$2 matrices $U$ and $V$ should fulfill $$\label{eq:props_UV}
\begin{split}
U^\dagger U-V^\dagger V&={\mathbb{I}}_2
\\
U^\mathrm{T}V-V^\mathrm{T}U&=0
\end{split}$$ in order for $\hat{b}_0$ and $\hat{b}_1$ to have the commutation relations for independent bosonic creation and annihilation operators \[see Eq. (\[eq:froz\_rad\_commu\])\]. Note that in general the Bogoliubov transform is *not* simply a choice of different spatial modes.
### Detuning dominated case $2|\chi|<|\hbar\omega_0+\hbar\omega_1|$ {#sec:detuned}
Depending on the strength of the coupling $|\chi|$ relative to the two-atom detuning $|\hbar\omega_0+\hbar\omega_1|$ the Hamiltonian can be written in one of two standard forms. If $2|\chi|<|\hbar\omega_0+\hbar\omega_1|$, $\hat{H}$ can be written as a sum of two independent oscillators $$\label{eq:H_real}
\hat{H}
=
\lambda_0 \left(
\hat{b}_0^\dagger \hat{b}_0 + \frac{1}{2}
\right)
+\lambda_1 \left(
\hat{b}_1^\dagger \hat{b}_1 + \frac{1}{2}
\right)
,$$ with $$\label{eq:lambdas_real}
\begin{split}
\lambda_{0,1}
=&
\pm\frac{1}{2}(\hbar\omega_0-\hbar\omega_1)
\\
& +
\frac{1}{2}(\hbar\omega_0+\hbar\omega_1)
\sqrt{1-\frac{4|\chi|^2}{|\hbar\omega_0+\hbar\omega_1|^2}}
\end{split}$$ The Bogoliubov transformation is given by: $$\label{eq:U_decoupled}
U
=
\begin{bmatrix}
\cosh r & 0 \\
0 & \cosh r
\end{bmatrix}$$ and $$\label{eq:V_decoupled}
V
=
\begin{bmatrix}
0 & \mp \frac{\chi^*}{|\chi|} \sinh r \\
\mp\frac{\chi^*}{|\chi|}\sinh r & 0
\end{bmatrix}
,$$ where the upper (lower) sign should be chosen for $\hbar\omega_0+\hbar\omega_1$ positive (negative). The squeezing parameter $r$ is defined through $$\label{eq:r_decoupled}
\tanh r =
\frac{2|\chi|}{|\hbar\omega_0+\hbar\omega_1|
+\sqrt{|\hbar\omega_0+\hbar\omega_1|^2-4|\chi|^2}}$$ We see that as $2|\chi|$ approaches $|\hbar\omega_0+\hbar\omega_1|$, the coefficients in the transformation diverge, i.e. the decoupled modes become infinitely squeezed.
Because of the diagonalized form of the Hamiltonian (\[eq:H\_real\]), the time evolution will be independent for the two Bogoliubov modes: Each will simply behave as a harmonic oscillator. This means that the dynamics of all quantities will be oscillatory with two fundamental frequencies given by $\lambda_{0,1}/\hbar$.
Note, that even if both mode energies $\hbar\omega_{0,1}$ are positive so that a pair of atoms actually has a higher energy than a molecule, one of the eigenvalues $\lambda_i$ can become negative. This happens if $|\chi|^2> \hbar\omega_0\hbar\omega_1$ and in that case the system will be thermodynamically unstable as it is energetically favourable to create the corresponding kind of quasi-particles.
### Coupling dominated case $2|\chi|>|\hbar\omega_0+\hbar\omega_1|$ {#sec:coupled}
If $2|\chi|>|\hbar\omega_0+\hbar\omega_1|$, the simplest form attainable by Bogoliubov transformations involves pair-creation (with real positive coefficient) into two symmetrically detuned modes: $$\label{eq:H_complex}
\hat{H}
=
{\mathrm{Re}}[\lambda] \left( \hat{b}_1^\dagger \hat{b}_1
- \hat{b}_0^\dagger \hat{b}_0 \right)
+{\mathrm{Im}}[\lambda]\left( \hat{b}_0^\dagger \hat{b}_1^\dagger
+ \hat{b}_0 \hat{b}_1 \right)$$ with $$\label{eq:lambdas_complex}
\begin{split}
\lambda
=&
\frac{1}{2}(\hbar\omega_1-\hbar\omega_0)
\\
&+ \frac{i}{2}(\hbar\omega_0+\hbar\omega_1)
\sqrt{\frac{4|\chi|^2}{|\hbar\omega_0+\hbar\omega_1|^²}-1}
.
\end{split}$$ The transformation is still of the form given in Eqs. (\[eq:U\_decoupled\]) and (\[eq:V\_decoupled\]), but now the squeezing parameter $r$ should be found from: $$\label{eq:r_pairs}
\tanh r =
\frac{|\hbar\omega_0+\hbar\omega_1|}
{2|\chi|+\sqrt{4|\chi|^2-|\hbar\omega_0+\hbar\omega_1|^2}}$$ We see that as before, $r$ diverges when $2|\chi|$ approaches $|\hbar\omega_0+\hbar\omega_1|$. At exact two-*atom* resonance, i.e. $\hbar\omega_0+\hbar\omega_1=0$, the Bogoliubov operators are simply the original $\hat{a}_0$ and $\hat{a}_1$.
The transformed Hamiltonian (\[eq:H\_complex\]) consist of two commuting terms and the first one even conserves the number of quasi-particles $\hat{N}_b=\hat{b}^\dagger_0 \hat{b}_0 +
\hat{b}^\dagger_1 \hat{b}_1$. The second term leads to unbound creation of quasi-particle pairs, in fact $\langle \hat{N}_b\rangle$ will grow as $\sinh^2 ({\mathrm{Im}}[\lambda] t)$. This production of quasi-particles translates to a similarly unbounded growth in the number of atoms (until depletion of the molecular condensate renders our simple Hamiltonian invalid), and because of the Bogoliubov tranformation, the atom numbers will show oscillations around their general growth.
Multi-mode Bogoliubov {#sec:many_modes_bogo}
---------------------
The generalization to many modes is relatively straightforward. By Bogoliubov transformations of the original set of creation and annihilation operators, $\hat{H}$ can be written as a sum of a number of independent oscillators and a number of pairs of symmetrically detuned modes into which quasi-particles are created: $$\label{eq:H_many_modes_diag}
\begin{split}
\hat{H}
= &
\sum_{j} \lambda_j
\left(
\hat{b}j^\dagger \hat{b}_j + \frac{1}{2}
\right)
\\
&+
\sum_{p} {\mathrm{Re}}[\lambda_p]
\left(
\hat{b}_{p2}^\dagger \hat{b}_{p2}
- \hat{b}_{p1}^\dagger \hat{b}_{p1}
\right)
\\
&\phantom{+\sum_{p}}
+{\mathrm{Im}}[\lambda_p]
\left(
\hat{b}_{p1}^\dagger \hat{b}_{p2}^\dagger
+ \hat{b}_{p1} \hat{b}_{p2}
\right) .
\end{split}$$ The dynamics in the decoupled modes is oscillatory while ever more quasi-particles will be created in the paired modes. Note that it is perfectly possible that all modes can be decoupled so that the unitary dynamics is purely oscillatory.
Consequences of the azimuthal symmetry {#sec:symmetries}
--------------------------------------
We consider a situation with axial symmetry: The single particle part of the Hamiltonian, $h({\ensuremath{\mathbf{r}}})$ of Eq. (\[eq:def\_h\]), is invariant under rotations around the $z$-axis. This leads to the usual block-diagonal form of the single particle Hamiltonian with respect to the quantum number $m$, the eigenvalue of the $z$-component of the angular momentum operator. At the same time, the molecular field \[$\chi$ of Eq. (\[eq:chi\_delta\])\] changes simply by a phase factor $\exp(iM\phi)$ when the system is rotated by an angle $\phi$. This leads to an exact selection rule on the pairs of modes that are populated in the dissociation process, namely $m+m'=M$. For odd $M$ this imposes an “off-diagonal” form on the pair creation, as the atoms created by a dissociation process must necessarily end up with different integer quantum numbers, $m$ and $M-m$, whereas for even $M$ the two atoms can both end up in an $m=M/2$ mode.
In summary, each $m$ subspace is coupled only to itself by the single particle part of the Hamiltonian and the dissociation can either couple it exclusively to itself or exclusively to one other $m$ subspace. In this way the full problem splits into a series of smaller problems consisting of either one or two $m$-subspaces.
Now consider the problem of two coupled subspaces. As pairs are always created with an atom in each subspace, there is an obvious conservation of the *difference* in the total number of atoms in the two subspaces. This can also be phrased as the invariance of the evolution under shifts in the *relative* energy of the two subspaces as long as their total energy is maintained. Among other consequences, this implies that there are no first-order coherences between the two subspaces and that the one-body density operator is correspondingly block-diagonal. It is, in fact, possible to show an even stronger, dynamical symmetry in the evolution: If the two subspaces are initially unpopulated, they will at all times remain unitarily equivalent. In particular, their one-body density matrices will have identical spectra of eigenvalues [@poulsen:_in_preparation].
Results {#sec:results}
=======
As discussed above, once the Hamiltonian has been brought to a standard form by a Bogoliubov transformation, a number of properties of the system are immediately clear. In particular, the oscillatory or unbounded behaviour depends on whether all modes can be decoupled. In Fig. \[fig:lambdas\] we show the real and imaginary parts of the $\lambda_j$’s as a function of the detuning $\Delta$.
![Bogoliubov eigenvalues as functions of the detuning $\Delta$ for the Hamiltonian (\[eq:H\_frozen\]) with a coupling field (\[eq:chi\_delta\]) with $M=1$. For concreteness, we assume a strength and a radial dependence of the coupling such that $\int\! d\!z\chi_\mathrm{cm}|\phi_z|^2 = 2 \hbar\omega \rho
\exp(-\rho^2/{\ensuremath{a_\text{osc}}}^2)/{\ensuremath{a_\text{osc}}}$. The lines show ${\mathrm{Re}}(\lambda)$, which are the mode energies while, in the coupling dominated regime, the shaded areas around them have a height of ${\mathrm{Im}}(\lambda)$ to indicate the coupling between pairs of modes. The darkness of the shading indicates the $m$-values in the pairs of coupled modes: the darkest shading is for $m=0,m'=1$ pairs, $m=-1,m'=2$ pairs are a tone brighter, etc. Note that members of coupled pairs lie symmetrically around $\lambda=0$.[]{data-label="fig:lambdas"}](lambdas)
When $\Delta$ is increased, all atomic modes move linearly down in energy according to the single atom Hamiltonian (7). As a pair of modes with $m+m'=M=1$ gets close to the two-atom resonance, i.e. to having equal and opposite energies, the coupling has the effect of bending the quasi-particle energy levels further downwards. The modes become increasingly squeezed but remain uncoupled – they are in the detuning dominated regime of Sec. \[sec:detuned\]. At the point of perfect two-quasi-particle resonance, it is no longer possible to decouple the two modes and the system moves into a regime with quasi-particle pair production, the coupling dominated case of Sec. \[sec:coupled\]. When a pair of quasi-particle modes become unstable, it is signalled by the corresponding $\lambda_j$’s attaining a finite imaginary part. In Fig. \[fig:lambdas\] this is plotted as a shaded region around the corresponding energy curves. The height of the shaded region signifies the magnitude of ${\mathrm{Im}}[\lambda]$. As $\Delta$ is further increased, the pair production strength also increases. It has its maximum approximately at the point of two-*atom* resonance, i.e., when the two *bare* atomic modes becomes resonant with the molecule energy.
As an example, let us follow the lowermost curve through the figure. At the left, at low values of $\Delta$, the curve corresponds to the atomic harmonic oscillator ground state in the trap with $n=0$ and $m=0$. When $\Delta$ gets close to 1, pair production together with the ($n=0,m=1$) state (second lowest curve) becomes resonant. Note that the ($n=0,m=1$) state is degenerate with the ($n=0,m=-1$) for very low $\Delta$. At approximately $\Delta=1.5\hbar\omega$ \[i.e.$E_{00}+E_{01}=0$, cf. Eq. (\[eq:E\_nm\])\] the pair production has its maximum. A small shift from the simple estimate is due to the multi-mode character of the problem. The next resonance for the ($n=0,m=0$) level is centered around at $\Delta=2.5\hbar\omega$ \[i.e.$E_{00}+E_{11}=0$\]. Note that around $\Delta=2.5\hbar\omega$, the production of pairs with ($n=0,m=-1$) and ($n=1,m=2$) is also resonant. The third resonance, where $E_{00}+E_{21}=0$, is at $\Delta
= 3.5\hbar\omega$. Here several other resonances are also present.
In Fig. \[fig:Natoms\] we plot the total number of atoms and in Fig. \[fig:cond\_frac\] the fractional distribution on single-particle modes as a function of time for three different values of $\Delta$.
![(color online) Total number of atoms as a function of time for three values of $\Delta$. The coupling is as in Fig. \[fig:lambdas\].[]{data-label="fig:Natoms"}](Natoms)
![(color online) Condensate fraction, i.e. relative weigth of the largest eigenvalue of the one-body density operator compared to the sum of all eigenvalues (total number of particles. Results are plotted for three different values of $\Delta$. As discussed in Sec. \[sec:symmetries\], the conservation of angular momentum dictates a block-diagonal form of the one-body density operator and that pairs of blocks for which $m+m'=M$ will have identical spectra. Fully drawn lines correspond to $m=1$ and $m'=0$, dashed lines to $m=2$ and $m'=-1$, and dotted lines to $m=3$ and $m'=-2$.[]{data-label="fig:cond_frac"}](cond_frac)
For $\Delta=0.80\hbar\omega$ the dynamics is detuning dominated and oscillatory. All modes can be decoupled and the resulting quasi-particle modes are only slightly squeezed resulting in a very modest production of atoms. This almost vanishing population mainly occupies two modes: one in the $m=0$ manifold and one in the $m'=1$ manifold. These modes constitute the pair that is closest to resonance and their oscillation period is of the same order of magnitude as the trap period. The fractional occupation of these modes is close to 50% except for times close to the minima in their oscillation (e.g.$\omega t\sim 7$ here). At $\Delta=1.40\hbar\omega$, the system can no longer be decoupled: A single $m=0$, $m'=1$ pair of modes has just become unstable and real pair production takes place. Therefore this pair of modes quickly becomes dominant. Finally, for $\Delta=2.15\hbar\omega$ the evolution is more complex. From Fig. \[fig:lambdas\] one can tell that there will be real pair production into a pair of modes with $m=-1$, $m'=2$. However, a pair of modes with $m=0,m'=1$ also becomes unstable if $\Delta$ is increased just a little bit. It turns out that the $m=0,m'=1$ pair dominates at early times, but later most atoms are produced in the $m=-1,m'=2$ pair of states.
The above results all concern one-body properties. An illustrative way to quantify some of the two-body correlations in the system is to calculate the *conditional* density distribution, i.e., the density distribution resulting after the detection of a single atom at some position $(x_d,y_d)$. In Fig. \[fig:conditonal\] we plot the result of such a calculation for $\Delta=2.15\hbar\omega$ and $\omega
t=2,4,6$.
The upper row shows the normalized density distribution, that is, the probability distribution for the detection of the first atom. In the lower row, an atom has been detected at $(x_d,y_d)=(1,0){\ensuremath{a_\text{osc}}}$ and the resulting conditional probability distribution for the next detection is shown. For all three times, this conditional distribution is dramatically different from the original one. Close to $(x_d,y_d)$, there is generally a significantly higher probabilty density, reminiscent of the familiar (thermal state) bunching of bosons. However, when higher angular momentum states are populated, additional peaks appear as the detection induces coherences between more $m$ values. From a calculation of the full one-body density operator we can also find the new “condensate fraction” and for all three cases this number is significantly increased compared to the situation before the detection (cf. caption of Fig. \[fig:conditonal\]). This is similar to what has been proposed in the case of number state condensates [@moelmer02:_macros].
Conclusion {#sec:conclusion}
==========
We have presented a qualitative discussion and a quantitative analysis of the break-up of a molecular condensate into an atomic system, in the case where the molecules occupy a vortex state with unit angular momentum. The rotational symmetry of the problem causes the fragmentation of the atomic system into (at least) two spatial components, and our multi-mode analysis quantified the dynamics of these components. Particle detection at random locations will generally break the rotational symmetry of the state, and both our simple two-mode model and our more general analysis show that such detections will indeed cause the system to approach a single macroscopically populated quantum state with a preferred angular dependence.
Several interesting possibilities exist for further studies. A more technical issue concerns our assumption of non-interacting atoms. As shown in [@poulsen01:_quant_states_bose_einst] interactions have non-trivial consequences for the quantum correlations in the system. Depletion of the molecular condensate [@heinzen00:_super] affects the quantum statistics of the atomic state, and moreover, the explicit inclusion of the molecular field will also enable processes where atoms in different fragments recombine and form molecules with new angular momenta, respecting always the overall rotational symmetry of the system, but introducing multi-atom correlations in the system. Within our simple model with non-interacting atoms and undepleted molecular systems, it is a natural question to ask, what atomic states will result from the dissociation of a molecular vortex *lattice*. Will two fragments be sufficient to describe such states? We assumed a cylindrical symmetry and referred extensively to angular momentum conservation in the arguments of this paper, but in an *interacting* system vortices also exist under non-symmetric confinement, and our symmetry argument should be adequately modified into an argument referring to the phase topology. Presumably one would still see the formation of a fragmented system with components with and without vorticity, but the need for interactions among the atoms for the stability of such a state is an interesting issue.
UVP acknowledges financial support from the Danish Natural Science Research Council.
Cylindrical symmetry {#sec:cyl_sym}
====================
The case of strict cylindrical symmetry can be treated in a very similar fashion as the one of frozen $z$ dynamics of Sec. \[sec:frozen\_z\]. Assume that potential is axially symmetric and flat along $z$: $\omega_z=0$. The coupling is also assumed to be $z$ independent, ${\ensuremath{\chi_\mathrm{cm}}}(\rho,z)=K(\rho)$. We then use a quantization box of length $2L$ along $z$ for the atoms. The field operator is expanded as follows $$\begin{gathered}
\label{eq:cyl_expan_oper}
\hat\Psi({\ensuremath{\mathbf{r}}})
=
\sum_{n=0}^\infty
\sum_{m=-\infty}^{\infty}
\sum_{k=-\infty}^{\infty}
\hat{a}_{nmk}
\\
\times
\Phi_{nm}(\rho)
\times
\sqrt{\frac{1}{2\pi}} e^{im\phi}
\times
\sqrt{\frac{1}{2L}}e^{i\pi k z/L}\end{gathered}$$ where the radial modefunctions where given in Eq. (\[eq:def\_phi\_nmk\]). The mode creation and annihilation operators satisfy usual commutation relations $$\label{eq:psi_rad_commu}
\begin{split}
\Bigl[\hat{a}_{nmk},\hat{a}_{n'm'k'}\Bigr]
&=
0
\\
\Bigl[\hat{a}_{nmk},\hat{a}^\dagger_{n'm'k'}\Bigr]
&=
\delta_{nn'}\delta_{mm'}\delta_{kk'}
.
\end{split}$$ Under the above assumptions, the Hamiltonian (\[eq:H\_many\_modes\]) then simplifies to the form: $$\begin{gathered}
\label{eq:H_cyl}
\hat{H}
=
\sum_{nmk}
E_{nmk} \hat{a}^\dagger_{nmk}\hat{a}_{nmk}
\\
+
\sum_{mk}
\sum_{nn'}
\left\{
K_{nn'm}
\hat{a}^\dagger_{nmk}\hat{a}^\dagger_{n'(M-m)(-k)}
+\text{h.c.}
\right\}
,\end{gathered}$$ where $$\label{eq:def_Enmk}
E_{nmk} = 2n + |m| + 1
+ \frac{1}{2}\left(\frac{\pi}{L}\right)^2k^2
-\Delta$$ and $$\label{eq:def_chi_nmk}
\begin{split}
&K_{nn'm}
=
\\
&\phantom{K}\int\! d\!z d\!\rho \rho \;
K(\rho) \;
\frac{1}{2L} \;
\Phi^*_{nm}(\rho) \Phi_{n'M-m}(\rho)
.
\end{split}$$
The main difference to the case of a single active $z$ mode treated above is that excess molecular energy can now be transformed to $z$ kinetic energy of the atoms. Thus the picture of isolated resonances breaks down, even at low coupling strengths. Many atomic modes will participate in the dynamics and it will take more detections to build up a sizable condensate fraction.
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[^1]: In principle, the dissociating fields could also contribute to $M$. Transfer of (orbital) angular momentum in Bragg scattering of BECs has been demonstrated in Ref. [@andersen:_quant_rot].
|
---
abstract: 'Recent advances in cold atom experimentation suggest that studies of quantum two-dimensional melting of dipolar molecules, with dipoles aligned perpendicular to ordering plane, may be on the horizon. An intriguing aspect of this problem is that two-dimensional *classical* aligned dipoles (already studied in great detail in soft matter experiments on magnetic colloids) are known to melt via a two-stage process, with an intermediate hexatic phase separating the usual crystal and isotropic fluid phases. We estimate here the effect of quantum fluctuations on this hexatic phase, for both dipolar systems and charged Wigner crystals. Our approximate phase diagrams rely on a pair of Lindemann criteria, suitably adapted to deal with effects of thermal fluctuations in two dimensions. As part of our analysis, we determine the phonon spectra of quantum particles on a triangular lattice interacting with repulsive $1/r^3$ and $1/r$ potentials. A large softening of the transverse and longitudinal phonon frequencies, due to both lattice effects and quantum fluctuations, plays a significant role in our analysis. The hexatic phase is predicted to survive down to very low temperatures.'
author:
- 'Georg M. Bruun'
- 'David R. Nelson'
title: 'Quantum hexatic order in two-dimensional dipolar and charged fluids'
---
The melting of crystals is a fundamental topic in condensed matter physics that has been studied for more than a century. Nevertheless, one lacks a quantitative understanding of the melting for many materials. This deficiency is partly due to imprecise knowledge of the particle interactions that control the relevant phase transitions on a microscopic scale. In two dimensions (2d), however, a defect-mediated theory of melting is available. Building on pioneering work proposing an entropically-driven proliferation of dislocations by Kosterlitz and Thouless and by Berezinski, [@Jose] and on unpublished work of Feynman, [@Feynman] a defect mediated theory was worked out. [@NelsonBook] The detailed theory actually invokes a *sequential* unbinding of dislocations and disclinations, with the usual latent heat of a first order melting transition spread out over an intermediate hexatic phase. The hexatic phase is characterised by extended bond orientational order at intermediate densities or temperatures. It has been observed in a series of impressive experiments using colloidal particles with a diameter of 4.5$\mu$m and an induced magnetic moment, with dynamics confined to an air-water interface. [@Gasser; @Strandburg] Although this system, characterised by long range $1/r^3$ dipole-dipole interactions, is thoroughly understood in the classical regime, it is presently unknown whether the hexatic phase exists when quantum fluctuations play a major role.
The advancing field of ultra-cold gases consisting of heteronuclear molecules with an electric dipole moment promises to change this situation. The dipolar gases are expected to exhibit qualitatively new physics, with several experimental groups reporting impressive progress towards achieving quantum degeneracy in these systems. [@Wu; @Ni; @Chotia; @Heo; @Pasquiou] Using dipolar gases, one should finally be able to probe experimentally the role of quantum fluctuations on the hexatic phase and study its stability at very low temperatures. Recent Monte-Carlo calculations predict that a 2d dipolar gas with the moments perpendicular to the 2d plane exhibits a quantum phase transition directly to a hexagonal crystal phase at zero temperature $T=0$. [@Astrakharchik; @Buchler] However, the effects of quantum zero point motion on the hexatic phase, accessible now for the first time via cold dipolar gases, have not been discussed.
In this paper, we explore this question by analysing the stability of crystal and hexatic order of a 2d system of dipoles including both thermal and quantum effects. First, we calculate the classical elastic coefficients of the crystal from the phonon spectrum. We then show how quantum effects soften the crystal by decreasing these coefficients. Using Lindemann criteria suitably modified to treat both 2d thermal fluctuations and quantum effects, we study the successive loss of translational and orientational order that lead to the melting of the crystal and hexatic phases. The relevant Lindemann numbers are extracted in the classical regime as well as for $T=0$ by comparing with Monte-Carlo and experimental results. Throughout the paper, we construct a useful comparison between the 2d dipolar system and a 2d fluid consisting of negatively charged particles immersed in a uniform background of positive charges – the Jellium model. An experimental realisation of charged systems in the classical limit with a Wigner crystal phase at low temperatures has been known for quite some time, in the form of a 2d electron gas trapped by a positively charged capacitor plate to the surface of liquid helium. [@Grimes] For computer simulation evidence that this system melts via a dislocation mechanism and may possess an intermediate hexatic phase, see Ref. and respectively. We obtain essentially the same Lindemann numbers for the dipolar and the charged systems, which suggests that the melting of these phases is a geometric phenomenon, insensitive to the detailed form of the interaction potential. Similar conclusions resulted from Monte-Carlo simulations of quantum hard sphere systems, [@Runge] and from a meta-analysis of Monte-Carlo results for the freezing of two-dimensional systems. [@Babadi] We show that quantum effects initially *increase* the temperature range where the hexatic phase is stable when the coupling strength is decreased from the strong coupling classical regime, provided the temperature dependence of the Lindemann numbers can be neglected. A tentative phase diagram is then provided showing that quantum effects are important even for very large interaction strengths where one would naively expect the system to be deep in the classical regime. Finally, we discuss the possible experimental observation of the hexatic phase in the quantum regime using cold dipolar gases.
We conclude this introduction with a few observations about the Lindemann criterion for melting of quantum and classical systems in 2d and the nature of quantum hexatics. As originally proposed by Lindemann, [@Lindemann] one first calculates the mean-square displacement $W=\langle|{\mathbf u}({\mathbf r})|^2\rangle$ of a single particle away from its equilibrium lattice position, where $\langle\ldots\rangle$ represents an ensemble average. Melting as a function of, say, the temperature or density then occurs when the root mean square displacement exceeds a fixed fraction of the lattice spacing $a$ , i.e. for $\sqrt W\ge c_La$, where the Lindemann number is typically in the range $c_L\approx0.1-0.3$. [@Blatter] This rough criterion fails, however, in 2d classical solids, because $W$ diverges logarithmically with system size. Here we use an alternative formulation that focuses on the stretching of a nearest neighbor distance, [@FisherFisher] namely $$\Delta({\mathbf r})=\sqrt{\langle |{\mathbf u}({\mathbf r}+{\mathbf b})-{\mathbf u}({\mathbf r})|^2\rangle}\ge\gamma_m a
\label{Lindemann}$$ where ${\mathbf b}$ connects nearest neighbor lattice sites ($|{\mathbf b}|=a$) and $\gamma_m$ is an alternative Lindemann number describing this new measure of the loss of translational order. The quantity $\Delta({\mathbf r})$ remains finite in the thermodynamic limit even in 2d and, as we show here, can also be computed in the quantum regime all the way down to $T=0$ for simple pair potentials. Moreover, a related criterion allows us to estimate where the order associated with the *rotational* broken symmetry of a 2d crystal is lost due to thermal or quantum fluctuations, namely $$\Delta_\theta({\mathbf r})=\sqrt{\langle\theta^2({\mathbf r})\rangle}=\frac 1 2 \sqrt{\left\langle \left|\partial_xu_y({\mathbf r})-\partial_yu_x({\mathbf r})\right|^2\right\rangle}\ge\gamma_i
\label{LindemannHexatic}$$ where $\gamma_i$ is a Lindemann number for the loss of bond orientational order. Here, $\theta({\mathbf r})=[\partial_xu_y({\mathbf r})-\partial_yu_x({\mathbf r})]/2$ is the local phonon-induced twist of the crystallographic axes, [@LandauLifshitz] a quantity whose fluctuations are known to remain finite even in the limit of infinite system size for a classical 2d crystal. [@Mermin] Note that $\Delta({\mathbf r})\approx \sqrt{\langle[({\mathbf b}\cdot\nabla){\mathbf u}({\mathbf r})]^2\rangle}$ has a similar gradient structure to $\Delta_\theta({\mathbf r})$. These two different Lindemann numbers $\gamma_m$ and $\gamma_i$ allow for two distinct melting temperatures, characterized by the successive loss of first translational and then orientational order, [@NelsonHalperin] a scenario we *know* occurs for classical colloidal particles interacting with repulsive long range $1/r^3$ dipole-dipole interaction. [@Gasser] Our evaluation of the criterion (\[LindemannHexatic\]) using *crystalline* phonon spectra to estimate the extent of the hexatic phase seems reasonable, provided local orientational order remains robust after long range translational order is lost, as is the case for the dislocation-disclination theory of classical 2d melting, [@NelsonHalperin] and in situations where a weakly first order transition leads to a hexatic phase. [@Engel] We also note that our use of phonon displacements from an underlying reference crystal implicitly treats quantum particles as distinguishable (a similar approximation is used in the Debye theory of the specific heat of crystals [@AshcroftMermin]), so we are effectively looking at the melting of quantum particles with Boltzmannian statistics. [@Clark] We assume that the exchange interactions that distinguish bosons from fermions play only a minor role in determining the locations of quantum melting transitions.
These Lindemann inequalities are *criteria*, and of course do not themselves constitute a *theory* of quantum or classical melting. We are not aware of a reliable microscopic theory of 2d quantum melting, and even the defect-mediated melting of classical particles in two dimensions could be preempted by a direct first order transition from a crystal to an isotropic liquid. [@NelsonBook] It is also worth noting the rather different nature of classical as opposed to quantum melting in two dimensions. This difference is particularly evident in the Feynman path integral formulation of nonrelativistic quantum statistical mechanics, [@FeynmanHibbs] where classical particles are replaced by configurations of particle world lines in imaginary time, see Fig. \[WorldlineFig\]. We allow, for simplicity, only the identity permutation with periodic boundary conditions across an imaginary time slab of thickness $\beta\hbar$. In the absence of interactions, these trajectories when projected down the imaginary time axis behave like two dimensional random walks as a function of the imaginary time variable, with a size given by the thermal de Broglie wavelength $\Lambda_T=\sqrt{2\pi\hbar^2\beta/m}$. In the classical limit $\Lambda_T\ll n_0^{-1/2}$, where $n_0$ is the areal particle density, the particle world lines are short and nearly straight; hence, the usual Lindemann picture for melting of point-like particles applies when interactions are turned on. However, in the highly quantum limit $\Lambda_T\gg n_0^{-1/2}$, quantum and thermal fluctuations act on a crystal of long wiggling *lines*: as illustrated in Fig. \[WorldlineFig\], a particle world line ${\mathbf r}_j(\tau)$ that makes a large excursion within its confining cage at imaginary time $\tau$ is connected by the kinetic energy interaction $m\int_0^{\hbar\beta}|d{\mathbf r}_j(\tau)/d\tau|^2/2$ to time slices above and below, and hence can more easily recover and return to its equilibrium position when the slab thickness is large. It is harder to melt arrays of lines in $2 + 1$ dimensions than point-like particles in two dimensions with the same pair potential. Thus, we should not be surprised if the Lindemann numbers $\gamma_m$ and $\gamma_i$ depend somewhat on $n_0\Lambda_T^2$, with larger Lindemann numbers required to produce melting when $n_0\Lambda_T^2\gg 1$. This is indeed what we find fitting to experiments on colloids and quantum simulations of power law potentials, with, e.g., $\gamma_m$ ranging from $\sim0.1$ in the classical limit to $\sim0.3$ when quantum effects predominate. Phonon nonlinearities can give rise to a weak temperature-dependence of the long wavelength elastic constants, even in the absence of quantum fluctuations. [@Morf] We neglect such effects here for simplicity. With these understandings, we believe the criteria sketched above can provide a rough map of where to look for quantum melting in the new arena provided by cold quantum gases.
What would a quantum hexatic look like, if next generation experiments were to discover such a thing? Roughly speaking, it would be a quantum liquid crystal, a cousin of the long-sought supersolid phase of $^4$He if the particles were bosons. [@Reppy; @Balibar] A fermionic analog would be the quantum nematic studied by Oganesyan *et. al.* in electronic systems, [@Oganesyan] which exhibits a two-fold rather than six-fold anisotropy. These authors also considered the possibility of an electronic quantum hexatic. A hexatic quantum fluid would display a fuzzy, six-fold-symmetric diffraction pattern, indicating extended orientational correlations, somewhat like a poorly averaged powder diffraction pattern. However, *unlike* a classical polycrystal, a quantum hexatic would be a fluid with zero shear modulus! If composed of bosons, it could develop a nonzero superfluid density and support supercurrents at sufficiently low temperatures. [@Mullen] The nature of hexatic order in real space, with extended correlations in the orientations of distant six-fold particle clusters, is discussed in Ref. .
In the crude phase diagrams constructed here, we have taken a conservative approach, and assumed the intermediate hexatic phase is squeezed out as $T\rightarrow0$, leaving behind a transition from a quantum solid directly to an isotropic quantum liquid. But this need not be the case: consider particles interacting with a screened, 2d Yukawa potential, $$V(r)=\epsilon_0K_0(\kappa r)
\label{Vpair}$$ where $K_0(x)$ is the modified Bessel function of the second kind, $K_0(x)\sim -\log x$ for $x\ll 1$, $K_0(x)\sim \exp(-\kappa x)$ for $x\gg 1$, and $\kappa^{-1}$ is a screening length. Such a potential describes interactions between vortex lines with weak thermal fluctuations in Type II superconductors with an external magnetic field, where $\kappa^{-1}$ is the London penetration depth. When the lines are very long compared to the vortex line spacing, and pinning is negligible, the classical statistical mechanics of these three dimensional lines at finite temperatures can be mapped via the transfer matrix method onto the quantum statistical mechanics of 2d bosons at $T=0$ interacting with the pair potential Eq. (\[Vpair\]).[@NelsonSeung] Here, the temperature $T$ of the 3d superconductor plays the role of $\hbar$ and the thickness of the bulk superconductor plays the role of $\hbar\beta$ in the equivalent 2d quantum system. A dislocation loop unbinding model then leads directly to an entangled liquid of vortex lines, with long range six-fold bond orientational order, equivalent via the path integral mapping to a zero temperature quantum hexatic.[@Marchetti] Although it has not yet been possible to check for hexatic order in melted vortex liquids in Type II superconductors (the signal from neutron diffraction is quite weak), something very like an entangled line hexatic has been seen in X-ray diffraction experiments off partially ordered arrays of aligned DNA molecules. [@Strey] When these charged, linear polymers are aligned by an external field, a screened, Debye-Hückel interaction arise in the perpendicular direction which has precisely the form (\[Vpair\]). The characteristic line hexatic diffraction pattern from this work is reproduced in Fig. \[ExperimentalFig\]. The 2d structure function $\langle|n({\mathbf q})|^2\rangle$ for particles in a quantum hexatic at low temperatures should look very similar.
Lamé coefficients
=================
In this section, we calculate the elastic coefficients from the phonon modes of a 2d hexagonal crystal consisting of charged particles with a neutralising background charge density, or particles with a dipole moment perpendicular to the 2d plane. The crystal lattice with lattice constant $a$ is spanned by the vectors ${\mathbf a}_1=a(\sqrt 3/2,1/2)$ and ${\mathbf a}_2=a(-\sqrt 3/2,1/2)$ corresponding to the reciprocal vectors ${\mathbf b}_1=2\pi a^{-1}(1/\sqrt 3,1)$ and ${\mathbf b}_2=2\pi a^{-1}(-1/\sqrt 3,1)$. The reciprocal lattice with the irreducible Brillouin zone is shown in Fig. \[LatticeFig\](left). The interaction between two particles separated by a distance $r$ in the plane is $$U(r)=\frac{D^2}{r^3}\text{ dipoles}\hspace{0.5cm}U(r)=\frac{Q^2}{r}\text{ charges}$$ where $D^2=d^2/4\pi\epsilon_0$ for electric dipoles with dipole moment $d$, and $Q^2=q^2/4\pi\epsilon_0$ for particles with charge $q$. We set $\hbar=k_B=1$ in the following.
Classical elasticity of point dipoles and point charges
-------------------------------------------------------
We find the phonon modes of the potential energy in the harmonic approximation in the usual way. To accelerate the convergence of the sums, we use the Ewald summation technique as detailed in Appendix \[Ewald\]. In Fig. \[LatticeFig\] we plot the resulting two phonon branches along the vector ${\mathbf b}_1$ for the dipoles (middle) and the charged particles (right).
The low energy mode is purely transverse and the high energy mode is purely longitudinal for long wavelengths where the hexagonal crystal is equivalent to an isotropic continuum system. [@LandauLifshitz] The characteristic phonon energy for the dipolar crystal is $\omega_D=\sqrt{D^2/ma^5}$, and a crystal of Coulomb charges, it is $\omega_C=\sqrt{Q^2/ma^3}$ where $m$ is the particle mass. For the dipoles, we find for long wave lengths the isotropic modes $$\omega_l(q)\simeq4.8\omega_D qa \hspace{0.4cm}\text{and}\hspace{0.4cm}\omega_t(q)\simeq1.4\omega_Dqa.
\label{SoundDipoles}$$ These sound velocities differ somewhat from what was reported recently, [@Lu] but as we shall demonstrate shortly, they accurately recover well established values for the $k=0$ Lamé coefficients [@Grunberg] which gives us confidence in our numerical calculations. For the charged particles, we have plotted the long wave length formulas, $$\omega_l(q)=\frac{2\sqrt{\pi}}{3^{1/4}}\omega_C\sqrt{qa}\hspace{0.3cm}\text{and}\hspace{0.3cm}\omega_t(q)=\frac{2^{1/4}\eta^{1/2}}{3^{1/8}}\omega_Cqa
\label{SoundCoulomb}$$ for the longitudinal and transverse mode respectively with $\eta=0.25$. [@Fisher] We see that the numerics reproduce these results confirming that the Ewald summation has converged. Note that the longitudinal mode scales as $\sqrt q$ for small momenta reflecting the long range nature of the Coulomb interaction.
The Lamé coefficients are defined by writing the elastic energy of the crystal as [@ChaikinBook] $$F_{\rm el}=\frac 1 {2L^2}\sum_{\mathbf k}\left\{\mu({\mathbf k})| u_t({\mathbf k})|^2
+[2\mu({\mathbf k})+\lambda({\mathbf k})]| u_l({\mathbf k})|^2\right\}k^2
\label{Fel}$$ with $|{\mathbf u}({\mathbf k})|^2={\mathbf u}({\mathbf k}){\mathbf u}(-{\mathbf k})$ and $L^2$ the area of the system. The longitudinal component of the displacement field is $u_l$, and $u_t$ is the transverse component. Note that “transverse” and “longitudinal” simply refers to the lowest and highest phonon mode for a given ${\mathbf k}$, since the eigenvectors are not in general parallel or perpendicular to ${\mathbf k}$ when lattice effects are taken into account. The relation between the Lamé coefficients and the phonon modes is then as usual $$\omega_t({\mathbf k})=\sqrt{\frac{\mu({\mathbf k})}{\rho}}k\hspace{0.5cm}\text{and}\hspace{0.5cm}\omega_l({\mathbf k})=\sqrt{\frac{2\mu({\mathbf k})+\lambda({\mathbf k})}{\rho}}k
\label{Lame}$$ where $\rho=m2/\sqrt3a^2$ is the mass areal density. The natural scale for the Lamé coefficients is $D^2/a^5$ for dipoles and $Q^2/a^3$ for charged particles, and they are ${\mathbf k}$ dependent due to lattice effects. Since $\omega_l({\mathbf k})\propto \sqrt k$ for $k\rightarrow 0$, the Lamé coefficient $\lambda({\mathbf k})$ diverges as $1/\sqrt k$ for the charged particles.
In Figs. \[LameFigDipoles\]-\[LameFigCoulomb\], we plot the classical Lamé coefficients along ${\mathbf b}_1$ for the dipoles and the charged particles. The elastic parameters display a significant $k$-dependent softening due to the discrete lattice symmetry. We also plot in Fig. \[LameFigDipoles\] the $k=0$ Lamé coefficients corresponding to Eq. (\[SoundDipoles\]), i.e. $$\mu({0})\simeq2.4\frac{D^2}{a^5}\hspace{0.3cm}\text{and}\hspace{0.3cm}2\mu({0})+\lambda({0})\simeq26\frac{D^2}{a^5}.
\label{LameDipoles}$$ These values agree very well those reported in Ref. . Likewise, we plot in Fig. \[LameFigCoulomb\] the $k=0$ value for the transverse mode corresponding to Eq. (\[SoundCoulomb\]), i.e. [@Fisher] $$\mu({0})=\eta\frac{2^{3/2}Q^2}{3^{3/4}a^3}
\label{LameCharges}$$ which is recovered by our numerics.
Using Eq. (\[Fel\]), it is straightforward to calculate the mean displacement of the particles from their equilibrium positions at a given temperature $T$, and we obtain $$\begin{aligned}
\langle u_l({\mathbf k})u_l({\mathbf k}')\rangle&=&\delta_{{\mathbf k},-{\mathbf k}'}L^2\frac{T}{[2\mu({\mathbf k})+\lambda({\mathbf k})]k^2}\nonumber\\
\langle u_t({\mathbf k})u_t({\mathbf k}')\rangle&=&\delta_{{\mathbf k},-{\mathbf k}'}L^2\frac{T}{\mu({\mathbf k})k^2}.
\label{uuk}\end{aligned}$$
Quantum softening of the Lamé coefficients
------------------------------------------
We now include quantum effects on the Lamé coefficients by quantizing the phonons. This can be done in several ways. Here, we use the path integral approach since it allows us to describe the quantum effects on the crystal melting in terms of a simple geometrical picture of wiggling particle trajectories described in the introduction. For completeness, we also present a canonical quantization approach in Appendix \[Canonical\].
The partition function $Z$ for the crystal can be written as an integral over all possible paths ${\mathbf u}({\mathbf r},\tau)$ of the particle displacements in imaginary time $\tau$ as [@FeynmanHibbs] $$Z=\int {\mathcal D}[{\mathbf u}({\mathbf k},\tau)]e^{-\int_0^\beta d\tau[F_{\rm kin}(\tau)+F_{\rm el}(\tau)]}
\label{Zpath}$$ where $F_{\rm kin}(\tau)=\rho \sum_{\mathbf k}|\partial_\tau{\mathbf u}({\mathbf k},\tau)|^2/2L^2$ is the kinetic energy, $\rho$ is the 2d mass density, and the elastic energy $F_{\rm el}(\tau)$ is given by Eq. (\[Fel\]) with the replacement ${\mathbf k}\rightarrow ({\mathbf k},\tau)$. We do not include permutations of the particle positions at the boundary $\tau=\beta$ of the imaginery time slab, so the boundary condition in Eq. (\[Zpath\]) is ${\mathbf u}({\mathbf r},\beta)={\mathbf u}({\mathbf r},0)$. At this level of approximation, we therefore cannot distinguish between bosonic and fermionic particles. We can from Eq. (\[Zpath\]) calculate the fluctuations of the particles including quantum effects, $$\begin{gathered}
\langle u_\sigma({\mathbf k},\tau) u_\sigma(-{\mathbf k},\tau)\rangle=
\frac{L^2\beta}{\rho}\sum_n\frac1{\omega_n^2+\omega_\sigma({\mathbf k})^2}\nonumber\\
=\frac{L^2}{2\rho\omega_\sigma({\mathbf k})}\coth[\beta\omega_\sigma({\mathbf k})/2)]
\label{Quantumuu}\end{gathered}$$ where the sum is over Matsubara frequencies $\omega_n=2n\pi T$ and phonon modes $\sigma=t,l$. Recasting this result in the form of Eq. (\[uuk\]) defines the quantum Lamé coefficients $\mu_Q({\mathbf k})$ and $\lambda_Q({\mathbf k})$ as $$\begin{gathered}
\mu_Q({\mathbf k})=\frac{2T\sqrt{\rho\mu}}{k}\tanh\left(\sqrt{\frac{\mu}{\rho}}\frac{k}{2T}\right)\nonumber\\
2\mu_Q({\mathbf k})+\lambda_Q({\mathbf k})=\frac{2T\sqrt{\rho(2\mu+\lambda)}}{k}\tanh\left(\sqrt{\frac{2\mu+\lambda}{\rho}}\frac{k}{2T}\right),
\label{LameQuantum}\end{gathered}$$ where we have suppressed the $k$-dependence of the classical Lamé coefficients $\mu({\mathbf k})$ and $\lambda({\mathbf k})$ for notational simplicity. Equation (\[LameQuantum\]) reveals that the magnitude of quantum effects on the Lamé coefficients is determined by the parameters $\sqrt{\mu/\rho}k/T=\omega_t/T$ and $\sqrt{(2\mu+\lambda)/\rho}k/T=\omega_l/T$. For $\omega_\sigma/T\rightarrow 0$ we recover the classical results given by Eq. (\[Lame\]), whereas the elastic coefficients are decreased due to quantum fluctuations whenever $\omega_\sigma/T\gtrsim 1$.
In Figs. \[LameFigDipoles\]-\[LameFigCoulomb\], we plot the quantum Lamé coefficients along ${\mathbf b}_1$ for $T/\omega_D=T/\omega_C=1$ and $T/\omega_D=T/\omega_C=0.2$. We see that quantum effects significantly soften the crystal for decreasing temperature, and that it is the high energy fluctuations which are reduced the most. Quantum softening is therefore greater for the longitudinal mode, reducing $[2\mu_Q({\mathbf k})+\lambda_Q({\mathbf k})]/[2\mu({\mathbf k})+\lambda({\mathbf k})]$ more than $\mu_Q({\mathbf k})/\mu({\mathbf k})$.
Modified Lindemann criteria for crystal and hexatic melting
===========================================================
Since there is algebraic, as opposed to long range translational order in a 2d crystal when $T>0$, [@MerminWagner] it is not possible to estimate the melting temperature of the crystal from a usual Lindemann criterion, as discussed in the introduction. We will therefore use a modified Lindemann criterium for the melting. Our basic assumption is that the melting of the crystal occurs in two steps with increasing temperature. [@HalperinNelson; @NelsonHalperin; @NelsonBook] First, the crystal melts at a temperature $T_m$ into a hexatic phase, characterised by long range bond angle order but short range translational order. Then, at a higher temperature $T_i$ the hexatic phase melts into an isotropic liquid. The existence of the hexatic phase is well established for classical systems [@Gasser], but our knowledge concerning its stability against quantum fluctuations is limited. A Monte-Carlo study supports the existence of such a phase in the quantum regime in the case of distinguishable particles with Coulomb interactions. [@Clark] We therefore focus on how quantum fluctuations affect on the stability of the hexatic phase.
Melting of the crystal phase
----------------------------
To calculate the temperature $T_m$ where the crystal melts into the hexatic phase, we will use the modified Lindemann criterion given by Eq. (\[Lindemann\]). It states that the crystal melts when the *relative* fluctuations of the particle positions of two nearest neighbours are larger than the lattice constant $a$. Using the quantum Lamé coefficients in Eq. (\[uuk\]) yields $$\begin{gathered}
\delta u_i^2=\langle |u_i({\mathbf r})-u_i(0)|^2\rangle=T\frac2{L^2}\sum_{\mathbf k}(1-\cos{\mathbf k}\cdot{\mathbf r})\nonumber\\
\times\left[\frac{\epsilon_{l,i}(\mathbf{k})^2}{[2\mu_Q({\mathbf k})+\lambda_Q({\mathbf k})]k^2}+
\frac{1-\epsilon_{l,i}(\mathbf{k})^2}{\mu_Q({\mathbf k})k^2}\right]
\label{uur}\end{gathered}$$ for the fluctuations along a specific direction $i=x$ or $i=y$. Here, $\epsilon_{l,i}(\mathbf{k})$ is $i$’th component of the eigenvector of the highest mode.
The melting temperature for the crystal phase in the classical limit was reported to be $T_m\simeq 0.0907{D^2}/{a^3}$ for dipoles, [@Kalia; @Grunberg] and $T_m\simeq 0.0136{Q^2}/{a}$ for charged particles. [@Muto] From these results, we can determine the Lindemann number in the classical regime. Using Eqs. (\[Lindemann\]) and (\[uur\]) with the classical Lamé coefficients to calculate the particle fluctuations at the classical melting temperature yields $$\gamma_{m,cl}=0.14\text{ dipoles}\hspace{0.5cm}\gamma_{m,cl}=0.15\text{ charges}.
\label{LindemannClass}$$ We see that the classical Lindemann numbers are essentially the same for the dipoles and the charged particles. This result suggests that melting is mainly determined by the geometry of the crystal, depending very weakly on the detailed form of the interaction potential so that the Lindemann number is almost universal.
As discussed in the introduction, we expect that the Lindemann numbers in general depend on temperature due to quantum effects. We can determine the Lindemann number $\gamma_m$ at $T=0$ using recent quantum Monte-Carlo results suggesting a $T=0$ quantum phase transition between the crystal phase and the liquid phase: For dipoles, two independent calculations give the value $r_D\simeq 18\pm 4$ [@Astrakharchik; @Buchler], and for distinguishable charged particles one obtains the value $r_C\simeq 127$ for the critical value of this quantum phase transition. [@Clark] Here, the $r$-parameters $$r_D=\frac{mD^2}{a}\text{ dipoles}\hspace{0.5cm}r_C=mQ^2a\text{ charges}$$ are the ratios between the nearest neighbour interaction energy and the quantum kinetic energy. Using the $T\rightarrow 0$ limit of Eq. (\[LameQuantum\]) in Eq. (\[uur\]), we obtain $$\begin{gathered}
\langle |u_i({\mathbf r})-u_i(0)|^2\rangle=\frac1{\rho L^2}\sum_{\mathbf k}(1-\cos{\mathbf k}\cdot{\mathbf r})\nonumber\\
\times\left[\frac{\epsilon_{l,i}(\mathbf{k})^2}{\omega_l({\mathbf k})}+
\frac{1-\epsilon_{l,i}(\mathbf{k})^2}{\omega_t({\mathbf k})}\right]
\label{uurQ}\end{gathered}$$ for $T=0$. Equation (\[uurQ\]) predicts as expected that the zero point motion of the particles scale as the typical harmonic oscillator length for the phonons, i.e. $\delta u^2\sim 1/m\omega_D$ for dipoles and $\delta u^2\sim 1/m\omega_C$ for charged particles. Using Eq. (\[uurQ\]), we can determine the Lindemann number for $T=0$ at the critical coupling strength for the quantum melting transition, obtaining $$\gamma_{m,0}=0.31\text{ dipoles}\hspace{0.5cm}\gamma_{m,0}=0.31\text{ charges}.
\label{LindemannT0}$$ The Lindemann numbers are again the same for the dipoles and the charges at this level of accuracy, indicating that the melting of the crystal phase is primarily determined by geometry also at $T=0$. The $T=0$ value of $\gamma_m$ at the critical point is consistent with what was obtained using perturbation theory. [@Lozovik2]
Comparing Eqs. (\[LindemannClass\]) and (\[LindemannT0\]) shows that the Lindemann numbers are significantly larger at $T=0$ than in the classical regime. As discussed in the introduction, the path integral approach provides a simple geometrical interpretation of this result. Indeed, from Eq. (\[Zpath\]) it is clear that the quantum problem corresponds to the melting of a crystal of lines in a 3d slab of thickness $\beta$, see Fig. \[WorldlineFig\]. Only when $\beta\rightarrow 0$ does one recover the classical problem of the melting of a 2d crystal. Since the lines can wiggle significantly along the $\beta$-direction without melting the crystal, it is natural to expect that the Lindemann number is larger in the quantum regime as compared to the classical regime.
Melting of the hexatic phase
----------------------------
In the two-step melting scenario, the system is in a hexatic phase characterised by extended bond angle order for temperatures $T_m<T<T_i$. We therefore use the Lindemann criterion based on the fluctuations in the bond angle $\theta$ given by Eq. (\[LindemannHexatic\]). Fourier transforming Eq. (\[uuk\]) using the quantum Lamé coefficients gives after some algebra $$\langle\theta^2\rangle=\frac T{4L^2}\sum_{\mathbf k}\left[\frac{(\mathbf{k}\times\mathbf{\epsilon}_l(\mathbf{k}))^2}{[2\mu_Q({\mathbf k})+\lambda_Q({\mathbf k})]k^2}+
\frac{(\mathbf{k}\times\mathbf{\epsilon}_t(\mathbf{k}))^2}{\mu_Q({\mathbf k})k^2}\right]
\label{AngleFluc}$$ where $\epsilon_t(\mathbf{k})$ is the eigenvector of the lowest mode. For an isotropic medium where $\epsilon_l\parallel{\mathbf k}$ and $\epsilon_t\perp{\mathbf k}$, Eq. (\[AngleFluc\]) reduces to $$\delta\theta^2=\frac T{4L^2}\sum_{\mathbf k}\frac 1{\mu_Q({\mathbf k})},
\label{AngleFlucSimple}$$ i.e. the bond angle fluctuations are determined by the transverse mode only. Since a hexagonal crystal is equivalent to an isotropic medium for long wave lengths,[@LandauLifshitz] and since it is these low energy modes which contribute most to the fluctuations, Eq. (\[AngleFlucSimple\]) turns out to be a very good approximation to Eq. (\[AngleFluc\]).
To determine the Lindemann numbers for the hexatic phase, we again use results for the melting temperatures reported in the literature. The melting temperature of the hexatic phase in the classical limit was found to be $T_i\simeq 0.0968U(a)$ with $U(a)=D^2/a^3$ for dipoles [@Grunberg], and $T_i\simeq 0.0159U(a)$ with $U(a)=Q^2/a$ for charged particles. [@Muto] Using these temperatures and the classical Lamé coefficients in Eq. (\[AngleFluc\]) yields $$\gamma_{i,cl}=0.12\text{ dipoles}\hspace{0.5cm}\gamma_{i,cl}=0.13\text{ charges}
\label{LindemannHexaticClass}$$ for the Lindemann numbers determining the melting of the hexatic phase in the classical regime. As for the crystal phase, the Lindemann numbers are essentially the same for the dipoles and the charged particles, suggesting again that melting of the hexatic phase is a geometric phenomenon, largely independent of the precise form of the interaction.
The $T=0$ Monte-Carlo calculations for the dipoles did not examine the quantum hexatic phase, [@Astrakharchik; @Buchler] so it is presently not known whether it exists all the way down to $T=0$. In the case of distinguishable “Boltzmannian” charged particles, it was found that the hexatic phase persists to quite low temperatures where quantum effects are significant, disappearing in a tricritical point at $T\simeq 0.04U(a)$. [@Clark] Since our analysis indicates that, for a given value of $n_0\Lambda_T^2$, the melting is insensitive to the detailed form of the interaction potential, this Monte-Carlo result suggests that the hexatic phase persist deep into the low temperature regime both for dipoles and for charged particles. It is therefore interesting to evaluate the Lindemann numbers at $T=0$ for the bond angle fluctuations. Using the $T=0$ limit of Eq. (\[AngleFluc\]), $$\delta\theta^2=\frac 1{8\rho L^2}\sum_{\mathbf k}\left[\frac{(\mathbf{k}\times\mathbf{\epsilon}_l(\mathbf{k}))^2}{\omega_l({\mathbf k})}+
\frac{(\mathbf{k}\times\mathbf{\epsilon}_t(\mathbf{k}))^2}{\omega_t({\mathbf k})}\right],
\label{AngleFlucQ}$$ yields $$\gamma_{i,0}=0.23\text{ dipoles}\hspace{0.5cm}\gamma_{i,0}=0.24\text{ charges}$$ at the quantum quantum transition points $r_D\simeq 18$ and $r_C\simeq 127$ for dipoles and charged particles respectively. Again, the angle fluctuations differ very little between the dipolar and the charged systems. The precise values of the Lindemann numbers may, of course, differ at the exact boundaries of the hexatic phase in the quantum regime, which are presently unknown.
As we discussed, the melting of the hexatic phase is determined almost exclusively by the lowest phonon mode in the sense that Eq. (\[AngleFlucSimple\]) is an excellent approximation to Eq. (\[AngleFluc\]). In addition, the lowest phonon mode is less affected by quantum fluctuations than the highest phonon mode. It follows from this that the hexatic phase is more robust towards quantum fluctuations compared to the crystal phase, and as a result the hexatic region in the phase diagram should *increase* with decreasing potential/quantum-kinetic energy ratio $r$. This conclusion depends of course on the assumption that the Lindemann numbers are independent of temperature, so we expect it to be valid only when the quantum corrections are small. When the temperature dependence of the Lindemann numbers is significant we cannot determine the fate of the hexatic without further information.
Phase diagrams
==============
In this section, we provide approximate phase diagrams, using the fact that the melting of the crystal and hexatic phases are insensitive to the detailed form of the interaction potential. However, by comparing the values of the Lindemann numbers in the classical regime and at $T=0$ given by Eqs. (\[LindemannClass\]) and (\[LindemannT0\]) we see that they depend on temperature. To provide a tentative phase diagram, we therefore write the Lindemann number determining the crystal melting on the phenomenological form $$\gamma_m(T)=\gamma_{m,0}+(\gamma_{m,cl}-\gamma_{m,0})\left(\frac{T}{T_m}\right)^n,
\label{GammaT}$$ which interpolates between the $T=0$ value and the classical value for $T=T_m$. We write $\gamma_i(T)$ in the same phenomenological form. To determine $n$, we compare the phase diagram produced by these phenomenological forms with the phase diagram obtained by Monte-Carlo calculations for distinguishable charged particles in Ref. . It turns out that $n=6$ yields a reasonable good fit as is shown in Fig. \[CeperleyPhase\]. It must be emphasised that we have not performed a systematic fit to determine the optimal value of $n$, since this is not relevant at this level of approximation, where our goal is simply to provide a qualitatively reliable phase diagram. The detailed form of the actual phase diagram could be different. For instance, our analysis does not reproduce the tri-critical point at $T\simeq0.004U(a)$ found in the Monte-Carlo calculations. Instead, we have chosen to let the hexatic and crystal phases continue down to $r_C\simeq95$ for $T=0$ corresponding to the Lindemann numbers $\gamma_{i,0}=0.25$ and $\gamma_{m,0}=0.33$. Moreover, as stressed in the introduction, for the case of a Yukawa potential, a quantum hexatic phase could exist for a finite parameter range even at $T=0$.
Note that there is no significant increase in the temperature range for which the hexatic phase is stable with decreasing $r_C$ in Fig. \[CeperleyPhase\]. This is because the phenomenological form for the temperature dependence of the Lindemann numbers given by Eq. (\[GammaT\]) with $n=6$ essentially cancels this effect for the temperatures shown.
In Fig. \[CeperleyPhase\], we also plot as dashed lines the critical temperatures in the classical limit. We see that the quantum suppression of the critical temperatures is significant even for $r_C\sim {\mathcal O}(100)$, where one would naively expect the system to be well within the classical regime. The enhanced importance of quantum fluctuations arises because the classical melting temperatures of the crystal and hexatic phases are so low with $T_m/U(a)=0.0136\ll 1$ and $T_i/U(a)=0.0159\ll 1$. Since quantum softening of the Lamé coefficients sets in for $T/\omega_C\lesssim 1$, and $T/\omega_C=\sqrt{r_C} T/U(a)$ with $T_m/U(a)\ll 1$ in the hexatic and crystal phases, the classical melting temperature is only recovered for $r_C\gg 1$.
In Fig. \[DipolePhase\], with applications to cold dipolar gases in mind, we plot our tentative phase diagram for dipolar charges obtained from the Lindemann criteria using temperature dependent Lindemann numbers given by Eq. (\[GammaT\]) with $n=6$. As for the charged case, quantum effects on the melting are significant even for $r_D\gg 1$ since the classical critical temperatures are so low with $T_m/U(a)\ll 1$ and $T_i/U(a)\ll 1$. As for the case of charged particles, there is no significant increase in the temperature range where the hexatic phase is stable with decreasing $r_D$ due to the chosen temperature dependence of the Lindemann numbers. Again, this phase diagram is only qualitatively reliable but it suggests that the hexatic phase extends well into the quantum regime.
Experimental considerations
===========================
The hexatic phase and other aspects of melting for aligned 2d dipolar systems have been observed with great precision in the classical regime in experiments using colloidal particles with a magnetic moment, confined to an air-water interface. [@Gasser] On the other hand, the quantum analogue of the hexatic phase is yet to be explored. The cold assemblies of dipolar molecules seem well suited to study both the classical and the quantum regimes of 2d melting. The typical dipole moment of these molecules is of the order of one Debye. Taking as an example the recently trapped $^{23}$Na$^{40}$K molecule which has a permanent dipole moment of $d=2.7$ Debye, [@Wu; @DipoleMoment] one gets $r_D\simeq 24$ for an average inter particle spacing of $300$nm, which is well inside the quantum regime as can be seen from Fig. \[DipolePhase\]. The critical temperature for the hexatic phase in the classical regime is $T\simeq 0.2\mu$K for this set of parameters. Even though the critical temperature will be lower in the quantum regime we estimate that the quantum hexatic phase should be within experimental reach once the cooling techniques for the dipolar gases have been optimised. One can furthermore reach much higher critical temperatures using molecules with larger dipole moments such as SrO with $d=8.9$ Debye. The presence of the hexatic phase can be detected by measuring the static structure factor via Bragg spectroscopy which is a well proven experimental probe for quantum gases. [@StamperKurn; @Steinhauer; @Veeravalli] In the hexatic phase, the structure factor will exhibit a six fold symmetry with no sharp peaks similar to what it is shown in Fig. \[ExperimentalFig\]. Recent impressive experiments have reported single atom resolution in optical lattices, [@Sherson; @Bakr] and if one is able to achieve the same resolution with dipolar systems, the hexatic phase can be seen directly by the characteristic presence of lattice defects consisting of tightly bound disclination pairs, i.e., particles with 5 and 7 neighbors respectively. [@NelsonBook]
Conclusions
===========
In this paper, we analysed the stability of the crystal and hexatic phases of 2d systems consisting of either dipoles or charges. The classical elastic coefficients were calculated from the phonon spectra of the triangular crystal, and we then demonstrated how quantum effects decrease these coefficients thereby softening the crystal. Using Lindemann criteria suitably adapted to deal with the large fluctuations in 2d systems, we calculated approximate phase diagrams for the existence of the hexatic and crystal phases, predicting that the hexatic phase is stable to very low temperatures. The relevant Lindemann numbers were extracted from experiments in the classical regime, and from Monte-Carlo calculations for $T=0$. The Lindemann numbers depend strongly on temperature, but they turn out to be essentially the same for the charged and the dipolar system for the same value of $n_0\Lambda_T^2$. This suggests that the two-step melting of the crystal phase with an intermediate hexatic phase is a geometric phenomenon, insensitive to the detailed form of the particle interaction. Finally, we discussed the exciting prospect of finally being able to probe the existence hexatic phase in the quantum regime using ultra-cold dipolar gases.
Acknowledgements
================
It is a pleasure to acknowledge the Kavli Institute for Theoretical Physics, UCSB, where this work began. DRN would like to acknowledge the hospitality of the Niels Bohr Institute, Copenhagen, and the support of the National Science Foundation, through grant DMR 1306367 and through the Harvard Material Research Science and Engineering Center via grant DMR-0820484. GMB would like to acknowledge the support of the Carlsberg Foundation via grant 2011010264 and the Villum Foundation via grant VKR023163.
Ewald summation {#Ewald}
===============
The phonon modes are as usual found by solving the matrix equation [@AshcroftMermin] $$m\omega^2{\mathbf \epsilon}({\mathbf k})={\mathbf D}({\mathbf k}){\mathbf \epsilon}({\mathbf k})$$ where ${\mathbf D}({\mathbf k})=\sum_{\mathbf R}{\mathbf D}({\mathbf R})\exp(-{\mathbf k}\cdot{\mathbf R})$ is the dynamical matrix with ${\mathbf R}$ the lattice vectors. We have $${\mathbf D}_{ij}({\mathbf R})=E_0\times\begin{cases}
\sum_{{\mathbf R}\neq 0}\left[(n+2)n\frac{R_iR_j}{R^{n+4}}-n\frac{\delta_{ij}}{R^{n+2}}\right]& \mathbf{R}=0\\
n\frac{\delta_{ij}}{R^{n+2}}-(n+2)n\frac{R_iR_j}{R^{n+4}}& \mathbf{R}\neq0
\end{cases}$$ for repulsive power law potentials $V(r)=E_0/r^n$ with $n=1$ and $E_0=Q^2$ for the charged particles, and $n=3$ and $E_0=D^2$ for the dipoles. We can write ${\mathbf D}({\mathbf k})$ as $${\mathbf D}_{ij}({\mathbf k})=E_0\lim_{u\rightarrow 0}\frac{\partial^2}{\partial u_i\partial u_j}\sum_{{\mathbf R}\neq 0}\frac1{|{{\mathbf R}+{\mathbf u}}|^n}(1-e^{-i{\mathbf k}\cdot{\mathbf R}}).
\label{EwaldSum}$$ Using $r^{-n}=(n+1)\pi^{-1/2}\int_0^\infty dy y^{n-1}e^{-r^2y^2}$ with $n=1,3$ and splitting the integral into a short range and a long range part, i.e. $\int_0^\infty dy\ldots=\int_0^{y_0} dy\ldots+\int_{y_0}^\infty dy\ldots$, the sum in Eq. (\[EwaldSum\]) is split into a short range and a long range part, ${\mathbf D}({\mathbf k})={\mathbf D}^<({\mathbf k})+{\mathbf D}^>({\mathbf k})$. Upon defining the function $$\varphi_n(x)=\frac 2{\sqrt\pi}\int_1^\infty dtt^ne^{-tx^2}$$ we get $$\begin{gathered}
{\mathbf D}_{ij}^>({\mathbf k})=E_0\frac{n+1}{2}y_0^{n+2}\sum_{{\mathbf R}\neq 0}(1-e^{-i{\mathbf k}\cdot{\mathbf R}})\nonumber\\
\times\left[2y_0^2R_iR_j\varphi_{n/2+1}(y_0R)-
\delta_{ij}\varphi_{n/2}(y_0R)\right].
\label{EwaldSumMore}\end{gathered}$$ The short range part of the sum is evaluated by Fourier transforming. Writing $\sum_{{\mathbf R}\neq 0}(1-e^{-i{\mathbf k}\cdot{\mathbf R}})\exp(-|{\mathbf R}+{\mathbf u}|^2y^2)=F_1({\mathbf u},y)-\exp({\mathbf k}\cdot{\mathbf u})F_2({\mathbf u},y)$ and Fourier transforming the functions $F_i({\mathbf u},y)$ which have the same periodicity as the lattice yields after some algebra $$\begin{gathered}
{\mathbf D}_{ij}^<({\mathbf k})=E_0\frac{n+1}{4\pi v}y_0^{n-2}\sum_{{\mathbf K}}
\left[({\mathbf K}+{\mathbf k})_i({\mathbf K}+{\mathbf k})_j\right.\nonumber\\
\left.
\times\varphi_{-n/2}(|{\mathbf K}+{\mathbf k}|/2y_0)-
{\mathbf K}_i{\mathbf K}_j\varphi_{-n/2}(|{\mathbf K}|/2y_0)
\right]
\label{EwaldSumLess}\end{gathered}$$ where $v=\sqrt 3 a^2/2$ is the area of the primitive cell of the lattice, and ${\mathbf K}$ are reciprocal lattice vectors. With Eqs. (\[EwaldSumMore\])-(\[EwaldSumLess\]) we have split the expression for ${\mathbf D}({\mathbf k})$ into two fast converging sums. These expressions agree with what is found in Refs. . We pick $y_0=1/a$ for the numerical calculations.
Canonical quantisation of the phonons {#Canonical}
=====================================
For clarity, we briefly discuss how quantum effects on the Lamé coefficients are included via canonical quantisation. In this approach, we introduce the bosonic annihilation operators $\hat b_{{\mathbf k}\sigma}$ for the phonons via $$\hat u_\sigma(\mathbf k)=\frac L{\sqrt{2\rho\omega_\sigma({\mathbf k})}}(\hat b_{{\mathbf k}\sigma}+\hat b^\dagger_{-{\mathbf k}\sigma})$$ where $\sigma=l,t$. Using $\langle\hat b^\dagger_{{\mathbf k}\sigma}\hat b_{{\mathbf k}\sigma}\rangle=[e^{\beta\hbar\omega_\sigma({\mathbf k})}-1]^{-1}$, we obtain $$\begin{gathered}
\langle \hat u_\sigma({\mathbf k})\hat u_\sigma(-{\mathbf k})\rangle=\frac{L^2}{\rho\omega_\sigma({\mathbf k})}(\frac{1}{e^{\beta\omega_\sigma({\mathbf k})}-1}+\frac 1 2)
\nonumber\\
=\frac{L^2}{2\rho\omega_\sigma({\mathbf k})}\coth[\beta\omega_\sigma({\mathbf k})/2)]\end{gathered}$$ which is identical to Eq. (\[Quantumuu\]).
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|
---
address: |
$^{1}$Departamento de F[í]{}sica Te[ó]{}rica, Universidad Complutense, Madrid, Spain.\
$^{2}$Instituto de Matem[á]{}ticas y F[í]{}sica Fundamental, C.S.I.C., Madrid, Spain.\
$^{3}$Max-Planck-Institut für Physik Komplexer Systeme, Dresden, Germany\
$^{4}$Fachbereich Physik, Philipps-Universität Marburg, Marburg, Germany
author:
- 'M.A. Martin-Delgado$^{1}$, G. Sierra$^{2}$, S. Pleutin$^{3}$ and E. Jeckelmann$^{4}$'
title: Matrix Product Approach to Conjugated Polymers
---
50000
Introduction
============
The study of conjugated polymers has been a subject of great interest for over two decades. There are both theoretical and technological reasons for this interest [@review]. On the theoretical side, there exists a controversy within the scientific community over how to explain, understand and describe the photophysics/photochemistry of this class of materials. This controversy is of such a fundamental nature that the solution of the problem might be in a unification of the semiconductor and metal physics with the molecular quantum chemistry. On the technological side, pi-conjugated polymers behave as semiconductors and this has prompted several research groups to investigate the physics of these materials in an effort to determine their potential for improving the performance and efficiency and reducing the cost of light-emitting diodes (LEDs). More recently, they are also considered to make an entrance in the field of photovoltaics, where they could be used as solar cells.
Saturated polymers are long chains of molecules, generally made of carbon with hydrogen on the sides, all attached to one another by single bonds. This constitutes the backbone of the macromolecule. The most relevant feature of these structures is the fact that the bonds are all single bonds or, in other words, that all the binding are of $\sigma$– type. Saturated polymers are then all very insulating; they are not electronically interesting but are known for their flexibility although they are also quite mechanically strong materials. The most familiar of these compounds is the polyethylene.
On the contrary, conjugated polymers show very interesting electronic properties together with remarkable mechanical properties; for instance, they can emit light and conduct electricity [@review]. In these compounds, two of the three $2p$ orbitals on each carbon atom hybridize with the $2s$ orbital to form three $sp^2$ molecular orbitals. These orbitals are responsible for the backbone of the molecular chain; these are the so-called $\sigma$–orbitals. The third carbon orbital is $p_z$ and points perpendicular to the chain. There exist a strong overlapping between nearest-neighbours $p_z$ orbitals so that the corresponding electrons are fully delocalized on the whole molecule; these are the $\pi$ electrons responsible for all the interesting electronic properties of low-energy. For instance, because of these electrons, the linear chain, the polyacetylene, which is considered in this work, is dimerized: its backbone shows an alternance between double and single bonds. Quite generally , despite a huge amount of works, the electronic properties of these compound stay rather controversial [@review].
The delocalization character of the electrons in the $\pi$ molecular orbitals of the conjugate polymer chains led to the introduction of model Hamiltonians to study and predict their electronic properties. The initially simplest possible model is a tight-binding approximation or Hückel model [@huckel] to describe the motion of $\pi$ electrons in a free way. This is a very crude approximation which has been improved in several fashions. One of them is the inclusion of electron-phonon interactions [@ssh; @holstein]. However, this is not enough as the electronic properties of $\pi$-conjugated polymers derive from a true many-body problem were electron-electron interactions are equally important as the electron-phonon interactions. Then, the PPP (Pariser-Parr-Pople) Hamiltonian [@ppp1; @ppp2; @review] is used to model these electron-electron effects in a first approximation without taking into account phonon effects, to make simpler a first analysis of the electronic properties. In the PPP Hamiltonian, the alternating single-double bonds of the backbone polymer structure is realized by means of a dimerization term in the hopping kinetic energy. The most general form of the PPP Hamiltonian reads as follows:
$$\begin{aligned}
H_{PPP} &=& H_{\rm K} + H_{\rm I}, \\
H_{\rm K} &=& - \sum_{\la i,j \ra,\sigma} t ( 1 - \delta (-1)^i)
( c_{i,\sigma}^\dagger c_{j,\sigma} + {\rm h.c.} ) \\
H_{\rm I} &=& U \sum_{i}
(c_{i,\uparrow}^\dagger c_{i,\uparrow} - \frac{n_{\rm el}}{2})
(c_{i,\downarrow}^\dagger c_{i,\downarrow}
- \frac{n_{\rm el}}{2}) \nonumber \\
&+& \sum_{i,j} V(r_{i,j})
(\sum_\sigma c_{i,\sigma}^\dagger c_{i,\sigma} - n_{\rm el})
(\sum_\tau c_{j,\tau}^\dagger c_{j,\tau} - n_{\rm el})\end{aligned}$$
where $H_K$ is the dimerized tight binding kinetic part and $H_I$ represents the Coulomb interactions among the electrons. Here the operators $c_{i,\sigma}$, $c_{i,\sigma}^\dagger$ are standard creation and annihilation operators for $\pi$ electrons at carbon site $i$ with spin $\sigma$. The parameter $t$ is the hopping overlapping integral between the nearest- neighbour carbon atoms, $\delta$ measures the dimerization of the chain, $n_{\rm el}$, is the average number of electrons per site, $U$ is the on-site Coulomb repulsion between the electrons, $r_{i,j}$ is the distance between sites $i$ and $j$ along the chain and $V(r_{i,j})$ is the long range contribution of the Coulomb repulsion.
The PPP Hamiltonian has been the subject of extensive studies using a great variety of techniques such as Hartree-Fock, CI calculations, small cluster exact diagonalization, Quantum Monte Carlo and so on and so forth [@review]. Only recently it has become possible to apply a new numerical technique, the Density Matrix Renormalization Group (DMRG) [@white], which allows us to obtain highly accurate results both for small, intermediate and large polymer chains [@Ramashesa1; @ramasesha; @bb; @yaron; @fano]. These DMRG studies have helped to clarify the correct ordering of excited states in the low energy part of the spectrum which are relevant for the nonlinear spectroscopic experiments.
In this paper we will concentrate on the study of the PPP Hamiltonian and leave the effect of interaction with phonons for future studies. The PPP Hamiltonian has been studied using an excitonic method based on a local description of the polymer in terms of monomers [@pleutin]. The relevant electronic configurations are built on a small number of pertinent local excitations. This has provided a simple and microscopic physical approximate picture of the model. Recently we have extended this local configuration studies using the Recurrent Variational Approach (RVA) method [@RVA-poly] in order to study larger polymer chains in a systematic way while retaining the previous intuitive physical picture. The RVA [@RVA] is a non-perturbative variational method in which one retains a single state as the best candidate for the ground state of the system. This reduction of degrees of freedom is initially done in order to keep the method manageable analytically. The aim of this analytical approach is to try to understand the relevant physical degrees of freedom so that we can figure out what the underlying physics is in a strongly correlated system. This initial analytical goal has been also developed in order to later acquire more numerical precision. To do this, the method becomes more numerical and somehow stands in between an analytical formulation of the DMRG and a numerical one.
This effort of understanding the relevant electronic configurations in conjugated polymers has been also carried out in exact small cluster calculations using excitonic Valence Bond basis [@exciton-basis] for polymer chains of length up to 10 sites, arranged into diatomic ethylene molecules.
A first comparison of RVA results with DMRG gave us promising perspectives to improve these variational calculations [@RVA-poly] by incorporating more local configurations and variational parameters. In this paper we undertake this project by using a Matrix Product ansatz for the ground state (GS) wave functions [@matrix-product]. This ansatz is a variational approach based on first order Recursion Relations (RR’s) instead of second order RR’s as in the RVA [@RSDM]. With this RR’s we construct the GS of the polymer chain in different symmetry sectors based on the 16 local configurations of the diatomic ethylene molecule within the PPP approximation. Thus, the chain is built up by adding one ethylene at each step of the variational process.
This paper is organized as follows. In Section 2 we introduce a Matrix Product ansatz specially adapted for the PPP Hamiltonian. In Section 3 we set up the Recurrent Relations to compute the GS energies in several sectors according to prescribed symmetries. In Section 4 we present variational and DMRG results and make a comparison obtaining a very good agreement between them. Section 5 is devoted to prospects and conclusions.
The Matrix Product Ansatz
=========================
The main idea of the MP method is to generate the ground state (GS) of a quasi-one dimensional system in terms of a set of states $ |\alpha \rangle_N$ generated by the following recursion formula [@matrix-product; @RSDM],
$$|\alpha \rangle_N = \sum_{m, \beta} \;
A_{\alpha, \beta}[m] \; |m \rangle_N \,
|\beta \rangle_{N-1}
\label{1}$$
where $N$ denotes the number of lattice sites and $|m \rangle_N $ is a set of states located at the site $N$. For conjugated polymers each lattice site in (\[1\]) refers to a monomer unit, and hence $|m \rangle_N$ describes the 16 possible states associated to a single monomer. In table 1 we show the basis of local monomer states $|m \rangle_N$ used in our construction. We have adopted a valence bond basis which is more convenient to our purposes although it can be easily related to the exciton-valence bond basis of references [@RVA-poly; @exciton-basis]. The states $|\alpha \rangle_N$ have to be regarded as block states made of intricate combinations of $N$ monomeric states whose structure depends of the MP amplitudes $A_{\alpha, \beta}[m]$, which in fact are the variational parameters of the method. The latter parameters can be made to depend on the step $N$ of the RR, but in the thermodynamic limit one expect them to reach a fixed point value. Below we shall assume the thermodynamic limit, i.e. independence of $A_{\alpha,\beta}[m]$ on $N$, although computations can be done for any finite value of $N$. The choice of the block states $|\alpha \rangle_N$ is mainly dictated by physical considerations and they are characterized by a set of quantum numbers as spin, charge, etc. In the case of conjugated polymers we shall keep 6 block states which are to be thought as the GS’s in the following sectors of the Hilbert space: i) singlet state at half filling with symmetry $^1A^+_g$, ii) singlet state at half filling with symmetry $^1B^-_u$, iii) a spin 1/2 doublet corresponding to making a hole to the half filled GS and iv) a spin 1/2 doublet corresponding to the addition of one electron to the half filled GS. The last two cases iii) and iv) describe localized charge transfer excitations between monomers, which play an important role in the GS of the polymer. In table 2 we give the 6 blocks used in the MP-ansatz.
Altogether we have a total of $6 \times 16 \times 6 = 576$ possible MP amplitudes, but further constraints greatly reduce this number. First of all and without loose of generality one can impose that the block states $|\alpha \rangle_N$ are orthonormal . This is guaranteed, for any value of $N$, by the following normalization conditions on the $A's$,
$$\sum_{m,\beta} A_{\alpha,\beta}[m] \; A_{\alpha',\beta}[m]
= \delta_{\alpha, \alpha'}
\label{2}$$
Moreover, the RR (\[1\]) should preserve the charge and the spin of the states, reflected in the equations,
$$\begin{aligned}
& h_\alpha = h_m + h_\beta & \label{3} \\
& S^z_\alpha = S^z_m + S^z_\beta & \nonumber \end{aligned}$$
where $h_\alpha, h_m, h_\beta$ denote the number of holes and $S^z_\alpha, S^z_m, S^z_\beta$ denote the third component of the spin of the corresponding states.
Finally, we can impose the conservation of the electron-hole and spin-parity symmetries generated by the operators $\hat{J}$ and $\hat{P}$, whose action on a $i^{\rm th}$- monomer is given by [@Ramashesa1],
$$\begin{aligned}
& \hat{J}_i | \circ \rangle = - | \times \rangle, \;\;
\hat{P}_i | \circ \rangle = - | \circ \rangle & \nonumber \\
& \hat{J}_i | \times \rangle = | \circ \rangle, \;\;
\hat{P}_i | \times \rangle = | \times \rangle & \label{4} \\
& \hat{J}_i | \uparrow \rangle = (-1)^{i+1} | \uparrow \rangle,
\;\;
\hat{P}_i |\uparrow \rangle = - | \downarrow \rangle & \nonumber \\
& \hat{J}_i | \downarrow \rangle = (-1)^{i+1} | \downarrow \rangle,
\;\;
\hat{P}_i | \downarrow \rangle = - | \uparrow \rangle & \nonumber \end{aligned}$$
The action of $\hat{J}$ and $\hat{P}$ for a polymer with $N$ units is simply the tensor product of their actions on each monomer. In the eqs.(\[4\]) we use the convention according to which a state with symmetry $^1A^+_g$ has $\hat{J}=\hat{P}=1$, a state with symmetry $^1B^-_u$ has $\hat{J}=-\hat{P}= -1$ while a state with symmetry $^3B^+_u$ has $\hat{J}=-\hat{P}= 1$ ( this differs in an overall sign to that used in [@Ramashesa1]). The labels $A$ and $B$ refer to the reflection symmetry of the polymer, which shall not be imposed explicitely. Both the monomer states $|m\rangle$ and the block states $|\alpha \rangle$ transform as follows under charge-transfer and spin-parity,
$$\begin{aligned}
& \hat{J} |\alpha \rangle = \eta^J_\alpha\; |\alpha_J\rangle,
\;\; \hat{J} |m \rangle = \eta^J_m \; |m_J\rangle & \label{5} \\
& \hat{P} |\alpha \rangle = \eta^P_\alpha \; |\alpha_P\rangle,
\;\; \hat{P} |m \rangle = \eta^P_m \; |m_P\rangle & \nonumber \end{aligned}$$
where $\eta^J_m$ and $\eta^P_m$ can be derived from eqs.(\[4\]), while $\eta^J_\alpha$ and $\eta^P_\alpha$ are the appropriated ones corresponding to the type of block chosen. In eq.(\[5\]) $\alpha_J$ and $m_J$ denote the states obtained after the application of $\hat{J}$ on the states $\alpha$ and $m$ respectively. All these quantities are given in tables 1 and 2. The MP equation (\[1\]) preserves the electron-hole and spin-parity symmetries provided the MP-amplitudes $A_{\alpha,\beta}[m]$ satisfy the following constraints,
$$\begin{aligned}
& A_{\alpha_J,\beta_J}[m_J] = \eta^J_\alpha \eta^J_m \eta^J_\beta
\; A_{\alpha,\beta}[m] & \label{6} \\
& A_{\alpha_P,\beta_P}[m_P] = \eta^P_\alpha \eta^P_m \eta^P_\beta
\; A_{\alpha,\beta}[m] & \nonumber \end{aligned}$$
Imposing the spin and charge conservation (\[3\]), the electron-hole and the spin-parity symmetries (\[6\]) we are left with a total of 62 non vanishing MP-amplitudes $A_{\alpha, \beta}[m]$ out of 576 possible ones. Moreover only 20 of these 62 parameters are independent. In table 3 we give a choice for these parameters in terms of the MP-amplitudes, which we shall call hereafter $x_i ( i=1, \dots, 20)$. Finally, the normalization conditions (\[2\]) yield three more conditions on the set $x_i$ given by,
$$\begin{aligned}
& x_1^2 + x_2^2 + x_3^2 + 4 x_4^2 + 4 x_5^2 = 1 & \nonumber \\
& x_6^2 + x_7^2 + x_8^2 + 4 x_9^2 + 4 x_{10}^2 = 1 & \label{7} \\
& \sum_{i=11}^{20} x_i^2 = 1 & \nonumber\end{aligned}$$
Hence altogether we are left with 17 independent variational parameters $y_j (j=1, \dots, 17)$ which will be determined by minimization of the GS energy. Before we do that it is convenient to parametrized the $x_i$ parameters in terms of the $y_j$ ones (see below). From physical reasons we expect that the most important MP-amplitudes will be given by $x_1 = A_{1,1}[{1}], x_8 = A_{2,1}[{3}] $ and $x_{17} = A_{3,1}[{9}]$. Indeed, $x_1, x_8$ and $x_{17}$ correspond to the addition of a singlet, a local $^1B_u^-$ state, and a bonding spin $1/2$ state to the GS block $|1 \rangle$ , yielding a block state with the same type of symmetry as the monomeric state added. From this observation the parametrization we are looking for is given by
$$\begin{aligned}
& x_1 = s_1 , x_2 = y_1 s_1, x_3 = y_2 s_1 , x_4 = y_3 s_1 , x_5 = y_4 s_1 &
\nonumber \\
& x_6 = y_5 s_2 , x_7 = y_6 s_2, x_8 = s_2 , x_9 = y_7 s_2 ,
x_{10} = y_8 s_2 & \label{8} \\
& x_{11} = y_9 s_3 , x_{12} = y_{10} s_3,
x_{13} = y_{11} s_3 , x_{14} = y_{12} s_3 , x_{15} = y_{13} s_3 &
\nonumber \\
& x_{16} = y_{14} s_3 , x_{17} = s_3,
x_{18} = y_{15} s_3 , x_{19} = y_{16} s_3 , x_{20} = y_{17} s_3 &
\nonumber \\
& s_1 = 1/\sqrt{ 1 + y_1^2 + y_2^2 + 4 y_3^2 + 4 y_4^2} & \nonumber \\
& s_2 = 1/\sqrt{ 1 + y_5^2 + y_6^2 + 4 y_7^2 + 4 y_8^2} & \nonumber \\
& s_3 = 1/\sqrt{ \sum_{j=9}^{17} y_j^2 } & \nonumber \end{aligned}$$
The normalization conditions (\[7\]) are automatically satisfied by the parametrization (\[8\]), which on the other hand is quite convenient for numerical purposes [@RSDM].
If we choose $y_j= 0 ( \forall j)$ then the state $|1 \rangle_N$ generated by (\[1\]) consists in the coherent superposition of singlets bonds on each monomer. On the other hand the RR’s (\[1\]) also contain the Simpson state [@simpson], which is the coherent superposition
$$|{\rm Simpson} \rangle_N = \prod_{n=1}^N \; ( x_1 |1 \rangle_n
+ x_2 |2 \rangle_n )
\label{9}$$
With this state, the dimerized chain is viewed as a simple one-dimensional crystal of ethylene where, moreover, the electron correlations are ignored; this state was the reference state in [@RVA-poly]. It corresponds to $y_1 = x_2/x_1 \neq 0$ and $y_j = 0 ( {\rm \mbox{for}} \;\; j > 2)$
Ground state energy
===================
In this section we shall briefly present the method for finding the GS energy of the MP ansatz whose minimization determines the MP parameters ( see references [@RSDM] for more details on the method).
Conjugated polymers are customarily described by the Pariser-Parr-Pople (PPP) Hamiltonian, however in our study we shall use a simplified version of it given by the U-V Hamiltonian defined as,
$$\begin{aligned}
& H= - t \sum_{i,s} [1 + (-1)^i \delta] (c^\dagger_{i,s} c_{i+1,s}
+ h.c.) & \label{10} \\
& + U \sum_{i} \; n_{i,\uparrow} n_{i,\downarrow} +
V \sum_i (n_i -1) (n_{i+1} -1) & \nonumber\end{aligned}$$
where $c_{i,s}^\dagger$ and $c_{i,s}$ are fermionic creation and destruction operators at the site $i$ and spin $s$, $n_{i,s} = c^\dagger_{i,s} c_{i,s}$ and $n_i = n_{i,\uparrow} + n _{i,\downarrow} $. We shall work in units where the hopping amplitude $t$ is set equal to one. The important parameters are therefore the dimerization $\delta$, the on-site Hubbard coupling $U$ and the nearest neighbour Coulomb interaction $V$. Since we are working in the monomer basis it is convenient to write the Hamiltonian (\[10\]) as follows,
$$\begin{aligned}
& H_N = \sum_{j=1}^N h^{(1)}_j + \sum_{j=1}^{N-1} h^{(2)}_{j,j+1}&
\label{12} \\
& h^{(1)}_j = - t \sum_{s} [1 + \delta] (c^\dagger_{2j-1,s} c_{2j,s}
+ h.c.) + U ( n_{2j-1,\uparrow} n_{2j-1,\downarrow} +
n_{2j,\uparrow} n_{2j,\downarrow} ) & \nonumber \\
& +
V (n_{2j -1}-1) (n_{2j} -1) & \nonumber \\
& h^{(2)}_{j,j-1} = - t \sum_{s} [1 - \delta]
(c^\dagger_{2j,s} c_{2j+1,s}
+ h.c.) +
V (n_{2j} -1) (n_{2j+1} -1) & \nonumber\end{aligned}$$
where $h^{(1)}_j$ is the intramonomer Hamiltonian of the $j^{\rm th}$ monomer and $h^{(2)}_{j,j+1}$ is the intermonomer Hamiltonian coupling the monomers $j$ and $j+1$. $N$ denotes the total number of monomers.
The block states $|\alpha\rangle_N$ belong to different Hilbert spaces of the Hamiltonian (\[12\]), therefore the vacuum expectation value of $H_N$ will be diagonal with entries,
$$E^{N}_\alpha = _N\langle \alpha| H_N | \alpha \rangle_N
\label{13}$$
The RR (\[1\]) yields a RR for these energies given by [@RSDM]
$$E^{(N+1)}_\alpha = \sum_{\beta} \; T_{\alpha, \beta} E^{(N)}_\beta
+ \widehat{h^{(1)}_\alpha} + \widehat{h^{(2)}_\alpha} \label{14}$$
where
$$\begin{aligned}
& T_{\alpha, \beta} = \sum_{m} ( A_{\alpha, \beta}[m])^2 &
\label{15} \\
& \widehat{h^{(1)}_\alpha} = \sum_{m,m',\beta} A_{\alpha,\beta}[m]
A_{\alpha,\beta}[m'] \; \epsilon_1(m,m') & \nonumber \\
& \widehat{h^{(2)}_\alpha} = \sum_{m's,\beta,\beta',\gamma}
A_{\alpha,\beta}[m_1] A_{\alpha,\beta'}[m'_1]
A_{\beta,\gamma}[m_2] A_{\beta',\gamma}[m'_2]
\; \epsilon_2(m_1,m_2;m'_1,m'_2) & \nonumber \end{aligned}$$
$$\begin{aligned}
& \epsilon_1(m,m') = \langle m | h^{(1)} | m' \rangle & \label{16} \\
& \epsilon_2(m_1,m_2;m'_1,m_2) = \langle m_2, m_1 | h^{(2)} |
m'_1, m'_2 \rangle & \nonumber \end{aligned}$$
The last two expressions are the intramonomer, i.e. $\epsilon_1$, and intermonomer, i.e. $\epsilon_2$, matrix elements in the monomer basis, which can be computed either analytically or numerically. For the case of the PPP Hamiltonian, the number of these energy matrix elements (\[16\]) is huge and it is very lengthy the analytical computations of so many quantities. Instead, we have used [*numerical*]{} exact diagonalization techniques in order to compute them numerically once the PPP coupling constants are specified. This numerical coding is divided into two parts: 1) We construct the Hilbert space of states for the one- and two-monomer basis. This is done in a binary notation using a string of bits of length $4$ for the one-monomers and $8$ for the two-monomers. In the first half of each string of bits we encode the spin-up states and in the second half we encode the spin-downs. This representation we call it the [*tensorial basis.*]{} 2) We represent numerically the action of the PPP Hamiltonian in the tensorial basis. This facilitates the computation of the energy matrix elements (\[16\]). Lastly, we perform several change of basis to bring the previous matrix elements to the Valence Bond basis employed in the variational recurrence relations.
The RR (\[14\]) can be iterated to give $E^{N}_\alpha$ once $E^1_\alpha$ is known. Actually, the same is true for eq.(\[1\]) which gives the MP states $|\alpha\rangle_N$ once $|\alpha \rangle_1$ is given. We shall choose as initial states $|\alpha \rangle_1$ the lowest states of the monomer hamiltonian $h^{(1)}$ in the corresponding Hilbert space sector. Hence the computation of $E^1_\alpha$ requires the diagonalization of $\epsilon_1(m,m')$.
Now the procedure goes as follows. Using eq.(\[14\]) we find the value of $E^N_1$ for a given set of variational parameters $y_j$ and look for the lowest possible value. This determines the value of these parameters and correspondingly that of the MP-amplitudes. One also finds in this way the value of the GS energy density per monomer in the thermodynamic limit,
$$e_\infty = \lim_{N \rightarrow \infty} E^N_1/N
\label{17}$$
Results
=======
In figure 1 we present the GS energies per monomer obtained with the MP method outlined above and the DMRG for the cases i) $ U=4,\; V=1, \; 2 \delta= 0.1, 0.3, 0.5, 1.5$ and ii) $ U=3, \; V= 1.2, \; 2 \delta= 0.1, 0.3, 0.5, 1.5$. For small dimerizations the relative error of the MP results as compared with the DMRG is around $4 \%$, while for strong dimerization it is around $2 \%$.
In figure 2 we plot the absolute value of the 20 amplitudes $x_i$ described in table 3 for weak dimerization ($\delta = 0.05$) and strong dimerization ($\delta = 0.75$) and couplings $U= 3, V= 1.2$ in both cases. It is clear from fig. 2 that for strong dimerization the MP state is very close to the Simpson state for the most important amplitudes are $x_1, x_2, x_8$ and $x_{17}$. For weak dimerization we observe a transfer of weight from these parameters to the remaining ones which show that the charge transfer excitations begin to play a more important role. This is specially clear in the behaviour of $x_4$ which involves the monomer configurations $(\circ \uparrow + \uparrow \circ),
(\circ \downarrow + \downarrow \circ),
(\times \uparrow - \uparrow \times),
(\times \downarrow - \downarrow \times)$, which are the typical local CT configurations appearing in the GS. On the contrary the parameter $x_5$ remains very small showing that the monomer configurations $(\circ \uparrow - \uparrow \circ),
(\circ \downarrow - \downarrow \circ),
(\times \uparrow + \uparrow \times),
(\times \downarrow + \downarrow \times)$ are very unlikely in the GS.
These results are encouraging since they show that the MP approach gives a reasonable representation of the GS of the conjugated polymers in terms of a small number of variational parameters. They also show the possible improvements which can be achieved by first rejecting those monomer configurations which have small weight in the GS. One could also include blocks with spin 1 and singlet blocks with degeneracy. The latter type of blocks is needed in order to discuss the interesting crossing between the energy levels $1 ^1B^-_u$ and $ 2 ^1A^+_g$ [@Ramashesa2].
Conclusions
===========
This paper represents the first attempt to generate a MP ansatz of the GS of conjugated polymers. Our results are rather encouraging since they show that we can get new insights and good numerical accuracy by improving the ansatz. Unlike other variational methods the MPM allows for a systematic improvement, becoming eventually exact when keeping a sufficient number of block states. Of course in the latter case the method becomes equivalent to the DMRG one [@DMNS]. The usefulness of the MPM thus lies in a certain compromise between the desired numerical accuracy and the physical insight usually associated with the analytic nature of the method. The MPM also demands much less computing effort, an aspect which is certainly non negligible
.
[**Acknowledgements**]{} We would like to thank useful correspondence with S.K. Pati and conversations with S. Ramasesha at the Max Planck Institute for the Physics of Complex Systems in Dresden during the DMRG98 Seminar/Workshop, at which the present work was initiated.
M.A.M.D. and G.S. acknowledge support from the DIGICYT under contract No. PB96/0906, S.P aknowledges support from the European Commission through the TMR network contract ERBFNRX-CT96-0079 (QUCEX) and support from the DIGICYT under contract No. PB96/0906 which permits his stay in Madrid for a short period.
State $m$ $h$ $2 S^z$ ${m}_J$ $\eta^J_m$ $m_P$ $\eta^P_m$
---------------------------------------------- ----- ----- --------- --------- ------------ ------- ------------
$\bullet - \bullet $ 1 0 0 1 1 1 1
$\times \circ + \circ \times$ 2 0 0 2 1 2 1
$\times \circ - \circ \times$ 3 0 0 3 -1 3 1
$\uparrow \uparrow$ 4 0 2 4 1 6 -1
$\uparrow \downarrow + \downarrow \uparrow $ 5 0 0 5 1 5 -1
$\downarrow \downarrow $ 6 0 -2 6 1 4 -1
$\circ \circ $ 7 2 0 8 -1 7 -1
$\times \times $ 8 -2 0 7 -1 8 -1
$\circ \uparrow + \uparrow \circ$ 9 1 1 15 -1 10 -1
$\circ \downarrow + \downarrow \circ$ 10 1 - 1 16 -1 9 -1
$\circ \uparrow - \uparrow \circ$ 11 1 1 13 -1 12 -1
$\circ \downarrow - \downarrow \circ$ 12 1 - 1 14 -1 11 -1
$\times \uparrow + \uparrow \times$ 13 -1 1 11 1 14 1
$\times \downarrow + \downarrow \times$ 14 -1 - 1 12 1 13 1
$\times \uparrow - \uparrow \times$ 15 -1 1 9 1 16 1
$\times \downarrow - \downarrow \times$ 16 -1 - 1 10 1 15 1
Table 1.- States forming the monomer basis of the MP ansatz. The states given in the first column have to be normalized. $\bullet - \bullet$ represents a singlet valence bond state, $\times $ represents a double occupied site, $\circ$ symbolizes an empty site and $\uparrow, \downarrow$ symbolizes singly occupied sites with spin up and down. $h$ denotes the excess or defect of holes as compared to the half filling situation. $2 S^z$ is twice the third component of the spin. $m_J$ and $m_P$ are the states obtained upon applying the operators $\hat{J}$ and $\hat{P}$ on the monomer state $m$ defined in eqs. (\[4\]). $\eta^J_m$ and $\eta^P_m$ are the corresponding signs appearing in eqs.(\[5\]).
State $\alpha$ $h$ $2 S^z$ ${\alpha}_J$ $\eta^J_\alpha$ $\alpha_P$ $\eta^P_\alpha$
----------------------------------------- ---------- ----- --------- -------------- ----------------- ------------ -----------------
$\bullet - \bullet $ 1 0 0 1 1 1 1
$\times \circ - \circ \times$ 2 0 0 2 -1 2 1
$\circ \uparrow + \uparrow \circ$ 3 1 1 5 -1 4 -1
$\circ \downarrow + \downarrow \circ$ 4 1 - 1 6 -1 3 -1
$\times \uparrow - \uparrow \times$ 5 -1 1 3 1 6 1
$\times \downarrow - \downarrow \times$ 6 -1 - 1 4 1 5 1
Table 2. The notations are as in table 1. The states appearing in the first column are for illustration purposes. They simply show the type of symmetry of the block state as compared with the monomer states defined in table 1.
$x_i$ $\alpha$ $m$ $\beta$ $x_i$ $\alpha$ $m$ $\beta$
---------- ---------- ----- --------- ---------- ---------- ----- ---------
$x_1$ 1 1 1 $x_{11}$ 3 1 3
$x_{2}$ 1 2 1 $x_{12}$ 3 2 3
$x_3$ 1 3 2 $x_{13}$ 3 3 3
$x_4$ 1 9 6 $x_{14}$ 3 4 4
$x_5$ 1 11 6 $x_{15}$ 3 5 3
$x_6$ 2 1 2 $x_{16}$ 3 7 5
$x_7$ 2 2 2 $x_{17}$ 3 9 1
$x_8$ 2 3 1 $x_{18}$ 3 9 2
$x_9$ 2 9 6 $x_{19}$ 3 11 1
$x_{10}$ 2 11 6 $x_{20}$ 3 11 2
Table 3. List of the variational parameters $x_i$ in terms of the MP-amplitudes $A_{\alpha,\beta}[m]$. The total of non vanishing amplitudes $A_{\alpha,\beta}[m]$ is 62. The remaining 42 amplitudes can be computed using eqs.(\[6\]).
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|
---
abstract: 'The total kinetic energy release in the neutron induced fission of $^{235}$U was measured (using white spectrum neutrons from LANSCE) for neutron energies from E$_{n}$ = 3.2 to 50 MeV. In this energy range the average post-neutron total kinetic energy release drops from 167.4 $\pm$ 0.7 to 162.1 $\pm$ 0.8 MeV, exhibiting a local dip near the second chance fission threshold. The values and the slope of the TKE vs. E$_{n}$ agree with previous measurements but do disagree (in magnitude) with systematics. The variances of the TKE distributions are larger than expected and apart from structure near the second chance fission threshold, are invariant for the neutron energy range from 11 to 50 MeV. We also report the dependence of the total excitation energy in fission, TXE, on neutron energy.'
author:
- 'R. Yanez'
- 'L. Yao'
- 'J. King'
- 'W. Loveland'
- 'F. Tovesson'
- 'N. Fotiades'
title: 'The Excitation Energy Dependence of the Total Kinetic Energy Release in $^{235}$U(n,f)'
---
Introduction
============
Most of the energy released in the nuclear fission process appears in the kinetic energy of the fission fragments. A first order estimate of the magnitude of the total kinetic energy release is that of the Coulomb energy of the fragments at scission, i.e., $$V_{Coul}=\frac{Z_{1}Z_{2}e^{2}}{r_{1}+r_{2}}$$ where Z$_{n}$, r$_{n}$ are the atomic numbers and radii of fragments 1 and 2. Recognizing that the fragments are deformed at scission, one can re-write equation 1 as $$TKE=\frac{Z_{1}Z_{2}e^{2}}{1.9(A_{1}^{1/3}+A_{2}^{1/3})}$$ where the coefficient 1.9 (instead of the usual 1.2 - 1.3) represents the fragment deformation. For symmetric fission, Z$_{1}$=Z$_{2}$=Z/2 and A$_{1}$ =A$_{2}$=A/2, then we have $$TKE = (0.119)\frac{Z^{2}}{A^{1/3}}MeV$$ Trajectory calculations [@raja] for alpha particle emission in fission have shown that the fission fragments are in motion at scission with a pre-scission kinetic energy of 7.3 MeV and an additive term representing this motion is needed. Thus we have the “Viola systematics" [@vic] that say $$TKE = (0.1189\pm 0.0011)\frac{Z^{2}}{A^{1/3}}+7.3(\pm 1.5)MeV$$
The deformed scission point fragments will contract to their equilibrium deformations and the energy stored in deformation will be converted into internal excitation energy. Thus we can define a related quantity, the total excitation energy , TXE, in fission as $$TXE=Q-TKE$$where Q is the mass-energy release. One quickly realizes that these quantities depend on the mass split in fission which in turn, at low excitation energies, may reflect the fragment nuclear structure. The TXE is the starting point for calculations of the prompt neutron and gamma emission in fission, the yields of beta emitting fission fragments, reactor anti-neutrino spectra, etc. As such, it is a fundamental property of all fissioning systems and sadly not very well known.
As a practical matter, one needs to know the dependence of the TKE and TXE on neutron energy for the neutron induced fission of technologically important actinide fissioning systems like $^{233}$U(n,f),$^{235}$U(n,f), and $^{239}$Pu(n,f). The first question we might pose is whether the TKE should depend on the excitation energy of the fissioning system. Does the energy brought in by an incident neutron in neutron induced fission appear in the fragment excitation energy or does it appear in the total kinetic energy? In a variety of experiments, one finds that increasing the excitation energy of the fissioning system does not lead to significant increases in the TKE of the fission fragments or changes in the fragment separation at scission. [@VH]. However, there may be more subtle effects that render this statement false in some circumstances. For example, we expect, on the basis of the Coulomb energy systematics given above, that the TKE will be proportional to changes in the fission mass splits which in turn can depend on the excitation energy.
For the technologically important reaction $^{235}$U(n,f), Madland [@dave] summarizes the known data [@straede; @meadows; @muller]with the following equations $$\left\langle T_{f}^{tot}\right\rangle =\left( 170.93\pm 0.07\right) -\left(
0.1544\pm 0.02\right) E_{n}(MeV)$$ $$\left\langle T_{p}^{tot}\right\rangle =\left( 169.13\pm 0.07\right) -\left(
0.2660\pm 0.02\right) E_{n}(MeV)$$ where E$_{n}$ is the energy of the incident neutron and T$_{f}^{tot}$and T$_{p}^{tot}$ are the average total fission fragment kinetic energy (before neutron emission) and the average fission product kinetic energy after neutron emission, respectively. These quantities are related by the relation $$\left\langle T_{p}^{tot}(E_{n}\right\rangle =\left\langle
T_{f}^{tot}(E_{n}\right\rangle \left[ 1-\frac{\overline{\nu _{p}}(E_{n})}{2A}%
\left( \frac{\left\langle A_{H}\right\rangle }{\left\langle
A_{L}\right\rangle }+\frac{\left\langle A_{L}\right\rangle }{\left\langle
A_{H}\right\rangle }\right) \right]$$ These data show a modest decrease in TKE with increasing excitation energy for the neutron energy interval E$_{n}$ =1-9 MeV. There is no clearly identified changes in the TKE values near the second chance fission threshold, a feature that is important in semi-empirical models of fission such as represented by the GEF code.[@khs]
In this paper, we report the results of measuring the total kinetic energy release in the neutron induced fission of $^{235}$U for neutron energies E$_{n}$ = 3.2 -50 MeV. The method used for the measurement is the 2E method, i.e., measurement of the kinetic energies of the two coincident fission products using semiconductor detectors. The time of flight of the neutrons inducing fission was measured, allowing deduction of their energy. The details of the experiment are discussed in Section II while the experimental results and a comparison of the results with various models and theories is made in Section III with conclusions being summarized in Section IV.
Experimental
============
This experiment was carried out at the Weapons Neutron Research Facility (WNR) at the Los Alamos Neutron Science Center (LANSCE) at the Los Alamos National Laboratory [@Lis; @Liso]. “White spectrum" neutron beams were generated from an unmoderated tungsten spallation source using the 800 MeV proton beam from the LANSCE linac. The experiment was located on the 15R beam line (15$^{\circ}$-right with respect to the proton beam). The calculated (MCNPX) “white spectrum " at the target position is shown in figure 1. [@snow] The proton beam is pulsed allowing one to measure the time of flight (energy) of the neutrons arriving at the experimental area.
A schematic diagram of the experimental apparatus is shown in figure 2. The neutron beam was collimated to a 1 cm diameter at the entrance to the experimental area. At the entrance to the scattering chamber, the beam diameter was measured to be 1.3 cm. A fission ionization chamber [@steve] was used to continuously monitor the absolute neutron beam intensities. The $^{235}$U target and the Si PIN diode fission detectors were housed in an evacuated, thin-walled aluminum scattering chamber. The scattering chamber was located $\sim$ 3.1 m from the collimator, and $\sim$ 11 m from the neutron beam dump. The center of the scattering chamber was located 16.46 m from the production target.
The $^{235}$U target consisted of a deposit of $^{235}$UF$_{4}$ on a thin C backing. The thickness of the $^{235}$U was 175.5 $\mu$g $^{235}$U/cm$^{2}$ while the backing thickness was 100 $\mu$g/cm$^{2}$. The isotopic purity of the $^{235}$U was 99.91 $\%$. The target was tilted at 50 $^{\circ}$ with respect to the incident beam.
Fission fragments were detected by two arrays of Si PIN photodiodes (Hamamatsu S3590-09) arranged on opposite sides of the beam. The area of the individual PIN diodes was 1 cm$^{2}$. The distance of the detectors from the target varied with angle from 2.60 cm to 4.12 cm. The coincident detector pairs were at approximately 45, 60, 90, 115, and 135 $^{\circ}$. The alpha particle energy resolution of the diodes was 18 keV for the 5475 keV line of $^{241}$Am.
The time of flight of each interacting neutron was measured using a timing pulse from a Si PIN diode and the accelerator RF signal. Absolute calibrations of this time scale were obtained from the photofission peak in the fission spectra and the known flight path geometry.
The energy calibration of the fission detectors was done with a $^{252}$Cf source. We have used the traditional Schmitt method [@hal]. Some have criticized this method especially for PIN diodes. However with our limited selection of detectors, we were unable to apply the methods of [@moz] to achieve a robust substitute for the Schmitt method.
The measured fragment energies have be to be corrected for energy loss in the $^{235}$UF$_4$ deposit and the C backing foil. This correction was done by scaling the energy loss correction given by the Northcliffe-Schilling energy loss tables [@NS] to a measured mean energy loss of collimated beams of light and heavy $^{252}$Cf fission fragments in 100 $\mu$ g/cm$^{2}$ C foils. The scaling factor that was used was a linear function of mass using the average loss of the heavy and light fission fragments as anchor points. The correction factors at the anchor points were 1.24 and 1.45 for the heavy and light fragments, respectively. Similar factors were obtained if the SRIM code [@srim] was used to calculate dE/dx. These large deviation factors from measured to calculated fission fragment stopping powers have been observed in the past [@Knyazheva], and represent the largest systematical uncertainty in the determination of the kinetic energies.
Results and Discussion
======================
The measured average post-neutron emission fission product total kinetic energy release for the $^{235}$U(n,f) reaction(Table 1) is shown in Figure 3 along with other data and predictions [@gunn; @kapoor; @stevenson]. The evaluated post-neutron emission data from Madlund [@dave] are shown as a dashed line while the individual pre-neutron emission measurements of [@muller] are shown as points. The point at E$_{n}$ =14 MeV is the average of [@gunn] and [@stevenson]. The slope of the measured TKE release (this work) is in rough agreement with the previous measurements [@dave] at lower energies. Also shown are the predictions of the GEF model [@khs]. GEF is a semi-empirical model of fission that provides a good description of fission observables using a modest number of adjustable parameters. The dashed line in Figure 1 is a semi-empirical equation (TKE = 171.5 -0.1E\* for E\* $>$ 9 MeV) suggested by Tudora et al. [@tudy] Qualitatively the decrease in TKE with increasing neutron energy reflects the increase in symmetric fission (with its lower associated TKE release) with increasing excitation energy. This general dependence is reflected in the GEF code predictions with the slope of our data set being similar to the predictions of the GEF model but with the absolute values of the TKE release being substantially less.
In Figure 4, we show some typical TKE distributions along with Gaussian representations of the data. In general, the TKE distributions appear to be Gaussian in shape. This is in contrast to previous studies [@PR; @D] which showed a sizable skewness in the distributions.
In Figure 5, we show the dependence of the measured values of the variance of the TKE distributions as a function of neutron energy along with the predictions of the GEF model of the same quantity. The measured variances are larger than expected. At low energies (near the second chance fission threshold) the observed variances show a dependence on neutron energy similar to that predicted by the GEF model, presumably reflecting the changes in variance with decreasing mass asymmetry. At higher energies (11-50 MeV) the variances are roughly constant with changes in neutron energy. Models [@poop] would suggest that most of the variance of the TKE distribution is due to fluctuations in the nascent fragment separation at scission. The constancy of the variances is puzzling.
Using the Q values predicted by the GEF code, one can make a related plot (Fig. 6) of the TXE values in the $^{235}$U(n,f) reaction. The “bump" in the TXE at lower neutron energies is pronounced and the dependence of the TXE upon neutron energy agrees with the GEF predictions although the absolute values are larger.
Conclusions
===========
We conclude that : (a) For the first time, we have measured the TKE release and its variance for the technologically important $^{235}$U(n,f) reaction over a large range of neutron energies (3.2 - 50 MeV). (b) The dependence of the TKE upon E$_{n}$ seems to agree with semi-empirical models although the absolute value does not. (c) Understanding the variance and its energy dependence for the TKE distribution remains a challenge.
This work was supported in part by the Director, Office of Energy Research, Division of Nuclear Physics of the Office of High Energy and Nuclear Physics of the U.S. Department of Energy under Grant DE-FG06-97ER41026. One of us (WL) wishes to thank the \[Department of Energy’s\] Institute for Nuclear Theory at the University of Washington for its hospitality and the Department of Energy for partial support during the completion of this work. This work has benefited from the use of the Los Alamos Neutron Science Center at the Los Alamos National Laboratory. This facility is funded by the U. S. Department of Energy under DOE Contract No. DE-AC52-06NA25396.
[10]{} P. W. Lisowski, C. D. Bowman, G. J. Russell, and S. A. Wender, Nucl. Sci. Eng. [**106**]{}, 208 (1990). P. W. Lisowski and K. F. Schoenberg, Nucl. Instr. Meth. in Phys. Res. A [**562**]{}, 910 (2006). J.F. Ziegler, M.D. Ziegler b, J.P. Biersack, Nucl. Instr. Meth. Phys. Res. [**B268**]{}, 1818 (2010). G.N. Knyazheva [*et. al*]{}, Nucl. Instr. Meth. Phys. Res. [**B248**]{}, 7 (2006).
E$_{n}$ (MeV) $ \overline{TKE}$(MeV) Uncertainty ($\overline{TKE}$)(MeV)
--------------- ------------------------ -------------------------------------
3.7 167.4 0.7
4.7 165.7 0.8
5.8 167.7 0.8
7.2 166.5 0.8
9.0 166.2 0.8
11.8 165.1 0.7
16.8 163.4 0.7
24.2 162.9 0.7
34.2 161.5 0.8
45.0 162.1 0.8
: Measured TKE release for $^{235}$U(n,f)
![The calculated neutron spectrum in the 15R beam area [@snow][]{data-label="fig1"}](nspec.eps){width="100mm"}
![(Color-online) Schematic diagram of the experimental apparatus. []{data-label="fig2"}](setup.eps){width="100mm"}
![(Color-online) TKE release data for $^{235}$U(n,f) []{data-label="fig3"}](235Uresults-fuf.eps){width="100mm"}
![(Color-online) Typical TKE distributions for $^{235}$U(n.f) []{data-label="fig4"}](tke-dists.eps){width="100mm"}
![(Color-online) Variance of the TKE distribution data for $^{235}$U(n,f) []{data-label="fig5"}](TKEvariance.eps){width="100mm"}
![(Color-online) TXE data for $^{235}$U(n,f) []{data-label="fig6"}](TXE.eps){width="100mm"}
|
---
abstract: |
Theoretical molecular descriptors alias topological indices are a convenient means for expressing in a numerical form the chemical structure encoded in a molecular graph. The structure descriptors derived from molecular graphs are widely used in Quantitative Structure-Property Relationship (QSPR) and Quantitative Structure-Activity Relationship (QSAR).
In this paper, we are interested in the Graovac-Pisanski index (also called modified Wiener index) introduced in 1991 by Graovac and Pisanski, which encounters beside the distances in a molecular graph also its symmetries. In the QSPR analysis we first calculate the Graovac-Pisanski index for different families of hydrocarbon molecules using a simple program and then we show a correlation with the melting points of considered molecules. We show that the melting points of the alkane series can be very effectively predicted by the Graovac-Pisanski index and for the rest of considered molecules (PAH’s and octane isomers) the regression models are different, but we establish some correlation with the melting points for them as well.
author:
- 'Matevž Črepnjak [^1], Niko Tratnik[^2], Petra Žigert Pleteršek [^3]'
title: 'Predicting Melting Points by the Graovac-Pisanski Index'
---
biblabel\[1\][\#1.]{}
INTRODUCTION {#introduction .unnumbered}
============
Theoretical molecular descriptors (also called topological indices) are graph invariants that play an important role in chemistry, pharmaceutical sciences, materials science and engineering, etc. The value of a molecular descriptor must be independent of the particular characteristics of the molecular representation, such as atom numbering or labeling. We model molecules of hydrocarbons by the corresponding molecular graph, where the vertices are the carbon atoms and the edges of the graph are the bonds between them.
One of the most investigated topological indices is the Wiener index introduced in 1947 [@wiener]. This index is defined as the sum of distances between all the pairs of vertices in a molecular graph. Wiener showed that the Wiener index is closely correlated with the boiling points of alkane molecules. In order to take into account also the symmetries of a molecule, Graovac and Pisanski in 1991 introduced the modified Wiener index [@graovac]. However, the name modified Wiener index was later used for different variations of the Wiener index and therefore, Ghorbani and Klavžar suggested the name Graovac-Pisanski index[@ghorbani], which is also used in this paper.
Very recently, the Graovac-Pisanski index of some molecular graphs and nanostructures was extensively studied[@ashrafi_koo_diu; @ashrafi_sha; @koo_ashrafi3; @koo_ashrafi; @koo_ashrafi2; @sha_ashrafi]. Moreover, the closed formulas for the Graovac-Pisanski index of zig-zag nanotubes were computed[@tratnik]. The only known connection of the Graovac-Pisanski index with some molecular properties is the correlation with the topological efficiency [@ashrafi_koo_diu1]. Therefore, it was pointed out by Ghorbani and Klavžar [@ghorbani] that the QSPR or QSAR analysis should be performed in order to establish correlation with some other physical or chemical properties of molecules.
On the other hand, there is no known molecular descriptor well correlated to the melting points of molecules, since graph-theoretical abstraction in most cases disregards many information that are relevant for the value of the melting point[@ro-king; @vuk-gas].
The symmetries of a molecule play an important role in the process of melting[@pinal]. The more symmetrical the molecules are, easier it is for them to stack together. Consequently, the fewer spaces there are between them and so the melting point is expected to be higher. Therefore, since the Graovac-Pisanski index considers the symmetries of a molecule, in this paper we investigate its correlation with the melting points of molecules.
We work with the data set of molecules proposed by the International Academy of Mathematical Chemistry, in particular alkane series, polyaromatic hydrocarbons (PAH’s) and octane isomers. We show that for alkane series the melting point is very well correlated with the Graovac-Pisanski index and for PAH’s the correlation is a little bit weaker. For octane isomers, the melting points are good correlated with the number of symmetries. However, we conclude that for alkanes the size of a molecule contributes the most to its melting point.
THE GRAOVAC-PISANSKI INDEX {#the-graovac-pisanski-index .unnumbered}
==========================
A *graph* $G$ is an ordered pair $G = (V, E)$ of a set $V$ of *vertices* (also called nodes or points) together with a set $E$ of *edges*, which are $2$-element subsets of $V$ (more information about some basic concepts in graph theory can be found in a book written by West[@west]). Having a molecule, if we represent atoms by vertices and bonds by edges, we obtain a *molecular graph*.
The graphs considered in this paper are all finite and connected. The [*distance*]{} $d_G(x,y)$ between vertices $x$ and $y$ of a graph $G$ is the length of a shortest path between vertices $x$ and $y$ in $G$ (we often use $d(x,y)$ for $d_G(x,y)$).
The [*Wiener index*]{} of a graph $G$ is defined as $\displaystyle{W(G) = \frac{1}{2} \sum_{u \in V(G)} \sum_{v \in V(G)} d_G(u,v)}$. Moreover, if $S \subseteq V(G)$, then $\displaystyle{W(S) = \frac{1}{2} \sum_{u \in S} \sum_{v \in S} d_G(u,v)}$.
An *isomorphism of graphs* $G$ and $H$ with $|E(G)|=|E(H)|$ is a bijection $f$ between the vertex sets of $G$ and $H$, $f: V(G)\to V(H)$, such that for any two vertices $u$ and $v$ of $G$ it holds that if $u$ and $v$ are adjacent in $G$ then $f(u)$ and $f(v)$ are adjacent in $H$. When $G$ and $H$ are the same graph, the function $f$ is called an *automorphism* of $G$. The composition of two automorphisms is again an automorphism, and the set of automorphisms of a given graph $G$, under the composition operation, forms a group ${{{\rm Aut}}}(G)$, which is called the *automorphism group* of the graph $G$.
The *Graovac-Pisanski index* of a graph $G$, $GP(G)$, is defined as $$GP(G) = \frac{|V(G)|}{2 |{{{\rm Aut}}}(G)|} \sum_{u \in V(G)} \sum_{\alpha \in {{{\rm Aut}}}(G)} d_G(u, \alpha(u)).$$
Next, we mention some important concepts of group theory. If $G$ is a group and $X$ is a set, then a *group action* $\phi$ of $G$ on $X$ is a function $\phi :G \times X \to X$ that satisfies the following: $\phi(e,x) = x$ for any $x \in X$ (where $e$ is the neutral element of $G$) and $\phi(gh,x)=\phi(g,\phi(h,x))$ for all $g,h \in G$ and $x \in X$. The *orbit* of an element $x$ in $X$ is the set of elements in $X$ to which $x$ can be moved by the elements of $G$, i.e. the set $\lbrace \phi(g,x) \, | \, g \in G \rbrace$. If $G$ is a graph and ${{{\rm Aut}}}(G)$ the automorphism group, then $\phi: {{{\rm Aut}}}(G) \times V(G) \to V(G)$, defined by $\phi(\alpha,u) = \alpha(u)$ for any $\alpha \in {{{\rm Aut}}}(G)$, $u \in V(G)$, is called the *natural action* of the group ${{{\rm Aut}}}(G)$ on $V(G)$.
It was shown by Graovac and Pisanski[@graovac] that if $V_1, \ldots, V_t$ are the orbits under the natural action of the group ${{{\rm Aut}}}(G)$ on $V(G)$, then $$\label{formula}
GP(G) = |V(G)| \sum_{i=1}^t \frac{1}{|V_i|}W(V_i).$$
As an example, we calculate the Graovac-Pisanski index for one of the octane isomers, more precisely for 2-methyl-3-ethyl-pentane, see Figure \[primer\_drevo\].
![\[primer\_drevo\] Molecular graph $G$ of 2-methyl-3-ethyl-pentane.](primer_drevo.eps){width="0.5\columnwidth"}
We do the calculation in two different ways. First, we calculate directly by the definition. There are all together 3 nontrivial (different from the identity) automorphisms of the considered molecular graph $G$: $$\alpha_1 = (1\ 6)(2)(3)(4)(5)(7)(8),\ \alpha_2 = (1)(2)(3)(4\ 7)(5\ 8)(6),\ \alpha_3 = (1\ 6)(2)(3)(4\ 7)(5\ 8).$$ Therefore, $$\begin{aligned}
GP(G) & = d(1,\alpha_1(1)) + d(6,\alpha_1(6)) + d(4,\alpha_2(4)) + d(7,\alpha_2(7)) + d(5,\alpha_2(5)) + d(8,\alpha_2(8)) \\
& + d(1,\alpha_3(1)) + d(6,\alpha_3(6)) + d(4,\alpha_3(4)) + d(7,\alpha_3(7)) + d(5,\alpha_3(5)) + d(8,\alpha_3(8)) \\
& = 2+2+2+2+4+4+2+2+2+2+4+4 \\
& = 32. \end{aligned}$$
Now, we calculate the index by using orbits. The vertex set is partitioned into 5 orbits $$V_1=\{1,6\},\,V_2=\{2\},\,V_3=\{3\},\,V_4=\{4,7\},\,V_5=\{5,8\}$$ and the Graovac-Pisanski index of 2-methyl-3-ethyl-pentane is calculated as $$GP(G) = |V(G)| \sum_{i=1}^5 \frac{1}{|V_i|}W(V_i)
=8\left( \frac{2}{2}+\frac{2}{2}+\frac{4}{2}\right)=32\,.$$
COMPUTATIONAL DETAILS {#computational-details .unnumbered}
=====================
In this section we present an algorithm which was used to compute the Graovac-Pisanski index and the number of automorphisms of a graph. The algorithm contains two special functions, i.e. [calculateAutomorphisms]{} and [calculateDistances]{}.
Let $G$ be a graph represented by a adjacency matrix with vertices $1,2,\ldots,n$. The function [calculateAutomorphisms]{} determines all the automorphisms of graph $G$ and saves them in a set $A$. One possibility is to go through all the permutations of the set $\lbrace 1,2,\ldots, n \rbrace$ and check whether it is an automorphism. Note that for the graph automorphism problem (which is the problem of testing whether a graph has a nontrivial automorphism) it is still unknown whether it has a polynomial time algorithm or it is NP-complete[@lubiw].
The function [calculateDistances]{} computes the matrix $M$ from the adjacency matrix of $G$. The element $M_{i,j}$, $i,j \in \lbrace 1,2,\ldots, n \rbrace$ of $M$ represents the distance between vertices $i$ and $j$ in $G$. Note that this algorithm is known as Floyd-Warshall algorithm [@floyd] and has the time complexity $O(n^3)$.
\[alg:edini\]
$A \leftarrow$ ($G$) $M \leftarrow$ ($G$) $X \leftarrow 0$
$GP(G) \leftarrow \frac{n}{2|A|}X$ $|{{\rm Aut}}(G)| \leftarrow |A|$
We notice that the time complexity of the algorithm is not polynomial if we go through all the permutations. However, for many chemical graphs the set of all the automorphisms of a graph can be easily obtained by hand. In such a case, we can skip the first line of the algorithm and consequently, it becomes much more efficient.
RESULTS AND DISCUSSION {#results-and-discussion .unnumbered}
======================
Alkanes {#alkanes .unnumbered}
=======
In this section, we have 31 alkane molecules, which are divided into two sets. The training set contains 26 molecules and the test set has 5 molecules (in Table \[tab1\] the molecules in the test set are written in bold style).
The data show that a logarithmic function $f(x)= a \ln x + b$ fits the best among all elementary functions. After performing nonlinear regression on the training set, we obtain $a=34,196$ and $ b=68,575$, see Figure \[figu1\]. Therefore, the following equation can be used to predict the melting points of the alkane series $$MP = 34,\!196 \ln GP + 68,\!575.$$
![\[figu1\] Nonlinear regression on the training set.](alkani_nelin.eps){width="0.9\columnwidth"}
We use the above formula to compute the predicted melting point $\widehat{MP}$ for every molecule from the test set, see Table \[tab2\]. From Table \[tab2\] we can see that the average error on the test set is less than 2%. Moreover, the statistics shows very good correlation since $R^2=0,\!9847$.
To conclude the section, we test the obtained formula on all the molecules from Table \[tab1\] and we obtain Table \[tab3\]. We can see that the average error is around 4%. Figure \[figu2\] shows the comparison between the melting points and predicted melting points.
![\[figu2\] Melting points and predicted melting points for alkane molecules.](alkani_mel_predmel.eps){width="0.9\columnwidth"}
Polyaromatic hydrocarbons (PAH’s) {#polyaromatic-hydrocarbons-pahs .unnumbered}
=================================
In this section, we consider 20 PAH molecules. In the first part, we divide these molecules into two sets (the training set contains 16 molecules and the test set has 4 molecules) and perform linear regression with respect to the Graovac-Pisanski index. In the second part, we improve the correlation by performing multilinear regression with respect to the Graovac-Pisanski index, the Wiener index and the number of automorphisms. The data for the PAH molecules is collected in Table \[tab4\]. The linear regression results in the function (see Figure \[figu3\]) $$MP = 0,\!6501 \, GP + 10,\!926.$$
![\[figu3\] Linear regression on the training set for PAH’s.](pah_gp_mel.eps){width="0.9\columnwidth"}
We use the above formula to compute the predicted melting point $\widehat{MP}$ for every molecule from the test set, see Table \[tab5\]. From Table \[tab5\] we can see that the average error on the test set is around 11% and the statistics shows quite good correlation since $R^2=0,\!8388$.
To improve the correlation, we perform multilinear regression on the whole set of PAH’s as described in the beginning of the section. It results in the formula
$$MP = -46,\!248 + 13,\!038\,(\# {\rm Aut}) + 0,\!446\,{GP} + 0,\!235\,{W}.$$
The regression statistics (multiple $R$ is 0,946; $R^2$ is 0,894; adjusted $R^2$ is 0,874; and standard error is 30,665) shows better correlation than the linear regression.
14 octane isomers {#octane-isomers .unnumbered}
=================
As the last one, we consider the set of 14 octane isomers. However, all together there are 18 octane isomers, but for 4 of them, the data for the melting point was unavailable. We compute the Graovac-Pisanski index and the number of automorphisms for these molecules, see Table \[tab6\].
It turns out that the correlation between the Graovac-Pisanski index and the melting point is not that good ($R^2$ is 0,2423), but there is good correlation ($R^2$ is 0,9687) between the number of automorphisms and the melting point, see Figure \[figu4\]. However, we can see that the data for 2,2,3,3-tetramethylbutane is standing out. If we exclude this molecule from our observation, the correlation between the number of automorphisms and the melting points becomes much weaker (close to zero), but the correlation between the Graovac-Pisanski index and the melting points becomes slightly stronger ($R^2$ is 0,4537). Therefore, we can conclude that the size of an alkane molecule contributes the most to its melting point.
![\[figu4\] Linear regression between the number of automorphisms and the melting point for octane isomers.](oktani_num_mel.eps){width="0.9\columnwidth"}
CONCLUSIONS {#conclusions .unnumbered}
===========
In recent years revived Graovac-Pisanki index was considered and calculated in different ways for some families of graphs, such as fullerene graphs or catacondensed benzenoid graphs, but almost no chemical application was known so far. In this paper we use the QSPR analysis and show that this index is well correlated with the melting points of the alkane series and polyaromatic hydrocarbon molecules. Beside that the connection between the number of graph automorphisms of the octane isomers and theirs melting point is established. In some sense this result in similar to the seminal paper in the field of molecular descriptors [@wiener] by Wiener from 1947, where it was shown that the boiling points of the alkane series can be predicted from the Wiener index. Therefore, these results might contribute to further development in the area of obtaining the Graovac-Pisanski index for different molecular graphs.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
---------------
The authors Matevž Črepnjak and Petra Žigert Pleteršek acknowledge the financial support from the Slovenian Research Agency, research core funding No. P1-0285 and No. P1-0297, respectively. The author Niko Tratnik was financially supported by the Slovenian Research Agency.
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Alkane $GP$ $MP \,(K)$
---------------- ------ ------------
ethane 1 91,39
propane 3 85,45
butane 8 134,75
pentane 15 143,35
hexane 27 178,15
heptane 42 182,54
**octane** 64 216,3
nonane 90 220,15
decane 125 245,25
undecane 165 247,15
dodecane 216 263,55
**tridecane** 273 268,15
tetradecane 343 278,65
pentadecane 420 283,05
hexadecane 512 291,15
heptadecane 612 294,15
**octadecane** 729 302,15
nonadecane 855 306,15
icosane 1000 309,85
henicosane 1155 313,65
docosane 1331 315,15
**tricosane** 1518 322,15
tetracosane 1728 325,15
pentacosane 1950 327,15
hexacosane 2197 329,55
heptacosane 2457 332,65
**octacosane** 2744 337,65
nonacosane 3045 336,85
triacontane 3375 338,95
hentriacontane 3720 341,05
dotriacontane 4096 342,15
: \[tab1\] The Graovac-Pisanski index $GP$ and the melting point $MP$ for 31 alkane molecules.
Alkane $GP$ $MP\,(K)$ $\widehat{MP}$ Residual % Residual
------------ ------ ----------- ---------------- ---------- ------------
octane 64 216,3 210,792 5,508 2,546
tridecane 273 268,15 260,396 7,754 2,891
octadecane 729 302,15 293,984 8,166 2,703
tricosane 1518 322,15 319,066 3,084 0,957
octacosane 2744 337,65 339,311 -1,661 0,492
average 1,918
: \[tab2\] Data for 5 alkane molecules in the test set.
Alkane $GP$ $MP\,(K)$ $\widehat{MP}$ Residual % Residual
---------------- ------ ----------- ---------------- ---------- ------------
ethane 1 91,39 68,575 22,815 24,964
propane 3 85,45 106,143 -20,693 24,217
butane 8 134,75 139,684 -4,935 3,661
pentane 15 143,35 161,179 -17,829 12,438
hexane 27 178,15 181,279 -3,129 1,757
heptane 42 182,54 196,388 -13,848 7,586
octane 64 216,3 210,792 5,508 2,546
nonane 90 220,15 222,450 -2,300 1,045
decane 125 245,25 233,684 11,566 4,716
undecane 165 247,15 243,178 3,972 1,607
dodecane 216 263,55 252,388 11,162 4,235
tridecane 273 268,15 260,396 7,754 2,891
tetradecane 343 278,65 268,202 10,448 3,749
pentadecane 420 283,05 275,128 7,922 2,7994
hexadecane 512 291,15 281,901 9,249 3,177
heptadecane 612 294,15 288,002 6,148 2,090
octadecane 729 302,15 293,984 8,166 2,703
nonadecane 855 306,15 299,436 6,714 2,193
icosane 1000 309,85 304,793 5,057 1,632
henicosane 1155 313,65 309,720 3,930 1,253
docosane 1331 315,15 314,570 0,580 0,184
tricosane 1518 322,15 319,066 3,084 0,957
tetracosane 1728 325,15 323,497 1,653 0,509
pentacosane 1950 327,15 327,630 -0,480 0,147
hexacosane 2197 329,55 331,708 -2,158 0,655
heptacosane 2457 332,65 335,533 -2,883 0,867
octacosane 2744 337,65 339,311 -1,661 0,492
nonacosane 3045 336,85 342,870 -6,020 1,787
triacontane 3375 338,95 346,388 -7,438 2,195
hentriacontane 3720 341,05 349,717 -8,667 2,541
dotriacontane 4096 342,15 353,009 -10,859 3,174
average 4,025
: \[tab3\] Results for all 31 alkane molecules.
Molecule \# Aut $W$ $GP$ $MP\,(K)$
---------------------------- -------- ----- ------ -----------
1-methylnaphthalene 1 140 0 -22
2-methylnaphthalene 1 144 0 35
1-ethylnaphthalene 1 182 0 -14
2-ethylnaphthalene 1 190 0 -7
2-6-dimethylnaphthalene 2 186 144 110
2-7-dimethylnaphthalene 2 185 108 97
1-7-dimethylnaphthalene 1 180 0 -14
1-5-dimethylnaphthalene 2 176 132 80
1-2-dimethylnaphthalene 1 178 0 -4
2-3-5-trimethylnaphthalene 1 224 0 25
anthracene 4 279 245 216
1-methylanthracene 1 334 0 86
2-7-dimethylanthracene 2 413 280 241
9-10-dimethylanthracene 4 378 320 183
phenanthrene 2 271 175 101
3-6-dimethylphenanthrene 2 396 256 141
naphtalene 4 109 95 81
1-3-7-trimethylnaphthalene 1 226 0 14
2-6-dimethylanthracene 2 414 336 250
4-5-methylenephenanthrene 2 300 165 116
: \[tab4\] Data for the training set and the test set of PAH’s.
Molecule $GP$ $MP\,(K)$ $\widehat{MP}$ Residual % Residual
--------------------------- ------ ----------- ---------------- ---------- ------------
naphtalene 95 81 72,686 8,315 10,265
1-3-7-trimethylnaphtalene 0 14 10,926 3,074 21,957
2-6-dimethylanthracene 336 250 229,360 20,640 8,256
4-5-methylenephenanthrene 165 116 118,193 -2,193 1,890
average 10,592
: \[tab5\] Predicted melting points on the test set for PAH’s.
Molecule \# Aut $GP$ $MP\,(K)$
--------------------------- -------- ------ -----------
octane 2 64 216,3
2-methyl-heptane 2 8 164,16
3-methyl-heptane 1 0 152,6
4-methyl-heptane 2 48 152
2,2-dimethyl-hexane 6 16 151,97
2,5-dimethyl-hexane 8 64 182
3,3-dimethyl-hexane 2 8 147
2-methyl-3-ethyl-pentane 4 32 158,2
3-methyl-3-ethyl-pentane 6 48 182,2
2,2,3-trimethyl-pentane 6 16 160,89
2,2,4-trimethyl-pentane 12 24 165,8
2,3,3-trimethyl-pentane 4 16 172,22
2,3,4-trimethyl-pentane 8 48 163,9
2,2,3,3-tetramethylbutane 72 56 373,8
: \[tab6\] Data for octane isomers.
[^1]: University of Maribor, Faculty of Natural Sciences and Mathematics, Koroška cesta 160, 2000 Maribor, Slovenia; University of Maribor, Faculty of Chemistry and Chemical Engineering, Smetanova ulica 17, 2000 Maribor, Slovenia; University of Primorska, Andrej Marušič Institute, Muzejski trg 2, 6000 Koper, Slovenia
[^2]: University of Maribor, Faculty of Natural Sciences and Mathematics, Koroška cesta 160, 2000 Maribor, Slovenia
[^3]: University of Maribor, Faculty of Natural Sciences and Mathematics, Koroška cesta 160, 2000 Maribor, Slovenia; University of Maribor, Faculty of Chemistry and Chemical Engineering, Smetanova ulica 17, 2000 Maribor, Slovenia
|
---
abstract: |
An automated magnetometer suitable for long lasting measurement under stable and controllable experimental conditions has been implemented. The device is based on Coherent Population Trapping (CPT) produced by a multi-frequency excitation. CPT resonance is observed when a frequency comb, generated by diode laser current modulation, excites Cs atoms confined in a $\pi/4\times(2.5)^2\times1~\textrm{cm}^3$, 2 Torr $N_2$ buffered cell. A fully optical sensor is connected through an optical fiber to the laser head allowing for truly remote sensing and minimization of the field perturbation. A detailed analysis of the CPT resonance parameters as a function of the optical detuning has been made in order to get high sensitivity measurements. The magnetic field monitoring performances and the best sensitivity obtained in a balanced differential configuration of the sensor are presented.
OCIS [120.4640, 020.1670, 300.6380.]{}
This work has been submitted to JOSA B for publication
\[Josa B (7) 2007\]
address: 'CNISM-Unità di Siena, Dipartimento di Fisica - Università di Siena, via Roma 56, 53100 Siena, Italy'
author:
- 'J.Belfi, G.Bevilacqua, V.Biancalana, Y.Dancheva, L.Moi'
title: All optical sensor for automated magnetometry based on coherent population trapping
---
Introduction
============
Atomic magnetometers, developed since 1960’s [@Blo62], have today a central role in the field of high sensitive magnetometry with important applications in geophysics, medicine, biology, testing of materials and of fundamental physics symmetries. Technical advances reached in the last years, will make optical magnetometers more suitable than SQUIDs in most of these applications because of the comparable - in some cases, even better [@Kom03] - sensitivity and the possibility to operate at room temperature with no need of cryogenic cooling.
Recently a direct measure of geophysical-scale field, with the impressive sensitivity of $60~\mbox {fT}/\sqrt{\mbox {Hz}}$ was demonstrated in a Non-linear Magneto-Optical Rotation experiment [@Aco06].
Detection of weak biological-scale magnetic fields (magneto-cardiometry) has been demonstrated using optically pumped magnetometers [@Wei03a; @Wei03b] based on the ‘double optical-RF excitation’. Direct RF magnetic excitation, anyway, does not permit the realization of fully optical sensors as RF coils have to be placed next to the vapor cell.
All-optical sensors based on Coherent Population Trapping (CPT) effect can be instead built. CPT effect occurs when two long-lived ground states are coupled to a common excited state by two coherent laser fields. When the frequency difference of the laser fields exactly matches the frequency separation of two not coupled ground levels, the population is trapped in the so called ‘dark’ state. This is a quantum-mechanical superposition of both ground states that is not coupled to the laser fields. An accumulation of population in such coherent state gives rise to a resonant transparency [@Alz76; @Ari76]. CPT resonances have line-widths much narrower than the natural line-width of the corresponding optical transitions. This makes them particularly suitable for precision spectroscopy applications in many other fields besides magnetometry [@Scu92; @Fle94] like metrology [@Tho82; @Hem93], detection of gravitational waves [@Cav81], laser physics [@Har89; @Koc92] and laser cooling[@Coh90].
In this work we present the characteristics of an all-optical magnetometer, working as a compact, automated device able to measure magnetic fields in wide range of amplitudes and time-scales, for example the daily Earth magnetic field variations or weak signals varying in the msec time-scales, superposed on the Earth magnetic field. The sensor works in the magnetically polluted environmental conditions typical of a scientific laboratory. Neither additional RF magnetic field excitation nor particular magnetic field shielding are necessary.
The principle of operation of our CPT magnetometer can be described as follows. Couples of laser fields, such that their frequency separation matches the energy splitting between the Zeeman sub-levels of a given hyperfine ground state of Cs, are produced by frequency modulation of the diode laser junction current in the $10$ kHz - $10$ MHz range. The obtained frequency modulated radiation is characterized by a very high modulation index ($\sim$ $10^3$) and the spectral structure of the laser can be seen as a comb of coherent modes with an overall width of the order of the Doppler broadened optical transition. With such a broad band excitation almost all the atomic velocity classes interact with resonant light and furthermore the power absorbed by each single class is very low. The first feature allows us to increase the resonance contrast, the second one reduces the power broadening. It is worth noting that this solution allows us to work without complex laser phase locking systems and, furthermore, without expensive and bulky highly stable microwave oscillators that are, instead, necessary in the case of CPT generation on Zeeman sub-levels of different hyperfine ground states[@Aff02].
Experimental setup and measurement procedure
============================================
A sketch of the experimental setup is presented in Fig.\[setup\]. The sensor measures the resonant absorption of laser light by Cs vapor contained in a cylindrical cell 2.5 cm in diameter and 1 cm in length. The cell is kept at room temperature and contains 2 Torr of $\mbox N_2$ as a buffer gas. $\mbox N_2$ minimizes the multiple, incoherent re-absorption thanks to the quenching of the resonant fluorescence. The laser radiation is tuned to excite the hyperfine transitions between the ground state with total angular momentum $F_g=3$ and the excited $^2P_{3/2}$ states with $F_e=2,
3, 4$.
![Experimental setup. OF: optical fiber, BC: beam collimator, IBS: intensity beam splitter, BE: beam expander, NF: neutral filters, PD: photo diode. \[setup\]](belfiF01.eps){width="7cm"}
The laser is a single-mode edge-emitting pigtail laser ($\lambda$= 852 nm) with 100 mW of laser power and an intrinsic line-width of less than 5 MHz. Optical feedback is avoided by means of 40 dB optical isolator and the laser light is coupled into a 10 m long single-mode polarization-maintaining fiber. The laser head containing the laser chip, the optical isolator and the fiber collimator is closed in a butterfly housing of only $\mbox
40~\textrm{cm}^3$. The beam coming from the fiber is collimated and its polarization is transformed from linear to circular using a quarter-wave plate. In order to increase the light-atom interaction time the beam waist is expanded to 4.3 mm and the laser intensity used for atom excitation is reduced to $36~\mbox{$ \rm \mu W/cm^2$}$ using a set of neutral filters. The transmitted laser light is detected and analyzed.
CPT resonance is created when two different Zeeman sublevels are coherently coupled to a third common Zeeman sublevel. The three-level system involved is called $\Lambda$ system in the case the third common coupled level is the highest in energy, $V$ system in the case it is lowest. In our case a number of $\Lambda$ and $V$ systems are created with circularly polarized light preserving the selection rules and . Chains of $\Lambda$ systems are formed on the $F_g=3 \rightarrow F_e=2,3$ group of transitions, while chains of V systems are formed on the $F_g=3
\rightarrow F_e=4$.
The magnetic field under measurement breaks the Zeeman degeneracy and makes adjacent ($|\Delta m_F|=1$) sublevels separated by $\hbar \mu_0 g_F B = \hbar \omega_L$, where $\hbar$ is the Planck constant, $\mu_0$ is the Bohr magneton, $g_F$ is the Landé factor of the considered ground-state, $\omega_L$ is the Larmor frequency and $B$ is the magnetic field strength. In Fig.\[levels\], as example, is represented the schematics of $\Lambda$-system chains formation for the $F_g=3 \rightarrow
F_e=2$ system of transitions.
![Representation of $\Lambda$-system chains for the $F_g=3
\rightarrow F_e=2$ system. The quantization axis is parallel to the magnetic field and perpendicular to the laser beam. Circular polarization is decomposed in two in-quadrature linearly polarized waves, one of which is in turn decomposed in two counter rotating fields circularly polarized around the quantization axis. The complete scheme would involve also the hyperfine components $F_g=3 \rightarrow F_e=3,4$. In the table are reported the gyromagnetic factors $\gamma$ for all the hyperfine levels of interest. \[levels\]](belfiF02.eps){width="7cm"}
The RF signal used to produce couples of suitably separated frequency components in the spectral profile of the laser emission is swept in a small interval around the resonant value $\omega_L$. A set of data acquired along the frequency sweep allows for visualizing the CPT resonance profile.
As the magnetic field strengths of interest range from few $\mu$T to few mT, the modulation frequency $\Omega_{RF}$ ranges from few tens of kHz to few MHz. The $\Omega_{RF}$ is generated by a waveform generator (Agilent 33250A 80-MHz Function Generator) that is coupled to the laser by means of a passive circuit specially designed to make the response of the laser rather flat in the frequency range of interest. The circuit uses capacitors to ac couple the RF to the laser junction, and resistors of rather large value (several hundreds Ohm) to convert the voltage to current at the output of the generator. A pair of inductive elements, oriented so that possible spurious magnetic pick-up is canceled, are connected in series with the dc supply in order to prevent the modulating signal from being significantly counteracted by the laser current driver. As an overall, this coupling allows for achieving very large modulation indexes. The envelope of the unresolved frequency components of the resulting comb of laser frequencies can be observed using a Fabry-Perot spectrometer (see Fig.\[pigtail\_sp\]).
![Pigtail laser spectrum recorded using a confocal Fabry-Perot interferometer with FSR = 1.5 GHz. The modulation frequency is 105 kHz. The inferred modulation index is .\[pigtail\_sp\].](belfiF03.eps){width="7cm"}
In order to increase the S/N of the detected transmitted light, a Phase Sensitive Detection (PSD) is performed. The $\Omega_{RF}$ is thus frequency-modulated at $\Omega_{PSD}$ and the atomic response in phase with a reference signal at $\Omega_{PSD}$ is extracted. Laser electric field can be expressed by: $$\begin{aligned}
\vec{E}&=&E_0~(\hat{x}+i\hat{y})~\exp\{i[\omega_0t+\varphi(t)+
\\\nonumber&+&
M_{RF}\cos(\Omega_{RF}t+M_{PSD}\cos(\Omega_{PSD}t))]\}+c.c.\end{aligned}$$ where $\hat{x}$ and $\hat{y}$ are the usual unitary vectors perpendicular to the laser wave-vector, $\omega_0$ is the optical frequency, $\varphi(t)$ accounts for the laser linewidth invoking for instance the celebrated phase-diffusion model, $\mbox M_{RF}$ is the modulation index of the RF modulation and $\Omega_{PSD}$ is the PSD modulation frequency with its modulation index $\mbox M_{PSD}$. One typical FM spectrum of the CPT profile is presented in Fig.\[FM\_sp\] where $\Omega_{PSD}=20~\textrm{kHz}$ and $\mbox M_{PSD}=1$. The central feature is used for magnetic field determination.
![FM spectroscopy of the CPT resonance at modulation frequency of 20 kHz with deviation of 20 kHz. \[FM\_sp\].](belfiF04.eps){width="7cm"}
An improvement in the noise rejection and hence in the sensitivity can be achieved using a differential sensor. Such arrangement is very appropriate when a registration of very weak magnetic fields is desired [@Aff02]. In this case, two identical sensors are assembled in parallel, at a distance of 11 cm. The light is coupled to the second arm of the sensor using an intensity beam splitter (see Fig.\[setup\]). When evaluating the efficiency of the differential setup in rejecting the noise, we distinguish three kinds of noise contribution. The first is due to the detection and amplification stages. This contribution, which is generally smaller than the others is increased (nominally by a factor $\sqrt2$). The second kind is due to intensity fluctuations of the laser emission (rather small in our case) and to frequency noise of its optical frequency, in turn discriminated by the Doppler profile. This kind of noise is effectively rejected. As a third kind one can consider the noise due to magnetic field fluctuations generated by magnetic sources different from the one under examination (e.g. consider the case of Earth magnetic field fluctuation while measuring weak biological fields generated by close sources). In this case, provided that both the Cs cells are in conditions of CPT resonance, the differential sensor responds to the (usually very small) gradients of the field generated by far-located sources (while their common mode field is canceled) and to the field generated by sources (if any) located very close to one of the two cells. In this sense depending on the application, the differential setup can be used either as a gradiometer or as detector of field variations produced by close sources placed in magnetically polluted environment.
The components of both arms of the differential sensor, [*i.e.*]{} the fiber collimator, quarter-wave plates, beam expanders, neutral filters, Cs-N$_2$ cells and the PDs (we use large area, low noise, non-magnetic photo diodes) including a reference Cs-vacuum cell are assembled in a separate plate and can be placed away from the instrumentation. All used materials are highly non-magnetic so that the sensor does not perturb the magnetic field to be measured. In the condition of our Laboratory an improvement of the signal to noise ratio (S/N) of a factor of 5 was obtained.
CPT profile and noise {#CPT profile and noise}
---------------------
The magnetic field measurement operatively consists in the determination of the central frequency $\nu_0$ of the resonance profile (2$\pi \nu_0$ is the estimate of $\omega_L$). The registered CPT profile, read at the output of the lock-in, reproduces the first derivative of the CPT (reduced absorption) resonance. One typical CPT profile is presented in Fig.\[CPT\_profile\]. The error bars are evaluated by the lock-in amplifier from the standard deviation of the output signal in steady conditions. Routinely, the noise measurement is done only once - after setting the lock-in operation parameters, because it is a time consuming operation, and can not be performed at each step.
![Typical CPT profile observed when scanning the modulation frequency around $\omega_L/2\pi$. The lock-in time constant is 30 ms with 12 dB/oct output filter which determines the detection bandwidth of 4.16 Hz. The linewidth resulting from the fit procedure is $\Gamma$ = 700 Hz. \[CPT\_profile\].](belfiF05.eps){width="7cm"}
The central frequency and the linewidth of the resonance are estimated by means of a best fit procedure. The fitting function is the first derivative of a Lorentzian profile with central frequency $\nu_0$ and FWHM $\Gamma$. It allows for achieving an excellent agreement of fit with the data detected in the experimental conditions described above, provided that the RF scan is performed in a narrow range around the $\nu_0$. Discrepancies appear for wider ranges of scan. In this condition we found that adding a secondary small, odd function with the same center as the principal derived Lorentzian, removes the discrepancy in the wings, making the fit result insensitive to the position of the center of the resonance with respect to the center of the scan. We used another (smaller and broader) Lorentzian derivative, to take into account such slower decays in the wings.
The uncertainty on the resonant frequency evaluated by the fit procedure, converted in magnetic field units, is consistent with the following estimation: $$\frac{\Delta B}{\sqrt{\Delta \nu}}=
\frac{1}{\gamma} \times
\frac{n\sqrt{\tau}}{\partial V/\partial \nu},
\label{deltaB_min}$$ where $n$ is the noise level at the output of the lock-in, $\tau$ is the measuring time (it depends on both the time-constant and the slope of the output filter in the lock-in), $\partial
V/\partial \nu$ is the slope of the CPT curve and $\gamma$ is the atomic gyromagnetic factor.
The noise level $n$ can be estimated, as written above, by direct measurement from the lock-in amplifier, while the resonance central slope is simply related to the ratio between the amplitude and the FWHM of the signal. In typical working conditions, $n$ amounts to about 3 times the photo-current shot-noise level, and is mainly due to fast laser frequency fluctuations (the measured noise decreases to the expected shot-noise level provided that the optical frequency is tuned out of resonance), while the FWHM line-width is about 700 Hz. The CPT parameters, 1/$\gamma$ factor in our configuration and the noise pattern of our registration system, including the magnetic noise in the Laboratory, set the ultimate sensitivity of the sensor to 260 pT/$\sqrt{\mbox{Hz}}$ according to Eq. \[deltaB\_min\]. This sensitivity limit is far above the very ultimate theoretical limit of the sensitivity, which considering only contribution of the light-atom interaction volume and Cs-Cs spin exchange collisional rate is 8 $\textrm{fT}/\sqrt{\mbox{Hz}}$ [@All02].
In general, in order to improve the sensitivity limit versus the theoretical one, one has to reduce the resonance line-width $\Gamma$, to increase the signal-to-noise ratio and to work in a highly shielded room.
The main CPT resonance broadening mechanism, in our case, is the limited light-atom interaction time and contributes to the total linewidth by an amount of the order of 600 Hz. Unshielded environmental conditions and, in particular, magnetic field gradients and AC magnetic fields, also contribute to the CPT line broadening. The maximum magnetic field gradient is estimated to be in the range of $80~\textrm{nT/cm}$, leading to, worst case, a broadening contribution of the order of 280 Hz. AC magnetic fields contribute, instead, mainly with the 50 Hz and its harmonics spectral components, with an overall intensity of 40 nT, determining a corresponding broadening of about 140 Hz. Smaller broadening contributions are furthermore given by the light shift and the non-linear Zeeman effect [@And03].
PC automated controls
=====================
The experimental procedure is totally controlled by a dedicated LabView program. The program is used to synchronously communicate with the lock-in amplifier and the RF wave generator so that the CPT response can be recorded in appropriate and reproducible conditions.
Automated registration of the CPT profile and subsequent fit
------------------------------------------------------------
The CPT profile is registered by querying the lock-in output after having set the RF and having waited the settling time. Alternatively, as described below, the program performs numerically PSD, using large data sets produced by a 16 bit ADC card. The program also contains routines devoted to an on-line analysis of the collected data. In particular, after that each RF scan is completed, a minimum $\chi^2$ fit procedure is launched to determine the parameters of CPT resonance profile.
When performing numerical PSD, in contrast with what reported in Section \[CPT profile and noise\], the noise level is evaluated at each $\Omega_{RF}$ step. In both cases, we obtain rather good values for the minimized $\chi^2$/DoF (degrees of freedom), demonstrating the reliability of the noise estimation and the suitability of the fitting procedure.
Numerical PSD
-------------
The external lock-in amplifier (bulky and expensive) can be replaced by a compact ADC card. We successfully tested and used a system based on a commercial 16 bit, 50 kS/s card USB interfaced to the PC and sets to operate at 40 kS/s. The principle of operation was slightly changed with respect to the one of the lock-in amplifier. Specifically, the RF is externally frequency modulated at 20 kHz using a square wave signal obtained by scaling by two the frequency of a clock signal generated by the ADC card. Consequently, the ADC data array $y$ corresponds to high and low values of the RF, accumulated in the even ($y_{2i}$) and odd ($y_{2i+1}$) elements, respectively. The PSD signal and its uncertainty are then obtained by considering the $N$-size array of differences $\delta_i=y_{2i+1}-y_{2i}$ and evaluating the average $\langle \delta \rangle= \Sigma_i \delta_i/ N$ and the standard deviation $[ \Sigma_i (\delta_i - \langle \delta
\rangle )^2/ N]^{1/2}$ scaled by $\sqrt {N}$, respectively.
The number $2N$ of the acquired data is chosen accordingly to the desired integration time of the PSD system, and its upper limit is set by the size of the data buffer in the ADC card (64 kB / 2 Bytes = 32767 readings which corresponds to 0.8 sec integration time). It is worth noting that the relatively large amount of data to be transferred makes the choice of USB 1.0 devices not very advantageous, because it introduces a relevant dead-time at each measurement.
Frequency stabilization on the Doppler profile
----------------------------------------------
Multi-frequency diode laser comb excitation is suitable for producing narrow and high contrast CPT signals in free-running lasing conditions. The optical frequency stabilization by the laser current and temperature controllers provides the needed short-term stability. Such passive stabilization method works well over time intervals of the order of 1 min, but over longer time-scales slow drifts of both temperature and current make it unsuitable. For this reason a long-term active stabilization system must be employed, allowing for relatively rough (accurate within some MHz) but reliable re-adjustments of the optical frequency.
We adopted a simple method based on a commercial USB ADC-DAC card with 12 bit resolution, which periodically and automatically (e.g. once per minute) performs a scan all over the Doppler profile, numerically determines center and width of the absorption curve, and finally provides a dc signal, which, sent to the modulation input of the laser current driver, establishes detuning with respect to the maximum of the absorption profile, in terms of the measured linewidth. In spite of the low cost and simplicity, such sub-system was demonstrated to be very effective and reliable for long-term compensation of the optical frequency drifts. Furthermore it made the whole system comfortable to be operated during the optimization as well as suitable for applicative long-lasting use.
Additionally, we note that such approach, at the expense of periodically suspending the CPT measurement for a few seconds, does not need any additional laser modulation (which would have effect on the CPT measurement) or external modulation elements (such as Electro optical modulators), which would make the set-up much more complex and expensive.
We have quantified the root mean square fluctuations $\Delta
\nu_{opt}^{rms}$ of the optical frequency, i.e. the center of the broadband laser spectrum used for CPT creation, observing the apparent fluctuation of the fitted Doppler maximum position when scanning the laser current in the same range of nominal values over the Doppler absorption. From the observed rms variation of the fitted maximum we get $\Delta \nu_{opt}^{rms}\simeq2 \rm{MHz}$ over time scales of the order of 1 sec.
Servo locking at the center of the CPT resonance
------------------------------------------------
When a higher rate in the magnetic field measurements is required the time needed to perform RF scan in order to determine the center of the CPT resonance can be avoided performing single readings of the lock-in output. This fast operation uses the central, essentially linear, slope of the CPT resonance (the Lorentzian derivative profile) resulting from the fit, to convert the lock-in output voltage in frequency units and hence in field units.
Initially, one complete $\Omega_{RF}$ scan is accomplished, the fitting procedure runs, and, provided that the $\chi^2$ /DoF is reasonably close to unity, the best-fit parameters are passed to the routine devoted to evaluate the field from single readings of the lock-in and to keep $\Omega_{RF}$ locked to the resonance center.
The fit procedure gives the values of both the slope $\frac{dV}{d\nu}$ and the offset $V_0$ at the center of resonance $\nu_0$. When the single reading procedure starts working, $\Omega_{RF}$ is set at $\nu_0$, and the lock-in output $V$ is queried. The deviation $\Delta
V=V-V_0$ is used in a linear approximation to obtain a new estimate of the central frequency $\nu_0+\Delta V / (dV/d\nu)$ and hence of the field. At each step, $\Omega_{RF}$ is updated to the new estimated $\nu_0$ in order to keep the system working at the center of the CPT resonance, with the double aim of maintaining the linear estimation appropriate, and preventing possible large drifts of the field from bringing the system out of the CPT resonance.
The lock-in time-constant can be selected with different values for the scan and the single-reading operations. Obviously, to obtain a comparable noise rejection in single-reading operation it is necessary to increase the time-constant. The lock-in settling time derived from the time-constant and the lock-in output filter slope is taken into account in the lock operation in order to update $\Omega_{RF}$ with a rate $R$ allowing for locking the system at the actual center of resonance with the maximum speed, but without risks of oscillations.
$\Omega_{RF}$ is actually updated at the rate $R^{\prime}$ of the readings (this value is limited by the RS232 communications, and generally exceeds $R$) consequently, the $\Omega_{RF}$ increment is scaled by a factor $R/R^{\prime}$.
The evaluated magnetic field is immediately saved on disk, possibly simultaneously with other reference signals (for instance, in the perspective of applications in magneto-cardiography, ECG signals will be saved as a reference, in view of offline analysis to be performed over long-lasting acquisitions).
Signal optimization, performances and limits
============================================
The dependence of the amplitude of the CPT resonance as a function of the laser optical detuning is shown in Fig.\[CPT\_ampl\_LS\]a, where the CPT amplitude, the Doppler broadened fluorescence line and the frequency position of the hyperfine transitions are reported. The CPT resonance amplitude shows a two-lobe structure that reflects the bridge-shaped spectral intensity profile shown in Fig.\[pigtail\_sp\], with the two maxima separated nearly by the same amount (about 600 MHz). The reason for the vanishing resonance at intermediate detunings lies in the opposite phase of the beating at $\Omega_{RF}$ of the FM laser spectrum. Actually, each couple of adjacent sidebands in the laser spectrum, with their amplitudes $J_m$, $J_{m+1}$, which produce a beating signal at $\Omega_{RF}$, contains one odd and one even Bessel function so that, due to the fact that $J_{-m}(M)=(-1)^m J_m(M)$ the beat phases is opposite for the couple (m, m+1) and (-m, -(m+1)) respectively, i.e. for the couples belonging to the right and left wings of the bridge. From this point of view, depending on the optical detuning, a synchronous excitation [@budk_rev02; @Aco06] of different velocity classes of atoms is performed, having a unique phase in the case where one side only of the bridge is in resonance with the Doppler profile, and two opposite phases when the bridge center coincides (about) with the Doppler center. A deeper analysis puts in evidence that the two lobes in Fig.\[CPT\_ampl\_LS\]a are different in amplitude, with higher values on the blue side. This is related to the dominance of the $F_g=3 \to F_e=3$ and $F_g=3 \to
F_e=4$ transitions on the blue wing of the Doppler line. On the other hand, the CPT resonance vanishes when the laser is tuned in the vicinity of the $F_g=3 \to F_e=2$ transition, and the maxima of the two lobes are symmetric with respect to such detuning, accordingly with the fact that this latter, closed transition gives the most relevant contribution to the CPT.
![a) CPT resonance amplitude versus optical detuning (zero frequency corresponds to the maximum of the Doppler profile). The frequency position of the three hyperfine transitions are marked with respect to the calculated Doppler profile. b) CPT HWHM versus optical detuning. []{data-label="CPT_ampl_LS"}](belfiF06.eps){width="7cm"}
Besides the evident variation of the resonance amplitude discussed above, changing the optical detuning determines also a variation both in the resonance linewidth (Fig. \[CPT\_ampl\_LS\] b) and in the resonance center frequency. The linewidth dependence on optical frequency affects mainly the sensitivity of the instrument according with Eq. \[deltaB\_min\]. The shift of the CPT resonance center versus optical detuning, also called light-shift or ac Stark effect, is due to the finite dephasing rate among ground states involved in the CPT preparation (see \[[@Ari96; @Coh92]\]). This effect represents an essential systematic error in the determination of the CPT resonance center and then affects both the accuracy and the sensitivity of the magnetometer. The CPT center frequency shift rate versus optical detuning is presented in Fig.\[CPT\_LS\] for laser intensity of 36 .
![CPT resonance center shift rate depending on optical detuning for laser intensity of 36 . The value given at each point is obtained by averaging the difference in the CPT resonance centers measured 30 MHz above and 30 MHz bellow the corresponding value of the detuning from the maximum absorption. Error bars represent the standard deviation of each data set consisting of about 150 measurements. \[CPT\_LS\]](belfiF07.eps){width="7cm"}
The optimal optical detuning, in view of best magnetometer operation, is around 400 MHz red detuning, where the light-shift rate passes through a minimum value and the CPT resonance has lower linewidth.
Magnetic field monitoring
-------------------------
The magnetometer performance was checked by registration of the Earth magnetic field variation in time. A record of few hours continuous magnetic field registration with our magnetometer [@noiweb] is shown in Fig.\[long\_run\] together with the corresponding data of the L’Aquila geomagnetic station [@aquilaweb].
![Long lasting monitor of the Earth magnetic field, comparison between two different independent measurements. Upper trace: acquisition of the fluctuations of the Earth magnetic field modulus measured by the 3-axes flux-gate magnetometer in the Geophysical Institute of L’Aquila. Sampling rate is one point per minute. Lower trace: acquisition of the modulus of the Earth magnetic field in the Physics Department of the University of Siena. In this case the sampling rate is about 1 point each 8 sec. Both the direction and the strength of the magnetic field are strongly influenced by the presence of ferromagnetic objects in the laboratory and in the structure of the building. \[long\_run\]](belfiF08.eps){width="7cm"}
Both the procedures, when fitting the CPT resonance for determination of its center and using fast acquisition, were considered and investigated. In the first one, the whole CPT profile is registered and fitted as described above. In this case the measuring time, limited by the time necessary for CPT profile registration together with the subsequent data analysis, is of the order of 8 sec. Such procedure does not make possible to register fast magnetic field variations and in this respect can find application in geophysics, archaeology, material science, etc.
A trace of the Earth magnetic field variation obtained in fast operation is shown in Fig.\[lockrun\_zoom\]. In this case the acquisition rate was increased by a factor of 40 (210 ms per reading).
![ Magnetic field variation registration in single reading operation. In the inset is sketched the zoom over 1 min acquisition.\[lockrun\_zoom\] ](belfiF09.eps){width="7cm"}
The magnetometer sensitivity to weak magnetic field variations superimposed on the Earth magnetic field was determined when working in the differential configuration. For this purpose, a calibrated variable magnetic field was applied using multi-turns coil, placed half a meter away from the sensor. The calibration of the magnetic field strengths on the two arms of the sensor produced by the coil was done using the magnetometer itself. The magnetometer response to a slow and weak variation of the magnetic field in time is presented in Fig.\[sensitivity\]. It can be seen that variations in the magnetic field difference of the order of 300 $\mbox{pT}_{\mbox{p-p}}$ are well resolved. The inferred magnetometer sensitivity in differential configuration is $45~\mbox{pT}/\sqrt{\mbox {Hz}}$.
![The magnetometer response to a 300 pT$_{p-p}$, 0.8 Hz square-wave magnetic signal registered in differential configuration with a band-width determined by the lock-in time constant of 3 msec, 12 dB/oct output filter and 10 averages. \[sensitivity\]](belfiF10.eps){width="7cm"}
Conclusions
===========
We have built an all-optical magnetometric sensor supplied with a PC-automated control of the experimental parameters and an absolute magnetic field measurement data acquisition system.
CPT resonance creation by a kHz-range frequency-modulation of a free running diode laser in totally unshielded environmental conditions is demonstrated and an accurate characterization of the optimal experimental parameters relative to the generation of broad band frequency comb spectrum and to the detection strategy is also given. We presented, furthermore, a detailed analysis of the dependence of CPT resonance amplitude and width on the optical frequency tuning, thus determining the optimal detuning from the central frequency of the single-photon absorption spectral profile. The magnetometer performs long term continuous monitoring of magnetic field in the Earth-field range providing a very sensitive tool for small magnetic field variation registration. We plan to routinely and systematically publish such data almost in real-time (preliminary sets are available in Ref.\[[@noiweb]\]), with the aim of making our system useful for remote Earth magnetic field continuous observations and possible comparisons with measurements performed elsewhere.
The best sensitivity, inside a totally unshielded environment, reached in the differential balanced configuration is $45~\mbox
{pT}/\sqrt{\mbox {Hz}}$.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank S. Cartaleva and A. Vicino for useful discussions and A.Barbini for the technical support. This work was supported by the Monte dei Paschi di Siena Foundation.
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http://magnetometer.fisica.unisi.it/lab
http://www.ingv.it/geomag/laquila.htm
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---
abstract: 'We present an algorithm for the efficient sampling of conditional paths of stochastic differential equations (SDEs). While unconditional path sampling of SDEs is straightforward, albeit expensive for high dimensional systems of SDEs, conditional path sampling can be difficult even for low dimensional systems. This is because we need to produce sample paths of the SDE which respect both the dynamics of the SDE and the initial and endpoint conditions. The dynamics of a SDE are governed by the deterministic term (drift) and the stochastic term (noise). Instead of producing conditional paths directly from the original SDE, one can consider a sequence of SDEs with modified drifts. The modified drifts should be chosen so that it is easier to produce sample paths which satisfy the initial and endpoint conditions. Also, the sequence of modified drifts converges to the drift of the original SDE. We construct a simple Markov Chain Monte Carlo (MCMC) algorithm which samples, in sequence, conditional paths from the modified SDEs, by taking the last sampled path at each level of the sequence as an initial condition for the sampling at the next level in the sequence. The algorithm can be thought of as a stochastic analog of deterministic homotopy methods for solving nonlinear algebraic equations or as a SDE generalization of simulated annealing. The algorithm is particularly suited for filtering/smoothing applications. We show how it can be used to improve the performance of particle filters. Numerical results for filtering of a stochastic differential equation are included.'
author:
- |
Panos Stinis\
Department of Mathematics\
University of Minnesota\
Minneapolis, MN 55455
title: Conditional path sampling for stochastic differential equations through drift relaxation
---
Introduction {#introduction .unnumbered}
============
The study of systems arising in different areas, from signal processing and chemical kinetics to econometrics and finance (see e.g. [@bolhuis; @stuart]) often requires the sampling of paths of stochastic differential equations (SDEs) subject to initial and endpoint conditions. While unconditional path sampling of SDEs is straightforward, albeit expensive for high dimensional systems of SDEs, conditional path sampling can be difficult even for low dimensional systems. This is because we need to produce sample paths of the SDE which respect both the dynamics of the SDE and the initial and endpoint conditions. An analogous situation arises in ordinary differential equations, where it can be considerably more difficult to create solutions to boundary value problems than it is to construct solutions to initial value problems (see e.g. Ch. 8 in [@deuflhard]). The problem of conditional path sampling of SDEs has been a subject of active research in recent years and some very interesting approaches have already been developed (see e.g. [@C08; @stuart; @W07]).
The dynamics of a SDE are governed by the deterministic term (drift) and the stochastic term (noise). Instead of producing conditional paths directly from the original SDE, one can consider a sequence of SDEs with modified drifts. The modified drifts should be chosen so that it is easier to produce sample paths which satisfy the initial and endpoint conditions. Also, the sequence of modified drifts converges to the drift of the original SDE. We construct a simple Markov Chain Monte Carlo (MCMC) algorithm which samples, in sequence, conditional paths from the modified SDEs, by taking the last sampled path at each level of the sequence as an initial condition for the sampling at the next level in the sequence.
We have used the drift relaxation algorithm to modify a popular filtering method called particle filter [@doucet]. A particle filter is a sequential importance sampling algorithm which is based on the recursive (online) Bayesian updating of the values of samples (called particles) to incorporate information from noisy observations of the state of a dynamic model. While the particle filter is a very versatile method it may require a very large number of samples to approximate accurately the conditional density of the state of the model. This has led to considerable research (see e.g. [@gilks; @W09]) into how one can modify a particle filter to make it more efficient (see also [@CT09] for a different approach to particle filtering). As an application of the drift relaxation algorithm we show in Section \[particle\_filtering\] how it can be used to construct a more efficient particle filter.
The paper is organized as follows. Section \[drift-relaxation\] presents the drift relaxation algorithm for an SDE conditional path sampling problem. Section \[particle\_filtering\] shows how to use the algorithm to modify a particle filter. Section \[numerical\] contains numerical results for the application of the modified particle filter to the standard example of filtering a diffusion in a double-well potential (more elaborate examples will be presented in [@MS10]). Finally, Section \[discussion\] discusses the results as well as current and future work.
Conditional path sampling and drift relaxation {#drift-relaxation}
==============================================
Suppose that we are given a system of stochastic differential equations (SDEs) $$\label{sdes}
dX_t= a(X_t)dt + \sigma(X_t) dB_t,$$ Suppose also that we want to construct, in the time interval $[0,T],$ sample paths from such that the endpoints are distributed according to the densities $h(X_0)$ and $g(X_T)$ respectively. Equation can be discretized in the interval $[0,T]$ by some numerical approximation scheme [@kloeden]. Suppose that we have discretized the interval $[0,T]$ using a stepsize $\Delta t= T/I.$ To construct conditional paths of we can start by sampling initial conditions from the density $h.$ For each initial condition $X_0$ we can then produce the desired conditional paths by sampling the density $$\label{path-density}
\prod_{i=1}^{I}p(X_{T_i}|X_{T_{i-1}})g(X_T),$$ where $p(X_{T_i}|X_{T_{i-1}})$ is the transition probability from $X_{T_{i-1}}$ at time $T_{i-1}$ to the point $X_{T_i}$ at time $T_i.$ The density can be sampled using MCMC assuming that the transition densities $p(X_{T_i}|X_{T_{i-1}})$ can be evaluated. However, the major issue with the MCMC sampling is whether it can be performed efficiently (see e.g. [@W07; @CT09]). Instead of MCMC sampling directly from the density i.e., starting from an arbitrary initial path and modifying it to become a path corresponding to , we can aid the MCMC sampling process by providing the MCMC sampler of the density with a better initial condition.
To this end, consider an SDE system with modified drift $$\label{sdes-simple}
dY_t= b(Y_t)dt + \sigma(Y_t) dB_t,$$ where $b(Y_t)$ can be suitably chosen to facilitate the conditional path sampling problem.
Also, consider the collection of $L+1$ modified SDE systems $$dY^l_t= (1-\epsilon_l) b(Y^l_t)dt + \epsilon_l a(Y^l_t) dt + \sigma(Y^l_t) dB_t,$$ where $\epsilon_l \in [0,1], \; l=0,\ldots,L,$ with $\epsilon_l < \epsilon_{l+1},$ $\epsilon_0=0$ and $\epsilon_{L}=1.$ Instead of sampling directly a (conditional) path from the SDE , one can sample a path from the modified SDE and gradually morph the path into a path of .
- ($l=0$) Begin with a sample path from the modified SDE .
- Sample through MCMC the density $\prod_{i=1}^{I}p(Y^0_{T_i}|Y^0_{T_{i-1}})g(Y^0_T).$
- For $l=1,...,L$ take the last sample path from the ($l-1$)st SDE and use it as in initial condition for MCMC sampling the density $$\prod_{i=1}^{I}p(Y^l_{T_i}|Y^l_{T_{i-1}})g(Y^l_T)$$ at the $l$th level.
- Keep the last sample path at the $L$th level.
The drift relaxation algorithm is similar to Simulated Annealing (SA) used in equilibrium statistical mechanics [@liu]. However, instead of modifying a temperature as in SA, here we modify the drift of the system.
Application to particle filtering {#particle_filtering}
=================================
We show in this section how the drift relaxation algorithm can be applied to particle filtering with the aim of bringing the samples closer to the observations.
Generic particle filter {#generic}
-----------------------
Suppose that we are given an SDE system and that we also have access to noisy observations $Z_{T_1},\ldots,Z_{T_K}$ of the state of the system at specified instants $T_1,\ldots,T_K.$ The observations are functions of the state of the system, say given by $Z_{T_k}=G(X_{T_k},\xi_k),$ where $\xi_k, k=1,\ldots,K$ are mutually independent random variables. For simplicity, let us assume that the distribution of the observations admits a density $g(X_{T_k},Z_{T_k}),$ i.e., $p(Z_{T_k}|X_{T_k} ) \propto
g(X_{T_k},Z_{T_k}).$
The filtering problem consists of computing estimates of the conditional expectation $E[f(X_{T_k})| \{Z_{T_j}\}^{k}_{j=1}],$ i.e., the conditional expectation of the state of the system given the (noisy) observations. Equivalently, we are looking to compute the conditional density of the state of the system given the observations $p(X_{T_k}|\{Z_{T_j}\}^{k}_{j=1}).$ There are several ways to compute this conditional density and the associated conditional expectation but for practical applications they are rather expensive.
Particle filters fall in the category of importance sampling methods. Because computing averages with respect to the conditional density involves the sampling of the conditional density which can be difficult, importance sampling methods proceed by sampling a reference density $q(X_{T_k}|\{Z_{T_j}\}^{k}_{j=1})$ which can be easily sampled and then compute the weighted sample mean $$E[f(X_{T_k})| \{Z_{T_j}\}^{k}_{j=1}] \approx \frac{1}{N} \sum_{n=1}^N
f(X^n_{T_k})\frac{p(X^n_{T_k}|\{Z_{T_j}\}^{k}_{j=1})}{q(X^n_{T_k}|\{Z_{T_j}\}^{k}_{j=1})}$$ or the related estimate $$\label{importance}
E[f(X_{T_k})| \{Z_{T_j}\}^{k}_{j=1}] \approx \frac{\sum_{n=1}^N
f(X^n_{T_k})
\frac{p(X^n_{T_k}|\{Z_{T_j}\}^{k}_{j=1})}{q(X^n_{T_k}|\{Z_{T_j}\}^{k}_{j=1})}}{\sum_{n=1}^N
\frac{p(X^n_{T_k}|\{Z_{T_j}\}^{k}_{j=1})}{q(X^n_{T_k}|\{Z_{T_j}\}^{k}_{j=1})}},$$ where $N$ has been replaced by the approximation $$N \approx \sum_{n=1}^N \frac{p(X^n_{T_k}|\{Z_{T_j}\}^{k}_{j=1})}{q(X^n_{T_k}|\{Z_{T_j}\}^{k}_{j=1})}.$$ Particle filtering is a recursive implementation of the importance sampling approach. It is based on the recursion $$\begin{aligned}
p(X_{T_k}|\{Z_{T_j}\}^{k}_{j=1}) & \propto g(X_{T_k},Z_{T_k}) p(X_{T_k}|\{Z_{T_j}\}^{k-1}_{j=1}),
\label{correct}\\
\text{where} \;\; p(X_{T_k}|\{Z_{T_j}\}^{k-1}_{j=1}) & = \int p(X_{T_k}|
X_{T_{k-1}})p(X_{T_{k-1}}|\{Z_{T_j}\}^{k-1}_{j=1}) dX_{T_{k-1}}.
\label{update}\end{aligned}$$ If we set $$q(X_{T_k}|\{Z_{T_j}\}^{k}_{j=1})=p(X_{T_k}|\{Z_{T_j}\}^{k-1}_{j=1}),$$ then from we get $$\frac{p(X_{T_k}|\{Z_{T_j}\}^{k}_{j=1})}{q(X_{T_k}|\{Z_{T_j}\}^{k}_{j=1})} \propto g(X_{T_k},Z_{T_k}).$$ The approximation in expression becomes $$\label{particle}
E[f(X_{T_i})| \{Z_{T_j}\}^{k}_{j=1}] \approx \frac{\sum_{n=1}^N
f(X^n_{T_k})g(X^n_{T_k},Z_{T_k})}{\sum_{n=1}^N
g(X^n_{T_k},Z_{T_k})}$$ From we see that if we can construct samples from the predictive distribution $p(X_{T_k}|\{Z_{T_j}\}^{k-1}_{j=1})$ then we can define the (normalized) weights $W^n_{T_k}=
\frac{g(X^n_{T_k},Z_{T_k})}{\sum_{n=1}^N g(X^n_{T_k},Z_{T_k})},$ use them to weigh the samples and the weighted samples will be distributed according to the posterior distribution $p(X_{T_k}|\{Z_{T_j}\}^{k}_{j=1}).$
In many applications, most samples will have a negligible weight with respect to the observation, so carrying them along does not contribute significantly to the conditional expectation estimate (this is the problem of degeneracy [@liu]). To create larger diversity one can resample the weights to create more copies of the samples with significant weights. The particle filter with resampling is summarized in the following algorithm due to Gordon [*et al.*]{} [@gordon].
1. Begin with $N$ unweighted samples $X^n_{T_{k-1}}$ from $p(X_{T_{k-1}}|\{Z_{T_j}\}^{k-1}_{j=1}).$
2. [**Prediction**]{}: Generate $N$ samples $X'^n_{T_k}$ from $ p(X_{T_k}| X_{T_{k-1}}).$
3. [**Update**]{}: Evaluate the weights $$W^n_{T_k}=
\frac{g(X'^n_{T_k},Z_{T_k})}{\sum_{n=1}^N
g(X'^n_{T_k},Z_{T_k})}.$$
4. [**Resampling**]{}: Generate $N$ independent uniform random variables $\{\theta^n\}_{n=1}^N$ in $(0,1).$ For $n=1,\ldots,N$ let $X^n_{T_k}=X'^j_{T_k} $where $$\sum_{l=1}^{j-1}W^l_{T_k} \leq \theta^j < \sum_{l=1}^{j}W^l_{T_k}$$ where $j$ can range from $1$ to $N.$
5. Set $k=k+1$ and proceed to Step 1.
The particle filter algorithm is easy to implement and adapt for different problems since the only part of the algorithm that depends on the specific dynamics of the problem is the prediction step. This has led to the particle filter algorithm’s increased popularity [@doucet]. However, even with the resampling step, the particle filter can still need a lot of samples in order to describe accurately the conditional density $p(X_{T_k}|\{Z_{T_j}\}^{k}_{j=1}).$ Snyder [*et al.*]{} [@snyder] have shown how the particle filter can fail in simple high dimensional problems because one sample dominates the weight distribution. The rest of the samples are not in statistically significant regions. Even worse, as we will show in the numerical results section, there are simple examples where not even one sample is in a statistically significant region. In the next subsection we present how drift relaxation can be used to push samples closer to statistically significant regions.
Particle filter with MCMC step {#MCMC_step}
------------------------------
Several authors (see e.g. [@gilks; @W09]) have suggested the use of a MCMC step after the resampling step (Step 4) in order to move samples away from statistically insignificant regions. There are many possible ways to append an MCMC step after the resampling step in order to achieve that objective. The important point is that the MCMC step must preserve the conditional density $p(X_{T_k}|\{Z_{T_j}\}^{k}_{j=1}).$ In the current section we show that the MCMC step constitutes a case of conditional path sampling.
We begin by noting that one can use the resampling step (Step 4) in the particle filter algorithm to create more copies not only of the good samples according to the observation, but also of the values (initial conditions) of the samples at the previous observation. These values are the ones who have evolved into good samples for the current observation (see more details in [@W09]). The motivation behind producing more copies of the pairs of initial and final conditions is to use the good initial conditions as starting points to produce statistically more significant samples according to the current observation. This process can be accomplished in two steps. First, Step 4 of the particle filter algorithm is replaced by
: Generate $N$ independent uniform random variables $\{\theta^n\}_{n=1}^N$ in $(0,1).$ For $n=1,\ldots,N$ let $(X^n_{T_{k-1}},X^n_{T_k})=(X'^j_{T_{k-1}},X'^j_{T_k}) $where $$\sum_{l=1}^{j-1}W^l_{T_k} \leq \theta^j < \sum_{l=1}^{j}W^l_{T_k}$$ Also, through Bayes’ rule [@W09] one can show that the posterior density $p(X_{T_k}|\{Z_{T_j}\}^{k}_{j=1})$ is preserved if one samples from the density $$g(X_{T_k},Z_{T_k}) p(X_{T_k}|X_{T_{k-1}})$$ where $X_{T_{k-1}}$ are given by the modified resampling step. This is a problem of conditional path sampling for (continuous-time or discrete) stochastic systems. The important issue is to perform the necessary sampling efficiently [@CT09; @W09]. We propose to do that here using drift relaxation (see Section \[drift-relaxation\]). The particle filter with MCMC step algorithm is given by
1. Begin with $N$ unweighted samples $X^n_{T_{k-1}}$ from $p(X_{T_{k-1}}|\{Z_{T_j}\}^{k-1}_{j=1}).$
2. [**Prediction**]{}: Generate $N$ samples $X'^n_{T_k}$ from $ p(X_{T_k}| X_{T_{k-1}}).$
3. [**Update**]{}: Evaluate the weights $$W^n_{T_k}=
\frac{g(X'^n_{T_k},Z_{T_k})}{\sum_{n=1}^N
g(X'^n_{T_k},Z_{T_k})}.$$
4. [**Resampling**]{}: Generate $N$ independent uniform random variables $\{\theta^n\}_{n=1}^N$ in $(0,1).$ For $n=1,\ldots,N$ let $(X^n_{T_{k-1}},X^n_{T_k})=(X'^j_{T_{k-1}},X'^j_{T_k}) $ where $$\sum_{l=1}^{j-1}W^l_{T_k} \leq \theta^j < \sum_{l=1}^{j}W^l_{T_k}$$ where $j$ can range from $1$ to $N.$
5. [**MCMC step**]{}: For $n=1,\ldots,N$ choose a modified drift (possibly different for each $n$). Construct a path for the SDE with the modified drift starting from $X^n_{T_{k-1}}.$ Construct through drift relaxation a Markov chain for $Y^{n}_{T_k}$ with stationary distribution $$g(Y^n,Z_{T_k}) p(Y^n | X^n_{T_{k-1}})$$
6. Set $X^n_{T_k}=Y^{n}_{T_k}.$
7. Set $k=k+1$ and proceed to Step 1.
Since the samples $X^n_{T_k}=Y^{n}_{T_k}$ are constructed by starting from different sample paths, they are independent. Also, note that the samples $X^n_{T_k}$ are unweighted. However, we can still measure how well these samples approximate the posterior density by comparing the effective sample sizes of the particle filter with and without the MCMC step. For a collection of $N$ samples the effective sample size $ess(T_k)$ is defined by $$ess(T_k) = \frac{N}{1+C_k^2}$$ where $$C_k = \frac{1}{W_k}
\sqrt{\frac{1}{N}\sum_{n=1}^N (g(X^n_{T_k},Z_{T_k}) -W_k)^2} \;\; \text{and} \;\; W_k = \frac{1}{N} \sum_{n=1}^N g(X^n_{T_k},Z_{T_k}).$$ The effective sample size can be interpreted as that the $N$ weighted samples are worth of $ess(T_k) = \frac{N}{1+C_k^2}$ i.i.d. samples drawn from the target density, which in our case is the posterior density. By definition, $ess(T_k) \le N.$ If the samples have uniform weights, then $ess(T_k)=N.$ On the other hand, if all samples but one have zero weights, then $ess(T_k)=1.$
Numerical results {#numerical}
=================
We present numerical results of the particle filter algorithm with MCMC step for the standard problem of diffusion in a double-well potential (more elaborate applications of the method will be presented elsewhere [@MS10]). Our objective here is to show how the generic particle filter’s performance can be significantly improved by incorporating the MCMC step via drift relaxation.
The problem of diffusion in a double well potential is described by the scalar SDE $$\label{double}
dX_t=-4X_t(X_t^2-1) + \frac{1}{2} dB_t.$$ The deterministic part (drift) describes a gradient flow for the potential $U(x)=x^4-2x^2$ which has two minima, at $x=\pm1.$ In the notation of Section \[particle\_filtering\] we have $a(X_t)=-4X_t(X_t^2-1)$ and $\sigma(X_t)=\frac{1}{2}.$ If the stochastic term is zero the solution wanders around one of the minima depending on the value of the initial condition. A weak stochastic term leads to rare transitions between the minima of the potential. We have chosen the coefficient $\frac{1}{2}$ to make the stochastic term rather weak. This is done because we plan to enforce the observations to alternate among the minima, and thus check if the particle filter can track these transitions.
The SDE is discretized by the Euler-Maruyama [@kloeden] scheme with step size $\Delta t=10^{-2}$ which is small enough to guarantee stability of the scheme. The initial condition is set to $-1$ and there is a total of $10$ observations at $T_k=k, k=1,\ldots,10.$ The observations are given by $Z_{T_k}=X_{T_k} + \xi_k,$ where $\xi_k \sim N(0,0.01)$ for $k=1,\ldots,10.$ For this choice of observation noise, the observation density (also called likelihood) is given by $$\label{double_observation}
g(X_{T_k},Z_{T_k}) \propto \exp \biggl[- \frac{( Z_{T_k}-X_{T_k})^2}{2*0.01} \biggr]$$ The observations alternate between $1$ and $-1.$ In particular, for $k=1,\ldots,10$ we have $Z_{T_k}=-1$ if $k$ is odd and $Z_{T_k}=1$ if $k$ is even.
In order to apply the MCMC step with drift relaxation we need to define the modified drift $b(Y_t)$ for the process $Y_t$ given by $$\label{double_simple}
dY_t=b(Y_t)+ \frac{1}{2}dB_t.$$ The modified drift can be the same for all the samples or different for each sample. Since the difficulty in tracking the observations comes from the inability of the original SDE to make frequent transitions between the two minima of the double well, an intuitively appealing choice for $b(Y_t)$ is $b(Y_t)=-\alpha4Y_t(Y_t^2-1),$ where $\alpha < 1.$ This drift corresponds to the potential $W(y)=\alpha(y^4-2y^2).$ The potential $W(y)$ has its minima also located at $y = \pm 1.$ However, the value of the potential at the minima is $-\alpha$ instead of $-1$ for the potential $U(x).$ This means that the wells corresponding to the minima of $W(y)$ are shallower than the wells corresponding to the minima of $U(x).$ This makes the transitions between the two wells for the process $Y_t$ more frequent than for the original process $X_t.$ For the numerical experiments we have chosen $\alpha=0.1.$
The sequence of modified SDEs for the drift relaxation algorithm with $L$ levels is given by $$\label{sdes-relaxation}
dY^l_t= (1-\epsilon_l) b(Y^l_t)dt + \epsilon_l a(Y^l_t) dt + \frac{1}{2} dB_t,$$ where $\epsilon_l \in [0,1], \; l=0,\ldots,L,$ with $\epsilon_l < \epsilon_{l+1},$ $\epsilon_0=0$ and $\epsilon_{L}=1.$ For our numerical experiments we chose $L=10$ and $\epsilon_l = l/10.$
Recall that the density we want to sample during the MCMC step is given by $$g(X_{T_k},Z_{T_k}) p(X_{T_k}|X_{T_{k-1}}),$$ where $p(X_{T_k}|X_{T_{k-1}})$ is the transition probability between $X_{T_{k-1}}$ and $X_{T_k}.$ For many applications, sampling directly from $p(X_{T_k}|X_{T_{k-1}})$ may be impossible. Thus, one needs to resort to some numerical approximation scheme which approximates the path between $X_{T_{k-1}}$ and $X_{T_k}$ by a discretized path. However (see [@W09] for details), even the evaluation of the discretized path’s density may not be efficient. Instead, by using the fact that each Brownian path in gives rise to a unique path for $X_t$ [@oksendal], we can replace the sampling of $g(X_{T_k},Z_{T_k}) p(X_{T_k}|X_{T_{k-1}})$ by sampling from the density $$\begin{gathered}
\label{density}
\exp \biggl[- \frac{( Z_{T}-X^n_{T}(\{\Delta B^n_{i} \}_{i=0}^{I-1}))^2}{2*0.01} \biggr] \prod_{i=0}^{I-1} \exp \biggl[ - \frac{(\Delta B^n_{i})^2}{2*\Delta t} \biggr] = \\
\exp \biggl[ - \biggl( \frac{( Z_{T}-X^n_{T}(\{\Delta B^n_{i} \}_{i=0}^{I-1}))^2}{2*0.01} + \sum_{i=0}^{I-1} \frac{(\Delta B^n_{i})^2}{2*\Delta t} \biggr) \biggr]\end{gathered}$$ where $\{\Delta B^n_{i} \}_{i=0}^{I-1}$ are the Brownian increments of the discretized path connecting $X_{T_{k-1}}$ and $X_{T_k}.$ Also, note that the final point $X_{T_k}$ has now become a function of the entire Brownian path $\{\Delta B^n_{i} \}_{i=0}^{I-1}.$ For the numerical experiments we have chosen $\Delta t= \frac{T_k-T_{k-1}}{I}=10^{-2}$ which, since $T_k-T_{k-1}=1,$ gives $I=100.$
We use drift relaxation to produce samples from the density . The Markov chain at each level of the drift relaxation algorithm is constructed using Hybrid Monte Carlo (HMC) [@liu]. At the $l$th level, we can discretize , say with the Euler-Maruyama scheme, and the points on the path will be given by $$Y^{l,n}_{i\Delta t}=Y^{l,n}_{(i-1)\Delta t} + (1-\epsilon_l) b(Y^{l,n}_{(i-1)\Delta t})\Delta t + \epsilon_l a(Y^{l,n}_{(i-1)\Delta t})\Delta t + \frac{1}{2} \Delta B^{l,n}_{i-1},$$ for $i=1,\ldots,I.$ We can use more sophisticated schemes than the Euler-Maruyama scheme for the discretization of the simplified SDE at the cost of making the expression for the density more complicated.
We can define a potential $V_{\epsilon_l}(\{\Delta B^{l,n}_{i} \}_{i=0}^{I-1})$ for the variables $\{\Delta B^{l,n}_{i} \}_{i=0}^{I-1}.$ The potential is given by $$V_{\epsilon_l} \bigl(\{\{\Delta B^{l,n}_{i} \}_{i=0}^{I-1} \bigr)=
\frac{( Z_{T}-Y^{l,n}_{I\Delta t}(\{\Delta B^{l,n}_{i} \}_{i=0}^{I-1}))^2}{2*0.01}
+ \sum_{i=0}^{I-1} \frac{(\Delta B^{l,n}_{i})^2}{2*\Delta t}$$ and the density to be sampled can be written as $$\exp\biggl[-V_{\epsilon_l} \bigl(\{\Delta B^{l,n}_{i} \}_{i=0}^{I-1} \bigr)\biggr].$$ The subscript ${\epsilon_l}$ is to denote the dependence of the potential on the drift relaxation parameter ${\epsilon_l}.$ In HMC one considers the variables on which the potential depends as the position variables of a Hamiltonian system. In our case we have $I$ position variables so we can define a $I$-dimensional position vector $\{q_i\}_{i=1}^{I}.$ The next step is to augment the position variables vector by a vector of associated momenta $\{p_i\}_{i=1}^{I}.$ Together they form a Hamiltonian system with Hamiltonian given by $$H_{\epsilon_l}\biggl(\{q_i\}_{i=1}^{I},\{p_i\}_{i=1}^{I}\biggr)=V_{\epsilon_l}\biggl(\{q_i\}_{i=1}^{I}\biggr) + \frac{p^T p}{2},$$ where $p=(p_1,\ldots,p_{I})$ is the momenta vector. Thus, the momenta variables are Gaussian distributed random variables with mean zero and variance 1. The equations of motion for this Hamiltonian system are given by Hamilton’s equations $$\frac{dq_i}{d\tau}=\frac{\partial H_{\epsilon_l}}{\partial p_i} \;\; \text{and} \;\; \frac{dp_i}{d\tau}=-\frac{\partial H_{\epsilon_l}}{\partial q_i} \;\; \text{for} \;\; i=1,\ldots,I.$$ Note that the Hamiltonian depends also on the initial condition for each sample $Y^n_0$ and we have written an equation for the evolution of $q_1=Y^n_0$ as well its associated momentum $p_1.$ Since the $Y^n_0$ are fixed by the resampling procedure we do not evolve them. However, note that the Brownian increment $\Delta B_0$ needs to be evolved because it affects the evolution of $Y_{\Delta t}.$
HMC proceeds by assigning initial conditions to the momenta variables (through sampling from $\exp(- \frac{p^T p}{2})$), evolving the Hamiltonian system in fictitious time $\tau$ for a given number of steps of size $\delta \tau$ and then using the solution of the system to perform a Metropolis accept/reject step (more details in [@liu]). After the Metropolis step, the momenta values are discarded. The most popular method for solving the Hamiltonian system, which is the one we also used, is the Verlet leapfrog scheme. In our numerical implementation, we did not attempt to optimize the performance of the HMC algorithm. For the sampling at each level of the drift relaxation process we used $10$ Metropolis accept/reject steps and $1$ HMC step of size $\delta \tau = 10^{-2}$ to construct a trial path. A detailed study of the drift relaxation/HMC algorithm for conditional path sampling problems outside of the context of particle filtering will be presented in a future publication.
For the chosen values of the parameters for the drift relaxation and HMC steps, the particle filter with MCMC step is about $500$ times more expensive per sample (particle) than the generic particle filter. However, we show that this increase in cost per sample is worthwhile. Figure \[plot\_comparison\_varying\] compares the performance of the particle filter with MCMC step with $10$ samples and the generic particle filter with $5000$ samples. It is obvious that the particle filter with MCMC step follows accurately all the transitions between the two minima of the double-well. On the other hand, the generic particle filter captures accurately only every other observation. It fails to performs the transitions between the two minima of the double-well. Of course, since the particle filter with MCMC step uses only $10$ samples the conditional expectation estimate of the hidden signal is not as smooth as the estimate of the generic particle filter which uses $5000$ samples. However, the particle filter with MCMC step allows a much better resolution of the conditional density (conditioned on the observations). This can be seen by computing the effective sample size for the two filters. Figure \[plot\_comparison\_ess\_varying\] shows the effective sample size for both filters. Because of the different number of samples used in the two filters we have plotted the effective sample size for each filter as a percentage of the respective sample size. We see that the particle filter with MCMC step has overall a much better sample size than the generic particle filter. The generic particle filter has a wildly fluctuating effective sample size. In particular, since the generic particle filter misses every other observation, the corresponding effective sample size dips down to $1$ sample for the missed observations. Note from Figure \[plot\_comparison\_varying\] that even this one sample which dominates the observation weight does not come close to the observation. For the observations that the generic particle filter does capture, its effective sample size is still lower than the effective sample size of the particle filter with MCMC step.
Discussion
==========
We have presented an algorithm for conditional path sampling of SDEs. The proposed algorithm is based on drift relaxation which allows to sample conditional paths from a modified drift equation. The conditional paths of the modified drift equation are then morphed into conditional paths of the original equation. We have called this process of gradually enforcing the drift of the original equation drift relaxation. The algorithm has been used to create a modified particle filter for SDEs. We have shown that the modified particle filter’s performance is significantly better than the performance of a generic particle filter.
In the current work, we have examined the application of drift relaxation to the filtering problem of diffusion in a double-well potential which is a standard example in the filtering literature. The same algorithm can be applied to the problem of tracking a single target. A problem of great practical interest is that of tracking not only one but multiple moving targets [@mahler; @godsill; @godsill2; @vo]. The multi-target tracking problem is much more difficult than the single-target problem due to the combinatorial explosion of the number of possible target-observation association arrangements. In this context, the accurate tracking of each target becomes crucial. Suppose that only one of the targets is of interest and the rest act as decoys [@mahler2]. The inability to track each potential target accurately can lead to ambiguity about the targets’ movement if the observations for different targets are close. We have already applied the drift relaxation modified particle filter to multi-target tracking problems with very encouraging results which will appear elsewhere [@MS10].
Acknowledgements {#acknowledgements .unnumbered}
================
I am grateful to Profs. V. Maroulas and J. Weare for many helpful discussions and comments. I would also like to thank for its hospitality the Institute for Mathematics and its Applications (IMA) at the University of Minnesota where the current work was completed.
[99]{}
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---
abstract: 'Using the critical Casimir force, we study the attractive-strength dependence of diffusion-limited colloidal aggregation in microgravity. By means of near field scattering we measure both the static and dynamic structure factor of the aggregates as the aggregation process evolves. The simultaneous measurement of both the static and dynamic structure factor under ideal microgravity conditions allows us to uniquely determine the ratio of the hydrodynamic and gyration radius as a function of the fractal dimension of the aggregate, enabling us to elucidate the internal structure of the aggregates as a function of the interaction potential. We find that the mass is evenly distributed in all objects with fractal dimension ranging from 2.55 for a shallow to 1.75 for the deepest potential.'
author:
- 'Marco A. C. Potenza'
- Andrea Manca
- 'Sandra J. Veen'
- Bart Weber
- Stefano Mazzoni
- Peter Schall
- 'Gerard H. Wegdam'
title: Dynamics of colloidal aggregation in microgravity by critical Casimir forces
---
Since long colloidal aggregation processes have been recognized as a class of phenomena that can be described by very general rules, often referred to as the universality of colloidal aggregation. After the fundamentals introduced by Verwey and Overbeek [@Verwey_Overbeek], the experimental discovery that colloidal clusters are endowed with fractal structures [@Weitz84] triggered a lot of work theoretically [@vanDongen85], by simulations [@Meakin85] and experimentally [@experimental] to describe aggregation phenomena that are ubiquitous in nature, foods, and many other consumer products. Colloidal aggregation is central to the formation of gels in systems with short-range attraction [@Lu08]. However, most of our understanding comes from limiting cases of infinitely strong particle attraction, where particles stick irreversibly. The current understanding can be roughly summed up by distinguishing two different regimes: diffusion limited aggregation (DLA) and reaction limited aggregation (RLA), depending on the slowest phenomenon limiting the cluster growth [@vanDongen85].
One of the big challenges lies in understanding the internal structure of the aggregates. Especially at low attractive strength, where particles can detach and rearrange, the situation is not clear. In this case, the internal structure would give important insight into the aggregate growth process and the nature of aggregation for a case that is most relevant for natural aggregation phenomena. As shown by Wiltzius [@Wiltzius87], this internal structure can be addressed by determining both the gyration radius, $R_g$, and the hydrodynamic radius, $R_h$, and their ratio, $\beta$, that is directly related to the density-density correlation function of the aggregate. However, the challenge in the determination of $\beta$ is that it requires simultaneous measurement of both static and dynamic light scattering (SLS and DLS) for large objects, which is a difficult task. Typically, for such large clusters, the Brownian motion becomes very slow and dynamic measurements are overshadowed by large-scale convective motion and sedimentation, making the measurement of hydrodynamic radii prohibitively difficult.
Here, we study the internal structure of colloidal aggregates formed at low to high attractive strength. We exploit data obtained on the International Space Station (ISS), where there is no convection or sedimentation, and pure diffusive motion of the aggregates is guaranteed. We realize an effective attractive potential of controllable strength and range by employing critical Casimir forces [@Bechinger08; @Bonn09]. These attractive forces result from the confinement of critical solvent fluctuations between the particle surfaces; their strength is determined by the correlation length of the solvent and hence by temperature. This effect thus offers an effective potential that adjusts with temperature on a molecular time scale, allowing us to study the aggregation process as a function of the attractive potential. The application of a recently developed method of near field scattering (NFS) provided the unique opportunity to have simultaneously both SLS and DLS measurements at many angles [@Mazzoni13] to determine both $R_g$ and $R_h$. This allowed us to elucidate the structure of the aggregates under ideal conditions and at finite potential. At low attractive strength, particles can be expected to detach and rearrange to form denser structures. Indeed, from the measured static form factors, the fractal dimension $d_f$ was observed to depend upon the interaction strength [@Veen12]. Here, we show how the particle attraction influences the internal structure of the aggregates. We elucidate the internal structure and present experimental results for the dependence of the ratio $\beta=R_h/R_g$ upon the fractal dimension $d_f$. These measurements provide a deeper insight into the influence of the potential on the aggregation process.
The experiment, named COLLOID, operated in the ESA Mirogravity Science Glovebox, under the ELIPS program. Charge stabilized fluorinated latex particles 400 nm in diameter, with a density of 1.6 g/mL and refractive index of $n_p$ =1.37 [@Koenderink01] were suspended in a mixture of 3-methyl pyridine ($3MP$) in water/heavy water. Weight fractions of $X_{hw}$ = 0.63 for $D_2O/H_2O$ and $X_{3MP}$ = 0.39 for $3MP$ have been used, the solvent refractive index being 1.40. Before adding colloids, the solvent mixture was purified by distillation under vacuum. Four different suspensions were prepared, each one with a colloid volume fraction of $\sim10^{-4}$, and with different salt concentrations of 0.31, 1.5, 2.7 mmol/L of sodium chloride. The corresponding Debye screening lengths are, respectively, $L_D$= 14 nm, $L_D$= 6.4 nm, $L_D$= 4.8 nm. The samples were filled into quartz cells under vacuum that were tightly sealed.
We used a collimated laser beam 8 mm in diameter with a wavelength of 930 nm to illuminate the sample cell, and a 0.25 NA, 20X microscope objective to project the transmitted and scattered light onto a CCD sensor. As detailed in [@Veen12], near-field scattering images result from the interference between the intense transmitted and the (fainter) scattered light; these provide directly the scattered field-field correlation functions of the objects within the range of scattering angles selected by the lens.
We first determined the aggregation temperature, $T_{agg}$, of each sample using the second moment $m_2$ of the scattered intensity. The second moment is proportional to the cross-section and the number of scatterers; aggregation is detected when the second moment passes a threshold of $10^{-3}$. We then followed the entire aggregation process in time after temperature jumps from below $T_{agg}$ to $T_{agg}$, $T_{agg}$ + 0.1, + 0.2, + 0.3 and + 0.4$K$, increasing the attractive strength with each jump. For each temperature, we monitored the aggregation process for one hour by acquiring NFS images with a frame rate of 1 $s^{-1}$ in batches of 100. The sample was then cooled down to a temperature far below $T_{agg}$ to split up the aggregates, followed by stirring for at least 3 hours before a new measurement was started.
![Evolution of the second moment and the kurtosis during the aggregation process at the lowest attractive strength at $T = T_{agg}$.[]{data-label="fig1"}](Kurtosi_NVI_Tagg+00_New.pdf){width="1.0\columnwidth"}
![Aggregate growth: Evolution of $R_g$ and $R_h$ with time. The radius of gyration was obtained from the Fisher-Burford fit of the static power spectra. The hydrodynamic radius was obtained using eq. \[eq1\].[]{data-label="fig2"}](RgRh_vs_t_New.pdf){width="1.0\columnwidth"}
A typical example of the evolution of $m_2$ is shown in Fig. \[fig1\]. Here, we have used the reduced time $t_r = (t-t_0)/t_s$, where $t_s$ is the time a single particle diffuses over its own diameter and $t_0$ is the start of the aggregation process, which we define by linear extrapolation of the data to vanishing $m_2$. Starting from $t_0$, the second moment rises continuously, indicating the growth of aggregates. In this regime, the $m_2$ curves of all samples at all temperatures can be scaled onto a single master curve, reflecting the universal growth of the aggregates. As a guide and measure of reliability, we also indicate the kurtosis, $\kappa = m_4/m_2^2$. The kurtosis varies wildly for $t < t_0$ and reaches a defined value of about 3 after that. We hypothesize that at these early times before $t_0$, subcritical nuclei may form and evaporate on the time scale of observation. After this, we can measure a reliable radius and diffusion coefficient of the scattering objects.
Microgravity conditions allow unique measurement of the internal structure of the aggregates: they permit the slow Brownian motion of large aggregates to be measured, until the aggregates exhibit the pronounced form factors of fractal objects. The NFS measurement technique then allows us to measure the intermediate scattering function $F(q,\tau)$ instantaneously with respect to the much slower aggregation process and diffusion time, enabling us to determine the evolution of both the hydrodynamic and the gyration radius as the aggregates grow.
We first determine the gyration radius from the static form factor $S(q)$. This is done using the Fisher-Burford fit $S(q,R_g) = \left(1+(2/3d_f)q^2R_g^2 \right)^{-d_f/2}$ for fractal aggregates that depends only on the radius of gyration and the fractal dimension, $d_f$ [@Veen12]. The resulting evolution of $R_g$ is shown by open symbols in Fig. \[fig2\]. In good approximation, $R_g$ grows as a power law, $R_g \sim t_r^{1/d_f}$, as expected for a pure DLA process. In pure DLA, the inverse exponent equals the fractal dimension of the aggregate. We find that, indeed, the slopes in Fig. \[fig2\] are in good agreement with the fractal dimension $d_f=2.4$ and $d_f=1.8$ determined from the static structure factor respectively at $T_{agg}$ and $T_{agg}+0.4K$ [@Veen12]. Hence, we can describe the entire aggregation processes within the framework of DLA with one consistent value of the fractal dimension.
To determine the hydrodynamic radius, we relate the decay of the intermediate scattering function $F(q,t)$ to the diffusion coefficient, $D$. In the case of monodisperse simple spherical objects, $D$ is related to the hydrodynamic radius, $R_h$, through the Stokes-Einstein relation $D=\frac{k_bT}{6\pi \eta R_h}$, in which $\eta$ is the solvent viscosity. No dependence of $R_h$ on $q$ is expected. For our aggregates, however, $R_h$ as obtained from the decay of $F(q,t)$ varies with $q$: the additional rotational degrees of freedom of objects with internal structure lead to a faster decay of $F(q,t)$. In this case, the effective diffusion coefficient $D_{eff}$ as defined from the decay time $\tau$ via $\tau=1/D_{eff}q^2$ is no longer related to the hydrodynamic radius via the Stokes-Einstein relation. To account for the rotations quantitatively, we define a class of spherical symmetries that determines how much an object needs to rotate before it decorrelates. The higher the symmetry, the smaller is the angle and thus the time needed to decorrelate the field [@Lattuada04]. Because the symmetry is connected to $qR_g$ via the scale-invariant fractal structure, $qR_g$ determines uniquely the correction to the decorrelation times. This allows us to relate $D_{eff}$ to $D$ via the static structure factor $S(q)$ [@Lin90]. For $S(q)$, we use the structure factor as measured at the same time. To be independent of fluctuations in $S(q)$, we again use the Fisher-Burford fit as detailed above; this expression provides a good fit to the measured average structure factor of the aggregates. Using it, we can rewrite eq. 7 from Lin et al. [@Lin90] into the following form:
$$2\left(\frac{R_h}{R_g}\right)^{2}\left(\frac{D_{eff}}{D(R_h)}-1\right) = 1-\frac{3d_{f}}{3d_{f}+2(qR_{g})^{2}},
\label{eq1}$$
where we have separated static and dynamic quantities. The right-hand side is given by the static data; it provides a master curve solely dependent on the product $qR_g$, taking into account both translations and rotations of the aggregates. The left-hand side is determined from the dynamics: taking $R_g$ from the static data, the only free parameter is the hydrodynamic radius. To determine it, we define $f(R_h)$ that equals the right and left-hand side of the equation, and determine diffusion coefficients $D_{eff}$ from the time decay of $F$ for 60 different $q$-values in the range 0.8 - 1.9 $\mu m^{-1}$. We then adjust $R_h$ so that $f(R_h)$ from the dynamic data provides the best fit to the master curve.
![Master curve according to eq. \[eq1\] of the scattering of aggregates growing by diffusion-limited aggregation at the lowest attractive strength at $T = T_{agg}$.[]{data-label="fig3"}](MC_Tagg+00.pdf){width="1.0\columnwidth"}
An example of the dynamic data compared to the master curve is shown in Fig. \[fig3\]. The weakest interaction is considered, namely at the temperature $T=T_{agg}$. The data is well fitted with the ratio $R_h/R_g$ = 1.05, one consistent value for the entire aggregation process. Fits of similar quality are obtained for all other interaction potentials as long as the aggregates are not too large. The situation changes for the highest interaction potential, when aggregates grow to larger sizes: the deviation from the master curve becomes larger due to polydispersity effects [@Wiltzius87]. By taking the polydispersity into account we can reduce the deviations and again obtain a good fit for one value of $\beta$. We accounted for polydispersity explicitly assuming a cluster mass distribution of the form:
$$N(M)=\frac{N_T}{\left\langle M\right\rangle}\left[1-\frac{1}{\left\langle M\right\rangle}\right]^{M-1}
\label{eq2}$$
where $N(M)$ is the number of clusters of mass $M$, $N_{T}$ is the total number of clusters, and $\langle M \rangle$ is the average cluster mass, which is related to the radius of gyration $R_g$ via $\langle M \rangle=(R_{g}/a)^{d_{f}}$, where $a$ is the monomer radius. This distribution represents a good description for the DLA case (see [@Lin90] and references therein for details). To determine the effective diffusion coefficient, for each cluster mass, we determined the individual diffusion coefficients using eq. \[eq1\], and we averaged the diffusion coefficients of all clusters, taking into account their statistical weight given by $N(M)$ [@Lin90; @Berne_Percora]. The results of this method agree with the earlier one that neglects polydispersity to within 2-3% for small values of $qR_g$; however, by incorporating polydispersity, we now obtain a similarly accurate fit over the entire range of $qR_g$, even for large values of $qR_g$. We note that this large-aggregate regime is important in microgravity measurements such as the one explored here, because large aggregates do not settle on the observation time scale and stay in the field of view. In contrast, this regime is rarely approached on earth, and is not reached in [@Lin90] because of the fast settling of large aggregates.
Thus, we have obtained the hydrodynamic radius upon growth of the aggregates for a large range of sizes. The resulting values of $R_h$ as a function of time are indicated by closed symbols in Fig. \[fig2\]. Comparison with $R_g$ in the same figure shows that the ratio $R_h/R_g$ becomes constant for larger aggregates, being $\beta=1.1\pm0.012$ for $T=T_{agg}$, and $\beta=0.85\pm0.015$ for $T=T_{agg}+0.4 K$. For smaller aggregates, the ratio $R_h/R_g$ appears not constant because the aggregates cannot be considered spherical as the monomers are 0.4 $\mu m$ in diameter and therefore aggregates consist of a few monomers only. This small size effect is also observed in the growth as a deviation from the expected power law $R_g=at^{1/d_{f}}$ dependence. By considering only sufficiently large aggregates that have a well-defined fractal dimension and structure, we can now elucidate the internal structure as a function of the attractive potential and fractal dimension.
![Ratio of hydrodynamic to gyration radius as a function of $d_f$ for three different salt concentrations (see text for details). Lines indicate the expected dependence obtained for unit step, Gaussian and exponential density-density correlation functions (from top to bottom). Insets show holographic reconstructions of aggregates grown at $T = T_{agg} + 0.4K$ (highest attraction, top) and $T = T_{agg}$ (lowest attraction, bottom). The length of the scale bar is 25$\mu$m.[]{data-label="fig4"}](Beta_vs_d_New.pdf){width="1.0\columnwidth"}
We plot the ratio $\beta = R_h / R_g$ as a function of $d_f$ in Fig. \[fig4\]. Measurements at all temperatures from $T_{agg}$ to $T_{agg}+0.4 K$, and for three different salt concentrations (0.31mmol/L, blue squares; 1.5 mmol/L, red circles; 2.7mmol/L, green triangles) are included. A systematic dependence of $\beta$ on $d_f$ is observed, indicating a consistent internal structure independent of the salt concentration. To interpret the $\beta$-values, we indicate by lines the dependence for a fully compact object with a unit-step density distribution (solid curve), for a fluffier object with a Gaussian density profile (dashed curve), and for an exponential profile (dotted curve) [@Wiltzius87]. The data lies closest to the unit step, indicating that the aggregates have fairly compact internal structure, regardless of the fractal dimension and thus the attractive strength. To illustrate the shape of the aggregates, as insets we show results obtained by holographic reconstruction of the aggregates at the latest stages of aggregation for the two conditions considered in Fig. \[fig2\]. The fluffier structure of aggregates formed at higher attractive interaction strength is clearly observable.
In summary, we measured the internal structure of DLA clusters over a wide range of attractive potential strength. This was possible owing to the peculiarity of the critical Casimir effect allowing us to induce tunable interactions that result in aggregates with a wide range of fractal dimensions. Notice that no systematic dependence on the added salt is observed, and the only dependence is on the fractal dimension $d_f$, set by the depth of the potential well [@Veen12]. The salt concentration i.e. the repulsive part of the potential has no effect on $d_f$, as it is only the depth of the potential well that determines the fractal dimension. While the aggregation temperature is set by the ratio of the solvent correlation length to the Debye screening length, it is the temperature increment from $T_{agg}$ that sets the depth of the potential well, and consequently determines the aggregate structure, i.e. its fractal dimension. This is the first time that a DLA process is studied with the widest range of possible fractal structures, owing to the variable critical Casimir potential, microgravity and the near field scattering technique that measures the intermediate structure function as a function of time.
We thank Matteo Alaimo for his support with the analysis code. This work was supported by the European Space Agency and the Dutch organization for scientific research NWO.
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---
abstract: 'Most of today’s work on knowledge graph completion is concerned with sub-symbolic approaches that focus on the concept of embedding a given graph in a low dimensional vector space. Against this trend, we propose an approach called AnyBURL that is rooted in the symbolic space. Its core algorithm is based on sampling paths, which are generalized into Horn rules. Previously published results show that the prediction quality of AnyBURL is on the same level as current state of the art with the additional benefit of offering an explanation for the predicted fact. In this paper, we are concerned with two extensions of AnyBURL. Firstly, we change AnyBURL’s interpretation of rules from $\Theta$-subsumption into $\Theta$-subsumption under Object Identity. Secondly, we introduce reinforcement learning to better guide the sampling process. We found out that reinforcement learning helps finding more valuable rules earlier in the search process. We measure the impact of both extensions and compare the resulting approach with current state of the art approaches. Our results show that AnyBURL outperforms most sub-symbolic methods.'
author:
- |
Christian Meilicke$^1$, Melisachew Wudage Chekol$^2$,\
Manuel Fink$^1$, Heiner Stuckenschmidt$^1$\
$^1$Research Group Data and Web Science, University Mannheim\
$^2$Utrecht University\
{christian,manuel,heiner}@informatik-uni-mannheim.de, m.w.chekol@uu.nl
bibliography:
- 'mybib.bib'
date:
title: |
Reinforced Anytime Bottom Up Rule Learning for\
Knowledge Graph Completion
---
Introduction {#sec:intro}
============
Bottom Up Rule Learning {#sec:learn}
=======================
Search Strategy {#sec:search}
===============
Experiments {#sec:experiments}
===========
Datasets and Settings {#sub:exp-datasets}
---------------------
Object Identity {#sub:exp-oi}
---------------
Reinforcement Learning {#sub:exp-reinforced}
----------------------
State of the Art {#sub:exp-stateoftheart}
----------------
Related Work {#sec:related}
============
Conclusion {#sec:conclusion}
==========
|
---
abstract: |
Feedback shift registers(FSRs) are a fundamental component in electronics and secure communication. An FSR $f$ is said to be reducible if all the output sequences of another FSR $g$ can also be generated by $f$ and the FSR $g$ has less memory than $f$. An FSR is said to be decomposable if it has the same set of output sequences as a cascade connection of two FSRs. It is proved that deciding whether FSRs are irreducible/indecomposable is [$\mathbf{NP}$]{}-hard.
*Key words*: feedback shift registers, irreducible, indecomposable, [$\mathbf{NP}$]{}-hard, Boolean circuit, cycle structure
author:
- |
[Lin W[ANG]{}]{}\
[*Science and Technology on Communication Security Laboratory*]{}\
[*Chengdu 610041, P. R. China*]{}\
[*Email: linwang@math.pku.edu.cn*]{}
title: 'Deciding Irreducibility/Indecomposability of Feedback Shift Registers is [$\mathbf{NP}$]{}-hard'
---
Introduction {#sect:intro}
============
Feedback shift registers are broadly used in spread spectrum radio, control engineering and confidential digital communication. Consequently, this subject has attracted substantial research over half a century. Particularly, feedback shift registers play a significant role in the stream cipher finalists of the eSTREAM project[@RB08].
(80,10.5) (0,0) (9,-3)[(0,0)[${x_{n-1}}$]{}]{} (22,-3)[(0,0)[${x_{n-2}}$]{}]{} (48,-3)[(0,0)[${x_{1}}$]{}]{} (61,-3)[(0,0)[${x_{0}}$]{}]{} (35,5.5)[(0,0)[$\crct{f_1}({x_0},{x_1},\dots,{x_{n-1}})$]{}]{} (70,-3)[(0,0)\[l\][output]{}]{}
As shown in Figure \[fig:FSR\], an $n$-stage *feedback shift register(FSR)* consists of $n$ bit registers ${x_0},{x_1},\dots,{x_{n-1}}$ and an $n$-input feedback logic $f_1$. The vector $\left({x_0}(t),{x_1}(t),\dots,{x_{n-1}}(t)\right)$ is called a *state* of this FSR, where ${x_i}(t)$ is the value of ${x_i}$ at clock cycle $t$, $0\leq i<n$. The state at clock cycle $0$ is called the *initial state*. Along with clock impulses the values stored in bit registers update themselves as $$\label{eqn:def-fsr-statetrans}
\left({x_0}(t+1),{x_1}(t+1),\dots,{x_{n-1}}(t+1)\right)=
\left({x_1}(t),\dots,{x_{n-1}}(t),f_1({x_0}(t),{x_1}(t),\dots,{x_{n-1}}(t))\right),$$ and the map defined by Eq.(\[eqn:def-fsr-statetrans\]) is called the *state transformation* of this FSR.
The $(n+1)$-input Boolean function $f(x_0,x_1,\dots,x_{n}) = x_{n}\oplus f_1(x_0,x_1,\dots,x_{n-1})$, where $\oplus$ denotes exclusive-or, is called the characteristic function of the FSR in Figure \[fig:FSR\], and without ambiguity we also denote this FSR by $f$. Let ${G\left(f\right)}$ denote the set of sequences generated by $f$, i.e., $${G\left(f\right)}=\set{{s}\in\BnrSet^*:\forall t, f({s}(t),{s}(t+1),\dots,{s}(t+n))=0},$$ where $\BnrSet^*$ is the set of binary sequences. If $f(x_0,x_1,\dots,x_{n})= x_{{n}}\oplus c_{{n}-1}x_{{n}-1}\oplus
\cdots \oplus c_1x_{1}\oplus c_0x_{0}$, where $c_0,c_1,\dots,c_{{n}-1}\in\BnrSet$, then $f$ is called a *linear feedback shift register(LFSR)*, and $p(x)=x^{{n}}\oplus c_{{n}-1}x^{{n}-1}\oplus \cdots \oplus c_1x \oplus
c_0$ is called its *characteristic polynomial*. Without ambiguity we also denote this LFSR by $p(x)$. An FSR which is not an LFSR is called a *nonlinear feedback shift register(NFSR)*.
If there exists an $m$-stage FSR $g$ such that $m<n$ and ${G\left(g\right)}\subset{G\left(f\right)}$, then $g$ is called a *subFSR* of $f$ and $f$ is said to be *reducible*. Otherwise, $f$ is said to be *irreducible*.
(150,10.5) (0,0) (9,-3)[(0,0)[$x_{n-1}$]{}]{} (22,-3)[(0,0)[$x_{n-2}$]{}]{} (48,-3)[(0,0)[$x_{1}$]{}]{} (61,-3)[(0,0)[$x_{0}$]{}]{} (35,5.5)[(0,0)[$\crct{f_1}(x_0,x_1,\dots,x_{n-1})$]{}]{} (70,-3.05)[(1,0)[3]{}]{} (71.5,-3.05) (71.45,-4.5)[(0,1)[3]{}]{} (71.45,3)[(0,-1)[4.9]{}]{} (71.5,0) (80.5,-3)[(0,0)[$y_{m-1}$]{}]{} (93.5,-3)[(0,0)[$y_{m-2}$]{}]{} (119.5,-3)[(0,0)[$y_{1}$]{}]{} (132.5,-3)[(0,0)[$y_{0}$]{}]{} (106.5,5.5)[(0,0)[$\crct{g_1}(y_0,y_1,\dots,y_{m-1})$]{}]{} (141.5,-3)[(0,0)\[l\][output]{}]{}
Let $f(x_0,x_1,\dots,x_{n})= x_n \oplus \crct{f_1}(x_0,x_1,\dots,x_{n-1})$ and $g(y_0,y_1,\dots,y_{m})= y_m \oplus \crct{g_1}(y_0,y_1,\dots,y_{m-1})$ be two FSRs. The finite state machine in Figure \[fig:cascade\] is called the *cascade connection* of $f$ into $g$. The Grain family ciphers use the cascade connection of an LFSR into an NFSR[@HJMM08]. Green and Dimond[@GD70] defined the *product FSR*[^1] of $f$ and $g$ to be $$(\crct{f}\cascade\crct{g})(x_0,x_1,\dots,x_{n+m})=
\crct{f}(\crct{g}(x_0,x_1,\dots,x_{m}),
\crct{g}(x_1,x_2,\dots,x_{m+1}),\dots,\crct{g}(x_{n},x_{n+1},\dots,x_{n+m})),$$ and showed ${G\left(f;g\right)}={G\left(f\cascade g\right)}$, where ${G\left(f;g\right)}$ is the set of output sequences of the cascade connection of $f$ into $g$. Given an FSR $h$, if there exist two FSRs $f$ and $g$ satisfying $h=f\cascade g$, then $h$ is said to be *decomposable*. Otherwise, $h$ is said to be *indecomposable*.
It is appealing to decide whether an FSR is (ir)reducible/(in)decomposable for the reasons below. First, it offers a new perspective on analysis of stream ciphers. Notice that all sequences generated by $g$ is also generated by $f\cascade g$ if $f$ can output the 0-sequence. A reducible/decomposable FSR in unaware use may undermine the claimed security of stream ciphers, e.g., causing inadequate period of the output sequences. Particularly, if $g$ is an LFSR and $f$ can output the 0-sequence, then $f\cascade g$ can generate a family of linear recurring sequences, vulnerable to the Berlekamp-Massey algorithm. Second, it potentially improves implementation of FSRs. On one hand, it costs less memory to replace an FSR with its large-stage subFSR, if there is one, while generating a great part of its output sequences. On the other hand, similar to the idea of Dubrova[@dbrv], substituting a decomposable FSR by its equivalent cascade connection as in Figure \[fig:cascade\] possibly reduces the circuit depth of the feedback logics, in favor of less propagation time and larger throughput. Third, an algorithm testing (ir)reducibility/(in)decomposability helps to design useful FSRs. Because Tian and Qi[@TQ13] proved that on average at least one among three randomly chosen NFSRs is irreducible, a great number of irreducible NFSRs can be found if deciding irreducibility of FSRs is feasible. Besides, FSRs generating maximal-length sequences were constructed based on inherent structure of decomposable FSRs[@MST79].
Two algorithms were proposed by [@TQ14] to find affine subFSRs of NFSRs. By [@JL16], if an NFSR $h$ is decomposed as the cascade connection of an LFSR $f$ into an NFSR $g$ and $f$ is primitive with stage no less than that of $g$, then all affine subFSRs of $h$ are actually those of $g$. (In)decomposability of LFSRs is completely determined by their characteristic polynomials. By [@GD70; @LN83; @TQ14s], an LFSR $h$, with its characteristic polynomial $p(x)$, is decomposed as $h=f\cascade g$ if and only if $f$ and $g$ are LFSRs and $p(x)=l_1(x)\cdot l_2(x)$, where $l_1(x)$ and $l_2(x)$ are characteristic polynomials of $f$ and $g$, respectively. In contrast, decomposing NFSRs seems much more challenging, though some progress has been made recently. Using the language of algebraic normal forms of Boolean functions, Ma *et al*[@MQT13] gave an algorithm to decompose NFSRs into the cascade connection of an NFSR into an LFSR, and Tian and Qi[@TQ14s] gave a series of algorithms to decompose NFSRs into the cascade connection of two NFSRs. Noteworthily, Zhang *et al*[@ZQTW15] gained an algorithm decomposing an NFSR $f$ into the cascade connection of an NFSR into an LFSR, and the complexity of their algorithm is polynomial in the size of the algebraic normal form of $f$ and the size of the binary decision diagram of $f$ if converting the algebraic normal form of $f$ to the binary decision diagram of $f$ is polynomial-time computable.
*Our contribution.* This correspondence studies irreducibility and indecomposability from the perspective of computational complexity. [$\mathbf{NP}$]{} is the class of all problems computed by polynomial-time nondeterministic Turing machines. A problem is [$\mathbf{NP}$]{}-*hard* if it is at least as hard as all [$\mathbf{NP}$]{} problems. This correspondence proves that deciding whether an FSR is irreducible(indecomposable) is [$\mathbf{NP}$]{}-hard.
The rest of this paper is organized as follows: In Section \[sect:preliminaries\] we prepare some notations, basic facts on Boolean circuits and some lemmas on the cycle structure of FSRs. [$\mathbf{NP}$]{}-hardness of FSR irreducibility and FSR indecomposability is shown in Sections \[sect:FSR-irreducibility\] and \[sect:FSR-indecomposability\], respectively. The last section includes a summary and a comment on future work.
Preliminaries {#sect:preliminaries}
=============
Notations
---------
Throughout this paper, $\Int$ denotes the set of integers, $+$ addition of integers, and $\oplus$ the exclusive-or(XOR) operation.
Denote ${\mathbf{1}^{m}}=(1,1,\dots,1)\in\BnrSet^{m}$, ${\mathbf{0}^{m}}=(0,0,\dots,0)\in\BnrSet^{m}$ and ${\bm{\iota}^{m}}=(1,0,\dots,0)\in\BnrSet^{m}$. For $\state{u}\in\BnrSet^m$, denote ${\overline{\state{u}}}=\state{u}\oplus{\mathbf{1}^{m}}$ and ${\widehat{\state{u}}}=\state{u}\oplus{\bm{\iota}^{m}}$.
For $\state{u}=(a_1,a_2,\dots,a_m)\in\BnrSet^m$ and $1\leq k<m$, let $$\begin{aligned}
{\lceil{\state{u}}\rceil_{k}}=&(a_1,a_2,\dots,a_k)\in\BnrSet^k;\\
{\lfloor{\state{u}}\rfloor_{k}}=&(a_{m-k+1},a_{m-k+2},\dots,a_{m-1},a_m)\in\BnrSet^k.\end{aligned}$$
For $\state{u}=(a_1,\dots,a_k)\in\BnrSet^k$ and $\state{v}=(b_1,\dots,b_m)\in\BnrSet^m$, denote $\state{u}\parallel\state{v}=(a_1,\dots,a_k,b_1,\dots,b_m)\in\BnrSet^{k+m}$.
Without ambiguity a vector $(a_0,a_1,\dots,a_{m-1})\in\BnrSet^{m}$ is uniquely taken as the nonnegative integer $\sum_{j=0}^{m-1}2^{j}a_j$. Thereby, the natural order relation on $\BnrSet^{m}$ is imposed, i.e., $(a_0,a_1,\dots,a_{m-1})<(b_0,b_1,\dots,b_{m-1})$ if and only if $\sum_{j=0}^{m-1}2^{j}a_j<\sum_{j=0}^{m-1}2^{j}b_j$.
Boolean circuits
----------------
An $m$-input *Boolean circuit* $f$ is a directed acyclic graph with $m$ sources and one sink [@AB12]. The value(s) of source(s) is(are) input(s) of the Boolean circuit; Any nonsource vertex, called a *gate*, is one of the logical operations OR($\myNOT$), AND($\myAND$) and NOT($\myNOT$), where the fan-in[^2] of OR and AND is $2$ and that of NOT is $1$; The value outputted from a gate is obtained by applying its logical operation on the value(s) inputted into it; The value outputted from the sink is the output of the Boolean circuit $f$. The size of the circuit $f$, denoted by $\sizeof{f}$, is the number of vertices in it. An $m$-input Boolean circuit $f$ is *satisfiable* if there exists $\state{v}\in\BnrSet^m$ such that $f(\state{v})=1$.
**PROBLEM**: CIRCUIT SATISFIABILITY
[INSTANCE]{}: A Boolean circuit $f$ with its size $\sizeof{f}$.
[QUESTION]{}: Is $f$ satisfiable?
A decision problem in [$\mathbf{NP}$]{} class is [$\mathbf{NP}$]{}-*complete* if it is not less difficult than any other [$\mathbf{NP}$]{} problem.
\[lemma:CKT-SAT-NP-C\] [@AB12] The CIRCUIT SATISFIABILITY problem is [$\mathbf{NP}$]{}-complete.
A decision problem $P$ is *polynomial-time Karp reducible* to a decision problem $Q$ if there is a polynomial-time computable transformation $T$ mapping instances of $P$ to those of $Q$ such that an instance $x$ of $P$ answers yes if and only if $T(x)$ answers yes[@AB12]. A decision problem is [$\mathbf{NP}$]{}-*hard* if a [$\mathbf{NP}$]{}-complete problem is polynomial-time Karp reducible to it[@AB12].
An FSR is completely characterized by its feedback logic. We use Boolean circuits to characterize the feedback logic of FSRs for the following two reasons[^3]. First, FSRs are mostly implemented with silicon chips, and the Boolean circuit is an abstract model of their feedback logic in silicon chips. Second, the Boolean circuit is a generalization of Boolean formula[@AB12]. Therefore, in this correspondence the size of an FSR is measured by the size of its feedback logic as a Boolean circuit.
The cycle structure of FSRs
---------------------------
A binary sequence ${s}$ is a map from $\Int$ to $\BnrSet$. If there exists some $\tau\in\Int$ such that ${s}(t+\tau)={s}(t)$ for any $t\in\Int$, ${s}$ is said to be *periodic* and the *period* of ${s}$ is defined to be $${\mathrm{per}\left({s}\right)} =
\min\set{\tau>0:{s}(t+\tau)={s}(t) \text{ for all }t\in\Int}.$$
Let $f$ be an $m$-stage FSR. The following three statements are equivalent [@Gol67]: (i) The state transformation of $f$ is bijective. (ii) Any sequence generated by $f$ is periodic. (iii) $\crct{f}(x_0,x_1,\dots,x_m)=x_m\oplus\crct{g}(x_1,x_2,\dots,x_{m-1})\oplus x_0$ for some $(m-1)$-input Boolean function $g$. If any of (i)-(iii) holds, $f$ is said to be *nonsingular*.
In the rest of this section we only consider nonsingular FSRs.
A sequence ${s}$ of period $m$ determines a cyclic sequence ${\theta\left({s}\right)}=[{s}(0),{s}(1),\dots,{s}(m-1)]$. We call ${\theta\left({s}\right)}$ to be an $m$-cycle and also denote $\lengthof{{\theta\left({s}\right)}}=m$. For the $m$-cycle ${\theta\left({s}\right)}$, define the set $${S_{k}\left({{\theta\left({s}\right)}}\right)}=
\set{\left({s}(i),{s}\left((i+1)\bmod m\right),\dots,{s}\left((i+k-1)\bmod m\right)\right)\in\BnrSet^k:0\leq i<m}.$$ Actually, any shift of a periodic sequence determines the same cycle, and $\set{{s}':{\theta\left({s}'\right)}={\theta\left({s}\right)}}$ is exactly the set of all shifts of ${s}$. Furthermore, if ${s}\in{G\left(f\right)}$ for a $k$-stage FSR $f$, then each vector in ${S_{k}\left({{\theta\left({s}\right)}}\right)}$ plays as a unique initial state and hence determines a unique sequence in $\set{{s}':{\theta\left({s}'\right)}={\theta\left({s}\right)}}$.
The *cycle structure* of an FSR ${f}$, denoted by ${\mathbf{CycStr}\left(f\right)}$, is $\set{{\theta\left({s}\right)}:{s}\in{G\left({f}\right)}}$. Following this definition, we have the lemma below.
\[lemma:subFSR-cyc-str\] Let $f$ and $g$ be FSRs. Then $g$ is a subFSR of $f$ if and only if ${\mathbf{CycStr}\left(g\right)}\subset{\mathbf{CycStr}\left(f\right)}$.
\[lemma:conjugate-nominkiss\] Let $f$ be an $m$-stage FSR. Suppose ${{\bm{c}}},{{\bm{d}}}\in{\mathbf{CycStr}\left(f\right)}$(including ${{\bm{c}}}={{\bm{d}}}$), $\state{u}\in{S_{m}\left({{{\bm{c}}}}\right)}$ and ${\widehat{\state{u}}}\in{S_{m}\left({{{\bm{d}}}}\right)}$. Then $\min\left({S_{m}\left({{{\bm{c}}}}\right)}\cup{S_{m}\left({{{\bm{d}}}}\right)}\right)<\min\set{\state{u},{\widehat{\state{u}}}}$ or $\state{u}\in\set{{\mathbf{0}^{m}},{\bm{\iota}^{m}}}$.
Let $F$ denote the state transformation of the FSR $f$. Then $\set{F(\state{u}),F({\widehat{\state{u}}})}
=\set{{\langle\state{u}/2\rangle},
{\langle\state{u}/2\rangle}+2^{m-1}}$, where ${\langle\state{u}/2\rangle}=\max\set{i\in\Int:i\leq\state{u}/2}$.
Notice that $F(\state{u})\in{S_{m}\left({{{\bm{c}}}}\right)}$, $F({\widehat{\state{u}}})\in{S_{m}\left({{{\bm{d}}}}\right)}$ and $\set{\state{u},{\widehat{\state{u}}}}=\set{2{\langle\state{u}/2\rangle},2{\langle\state{u}/2\rangle}+1}$. If ${\langle\state{u}/2\rangle}>0$, then ${\langle\state{u}/2\rangle}<\min\set{\state{u},{\widehat{\state{u}}}}$, implying $$\min\left({S_{m}\left({{{\bm{c}}}}\right)}\cup{S_{m}\left({{{\bm{d}}}}\right)}\right)
\le \min\set{F(\state{u}),F({\widehat{\state{u}}})} <\min\set{\state{u},{\widehat{\state{u}}}}.$$ If ${\langle\state{u}/2\rangle}=0$, then $\state{u}\in\set{{\mathbf{0}^{m}},{\bm{\iota}^{m}}}$.
\[lemma:cycle-structure-fsr\] Let ${\mathfrak{C}}$ be a set of cycles. Then there exists an $m$-stage FSR ${f}$ with ${\mathbf{CycStr}\left(f\right)}={\mathfrak{C}}$ if and only if the following two conditions hold: (i) $\sum_{{{\bm{c}}}\in{\mathfrak{C}}}\lengthof{{{\bm{c}}}} = 2^m$; (ii) The map $\state{v}\mapsto{\lfloor{\state{v}}\rfloor_{m}}$ is injective on ${\bigcup_{{{\bm{c}}}\in{\mathfrak{C}}}{S_{m+1}\left({{{\bm{c}}}}\right)}}$.
To prove Lemma \[lemma:cycle-structure-fsr\], we use the following Lemma.
\[lemma:cycle-structure-fsr-cnd\] Let ${\mathfrak{C}}$ be a set of finitely many cycles. Then the following three statements are equivalent: (i) $\cset{\bigcup_{{{\bm{c}}}\in{\mathfrak{C}}}{S_{m}\left({{{\bm{c}}}}\right)}}=\sum_{{{\bm{c}}}\in{\mathfrak{C}}}\lengthof{{{\bm{c}}}}$; (ii) The map $\state{v}\mapsto{\lfloor{\state{v}}\rfloor_{m}}$ is injective on ${\bigcup_{{{\bm{c}}}\in{\mathfrak{C}}}{S_{m+1}\left({{{\bm{c}}}}\right)}}$; (iii) The map $\state{v}\mapsto{\lceil{\state{v}}\rceil_{m}}$ is injective on ${\bigcup_{{{\bm{c}}}\in{\mathfrak{C}}}{S_{m+1}\left({{{\bm{c}}}}\right)}}$.
First we prove that Statements (i) and (ii) are equivalent.
Let ${\mathfrak{C}}=\set{{{\bm{c}}}_1,{{\bm{c}}}_2,\dots,{{\bm{c}}}_k}$ and ${{\bm{c}}}_i=[c_{i,0},c_{i,1},\dots,c_{i,p_i-1}]$, $1\leq i\leq k$, where $p_i=\lengthof{{{\bm{c}}}_i}$. In this proof, a tuple $(i,j)$ denotes a pair of integers satisfying $1\leq i\leq k$ and $0\leq j<p_i$. Denote $$\begin{aligned}
\state{x}_{i,j}=&(c_{i,(j+1)\bmod p_i},c_{i,(j+2)\bmod p_i},\dots,c_{i,(j+m)\bmod p_i});\\
\state{y}_{i,j}=&(c_{i,j},c_{i,(j+1)\bmod p_i},c_{i,(j+2)\bmod p_i},\dots,c_{i,(j+m)\bmod p_i}).\end{aligned}$$ Notice $\bigcup_{{{\bm{c}}}\in{\mathfrak{C}}}{S_{m}\left({{{\bm{c}}}}\right)}
=\bigcup_{i=1}^k \set{ \state{x}_{i,j}: 0\leq j< p_i}$ and $\bigcup_{{{\bm{c}}}\in{\mathfrak{C}}}{S_{m+1}\left({{{\bm{c}}}}\right)}
=\bigcup_{i=1}^k \set{ \state{y}_{i,j}: 0\leq j< p_i}$. It is sufficient to consider cases below.
- Case $\cset{\bigcup_{{{\bm{c}}}\in{\mathfrak{C}}}{S_{m}\left({{{\bm{c}}}}\right)}}=\sum_{{{\bm{c}}}\in{\mathfrak{C}}}\lengthof{{{\bm{c}}}}$. Then $\state{x}_{i,j} = \state{x}_{i',j'}$ if and only if $(i,j)=(i',j')$. Since $\state{x}_{i,j}={\lfloor{\state{y}_{i,j}}\rfloor_{m}}$, $\state{y}_{i,j} = \state{y}_{i',j'}$ occurs only if $(i,j)=(i',j')$. That is, the map $\state{y}_{i,j} \mapsto {\lfloor{\state{y}_{i,j}}\rfloor_{m}}=\state{x}_{i,j}$ is injective on ${\bigcup_{{{\bm{c}}}\in{\mathfrak{C}}}{S_{m+1}\left({{{\bm{c}}}}\right)}}$.
- Case $\cset{\bigcup_{{{\bm{c}}}\in{\mathfrak{C}}}{S_{m}\left({{{\bm{c}}}}\right)}}\neq\sum_{{{\bm{c}}}\in{\mathfrak{C}}}\lengthof{{{\bm{c}}}}$. Then $\state{x}_{i_0,j_0} = \state{x}_{i_0',j_0'}$ for some $(i_0,j_0)\neq(i_0',j_0')$.
*Claim:* If $\state{x}_{i,j_0} = \state{x}_{i',j_0'}$ for some $(i,j_0)\neq(i',j_0')$, then there exists $(i,j_1)$ and $(i',j_1')$ such that $\state{x}_{i,j_1} = \state{x}_{i',j_1'}$ and $\state{y}_{i,j_1} \neq \state{y}_{i',j_1'}$.
*Proof of the claim.* Assume that this claim does not hold. Then for any $(i,j_1)$ and $(i',j_1')$, if $\state{x}_{i,j_1} = \state{x}_{i',j_1'}$ then $\state{y}_{i,j_1} = \state{y}_{i',j_1'}$. Notice that $\state{y}_{i,j} = \state{y}_{i',j'}$ implies $\state{x}_{i,(j-1)\bmod p_i} = \state{x}_{i',(j'-1)\bmod p_{i'}}$. Then $\state{x}_{i,(j_0-t)\bmod p_i} = \state{x}_{i',(j_0'-t)\bmod p_{i'}}$ for any $t\geq0$. Hence, ${{\bm{c}}}_i={{\bm{c}}}_{i'}$ and $p_i\mid (j_0'-j_0)$, contradictory to $(i,j_0)\neq(i',j_0')$. Therefore, our assumption is absurd and the claim is proved.
Following this claim, we assume $\state{x}_{i_0,j_0} = \state{x}_{i_0',j_0'}$ and $\state{y}_{i_0,j_0} \neq \state{y}_{i_0',j_0'}$ for some $(i_0,j_0)\neq(i_0',j_0')$. Thus, the map $\state{v} \mapsto {\lfloor{\state{v}}\rfloor_{m}}$ is not injective on ${\bigcup_{{{\bm{c}}}\in{\mathfrak{C}}}{S_{m+1}\left({{{\bm{c}}}}\right)}}$.
The proof of equivalence of Statements (i) and (iii) is similar and we omit it here. By Lemma \[lemma:cycle-structure-fsr-cnd\], it is sufficient to prove this statement: ${\mathbf{CycStr}\left(f\right)}={\mathfrak{C}}$ if and only if $\cset{\bigcup_{{{\bm{c}}}\in{\mathfrak{C}}}{S_{m}\left({{{\bm{c}}}}\right)}}=
\sum_{{{\bm{c}}}\in{\mathfrak{C}}}\lengthof{{{\bm{c}}}}=
2^m$.
Suppose ${\mathfrak{C}}={\mathbf{CycStr}\left({f}\right)}$ for some $m$-stage FSR ${f}$. Then for any ${{\bm{c}}}\in{\mathfrak{C}}$, a vector in ${S_{m}\left({{{\bm{c}}}}\right)}$ is exactly an initial state and uniquely determines a sequence in ${G\left(f\right)}$. Thus, $\bigcup_{{{\bm{c}}}\in{\mathfrak{C}}}{S_{m}\left({{{\bm{c}}}}\right)}=\BnrSet^k$ and $\cset{\bigcup_{{{\bm{c}}}\in{\mathfrak{C}}}{S_{m}\left({{{\bm{c}}}}\right)}}=
\sum_{{{\bm{c}}}\in{\mathfrak{C}}}\lengthof{{{\bm{c}}}}$.
Suppose $\cset{\bigcup_{{{\bm{c}}}\in{\mathfrak{C}}}{S_{m}\left({{{\bm{c}}}}\right)}}=
\sum_{{{\bm{c}}}\in{\mathfrak{C}}}\lengthof{{{\bm{c}}}}=
2^m$. Then $\bigcup_{{{\bm{c}}}\in{\mathfrak{C}}}
{{S_{m}\left({{{\bm{c}}}}\right)}}=\BnrSet^{m}$. Define an $m$-input Boolean function $f_1$ as follows. By Lemma \[lemma:cycle-structure-fsr-cnd\], for any $\state{v}=(a_0,a_1,\dots,a_{m-1})\in\BnrSet^m$, there exists uniquely $b\in\BnrSet$ such that $(a_0,a_1,\dots,a_{m-1},b)\in
\bigcup_{{{\bm{c}}}\in{\mathfrak{C}}}{S_{m+1}\left({{{\bm{c}}}}\right)}$. We define $f_1(\state{v})=b$. Immediately, ${\mathfrak{C}}$ is the cycle structure of an FSR whose feedback logic is logically equivalent to $f_1$.
\[lemma:cycle-state-cycle\] Let $f$ be an $m$-stage FSR and $F$ the state transformation of $f$. Let ${{\bm{c}}}\in{\mathbf{CycStr}\left(f\right)}$ and ${\mathrm{per}\left({{\bm{c}}}\right)}=p$. Then for any $\state{v}\in{S_{m}\left({{{\bm{c}}}}\right)}$, $\min\set{i>0:F^i(\state{v})=\state{v}}=p$ and ${S_{m}\left({{{\bm{c}}}}\right)}=\set{\state{v},
F(\state{v}),\dots,F^{p-1}(\state{v})}$.
Let $\state{v}\in{S_{m}\left({{{\bm{c}}}}\right)}$ and $q=\min\set{i>0:F^i(\state{v})=\state{v}}$. Clearly, $q\leq p$. Then $${{{\bm{c}}}}=[{\lceil{\state{v}}\rceil_{1}},
{\lceil{F(\state{v})}\rceil_{1}},\dots,{\lceil{F^{q-1}(\state{v})}\rceil_{1}}],$$ and $q={\mathrm{per}\left({{\bm{c}}}\right)}=p$. Because $\set{F^i(\state{v}):i\in\Int}\subseteq {S_{m}\left({{{\bm{c}}}}\right)}$ and $\cset{{S_{m}\left({{{\bm{c}}}}\right)}}\leq {\mathrm{per}\left({{\bm{c}}}\right)}$, we conclude that $\cset{{S_{m}\left({{{\bm{c}}}}\right)}}=p$ and ${S_{m}\left({{{\bm{c}}}}\right)}=\set{\state{v},
F(\state{v}),\dots,F^{p-1}(\state{v})}$ is a set of $p$ vectors in $\BnrSet^m$.
\[lemma:cycle-join\] Let $g(x_0,x_1,\dots,x_m)$ be an $m$-stage FSR and $$f(x_0,x_1,\dots,x_m)=g(x_0,x_1,\dots,x_m)\oplus f_3(x_1,x_2,\dots,x_{m-1}),$$ where $f_3$ is an $(m-1)$-input Boolean logic. Let $\lambda:\BnrSet^m\rightarrow\BnrSet$ be a map satisfying $$\label{eqn:cycle-join-lambda-cnd}
\left\{
\begin{aligned}
&\cset{\set{{\state{v}}\in{S_{m}\left({{{\bm{c}}}}\right)}:{\lambda\left(\state{v}\right)}=1}}\leq1
\text{ for any }{{\bm{c}}}\in{\mathbf{CycStr}\left(g\right)};\\
&
{ {\lambda\left(\state{v}\right)}\cdot{\lambda\left({\widehat{\state{v}}}\right)}=0
\text{ for any }\state{v}\in\BnrSet^m;}\\
& \text{For any }\state{u}\in\BnrSet^{m-1}
\text{ with }
f_3({\state{u}})=1, \text{ there exists }b\in\BnrSet
\text{ satisfying }{\lambda\left(b\parallel\state{u}\right)}=1.
\end{aligned}
\right.$$ A directed graph $D_g^f$ is defined as follows: the set of vertices is ${\mathbf{CycStr}\left(g\right)}$, and an arc is incident from ${{\bm{c}}}_1$ to ${{\bm{c}}}_2$ if and only if $${\set{\state{v}\in{S_{m}\left({{{\bm{c}}}_1}\right)}:
f_3({\lfloor{\state{v}}\rfloor_{m-1}})=1,{\lambda\left(\state{v}\right)}=1,
{\widehat{\state{v}}}\in {S_{m}\left({{{\bm{c}}}_2}\right)}}}\neq\emptyset.$$ If $D_g^f$ is acyclic, then the following two statements hold: (i) Any ${{\bm{d}}}\in{\mathbf{CycStr}\left(f\right)}$ is joined by all cycles in a weakly connected component[^4] ${\mathfrak{C}}$ of $D_g^f$ and ${S_{m}\left({{{\bm{d}}}}\right)}=\bigcup_{{{\bm{c}}}\in{\mathfrak{C}}
}{S_{m}\left({{{\bm{c}}}}\right)}$. (ii) If $h$ is a subFSR of $f$, then ${\mathbf{CycStr}\left(h\right)}\subset{\mathbf{CycStr}\left(g\right)}$.
Statement (i) of this lemma follows from the idea of the cycle joining method[@Gol67], and we leave its proof in Appendix \[appnd:proof-cycle-join\]. Below we prove Statement (ii) of this lemma.
By Lemmas \[lemma:subFSR-cyc-str\] and \[lemma:cycle-structure-fsr\], it is sufficient to prove this statement: if ${\mathfrak{C}}\subset{\mathbf{CycStr}\left(f\right)}$ and ${\mathfrak{C}}\not\subset{\mathbf{CycStr}\left(g\right)}$, then for any $1\leq k< m$, the map $\state{v}\mapsto{\lfloor{\state{v}}\rfloor_{k}}$ is not injective on $\bigcup_{{{\bm{c}}}\in{\mathfrak{C}}}{S_{k+1}\left({{{\bm{c}}}}\right)}$. Suppose ${{\bm{d}}}\in{\mathfrak{C}}
\setminus{\mathbf{CycStr}\left(g\right)}$. As proved in Statement (i), ${{\bm{d}}}$ is joined by the cycles composing a weakly connected component ${\mathfrak{D}}$ of the graph $D_{g}^f$. Since ${\mathfrak{D}}\subset{\mathbf{CycStr}\left(g\right)}$ and ${{\bm{d}}}\notin{\mathbf{CycStr}\left(g\right)}$, we have $\cset{{\mathfrak{D}}}>1$. Hence, by Statement (i) and the definition of $D_{g}^{f}$, there exists $\state{v}\in\BnrSet^{m}$ satisfying $\set{\state{v},{\widehat{\state{v}}}}\subset{S_{m}\left({{{\bm{d}}}}\right)}$. Then for any $1\le k<m$, ${\lceil{\state{v}}\rceil_{k+1}},{\lceil{{\widehat{\state{v}}}}\rceil_{k+1}}
\in{S_{k+1}\left({{{\bm{d}}}}\right)}$ satisfy ${\lceil{\state{v}}\rceil_{k+1}}\neq{\lceil{{\widehat{\state{v}}}}\rceil_{k+1}}$ and ${\lfloor{{\lceil{\state{v}}\rceil_{k+1}}}\rfloor_{k}}=
{\lfloor{{\lceil{{\widehat{\state{v}}}}\rceil_{k+1}}}\rfloor_{k}}$. Therefore, the map $\state{v}\mapsto{\lfloor{\state{v}}\rfloor_{k}}$ is not injective on ${S_{k+1}\left({{{\bm{d}}}}\right)}$, and hence is not injective on $\bigcup_{{{\bm{c}}}\in{\mathfrak{C}}}{S_{k+1}\left({{{\bm{c}}}}\right)}$.
Given an $m$-cycle ${{\bm{c}}}=[b_0,b_1,\dots,b_m]$, let ${\overline{{{\bm{c}}}}}$ denote the cycle $[b_0\oplus1,b_1\oplus1,\dots,b_m\oplus1]$.
The cycle structure of LFSRs is well understood.
\[lemma:poly-irreducible\] \[lemma:cycle-structure-lfsr-p0\] \[lemma:cycle-structure-lfsr-p0x\] Let ${n}=3^{k}$, $0\leq k\in\Int$. Let $p_0(x)=x^{2{n}} \oplus x^{{n}} \oplus1$, $p_1(x)=(x\oplus1)\cdot p_0(x)$, and $p_2(x)=x^{4{n}}\oplus x^{2{n}}\oplus 1$ be polynomials over the binary field $\GF{2}$. Then $p_0$ is irreducible over $\GF{2}$ and $$\begin{aligned}
{\mathbf{CycStr}\left(p_0\right)}=&\set{[0],
{{\bm{\beta}}}_1,{{\bm{\beta}}}_2,\dots,{{\bm{\beta}}}_{\frac{2^{2{n}}-1}{3{n}}}},\\
{\mathbf{CycStr}\left(p_1\right)}=&\set{[0],{{\bm{\beta}}}_1,{{\bm{\beta}}}_2,\dots,{{\bm{\beta}}}_{\frac{2^{2{n}}-1}{3{n}}},
[1],{\overline{{{\bm{\beta}}}_1}},{\overline{{{\bm{\beta}}}_2}},\dots,{\overline{{{\bm{\beta}}}_{\frac{2^{2{n}}-1}{3{n}}}}}},\\
{\mathbf{CycStr}\left(p_2\right)}=&\set{[0],
{{\bm{\beta}}}_1,{{\bm{\beta}}}_2,\dots,{{\bm{\beta}}}_{\frac{2^{2{n}}-1}{3{n}}},
{{\bm{\gamma}}}_1,{{\bm{\gamma}}}_2,\dots,{{\bm{\gamma}}}_{\frac{2^{4{n}}-2^{2{n}}}{6{n}}}},\end{aligned}$$ where ${\mathrm{per}\left({{\bm{\beta}}}_i\right)}={\mathrm{per}\left({\overline{{{\bm{\beta}}}_i}}\right)}=3{n}$ for $1\leq i\leq \frac{2^{2{n}}-1}{3{n}}$, and ${\mathrm{per}\left({{\bm{\gamma}}}_i\right)}=6{n}$ for $1\leq i\leq \frac{2^{4{n}}-2^{2{n}}}{6{n}}$.
Since $p_0(x)\cdot(x^{3^k}\oplus1)=
x^{3^{k+1}}\oplus 1$ and $\gcd(p_0,x^{3^k}\oplus1)=1$, the roots of $p_0$ are exactly primitive $3^{k+1}$-th roots of unity. Thus, $p_0$ is irreducible and $\min\set{0<t\in\Int:p_0\mid(x^t-1)}=3{n}$ is the order of any primitive $3^{k+1}$-th root of unity in the multiplicative group of the finite field $\GF{2}[x]/(p_0(x))$.
The rest of this lemma directly follows from [@LN83 Theorem 8.53, 8.55, 8.63].
[$\mathbf{NP}$]{}-hardness of deciding irreducible FSRs {#sect:FSR-irreducibility}
=======================================================
Below Algorithm \[alg:fsr-red\] transforms a given Boolean circuit to an FSR.
An $r$-input Boolean circuit $\crct{f_0}$. \[line:len-red\] A $4{n}$-stage FSR $f$, where $k=\min\set{i\in\Int:i\geq\log_3(r/2)}$ and ${n}=3^{k}$. \[line:C1-begin\] Let $\state{x}
\in\BnrSet^{4{n}-1}$ be the input of $\crct{f_3}$. Let $\state{u}_0=0\parallel\state{x}
$ and $\state{v}_0=1\parallel\state{x}
$. $\state{u}_i=L(\state{u}_{i-1})$ and $\state{v}_i=L(\state{v}_{i-1})$. $a_i={f_0}({\lfloor{\state{u}_i}\rfloor_{r}})$ and $b_i={f_0}({\lfloor{\state{v}_i}\rfloor_{r}})$. $c_i=1$. $c_i=0$. $d_i=1$. $d_i=0$. \[line:max\] $\state{u}_{\min}=\min\set{{\state{u}}_i:1\leq i\leq 6{n}}$ and $\state{v}_{\min}=\min\set{{\state{v}}_i:1\leq i\leq 6{n}}$. $q(\state{u}_0)=1$. $q(\state{u}_0)=0$. $q(\state{v}_0)=1$. $q(\state{v}_0)=0$. The Boolean circuit $\crct{f_3}$ returns $1$. The Boolean circuit $\crct{f_3}$ returns $1$. The Boolean circuit $\crct{f_3}$ returns $1$. The Boolean circuit $\crct{f_3}$ returns $0$. \[line:C1-end\] the FSR $f(x_0,\dots,x_{4{n}})=x_{4{n}}\oplus x_{2{n}}\oplus x_0
\oplus \crct{f_3}(x_1,x_2,\dots,x_{4{n}-1})$.
In the rest of this section, we use notations $\crct{f_0}$, $\crct{f_3}$ and $f$ defined in Algorithm \[alg:fsr-red\].
Clearly, $f$ is a nonsingular FSR.
Following Algorithm \[alg:fsr-red\], the Boolean circuit $f_3$ is described with Figures \[fig:subcircuitCP\], \[fig:subcircuitCMP\], \[fig:subcircuitMQ\], \[fig:subcircuitPS\] and \[fig:circuit-f3\]. To ease our presentation, from now on we also use operations with finite fan-in and fan-out for sketching a Boolean circuit. For example, as $x\oplus y=((\myNOT x)\myAND y)\myOR((\myNOT y)\myAND x)$, we allow XOR($\myXOR$), logically equivalent to a subcircuit consisting of five gates.
(17,29)(0,10)
In Figures \[fig:subcircuitCP\], \[fig:subcircuitCMP\], \[fig:subcircuitMQ\] and \[fig:subcircuitPS\], the operation ${\stackrel{\text{\tiny ?}}{=}}$ decides whether two $4{n}$-bit inputs are equal or not.
(26,53)(0,21)
In Figures \[fig:subcircuitCMP\] and \[fig:subcircuitMQ\], the operation $\min$ computes the minimum of two $4{n}$-bit integers.
(32,58) (0,30) (0,2)[(0,0)\[l\][Input: $\state{x}\in\BnrSet^{4{n}}$]{}]{} (0,-1)[(0,0)\[l\][Output: $m(\state{x}),q(\state{x})\in\BnrSet$]{}]{} (16,-4)[(0,0)[The subcircuits CP and CMP are given in Figures \[fig:subcircuitCP\] and \[fig:subcircuitCMP\], respectively.]{}]{}
(28,46)(0,23) (0,-1)[(0,0)\[l\][Input: $\state{x}\in\BnrSet^{4{n}}$]{}]{} (0,-4)[(0,0)\[l\][Output: $p(\state{x}),s(\state{x})\in\BnrSet$]{}]{}
(44,28)(22,10.5) (22,-4)[(0,0)[The subcircuits MQ and PS are given in Figures \[fig:subcircuitMQ\] and \[fig:subcircuitPS\], respectively.]{}]{}
\[lemma:alg-red-poly-time\] Let $f_1$ be the feedback logic of the FSR $f$ given by Algorithm \[alg:fsr-red\]. Then $\sizeof{{f_1}}< 37908\cdot\sizeof{f_0}^4$ and Algorithm \[alg:fsr-red\] is polynomial-time computable.
The operation $\state{x}\mapsto{\widehat{\state{x}}}$ uses one NOT gate on ${\lceil{\state{x}}\rceil_{1}}$. Given the input $(x_0,x_1,\dots,x_{4{n}-1})$ and $(y_0,y_1,\dots,y_{4{n}-1})$, the operation ${\stackrel{\text{\tiny ?}}{=}}$outputs $\myNOT((x_0\oplus y_0)\myOR(x_1\oplus y_1)\myOR\dots\myOR(x_{4{n}-1}\oplus y_{4{n}-1}))$ and costs at most $24{n}$ gates. The state transformation $L$ is performed by one XOR gate, i.e., $5$ gates. By Appendix \[appnd:min\], the operation $\min$ uses $104{n}^2 + 66{n}- 22$ gates.
Noticing $r\leq 2{n}\leq 3r-1$, $r\leq \sizeof{\crct{f_0}}$ and $$f_1(x_0,\dots,x_{4{n}-1})=x_0\oplus x_{2{n}}\oplus \crct{f_3}(x_1,x_2,\dots,x_{4{n}-1}),$$ we count gates in Figure \[fig:circuit-f3\] and obtain $$\begin{aligned}
\sizeof{\crct{f_1}}=&11+\sizeof{\crct{f_3}}\\
=&12{n}\cdot\sizeof{f_0} + 7488{n}^4 + 4752{n}^3 - 856{n}^2 + 274{n}+ 69\\
<& 37908\cdot\sizeof{f_0}^4.\end{aligned}$$
The Boolean circuit $f_0$ has $\sizeof{f_0}$ vertices and less than $2\cdot\sizeof{f_0}$ arcs; The feedback logic $f_1$ has at most $37908\cdot\sizeof{f_0}^4$ vertices and at most $75816\cdot\sizeof{f_0}^4$ arcs. The FSR $f$ uses $f_0$ and basic polynomial-time computable operations for at most $37908\cdot\sizeof{f_0}^4$ times and its main architecture is given by Figures \[fig:subcircuitCP\], \[fig:subcircuitCMP\], \[fig:subcircuitMQ\], \[fig:subcircuitPS\] and \[fig:circuit-f3\]. Therefore, Algorithm \[alg:fsr-red\] is polynomial-time computable.
In the rest of this section, ${n}$ is as given in Algorithm \[alg:fsr-red\], $p_0$ and $p_2$ are the polynomials as defined in Lemma \[lemma:poly-irreducible\], we also denote ${\mathfrak{C}}_{6{n}}={\mathbf{CycStr}\left(p_2\right)}\setminus{\mathbf{CycStr}\left(p_0\right)}$.
\[lemma:ell-conjugate-2ell\] Let $\state{v}\in{S_{4{n}}\left({{{\bm{\beta}}}}\right)}$, where ${{\bm{\beta}}}\in{\mathbf{CycStr}\left(p_0\right)}$. Then ${\widehat{\state{v}}}\in{S_{4{n}}\left({{{\bm{\gamma}}}}\right)}$ for some ${{\bm{\gamma}}}\in{\mathfrak{C}}_{6{n}}$.
Suppose ${\widehat{\state{v}}}\in{S_{4{n}}\left({{{\bm{\gamma}}}}\right)}$ for some ${{\bm{\gamma}}}\in{\mathbf{CycStr}\left(p_0\right)}$. By Lemmas \[lemma:cycle-state-cycle\] and \[lemma:cycle-structure-lfsr-p0\], $L^{3{n}}({\widehat{\state{v}}})={\widehat{\state{v}}}$ and $L^{3{n}}(\state{v})=\state{v}$. Since $L$ is a linear transformation and ${\bm{\iota}^{4{n}}}=\state{v}\oplus
{\widehat{\state{v}}}$, we have $L^{3{n}}({\bm{\iota}^{4{n}}})={\bm{\iota}^{4{n}}}$, contradictory to $L^{3{n}}({\bm{\iota}^{4{n}}})=
({\mathbf{0}^{{n}}}1{\mathbf{0}^{2{n}-1}}1{\mathbf{0}^{{n}-1}})$, where this vector is written without commas between bits. Therefore, the supposition above is absurd and ${{\bm{\gamma}}}\in{\mathfrak{C}}_{6{n}}$.
Because any $\state{v}\in\BnrSet^{4{n}}$, as an initial state of the $4{n}$-stage LFSR $p_2$, determines a unique cycle, in the rest of this section we denote ${\xi\left(\state{v}\right)}={{\bm{c}}}\in{\mathbf{CycStr}\left(p_2\right)}$ such that $\state{v}\in{S_{4{n}}\left({{{\bm{c}}}}\right)}$.
\[lemma:cycle-structure-fsr-red-f-Vset\] Let $$\begin{aligned}
{\mathfrak{D}}=&\left\{
{{\bm{c}}}\in{\mathbf{CycStr}\left(p_2\right)}:
{\xi\left(
\min{S_{4{n}}\left({{{\bm{c}}}}\right)}\oplus{\bm{\iota}^{4{n}}}
\right)}\in{\mathbf{CycStr}\left(p_0\right)},
\right.\\
&\qquad\qquad\qquad\qquad\left.\left\{\state{v}\in{S_{4{n}}\left({{{\bm{c}}}}\right)}:
{\xi\left({\widehat{\state{v}}}\right)}\in{\mathfrak{C}}_{6{n}}
\text{ and }
{\widehat{\state{v}}}
=\min{S_{4{n}}\left({{\xi\left({\widehat{\state{v}}}\right)}}\right)}\right\}=\emptyset\right\}\end{aligned}$$ and define a map $\rho:{\mathbf{CycStr}\left(p_2\right)}\rightarrow\BnrSet^{4{n}}$ as $$\label{eq:1}
{\rho\left({{\bm{c}}}\right)}=
\left\{
\begin{array}{l@{\text{ if }}l}
\min {S_{4{n}}\left({{{\bm{c}}}}\right)},& {{\bm{c}}}\in{\mathbf{CycStr}\left(p_2\right)}\setminus{\mathfrak{D}};\\
L^{5{n}}\left(\min{S_{4{n}}\left({{{\bm{c}}}}\right)}\right),&{{\bm{c}}}\in{\mathfrak{D}}.
\end{array}
\right.$$ Then the following two statements hold: (i) ${\mathfrak{D}}\subset{\mathfrak{C}}_{6{n}}$ and for any ${{\bm{c}}}\in{\mathfrak{D}}$, ${\xi\left({\widehat{{\rho\left({{\bm{c}}}\right)}}}\right)}\in {\mathfrak{C}}_{6{n}}\setminus{\mathfrak{D}}$. (ii) If ${{\bm{c}}}\in{\mathbf{CycStr}\left(p_2\right)}$ and ${\widehat{{\rho\left({{\bm{c}}}\right)}}}\in
\set{{\rho\left({{\bm{e}}}\right)}:{{\bm{e}}}\in{\mathbf{CycStr}\left(p_2\right)}}$, then ${{\bm{c}}}\in\set{[0],{\xi\left({\bm{\iota}^{4{n}}}\right)}}$.
For convenience in this proof we may write a cycle or vector without commas between its bits.\
*Claim:* If ${{\bm{c}}}\in{\mathbf{CycStr}\left(p_2\right)}$ satisfies ${\xi\left(\min{S_{4{n}}\left({{{\bm{c}}}}\right)}\oplus{\bm{\iota}^{4{n}}}
\right)}\in{\mathbf{CycStr}\left(p_0\right)}$, then ${{\bm{c}}}=[1\state{u}_{0}
0\state{u}_{1}0\state{u}_{2}
0\state{u}_{0}1\state{u}_{1}
0\state{u}_{2}]$, where $\state{u}_0,\state{u}_1,\state{u}_2\in\BnrSet^{{n}-1}$, $\state{u}_2=\state{u}_0\oplus \state{u}_1$ and $(1\state{u}_{0}
0\state{u}_{1}0\state{u}_{2}
0\state{u}_{0})=\min{S_{4{n}}\left({{{\bm{c}}}}\right)}$.\
*Proof of the claim.* By Lemma \[lemma:ell-conjugate-2ell\], ${{\bm{c}}}\in{\mathfrak{C}}_{6{n}}$. Denote ${{\bm{c}}}=[a_0\state{u}_0a_1\state{u}_1
a_2\state{u}_2a_3\state{u}_3
a_4\state{u}_4a_5\state{u}_5]$, where $$\left\{
\begin{aligned}
& a_i\in\BnrSet,0\leq i\leq 5;\\
&\state{u}_i\in\BnrSet^{{n}-1}, 0\leq i\leq 5;\\
&(a_0\state{u}_0a_1\state{u}_1
a_2\state{u}_2a_3\state{u}_3)=\min{S_{4{n}}\left({{{\bm{c}}}}\right)}.
\end{aligned}
\right.$$ Notice ${\xi\left({\bm{\iota}^{4{n}}}\right)}=[1{\mathbf{0}^{4{n}-1}}1{\mathbf{0}^{2{n}-1}}]$. Then ${\overline{a_0}}\state{u}_0a_1\state{u}_1
a_2\state{u}_2a_3\state{u}_3
{\overline{a_4}}\state{u}_4a_5\state{u}_5$ is concatenation of a same cycle in ${\mathbf{CycStr}\left(p_0\right)}$, implying where $a_2=a_0\oplus a_1\oplus1$ and $\state{u}_2=\state{u}_0\oplus \state{u}_1$. By Lemma \[lemma:cycle-structure-lfsr-p0\], $${{\bm{c}}}=[a_0\state{u}_0a_1\state{u}_1a_2\state{u}_2
{\overline{a_0}}\state{u}_0{\overline{a_1}}\state{u}_1a_2\state{u}_2].$$ By $$(a_0\state{u}_0a_1\state{u}_1
a_2\state{u}_2{\overline{a_0}}\state{u}_0)\leq
({\overline{a_0}}\state{u}_0{\overline{a_1}}\state{u}_1a_2\state{u}_2
a_0\state{u}_0),$$ we have $a_0=1$. By $$(a_0\state{u}_0a_1\state{u}_1
a_2\state{u}_2{\overline{a_0}}\state{u}_0)\leq
(\state{u}_0a_1\state{u}_1
a_2\state{u}_2{\overline{a_0}}\state{u}_0{\overline{a_1}}),$$ we have $a_1=0$. Then $a_2=0$. The proof of this claim is complete.
For a $k{n}$-cycle ${{\bm{c}}}=[b_0,b_1,\dots,b_{k{n}-1}]$, we call $$(b_{i},b_{(i+{n})\bmod k{n}},b_{(i+2{n})\bmod k{n}},\dots,b_{(i+(k-1){n})})$$ an *${n}$-sampling* of ${{\bm{c}}}$, $0\leq i< k{n}$.
Choose any ${{\bm{c}}}\in{\mathfrak{D}}$. Because of the claim above, let ${{\bm{c}}}=[1\state{u}_{0}
0\state{u}_{1}0\state{u}_{2}
0\state{u}_{0}1\state{u}_{1}
0\state{u}_{2}]$, where $\state{u}_0,\state{u}_1,\state{u}_2\in\BnrSet^{{n}-1}$, $\state{u}_2=\state{u}_0\oplus \state{u}_1$ and $(1\state{u}_{0}
0\state{u}_{1}0\state{u}_{2}
0\state{u}_{0})=\min{S_{4{n}}\left({{{\bm{c}}}}\right)}$. Then $${\widehat{{\rho\left({{\bm{c}}}\right)}}}={\bm{\iota}^{4{n}}}\oplus L^{5{n}}\left(\min{S_{4{n}}\left({{{\bm{c}}}}\right)}\right)=
(1\state{u}_{2}1\state{u}_{0}
0\state{u}_{1}0\state{u}_{2})$$ and hence $${\xi\left({\widehat{{\rho\left({{\bm{c}}}\right)}}}\right)}=
[
1\state{u}_{2}
1\state{u}_{0}
0\state{u}_{1}0\state{u}_{2}
1\state{u}_{0}1\state{u}_{1}
].$$ First, $3{n}\nmid\lengthof{{\xi\left({\widehat{{\rho\left({{\bm{c}}}\right)}}}\right)}}$. By Lemma \[lemma:cycle-structure-lfsr-p0\], ${\xi\left({\widehat{{\rho\left({{\bm{c}}}\right)}}}\right)}\in {\mathfrak{C}}_{6{n}}$. Second, as shown in the claim above, there is an ${n}$-sampling $(100010)$ of any cycle in ${\mathfrak{D}}$, while $(100010)$ is not an ${n}$-sampling of ${\xi\left({\widehat{{\rho\left({{\bm{c}}}\right)}}}\right)}$. Hence, ${\xi\left({\widehat{{\rho\left({{\bm{c}}}\right)}}}\right)}\notin {\mathfrak{D}}$. By Lemma \[lemma:ell-conjugate-2ell\], ${{\bm{c}}}\notin{\mathbf{CycStr}\left(p_0\right)}$. Till now Statement (i) of this lemma is proved.
Now we prove Statement (ii) of this lemma.
Denote $\state{v}_0=(01{\mathbf{0}^{4{n}-2}})$. Then ${\xi\left({\widehat{\state{v}_0}}\right)}
=[11{\mathbf{0}^{4{n}-2}}11{\mathbf{0}^{2{n}-2}}]$. By Lemma \[lemma:cycle-structure-lfsr-p0\], ${\xi\left({\widehat{\state{v}_0}}\right)}\in{\mathfrak{C}}_{6{n}}$. Seeing ${\widehat{\state{v}_0}}=\min
{S_{4{n}}\left({{\xi\left(
{\widehat{\state{v}_0}}\right)}
}\right)}$ and $\state{v}_0\in{S_{4{n}}\left({{\xi\left({\bm{\iota}^{4{n}}}\right)}}\right)}$, we have ${\xi\left({\bm{\iota}^{4{n}}}\right)}
\notin{\mathfrak{D}}$ and ${\bm{\iota}^{4{n}}}=
{\rho\left({\xi\left({\bm{\iota}^{4{n}}}\right)}\right)}$.
Denote $V_c=\set{{\rho\left({{\bm{e}}}\right)}:{{\bm{e}}}\in{\mathbf{CycStr}\left(p_2\right)}}$. Notice that ${\rho\left({{\bm{c}}}\right)}\in{S_{4{n}}\left({{{\bm{c}}}}\right)}$, ${{\bm{c}}}\in{\mathbf{CycStr}\left(p_2\right)}$. It is sufficient to consider the following cases.
1. \[case:cycextr-1\] If ${{\bm{c}}}\in{\mathfrak{D}}$, by Statement (i), ${\xi\left({\widehat{{\rho\left({{\bm{c}}}\right)}}}\right)}\in{\mathfrak{C}}_{6{n}}\setminus{\mathfrak{D}}$. By the definition of ${\mathfrak{D}}$, $${\widehat{{\rho\left({{\bm{c}}}\right)}}}\neq \min{S_{4{n}}\left({{\xi\left({\widehat{{\rho\left({{\bm{c}}}\right)}}}\right)}}\right)}
={\rho\left({\xi\left({\widehat{{\rho\left({{\bm{c}}}\right)}}}\right)}\right)}$$ and hence $ {\widehat{{\rho\left({{\bm{c}}}\right)}}}\notin V_c$.
2. \[case:cycextr-2\] If ${\xi\left({\bm{\iota}^{4{n}}}\right)}\neq{{\bm{c}}}\in{\mathfrak{C}}_{6{n}}\setminus{\mathfrak{D}}$, then ${\rho\left({{\bm{c}}}\right)}=\min{S_{4{n}}\left({{{\bm{c}}}}\right)}\notin
\set{{\mathbf{0}^{4{n}}},{\bm{\iota}^{4{n}}}}$. By the definition of ${\mathfrak{D}}$, ${\xi\left({\widehat{{\rho\left({{\bm{c}}}\right)}}}\right)}\notin
{\mathfrak{D}}$, yielding ${\rho\left({\xi\left({\widehat{{\rho\left({{\bm{c}}}\right)}}}\right)}\right)}=\min{S_{4{n}}\left({{\xi\left({\widehat{{\rho\left({{\bm{c}}}\right)}}}\right)}}\right)}$. By Lemma \[lemma:conjugate-nominkiss\], ${\widehat{{\rho\left({{\bm{c}}}\right)}}}\neq
\min{S_{4{n}}\left({{\xi\left({\widehat{{\rho\left({{\bm{c}}}\right)}}}\right)}}\right)}$ and hence ${\widehat{{\rho\left({{\bm{c}}}\right)}}}\notin V_c$.
3. \[case:cycextr-3\] If $[0]\neq{{\bm{c}}}\in{\mathbf{CycStr}\left(p_0\right)}$ and ${\xi\left({\widehat{{\rho\left({{\bm{c}}}\right)}}}\right)}\notin
{\mathfrak{D}}$, then similar to Case (\[case:cycextr-2\]), we also get ${\widehat{{\rho\left({{\bm{c}}}\right)}}}\notin V_c$.
4. \[case:cycextr-4\] If ${{\bm{c}}}\in{\mathbf{CycStr}\left(p_0\right)}$ and ${\xi\left({\widehat{{\rho\left({{\bm{c}}}\right)}}}\right)}\in
{\mathfrak{D}}$, then by the proved Statement (i) of this lemma, $${{\bm{c}}}\neq{\xi\left({\rho\left({\xi\left({\widehat{{\rho\left({{\bm{c}}}\right)}}}\right)}\right)}\oplus{\bm{\iota}^{4{n}}}\right)}\in{\mathfrak{C}}_{6{n}}.$$ Because ${\xi\left({\rho\left({{\bm{e}}}\right)}\right)}={{\bm{e}}}$ for any ${{\bm{e}}}\in{\mathbf{CycStr}\left(p_2\right)}$, we have ${\widehat{{\rho\left({{\bm{c}}}\right)}}}\neq {\rho\left({\xi\left({\widehat{{\rho\left({{\bm{c}}}\right)}}}\right)}\right)}$, yielding ${\widehat{{\rho\left({{\bm{c}}}\right)}}}\notin V_c$.
5. \[case:cycextr-5\] Besides, consider ${{\bm{c}}}\in\set{[0],{\xi\left({\bm{\iota}^{4{n}}}\right)}}$. We have ${\widehat{{\rho\left([0]\right)}}}={\bm{\iota}^{4{n}}}=
{\rho\left({\xi\left({\bm{\iota}^{4{n}}}\right)}\right)}$ since ${\xi\left({\bm{\iota}^{4{n}}}\right)}\in{\mathfrak{C}}_{6{n}}\setminus{\mathfrak{D}}$.
Till now all cases are listed and Statement (ii) of this lemma holds.
\[lemma:cycle-structure-fsr-red-f\] Let $\rho$ be given in Lemma \[lemma:cycle-structure-fsr-red-f-Vset\]. Let the map $\lambda:\BnrSet^{4{n}}\rightarrow\BnrSet$ be defined as $${\lambda\left(\state{v}\right)}=
\left\{
\begin{aligned}
1,& \text{ if }\state{v}\in\set{{\rho\left({{\bm{c}}}\right)}:{{\bm{c}}}\in{\mathbf{CycStr}\left(p_2\right)}}
\text{ and }{\xi\left({\widehat{\state{v}}}\right)}\in{\mathfrak{C}}_{6{n}};\\
0,& \text{ otherwise.}
\end{aligned}
\right.$$ Let $D_{p_2}^{f}$ be the graph defined as in Lemma \[lemma:cycle-join\](Recall that $f$ and $f_3$ are given in Algorithm \[alg:fsr-red\]). Then the following statements hold: (i) Statements (i) and (ii) of Lemma \[lemma:cycle-join\] hold, where $g$ in Lemma \[lemma:cycle-join\] is the LFSR $p_2$. (ii) Each ${{\bm{c}}}\in{\mathfrak{C}}_{6{n}}$ is not an isolated vertex in $D_{p_2}^{f}$. (iii) Every ${{\bm{c}}}\in{\mathbf{CycStr}\left(p_0\right)}$ is an isolated vertex in $D_{p_2}^{f}$ if and only if $f_0$ is unsatisfiable.
Since ${\rho\left({{\bm{c}}}\right)}\in{S_{4{n}}\left({{{\bm{c}}}}\right)}$ for any ${{\bm{c}}}\in {\mathbf{CycStr}\left(p_2\right)}$, we have $$\cset{\set{\state{v}\in{S_{4{n}}\left({{{\bm{c}}}}\right)}:{\lambda\left(\state{v}\right)}=1}}
\le
\cset{\set{{\rho\left({{\bm{c}}}\right)}}}=1.$$
Following from Statement (ii) of Lemma \[lemma:cycle-structure-fsr-red-f-Vset\] and ${\lambda\left({\bm{\iota}^{4{n}}}\right)}=0$, we have ${\lambda\left(\state{v}\right)}\cdot {\lambda\left({\widehat{\state{v}}}\right)}=0$ for any $\state{v}\in\BnrSet^{4{n}}$.
Use ${\mathfrak{D}}$ defined in Lemma \[lemma:cycle-structure-fsr-red-f-Vset\].
Let $q(\state{u}_0)$, $q(\state{v}_0)$, $\state{u}_i$ and $\state{v}_i$ be as in Algorithm \[alg:fsr-red\], $0\leq i\leq 6{n}$. Denote ${{\bm{e}}}_0={\xi\left(\state{u}_0\right)}$ and ${{\bm{e}}}_1={\xi\left(\state{v}_0\right)}$. By Lemmas \[lemma:cycle-state-cycle\] and \[lemma:cycle-structure-lfsr-p0\], ${S_{4{n}}\left({{{\bm{e}}}_0}\right)}=\set{\state{u}_i:1\leq i\leq 6{n}}$; ${S_{4{n}}\left({{{\bm{e}}}_1}\right)}=\set{\state{v}_i:1\leq i\leq 6{n}}$; $\state{u}_0=\state{u}_{6{n}}$; $\state{v}_0=\state{v}_{6{n}}$; $\state{u}_0=\state{u}_{3{n}}$(resp. $\state{v}_0=\state{v}_{3{n}}$) if and only if ${{\bm{e}}}_0\in{\mathbf{CycStr}\left(p_0\right)}$(resp. ${{\bm{e}}}_1\in{\mathbf{CycStr}\left(p_0\right)}$); $L^{3{n}}({\widehat{\state{u}_i}})={\widehat{\state{u}_i}}$(resp. $L^{3{n}}({\widehat{\state{v}_i}})={\widehat{\state{v}_i}}$) if and only if ${\xi\left({\widehat{\state{u}_i}}\right)}
\in{\mathbf{CycStr}\left(p_0\right)}$(resp. ${\xi\left({\widehat{\state{v}_i}}\right)}
\in{\mathbf{CycStr}\left(p_0\right)}$); $L^{6{n}}({\widehat{\state{u}_i}})\neq \min\set{L^j({\widehat{\state{u}_i}}):1\leq j\leq 6{n}}$(resp. $L^{6{n}}({\widehat{\state{v}_i}})\neq \min\set{L^j({\widehat{\state{v}_i}}):1\leq j\leq 6{n}}$) if and only if ${\widehat{\state{u}_i}}\neq \min{S_{4{n}}\left({{\xi\left({\widehat{\state{u}_i}}\right)}}\right)}$(resp. ${\widehat{\state{v}_i}}\neq \min{S_{4{n}}\left({{\xi\left({\widehat{\state{v}_i}}\right)}}\right)}$); $\state{u}_{n}=\state{u}_{\min}$(resp. $\state{v}_{n}=\state{v}_{\min}$) if and only if $\state{u}_0=L^{5{n}}\left(\min{S_{4{n}}\left({{{\bm{e}}}_0}\right)}\right)$ (resp. $\state{v}_0=L^{5{n}}\left(\min{S_{4{n}}\left({{{\bm{e}}}_1}\right)}\right)$); $L^{3{n}}({\widehat{\state{u}_{\min}}})={\widehat{\state{u}_{\min}}}$(resp. $L^{3{n}}({\widehat{\state{v}_{\min}}})={\widehat{\state{v}_{\min}}}$) is equivalent to ${\xi\left({\widehat{\state{u}_{\min}}}\right)}\in{\mathbf{CycStr}\left(p_0\right)}$(resp. ${\xi\left({\widehat{\state{v}_{\min}}}\right)}\in{\mathbf{CycStr}\left(p_0\right)}$). Then $q(\state{u}_0)=1$(resp. $q(\state{v}_0)=1$) if and only if ${{\bm{e}}}_0\in{\mathfrak{D}}$ and $\state{u}_0={\rho\left({{\bm{e}}}_0\right)}$(resp. ${{\bm{e}}}_1\in{\mathfrak{D}}$ and $\state{v}_0={\rho\left({{\bm{e}}}_1\right)}$). By Lemma \[lemma:ell-conjugate-2ell\], $\set{{{\bm{e}}}_0,{{\bm{e}}}_1}\not\subset{\mathbf{CycStr}\left(p_0\right)}$. Then $f_3({\lfloor{\state{u}_0}\rfloor_{4{n}-1}})=f_3({\lfloor{\state{v}_0}\rfloor_{4{n}-1}})=1$ if and only if one of the following cases holds:
1. \[case:1\] ${{\bm{e}}}_0\in{\mathbf{CycStr}\left(p_0\right)}$, $\state{u}_0 = \min{S_{4{n}}\left({{{\bm{e}}}_0}\right)}$ and $\set{\state{v}\in{S_{4{n}}\left({{{\bm{e}}}_0}\right)}: f_0({\lfloor{\state{v}}\rfloor_{r}})=1}\neq\emptyset$;
2. \[case:2\] ${{\bm{e}}}_1\in{\mathbf{CycStr}\left(p_0\right)}$, $\state{v}_0 = \min{S_{4{n}}\left({{{\bm{e}}}_1}\right)}$ and $\set{\state{v}\in{S_{4{n}}\left({{{\bm{e}}}_1}\right)}: f_0({\lfloor{\state{v}}\rfloor_{r}})=1}\neq\emptyset$;
3. \[case:3\] ${{\bm{e}}}_0\in{\mathfrak{C}}_{6{n}}$, $\state{u}_0 = \min{S_{4{n}}\left({{{\bm{e}}}_0}\right)}$ and ${\xi\left({\widehat{\state{u}_0}}\right)}={{\bm{e}}}_1\in{\mathfrak{C}}_{6{n}}$;
4. \[case:4\] ${{\bm{e}}}_1\in{\mathfrak{C}}_{6{n}}$, $\state{v}_0 = \min{S_{4{n}}\left({{{\bm{e}}}_1}\right)}$ and ${\xi\left({\widehat{\state{v}_0}}\right)}={{\bm{e}}}_0\in{\mathfrak{C}}_{6{n}}$;
5. \[case:5\] ${{\bm{e}}}_0,{{\bm{e}}}_1\in{\mathfrak{C}}_{6{n}}$, ${{\bm{e}}}_0\in{\mathfrak{D}}$ and $\state{u}_0={\rho\left({{\bm{e}}}_0\right)}$;
6. \[case:6\] ${{\bm{e}}}_0,{{\bm{e}}}_1\in{\mathfrak{C}}_{6{n}}$, ${{\bm{e}}}_1\in{\mathfrak{D}}$ and $\state{v}_0={\rho\left({{\bm{e}}}_1\right)}$.
Considering Statement (i) of Lemma \[lemma:cycle-structure-fsr-red-f-Vset\], we have $$\label{eqn:red-f3-equiv}
f_3(\state{x})=\left\{
\begin{array}{lll}
1,&\text{ if } \state{x}={\lfloor{\state{v}}\rfloor_{4{n}-1}}, \text{ where }
\state{v}={\rho\left({{\bm{c}}}\right)} \text{ and } {{\bm{c}}},
{\xi\left({\widehat{\state{v}}}\right)}\in{\mathfrak{C}}_{6{n}};
&\text{ by Cases \ref{case:3}, \ref{case:4}, \ref{case:5} and \ref{case:6}}\\
1,&\text{ if } \state{x}={\lfloor{\state{v}}\rfloor_{4{n}-1}}, \text{ where }
\state{v}={\rho\left({{\bm{c}}}\right)}, {{\bm{c}}}\in{\mathbf{CycStr}\left(p_0\right)}\\
&\text{ and }
\set{\state{u}\in{S_{4{n}}\left({{{\bm{c}}}}\right)}:f_0({\lfloor{\state{u}}\rfloor_{r}})=1}\neq\emptyset;
&\text{ by Cases \ref{case:1} and \ref{case:2}}\\
0,&\text{ otherwise.}
& \end{array}
\right.$$ By Eq.(\[eqn:red-f3-equiv\]) and Lemma \[lemma:ell-conjugate-2ell\], $ f_3(\state{x})=1$ implies that there exists $\state{v}\in\BnrSet^{4{n}}$ satisfying $\state{x}={\lfloor{\state{v}}\rfloor_{4{n}-1}}$ and ${\lambda\left(\state{v}\right)}=1$.
Hitherto we have shown that Eq.(\[eqn:cycle-join-lambda-cnd\]) holds, where $g$ in Eq.(\[eqn:cycle-join-lambda-cnd\]) is the LFSR $p_2$.
By Lemma \[lemma:conjugate-nominkiss\] and Statement (i) of Lemma \[lemma:cycle-structure-fsr-red-f-Vset\], $D_{p_2}^f$ is loopless. Assume that $D_{p_2}^f$ is not acyclic. Then in $D_{p_2}^f$ there is a walk $({{\bm{c}}}_0,{{\bm{c}}}_1,\dots,{{\bm{c}}}_{m-1},{{\bm{c}}}_{m})$ for some $m\ge2$. Specifically, ${{\bm{c}}}_i\in{\mathbf{CycStr}\left(p_2\right)}$, $0\leq i<m$, are pairwise distinct, ${{\bm{c}}}_{m}={{\bm{c}}}_0$, and for any $0\leq i< m$ there is an arc incident from ${{\bm{c}}}_{i}$ to ${{\bm{c}}}_{i+1}$. By Lemma \[lemma:ell-conjugate-2ell\] and the definition of $\lambda$, any ${{\bm{c}}}\in{\mathbf{CycStr}\left(p_0\right)}$ is a source in $D_{p_2}^f$. Additionally, for ${{\bm{c}}}\in{\mathfrak{D}}$, by Statement (i) of Lemma \[lemma:cycle-structure-fsr-red-f-Vset\] and $$\left\{\state{v}\in{S_{4{n}}\left({{{\bm{c}}}}\right)}:
{\xi\left({\widehat{\state{v}}}\right)}\in{\mathfrak{C}}_{6{n}}
\text{ and }
{\widehat{\state{v}}}
=\min{S_{4{n}}\left({{\xi\left({\widehat{\state{v}}}\right)}}\right)}\right\}=\emptyset,$$ no arc is incident from a cycle in ${\mathfrak{C}}_{6{n}}$ to ${{\bm{c}}}$, i.e., any arc entering ${{\bm{c}}}$ leaves from a source in ${\mathbf{CycStr}\left(p_0\right)}$. Besides, as shown in the proof of Lemma \[lemma:cycle-structure-fsr-red-f-Vset\], ${\rho\left({\xi\left({\bm{\iota}^{4{n}}}\right)}\right)}={\bm{\iota}^{4{n}}}$ and ${\xi\left({\widehat{{\bm{\iota}^{4{n}}}}}\right)}=[0]$, then ${\xi\left({\bm{\iota}^{4{n}}}\right)}$ is a sink in $D_{p_2}^f$. Therefore, ${\xi\left({\bm{\iota}^{4{n}}}\right)}\neq{{\bm{c}}}_i\in{\mathfrak{C}}_{6{n}}\setminus {\mathfrak{D}}$ and ${\rho\left({{\bm{c}}}_i\right)}=\min{S_{4{n}}\left({{{\bm{c}}}_i}\right)}$, $0\leq i<m$. Noticing ${\widehat{{\rho\left({{\bm{c}}}_i\right)}}}\in
{S_{4{n}}\left({{{\bm{c}}}_{i+1}}\right)}$, $0\leq i<m$, by Lemma \[lemma:conjugate-nominkiss\], we have $\min{S_{4{n}}\left({{{\bm{c}}}_{i+1}}\right)}<\min{S_{4{n}}\left({{{\bm{c}}}_{i}}\right)}$ for $0\leq i< m$, implying $\min{S_{4{n}}\left({{{\bm{c}}}_0}\right)}<
\min{S_{4{n}}\left({{{\bm{c}}}_0}\right)}$, which is ridiculous. Therefore, the assumption is absurd and $D_{p_2}^f$ is acyclic.
Till now we have proved that Eq.(\[eqn:cycle-join-lambda-cnd\]) holds and $D_{p_2}^{f}$ is acyclic, where $g$ is the LFSR $p_2$. By Lemma \[lemma:cycle-join\], Statement (i) of this lemma is proved.
Now we prove Statement (ii) of this lemma. Suppose ${{\bm{c}}}\in{\mathfrak{C}}_{6{n}}$.
- If ${\xi\left({\bm{\iota}^{4{n}}}\oplus\min{S_{4{n}}\left({{{\bm{c}}}}\right)}\right)}\in
{\mathfrak{C}}_{6{n}}$, then ${{\bm{c}}}\notin{\mathfrak{D}}$, ${\rho\left({{\bm{c}}}\right)}=\min{S_{4{n}}\left({{{\bm{c}}}}\right)}$ and ${\lambda\left({\rho\left({{\bm{c}}}\right)}\right)}=1$. By Eq.(\[eqn:red-f3-equiv\]), an arc leaves ${{\bm{c}}}$.
- If ${\xi\left({\bm{\iota}^{4{n}}}\oplus\min{S_{4{n}}\left({{{\bm{c}}}}\right)}\right)}\in
{\mathbf{CycStr}\left(p_0\right)}$ and $$\left\{\state{v}\in{S_{4{n}}\left({{{\bm{c}}}}\right)}:
{\xi\left({\widehat{\state{v}}}\right)}\in{\mathfrak{C}}_{6{n}}
\text{ and }
{\widehat{\state{v}}}
=\min{S_{4{n}}\left({{\xi\left({\widehat{\state{v}}}\right)}}\right)}\right\}\neq\emptyset.$$ Let $\state{v}_0\in{S_{4{n}}\left({{{\bm{c}}}}\right)}$ satisfy ${\xi\left({\widehat{\state{v}_0}}\right)}\in{\mathfrak{C}}_{6{n}}$ and ${\widehat{\state{v}_0}}
=\min{S_{4{n}}\left({{\xi\left({\widehat{\state{v}_0}}\right)}}\right)}$. Clearly, ${\xi\left({\widehat{\state{v}_0}}\right)}\notin{\mathfrak{D}}$. Then ${\rho\left({\xi\left({\widehat{\state{v}_0}}\right)}\right)}
=\min{S_{4{n}}\left({{\xi\left({\widehat{\state{v}_0}}\right)}}\right)}
={\widehat{\state{v}_0}}$ and ${\lambda\left({\widehat{\state{v}_0}}\right)}=1$. By Eq.(\[eqn:red-f3-equiv\]), an arc enters ${{\bm{c}}}$.
- If ${\xi\left({\bm{\iota}^{4{n}}}\oplus\min{S_{4{n}}\left({{{\bm{c}}}}\right)}\right)}\in
{\mathbf{CycStr}\left(p_0\right)}$ and $$\left\{\state{v}\in{S_{4{n}}\left({{{\bm{c}}}}\right)}:
{\xi\left({\widehat{\state{v}}}\right)}\in{\mathfrak{C}}_{6{n}}
\text{ and }
{\widehat{\state{v}}}
=\min{S_{4{n}}\left({{\xi\left({\widehat{\state{v}}}\right)}}\right)}\right\}=\emptyset,$$ then ${{\bm{c}}}\in{\mathfrak{D}}$. By Statement (i) of Lemma \[lemma:cycle-structure-fsr-red-f-Vset\] and Eq.(\[eqn:red-f3-equiv\]), an arc is incident from ${{\bm{c}}}$ to a cycle in ${\mathfrak{C}}_{6{n}}\setminus{\mathfrak{D}}$.
Till now Statement (ii) of this lemma is proved.
Now we prove Statement (iii) of this lemma. Since ${\lambda\left(\state{v}\right)}=1$ occurs only if ${\xi\left({\widehat{\state{v}}}\right)}\in{\mathfrak{C}}_{6{n}}$, in $D_{p_2}^f$ no arc enters any ${{\bm{c}}}\in{\mathbf{CycStr}\left(p_0\right)}$. Since $r\leq2{n}$ and $\bigcup_{{{\bm{c}}}\in{\mathbf{CycStr}\left(p_0\right)}}{S_{2{n}}\left({{{\bm{c}}}}\right)}=\BnrSet^{2{n}}$, we have $$\set{{\lfloor{\state{v}}\rfloor_{r}}:\state{v}\in\bigcup_{{{\bm{c}}}\in{\mathbf{CycStr}\left(p_0\right)}}{S_{4{n}}\left({{{\bm{c}}}}\right)}}=
\set{{\lfloor{\state{v}}\rfloor_{r}}:\state{v}\in\bigcup_{{{\bm{c}}}\in{\mathbf{CycStr}\left(p_0\right)}}{S_{2{n}}\left({{{\bm{c}}}}\right)}}
=\BnrSet^r.$$ By Lemma \[lemma:ell-conjugate-2ell\], ${\lambda\left({\rho\left({{\bm{c}}}\right)}\right)}=1$ for any ${{\bm{c}}}\in{\mathbf{CycStr}\left(p_0\right)}$. Then by Eq.(\[eqn:red-f3-equiv\]), in $D_{P_2}^f$ there exists an arc incident from some ${{\bm{c}}}\in{\mathbf{CycStr}\left(p_0\right)}$ to some ${{\bm{d}}}\in{\mathfrak{C}}_{6{n}}$ if and only if $f_0$ is satisfiable. Thus, Statement (iii) of this lemma is proved.
\[lemma:subFSR-order\] If $g$ is a subFSR of the FSR $f$, then $g$ is of stage $2{n}$.
Let $g$ be of stage $m$. By Lemmas \[lemma:cycle-structure-fsr\], $\sum_{{{\bm{d}}}\in{\mathbf{CycStr}\left(g\right)}}{\mathrm{per}\left({{\bm{d}}}\right)}=2^m$. By Lemmas \[lemma:subFSR-cyc-str\], \[lemma:cycle-structure-lfsr-p0\] and Statement (i) of \[lemma:cycle-structure-fsr-red-f\], for any ${{\bm{d}}}\in{\mathbf{CycStr}\left(g\right)}$, ${\mathrm{per}\left({{\bm{d}}}\right)}\equiv\cset{\set{{\mathbf{0}^{4{n}}}}\cap{S_{4{n}}\left({{{\bm{d}}}}\right)}}\bmod 3{n}$. Then we have an integer equation $a+3{n}b=2^m$, where $a\in\BnrSet$ and $0\leq b<2^{4{n}}/(3{n})$. Since $2{n}=\min\set{i>0:{3{n}}\mid(2^i-1)}$, where ${n}=3^k$ for some $1\leq k\in\Int$, we have $a=1$ and $2{n}\mid m$. Hence, $m=2{n}<4{n}$.
\[lemma:reduction-irreducible\] The FSR $f$ is irreducible if and only if the Boolean circuit $f_0$ is satisfiable.
Suppose $f_0$ to be unsatisfiable. By Statements (i) and (iii) of Lemma \[lemma:cycle-structure-fsr-red-f\], ${\mathbf{CycStr}\left(p_0\right)}\subset{\mathbf{CycStr}\left(f\right)}$. By Lemma \[lemma:subFSR-cyc-str\], $p_0$ is a subFSR of $f$ and hence $f$ is reducible.
Suppose $f_0$ to be satisfiable. Assume that $h$ is a subFSR of $f$. By Statement (i) of Lemma \[lemma:cycle-structure-fsr-red-f\], $$\label{eqn:red-irr-1}
{\mathbf{CycStr}\left(h\right)}\subset{\mathbf{CycStr}\left(p_2\right)}.$$ Furthermore, by Statements (i) and (ii) of Lemma \[lemma:cycle-structure-fsr-red-f\], any cycle in ${\mathfrak{C}}_{6{n}}$ joins with other cycles to combine a cycle in ${\mathbf{CycStr}\left(f\right)}$, implying $$\label{eqn:red-irr-2}
\left({\mathbf{CycStr}\left(p_2\right)}\setminus{\mathbf{CycStr}\left(p_0\right)}\right)\cap {\mathbf{CycStr}\left(f\right)}=\emptyset.$$ Similarly, by Statements (i) and (iii) of Lemma \[lemma:cycle-structure-fsr-red-f\], if $f_0$ is satisfiable, then $$\label{eqn:red-irr-3}
{\mathbf{CycStr}\left(p_0\right)}\not\subset{\mathbf{CycStr}\left(f\right)}.$$ By Eqs.(\[eqn:red-irr-1\]), (\[eqn:red-irr-2\]), (\[eqn:red-irr-3\]) and Lemma \[lemma:subFSR-cyc-str\], we get $${\mathbf{CycStr}\left(h\right)}\subset{\mathbf{CycStr}\left(f\right)}\cap{\mathbf{CycStr}\left(p_2\right)}
\subset{\mathbf{CycStr}\left(f\right)}\cap{\mathbf{CycStr}\left(p_0\right)}
\subsetneq{\mathbf{CycStr}\left(p_0\right)}.$$ By Lemma \[lemma:subFSR-order\], $h$ is of stage $2{n}$. However, by Lemma \[lemma:cycle-structure-fsr\], $$2^{2{n}}=\sum_{{{\bm{c}}}\in{\mathbf{CycStr}\left(h\right)}}{\mathrm{per}\left({{\bm{c}}}\right)}<
\sum_{{{\bm{c}}}\in{\mathbf{CycStr}\left(p_0\right)}}{\mathrm{per}\left({{\bm{c}}}\right)}=2^{2{n}},$$ which is absurd. Therefore, $f$ is irreducible.
**PROBLEM**: FSR IRREDUCIBILITY
[INSTANCE]{}: An FSR $f$ with its feedback logic $f_1$ as a Boolean circuit of size $\sizeof{f_1}$.
[QUESTION]{}: Is $f$ irreducible?
By Lemmas \[lemma:CKT-SAT-NP-C\], \[lemma:alg-red-poly-time\] and \[lemma:reduction-irreducible\], Algorithm \[alg:fsr-red\] is a polynomial-time Karp reduction from CIRCUIT SATISFIABILITY to FSR IRREDUCIBILITY. Therefore, we conclude that
\[thm:irreducibility-np-hard\] The FSR IRREDUCIBILITY problem is [$\mathbf{NP}$]{}-hard.
[$\mathbf{NP}$]{}-hardness of deciding indecomposable FSRs {#sect:FSR-indecomposability}
============================================================
\[lemma:f2-f0-satisfiability\] Let $f_0$ be an $r$-input Boolean logic and $$\label{eqn:f0-f2}
f_2(\state{x})=\left\{
\begin{array}{ll}
0, & \text{ if } \state{x}={\mathbf{0}^{r}}; \\
1, & \text{ if } \state{x}={\mathbf{1}^{r}} \text{ and } f_0({\mathbf{1}^{r}})=1; \\
f_0({\mathbf{0}^{r}}), &\text{ if }
\state{x}={\mathbf{1}^{r}} \text{ and } f_0({\mathbf{1}^{r}})=0; \\
f_0(\state{x}), & \text{ otherwise. }
\end{array}
\right.$$ Then the Boolean function $\crct{f_2}$ is satisfiable if and only if $\crct{f_0}$ is satisfiable.
Below Algorithm \[alg:fsr-dec\] transforms a given Boolean circuit to an FSR.
An $r$-input Boolean circuit $\crct{f_0}$. A $(2{n}+1)$-stage FSR $f$, where $k=\min\set{i\in\Int:i\geq\log_3(r/2)}$ and ${n}=3^{k}$. Construct an $r$-input Boolean circuit $\crct{f_2}$ defined by Eq.(\[eqn:f0-f2\]). \[line:Cdec-begin\] Let $(x_{1},x_{2},\dots,x_{2{n}})$ be the input of $\crct{f_3}$. \[line:u0\] $\state{u}_0=(x_{2{n}}\oplus x_{{n}}\oplus x_1,x_1\oplus x_2,
x_2\oplus x_3,\dots,x_{2{n}-1}\oplus x_{2{n}})$. $\state{u}_i=L(\state{u}_{i-1})$. $a_i=\crct{f_2}({\lfloor{\state{u}_i}\rfloor_{r}})$. The Boolean circuit $\crct{f_3}$ returns $1$. The Boolean circuit $\crct{f_3}$ returns $0$. \[line:Cdec-end\] the FSR $f(x_0,\dots,x_{2{n}+1})=x_{2{n}+1}
\oplus x_{2{n}}\oplus x_{{n}+1}\oplus x_{{n}}\oplus x_1 \oplus x_0
\oplus \crct{f_3}(x_1,x_2,\dots,x_{2{n}})$.
Figure \[fig:f2\] is a sketch of $f_2$.
(26,47.5) (1,18) (13,-4)[(0,0)[Here $\myNOT_r$(resp. $\myAND_r$) denotes the logical NOT(resp. AND) of $r$ bits.]{}]{}
Following Algorithm \[alg:fsr-dec\], we describe $f_3$ with Figure \[fig:DEC-f3\].
(30,41)(0,23)
(-12,1)[(0,0)\[l\][ $M$ is defined in Line \[line:u0\] of Algorithm \[alg:fsr-dec\];]{}]{} (-12,-1)[(0,0)\[l\][${\stackrel{\text{\tiny ?}}{=}}$ decides whether two $2{n}$-bit inputs are equal or not; ]{}]{} (-12,-4)[(0,0)\[l\][$\min$ computes the minimum of two $2{n}$-bit integers. ]{}]{}
In the rest of this section, we use notations $\crct{f_0}$, $f_2$, $\crct{f_3}$ and $f$ defined in Algorithm \[alg:fsr-dec\].
Clearly, $f$ is a nonsingular FSR.
Similar to Lemma \[lemma:alg-red-poly-time\], we count gates in Figure \[fig:DEC-f3\] and derive the lemma below.
\[lemma:alg-dec-poly-time\] Let $f_1$ be the feedback logic of the FSR $f$ given by Algorithm \[alg:fsr-dec\]. Then $\sizeof{{f_1}}\leq 264\cdot\sizeof{\crct{f_0}}^3$. Particularly, Algorithm \[alg:fsr-dec\] is polynomial-time computable.
In the rest of this section, ${n}$ is given in Algorithm \[alg:fsr-dec\], $p_0$ and $p_1$ are the polynomials as defined in Lemma \[lemma:poly-irreducible\], and we denote ${\overline{{\mathbf{CycStr}\left(p_0\right)}}}={\mathbf{CycStr}\left(p_1\right)}\setminus{\mathbf{CycStr}\left(p_0\right)}$. Moreover, let $L_1$ denote the state transformation of the LFSR $p_1$.
For $\state{v}=(v_0,v_1,\dots,v_{2{n}})\in\BnrSet^{2{n}+1}$, define the map ${\pi\left({\state{v}}\right)}=
(v_0\oplus v_{1},v_1\oplus v_{2},\dots,v_{2{n}-1}\oplus v_{2{n}})\in\BnrSet^{2{n}}$ and ${\chi\left({\state{v}}\right)}=v_0\oplus v_{{n}}\oplus v_{2{n}}$.
The maps $\pi$ and $\chi$ have the properties in Lemma \[lemma:conjugate-property\].
\[lemma:conjugate-property\]\[lemma:cycle-2-family\] The following statements hold. (i) For $\state{v}\in\BnrSet^{2{n}+1}$, ${\chi\left({{\widehat{\state{v}}}}\right)}
={\chi\left({{\overline{\state{v}}}}\right)}
={\chi\left({{\state{v}}}\right)}\oplus1$, ${\pi\left({{\widehat{\state{v}}}}\right)}={\widehat{{\pi\left({\state{v}}\right)}}}$ and $L({\pi\left({\state{v}}\right)})={\pi\left({L_1(\state{v})}\right)}$. (ii) For $\state{w}=(w_0,w_1,\dots,w_{2{n}-1})\in\BnrSet^{2{n}}$, $\set{\state{v}\in\BnrSet^{2{n}+1}:{\pi\left({\state{v}}\right)}=\state{w}}=
\set{\state{u},{\overline{\state{u}}}}$, where $$\state{u}=\left(0,w_0, w_0\oplus w_1,\dots,w_0\oplus w_1\oplus\cdots\oplus w_{2{n}-1}\right).$$ (iii) For any $\state{v}\in\BnrSet^{2{n}+1}$, $${\chi\left({\state{v}}\right)}=
\left\{
\begin{aligned}
0,&\text{ if }\state{v}\in\bigcup_{{{\bm{c}}}\in {\mathbf{CycStr}\left(p_0\right)}}{S_{2{n}+1}\left({{{\bm{c}}}}\right)},\\
1,&\text{ if }\state{v}\in\bigcup_{{{\bm{c}}}\in {\overline{{\mathbf{CycStr}\left(p_0\right)}}}}{S_{2{n}+1}\left({{{\bm{c}}}}\right)},
\end{aligned}
\right.$$ (iv) For $\state{v}\in\BnrSet^{2{n}+1}$, if $${\state{v}}\in\bigcup_{{{\bm{c}}}\in{{\mathbf{CycStr}\left(p_0\right)}}}{S_{2{n}+1}\left({{{\bm{c}}}}\right)},$$ then $${\widehat{\state{v}}}\in\bigcup_{{{\bm{c}}}\in{\overline{{\mathbf{CycStr}\left(p_0\right)}}}}{S_{2{n}+1}\left({{{\bm{c}}}}\right)}.$$ (v) For any $\state{v}\in\BnrSet^{2{n}+1}$, ${\lfloor{L_1(\state{v})}\rfloor_{2{n}}}=
L({\lfloor{\state{v}}\rfloor_{2{n}}})\oplus \state{w}_0$, where $\state{w}_0=(0,\dots,0,{\chi\left({\state{v}}\right)})\in\BnrSet^{2{n}}$.
Statements (i) and (ii) of this lemma can be proved by direct computation.
Denote $\state{v}=(v_0,v_1,\dots,v_{2{n}})$.
If $\state{v}\in\bigcup_{{{\bm{c}}}\in {\mathbf{CycStr}\left(p_0\right)}}{S_{2{n}+1}\left({{{\bm{c}}}}\right)}$. clearly, ${\chi\left({\state{v}}\right)}=v_0\oplus v_{n}\oplus v_{2{n}}=0$. Suppose $\state{v}\in\bigcup_{{{\bm{c}}}\in {\overline{{\mathbf{CycStr}\left(p_0\right)}}}}{S_{2{n}+1}\left({{{\bm{c}}}}\right)}$. By Lemma \[lemma:cycle-structure-lfsr-p0x\], ${\overline{\state{v}}}\in\bigcup_{{{\bm{c}}}\in {\mathbf{CycStr}\left(p_0\right)}}{S_{2{n}+1}\left({{{\bm{c}}}}\right)}$. Then by Statement (i), ${\chi\left({\state{v}}\right)}=1\oplus{\chi\left({{\overline{\state{v}}}}\right)}=1$. Statement (iii) is proved.
By Lemma \[lemma:cycle-structure-lfsr-p0x\], $$\left(\bigcup_{{{\bm{c}}}\in{{\mathbf{CycStr}\left(p_0\right)}}}{S_{2{n}+1}\left({{{\bm{c}}}}\right)}\right)
\bigcup
\left(\bigcup_{{{\bm{c}}}\in{\overline{{\mathbf{CycStr}\left(p_0\right)}}}}{S_{2{n}+1}\left({{{\bm{c}}}}\right)}\right)
=\BnrSet^{2{n}+1}.$$ Then Statement (iv) follows from Statement (i) and (iii).
Additionally, Statement (v) holds because $$\begin{aligned}
{\lfloor{L_1(\state{v})}\rfloor_{2{n}}}=&
(v_2,\dots,v_{2{n}},v_{2{n}}\oplus v_{{n}+1}\oplus v_{{n}}\oplus v_1 \oplus v_0)\\
=&(v_2,\dots,v_{2{n}}, v_{{n}+1}\oplus v_1\oplus {\chi\left({\state{v}}\right)})\\
=&L((v_1,v_2,\dots,v_{2{n}}))\oplus \state{w}_0\\
=&L({\lfloor{\state{v}}\rfloor_{2{n}}})\oplus \state{w}_0. \hfill\qedhere\end{aligned}$$
\[lemma:unique-subFSR-01-pre\] Let the map $\lambda:\BnrSet^{2{n}+1}\rightarrow\BnrSet$ be defined as $${\lambda\left(\state{v}\right)}=
\left\{
\begin{aligned}
1,&\text{ if } {\chi\left({\state{v}}\right)}=0 \text{ and }{\pi\left({\state{v}}\right)}=
\min\set{L^i({\pi\left({\state{v}}\right)}): 1\leq i\leq 3{n}};\\
0,&\text{ otherwise.}
\end{aligned}
\right.$$ Let $D_{p_1}^{f}$ be the graph defined as in Lemma \[lemma:cycle-join\](Recall that $f$ and $f_3$ are given in Algorithm \[alg:fsr-dec\]). Then the following statements hold: (i) Statements (i) and (ii) of Lemma \[lemma:cycle-join\] hold, where $g$ in Lemma \[lemma:cycle-join\] is the LFSR $p_1$. (ii) If $f_2$ is satisfiable, then ${\mathbf{CycStr}\left(p_0\right)}\not\subset{\mathbf{CycStr}\left(f\right)}$ and there exists $\state{v}\in\BnrSet^{2{n}+1}$ satisfying $f_3({\lfloor{\state{v}}\rfloor_{2{n}}})=1$ and ${\chi\left({\state{v}}\right)}=0$.
Suppose $\state{v}\in{S_{2{n}+1}\left({{{\bm{c}}}}\right)}$, where ${{\bm{c}}}\in{\mathbf{CycStr}\left(p_1\right)}$. By Statement (i) of Lemma \[lemma:conjugate-property\], Lemmas \[lemma:cycle-state-cycle\] and \[lemma:cycle-structure-lfsr-p0x\], we get $$\label{eqn:dec-cycle-min}
\set{L^i({\pi\left({\state{v}}\right)}):1\leq i\leq 3{n}}=
\set{{\pi\left({L_1^i(\state{v})}\right)}:1\leq i\leq 3{n}}=
\set{{\pi\left({\state{u}}\right)}:\state{u}\in{S_{2{n}+1}\left({{{\bm{c}}}}\right)}}.$$ Besides, by Statements (ii) of Lemma \[lemma:conjugate-property\], there exists a unique vector $\state{u}$ in ${S_{2{n}+1}\left({{{\bm{c}}}}\right)}$ satisfying ${\pi\left({\state{u}}\right)}=\min\set{{\pi\left({\state{u}}\right)}:\state{u}\in{S_{2{n}+1}\left({{{\bm{c}}}}\right)}}$. Thus, by Statement (iii) of Lemma \[lemma:cycle-2-family\], we have $$\label{eqn:dec-cycle-extr}
\cset{\set{\state{v}\in{S_{2{n}+1}\left({{{\bm{c}}}}\right)}:{\lambda\left(\state{v}\right)}=1}}=
\left\{
\begin{aligned}
1,&\text{ if }\state{c}\in{\mathbf{CycStr}\left(p_0\right)};\\
0,&\text{ if }\state{c}\in{\overline{{\mathbf{CycStr}\left(p_0\right)}}}.
\end{aligned}
\right.$$
By Statement (i) of Lemma \[lemma:conjugate-property\], ${\lambda\left(\state{v}\right)}\cdot{\lambda\left({\widehat{\state{v}}}\right)}=0$ for any $\state{v}\in\BnrSet^{2{n}+1}$.
In Algorithm \[alg:fsr-dec\], $\state{x}=(x_1,x_2,\dots,x_{2{n}})$ and $\state{u}_0={\pi\left({\state{y}}\right)}$, where $\state{y}=(x_{2{n}}\oplus x_{{n}},x_1,x_2,\dots,x_{2{n}})
$ is the unique vector in $\BnrSet^{2{n}+1}$ satisfying ${\chi\left({\state{y}}\right)}=0$ and ${\lfloor{\state{y}}\rfloor_{2{n}}}=\state{x}$. Let ${{\bm{c}}}$ be the cycle satisfying $\state{y}\in{S_{2{n}+1}\left({{{\bm{c}}}}\right)}$. By Lemmas \[lemma:cycle-state-cycle\], \[lemma:cycle-structure-lfsr-p0x\] and Eq.(\[eqn:dec-cycle-min\]), $\state{u}_{3{n}}=
\min\set{L^i(\state{u}_0):1\leq i\leq 3{n}}$ is equivalent to $\state{u}_0=\min\set{{\pi\left({\state{v}}\right)}:\state{v}\in{S_{2{n}+1}\left({{{\bm{c}}}}\right)}}$. By Eq.(\[eqn:dec-cycle-min\]), $$\set{1\leq i\leq 3{n}:f_2({\lfloor{L^i(\state{u}_0)}\rfloor_{r}})=1}\neq\emptyset$$ is equivalent to $$\set{\state{u}\in{S_{2{n}+1}\left({{{\bm{c}}}}\right)}:f_2({\lfloor{{\pi\left({\state{u}}\right)}}\rfloor_{r}})=1}\neq\emptyset.$$ Thus, by Algorithm \[alg:fsr-dec\], we have the following claim.\
*Claim.* $f_3(\state{x})=1$ if and only if ${\lambda\left(\state{y}\right)}=1$ and $\set{\state{v}\in{S_{2{n}+1}\left({{{\bm{c}}}}\right)}:f_2({\lfloor{{\pi\left({\state{v}}\right)}}\rfloor_{r}})=1}\neq\emptyset$.
If $f_3(\state{x})=1$, then ${\lambda\left(\state{y}\right)}=1$ and ${\lfloor{\state{y}}\rfloor_{2{n}}}=\state{x}$. Therefore, Eq.(\[eqn:cycle-join-lambda-cnd\]) holds, where $g$ in Lemma \[lemma:cycle-join\] is the LFSR $p_1$.
Furthermore, by Statement (iv) of Lemma \[lemma:conjugate-property\] and Eq.(\[eqn:dec-cycle-extr\]), any arc of $D_{p_1}^f$ is incident from a cycle in ${\mathbf{CycStr}\left(p_0\right)}$ to a cycle in ${\overline{{\mathbf{CycStr}\left(p_0\right)}}}$. Hence, $D_{p_1}^f$ is acyclic.
Till now we have proved that Eq.(\[eqn:cycle-join-lambda-cnd\]) holds and $D_{p_1}^{f}$ is acyclic, where $g$ in Eq.(\[eqn:cycle-join-lambda-cnd\]) is the LFSR $p_1$. By Lemma \[lemma:cycle-join\], Statements (i) and (ii) of Lemma \[lemma:cycle-join\] hold and Statement (i) of this lemma is proved, where $g$ in Lemma \[lemma:cycle-join\] is the LFSR $p_1$.
Now we prove Statement (ii) of this lemma. Suppoe that $f_2$ is satisfiable. Since $D_{p_1}^f$ is acyclic, by Statement (i) of Lemma \[lemma:cycle-join\], it is sufficient to prove that not every ${{\bm{c}}}\in{\mathbf{CycStr}\left(p_0\right)}$ is isolated in $D_{p_1}^f$.
Following from Eq.(\[eqn:dec-cycle-extr\]) and the claim above, for ${{\bm{c}}}\in{\mathbf{CycStr}\left(p_0\right)}$, there exists $\state{v}\in{S_{2{n}+1}\left({{{\bm{c}}}}\right)}$ satisfying ${\lambda\left(\state{v}\right)}=1$ and $f_3({\lfloor{\state{v}}\rfloor_{2{n}}})=1$ if and only if $\set{\state{v}\in{S_{2{n}+1}\left({{{\bm{c}}}}\right)}:f_2({\lfloor{{\pi\left({\state{v}}\right)}}\rfloor_{r}})=1}\neq\emptyset$.
By Lemma \[lemma:cycle-structure-lfsr-p0x\] and Statement (ii) of Lemma \[lemma:conjugate-property\], the map $\pi$ gives a bijection from $\bigcup_{{{\bm{c}}}\in{\mathbf{CycStr}\left(p_0\right)}}{S_{2{n}+1}\left({{{\bm{c}}}}\right)}$ to $\BnrSet^{2{n}}$. Thus, seeing $r\leq 2{n}$, we get $$\set{{\lfloor{{\pi\left({\state{v}}\right)}}\rfloor_{r}}:\state{v}\in\bigcup_{{{\bm{c}}}\in{\mathbf{CycStr}\left(p_0\right)}} {S_{2{n}+1}\left({{{\bm{c}}}}\right)}}
=\set{{\lfloor{\state{v}}\rfloor_{r}}:\state{v}\in\BnrSet^{2{n}}}
=\BnrSet^r.$$ Therefore, on one hand, there exists $\state{v}\in\BnrSet^{2{n}+1}$ satisfying $f_3({\lfloor{\state{v}}\rfloor_{2{n}}})=1$ and ${\chi\left({\state{v}}\right)}=0$; On the other hand, in $D_{p_1}^f$ there exists at least one arc incident from a cycle in ${\mathbf{CycStr}\left(p_0\right)}$, i.e., some ${{\bm{c}}}\in{\mathbf{CycStr}\left(p_0\right)}$ is not isolated in $D_{p_1}^f$. By Statement (i) of this lemma, ${{\bm{c}}}$ joins with other cycles in ${\mathbf{CycStr}\left(p_1\right)}$ to combine one cycle in ${\mathbf{CycStr}\left(f\right)}$, and hence ${{\bm{c}}}\notin{\mathbf{CycStr}\left(f\right)}$, yielding ${\mathbf{CycStr}\left(p_0\right)}\not\subset{\mathbf{CycStr}\left(f\right)}$.
\[lemma:unique-subFSR-01\] If $f_2$ is satisfiable and $g$ is a subFSR of $f$ satisfying $[0]\in{\mathbf{CycStr}\left(g\right)}$, then $g$ is the LFSR $x_1\oplus x_0$, i.e., ${\mathbf{CycStr}\left(g\right)}=\set{[0],[1]}$.
Let $g$ be an $m$-stage subFSR of $f$.
By Lemma \[lemma:cycle-structure-fsr\], we have $2^m=\sum_{{{\bm{d}}}\in{\mathbf{CycStr}\left(g\right)}}\lengthof{{{\bm{d}}}}$. Furthermore, by Lemma \[lemma:cycle-structure-lfsr-p0x\] and Statement (i) of \[lemma:unique-subFSR-01-pre\], we have $$\label{eqn:dec-cyc-cnt}
{\mathrm{per}\left({{\bm{d}}}\right)}\equiv
\cset{\set{{\mathbf{0}^{2{n}+1}},{\mathbf{1}^{2{n}+1}}}\cap{S_{2{n}+1}\left({{{\bm{d}}}}\right)}}
\bmod 3{n}.$$ Since for any $\state{v}\in\BnrSet^{2{n}+1}$, there exists a unique cycle ${{\bm{c}}}\in{\mathbf{CycStr}\left(f\right)}$ satisfying $\state{v}\in{S_{2{n}+1}\left({{{\bm{c}}}}\right)}$, we get an integer equation $$\label{eqn:order-cnt-dec}
3{n}a+b=2^m,$$ where $1\leq m\leq2{n}$, $0\leq a\leq 2(2^{2{n}}-1)/(3{n})$ and $b\in\set{0,1,2}$. Since $2{n}=\min\set{0<i\in\Int:3{n}\mid(2^i-1)}$, where ${n}=3^k$ for some $1\leq k\in\Int$, Eq.(\[eqn:order-cnt-dec\]) holds only if (i) $b=1$ and $m=2{n}$ or (ii) $b=2$ and $m=1$. So, we only have to consider two possible cases below.
Case (i): $g$ is of stage $2{n}$. By Statement (i) of Lemma \[lemma:unique-subFSR-01-pre\], ${\mathbf{CycStr}\left(g\right)}\subset{\mathbf{CycStr}\left(p_1\right)}$. Denote $$\begin{aligned}
V_0=&\set{\state{v}\in\BnrSet^{2{n}}:
\state{v}\in{S_{2{n}}\left({{{\bm{c}}}}\right)},{{\bm{c}}}\in{\mathbf{CycStr}\left(g\right)}\cap{\mathbf{CycStr}\left(p_0\right)}};\\
V_1=&\set{\state{v}\in\BnrSet^{2{n}}:
\state{v}\in{S_{2{n}}\left({{{\bm{c}}}}\right)},{{\bm{c}}}\in{\mathbf{CycStr}\left(g\right)}\cap{\overline{{\mathbf{CycStr}\left(p_0\right)}}}}.\end{aligned}$$ Since $b=1$ in Eq.(\[eqn:order-cnt-dec\]), by Lemma \[lemma:cycle-structure-lfsr-p0x\], ${\mathbf{CycStr}\left(g\right)}$ consists of $[0]$ and $({2^{2{n}}-1})/({3{n}})$ $3{n}$-cycles. Moreover, by Statement (ii) of Lemma \[lemma:unique-subFSR-01-pre\], we have $$\cset{{\mathbf{CycStr}\left(g\right)}\cap{\overline{{\mathbf{CycStr}\left(p_0\right)}}}}\ge1,$$ implying $V_1\neq\emptyset$. Besides, as the states of the $2{n}$-stage FSR $g$, $V_0\cup V_1=\BnrSet^{2{n}}$ and $V_0\cap V_1=\emptyset$. For $V\subset\BnrSet^{2{n}}$, denote $L(V)=\set{L(\state{v}):\state{v}\in V}$. On one hand, by Lemma \[lemma:cycle-state-cycle\], $L(V_0)=V_0$. Because $L$ is bijective on $\BnrSet^{2{n}}$, we have $L(V_1)=V_1$. Denote $\state{w}_0=(0,\dots,0,1)\in\BnrSet^{2{n}}$. On the other hand, by Statements (iii) and (v) of Lemma \[lemma:cycle-2-family\], we have $L(\state{v})\oplus \state{w}_0\in V_1$ for any $\state{v}\in V_1$. Thus, both $\state{v}\mapsto L(\state{v})$ and $\state{v}\mapsto L(\state{v})\oplus \state{w}_0$ are closed on $V_1$. Since the linear transformation $L$ has its irreducible minimal polynomial $p_0$ of degree $2{n}$, $L^i(\state{w}_0)$, $i=0,\dots,2{n}-1$, is a basis of the linear space $\BnrSet^{2{n}}$. Then for any $\state{v}_0\in V_1$, there exist $b_i\in\BnrSet$, $1\leq i\leq 3{n}$, satisfying $\state{v}_0=\bigoplus_{i=1}^{3{n}}b_i\cdot L^{3{n}-i}(\state{w}_0)$. Let $\state{v}_i=L(\state{v}_{i-1})\oplus (b_i\cdot\state{w}_0)$, $1\leq i\leq 3{n}$. Then $\state{v}_i\in V_1$, $1\leq i\leq 3{n}$. However, by Lemmas \[lemma:cycle-state-cycle\] and \[lemma:cycle-structure-lfsr-p0x\], $L^{3{n}}$ is an identity map. Hence, $\state{v}_{3{n}}=L^{3{n}}(\state{v}_0)\oplus
\left(\bigoplus_{i=1}^{3{n}}b_i\cdot L^{3{n}-i}(\state{w}_0)\right)={\mathbf{0}^{2{n}}}\in V_0$, yielding ${\mathbf{0}^{2{n}}}\in V_0\cap V_1=\emptyset$, which is ridiculous. Therefore, Case (i) does not occur.
Case(ii). $g$ is of stage $1$. Since $[0]\in{\mathbf{CycStr}\left(g\right)}$, we have ${\mathbf{CycStr}\left(g\right)}=\set{[0],[1]}$, i.e., $g$ is the the LFSR $x_1\oplus x_0$.
\[lemma:unique-factor-01\] If $f_2$ is satisfiable, then for any FSR $h$, $f\neq h\cascade(x_1\oplus x_0)$.
Assume $f= h\cascade(x_1\oplus x_0)$. Then $h$ is a $2{n}$-stage FSR and $h(x_0,x_1,\dots,x_{2{n}})=x_{2{n}}\oplus h_1(x_0,x_1,\dots,x_{2{n}-1})$, where $h_1$ is a $2{n}$-input Boolean logic. [ By Statement (ii) of Lemma \[lemma:unique-subFSR-01-pre\], if $f_2$ is satisfiable, then there exists $\state{v}_0\in
\BnrSet^{2{n}+1}
$ satisfying $f_3({\lfloor{\state{v}_0}\rfloor_{2{n}}})=1$ and ${\chi\left({\state{v}_0}\right)}=0$.]{} Let $f_1$ denote the feedback logic of $f$ and $\state{v}_0=(a_0,a_1,\dots,a_{2{n}})$. Then $f_1(\state{v}_0)=a_1\oplus a_{{n}+1}\oplus{\chi\left({\state{v}_0}\right)}\oplus
f_3({\lfloor{\state{v}_0}\rfloor_{2{n}}})
=a_1\oplus a_{{n}+1}\oplus1$. Thus, $f(\state{v}_0\parallel f_1(\state{v}_0))=
h\left({\pi\left({\state{v_0}}\right)}\parallel
(a_{2{n}}\oplus a_1\oplus a_{{n}+1}\oplus1)\right)=0$, yielding $$\label{eqn:unique-factor-01-1}
h_1({\pi\left({\state{v_0}}\right)})=a_{2{n}}\oplus a_1\oplus a_{{n}+1}\oplus1.$$
Let $\state{u}_0={\widehat{{\overline{\state{v}_0}}}}$. By Statements (i) and (ii) of Lemma \[lemma:conjugate-property\], ${\chi\left({\state{u}_0}\right)}=0$ and ${\pi\left({\state{u}_0}\right)}={\widehat{{\pi\left({\state{v}_0}\right)}}}$. If $${\pi\left({\state{u}_0}\right)}
\neq\min\set{L^i({\pi\left({\state{u}_0}\right)}):
1\leq i\leq 3{n}},$$ then $f_3({\lfloor{{\overline{\state{v}_0}}}\rfloor_{2{n}}})=f_3({\lfloor{\state{u}_0}\rfloor_{2{n}}})=0$. Otherwise, assume ${\pi\left({\state{u}_0}\right)}=\min\set{L^i({\pi\left({\state{u}_0}\right)}):
1\leq i\leq 3{n}}$. Since $f_3({\lfloor{\state{v}_0}\rfloor_{2{n}}})=1$, we get $${\pi\left({\state{v}_0}\right)}=
\min\set{L^i({\pi\left({\state{v}_0}\right)}):1\leq i\leq 3{n}}.$$ As ${\pi\left({\state{u}_0}\right)}={\widehat{{\pi\left({\state{v}_0}\right)}}}$, by Lemmas \[lemma:conjugate-nominkiss\] and \[lemma:cycle-structure-lfsr-p0\], we have $\set{{\pi\left({\state{v}_0}\right)},{\pi\left({\state{u}_0}\right)}}=\set{{\mathbf{0}^{2{n}}},{\bm{\iota}^{2{n}}}}$. Considering ${\chi\left({\state{v}_0}\right)}={\chi\left({\state{u}_0}\right)}=0$, we have $$\set{{\state{v}_0},{\state{u}_0}}=\set{{\mathbf{0}^{2{n}+1}},
{\overline{{\bm{\iota}^{2{n}+1}}}}}.$$ Because $f_3({\lfloor{\state{v}_0}\rfloor_{2{n}}})=1$ while $f_3({\mathbf{0}^{2{n}}})=0$, we have $\state{u}_0={\mathbf{0}^{2{n}+1}}$, yielding $f_3({\lfloor{{\overline{\state{v}_0}}}\rfloor_{2{n}}})=
f_3({\lfloor{\state{u}_0}\rfloor_{2{n}}})=0$.
We have proved $f_3({\lfloor{{\overline{\state{v}_0}}}\rfloor_{2{n}}})=0$. Then $F({\overline{\state{v}_0}})=L_1({\overline{\state{v}_0}})$, where $F$ is the state transformation of $f$. Using ${\chi\left({{\state{v}_0}}\right)}=0$ and Statements (i)-(ii) of Lemma \[lemma:conjugate-property\], we get $$f({\overline{\state{v}_0}}\parallel (a_1\oplus a_{{n}+1}\oplus{\chi\left({{\overline{\state{v}_0}}}\right)}\oplus f_3({\lfloor{{\overline{\state{v}_0}}}\rfloor_{2{n}}})))=
h({\pi\left({\state{v}_0}\right)}
\parallel(a_{2{n}}\oplus a_1\oplus a_{{n}+1}))=0,$$ implying $$\label{eqn:unique-factor-01-2}
h_1({\pi\left({\state{v}_0}\right)})
=a_{2{n}}\oplus a_1\oplus a_{{n}+1}
.$$
Our assumption $f= h\cascade(x_1\oplus x_0)$ leads to contradictory Eqs. (\[eqn:unique-factor-01-1\]) and (\[eqn:unique-factor-01-2\]). The proof is completed.
\[lemma:decomposable-reducible\][@GD70] Let ${h}$ be an $m$-stage decomposable FSR satisfying $h({\mathbf{0}^{m+1}})=0$. Then there exist two FSRs $h_1$ and $h_2$ such that $h=h_1\cascade h_2$, where $h_2$ is a $k$-stage FSR for some $1\le k<m$ and $[0]\in{\mathbf{CycStr}\left(h_2\right)}$. Particularly, $h_2$ is a subFSR of $h$ and $h$ is reducible.
Since $h$ is decomposable, we assume $h=h_1'\cascade h_2'$, where $h_2'$ is a $k$-stage FSR, $1\leq k<m$. If $h_2'({\mathbf{0}^{k+1}})=0$, let $h_1=h_1'$ and $h_2=h_2'$. Assume $h_2'({\mathbf{0}^{k+1}})=1$. Let $h_2=h_2'\oplus1$ and $h_1(x_0,x_1,\dots,x_{m-k})=h_1'(x_0\oplus1,x_1\oplus1,\dots,x_{m-k}\oplus1)$. Then $h=h_1'\cascade h_2'=h_1\cascade h_2$ and $h_2({\mathbf{0}^{k+1}})=h_2'({\mathbf{0}^{k+1}})\oplus1=0$. Besides, $h_2({\mathbf{0}^{k+1}})=0$ is equivalent to $[0]\in{\mathbf{CycStr}\left(h_2\right)}$.
Because $h_1({\mathbf{0}^{m-k+1}})=h_1(h_2({\mathbf{0}^{k+1}}),h_2({\mathbf{0}^{k+1}}),\dots,h_2({\mathbf{0}^{k+1}}))=
h({\mathbf{0}^{m+1}})=0$, we have ${G\left(h_2\right)}\subset{G\left(h_1;h_2\right)}={G\left(h\right)}$, where ${G\left(h_1;h_2\right)}$ is the set of sequences generated by the cascade connection of $h_1$ into $h_2$. Therefore, $h_2$ is a subFSR of $h$ and $h$ is reducible. The idea of Lemma \[lemma:decomposable-reducible\] was given by [@GD70] and here we reinterpret it for readability.
\[lemma:reduction-indecomposable\] The FSR $f$ is indecomposable if and only if the Boolean circuit $f_0$ is satisfiable.
Consider two cases below.
Case (i): $f_0$ is satisfiable. By Lemma \[lemma:f2-f0-satisfiability\], $f_2$ is satisfiable. Assume $f$ to be decomposable. Since $f_2({\mathbf{0}^{r}})=0$, by Algorithm \[alg:fsr-dec\], we have $f_3({\mathbf{0}^{2{n}}})=0$ and $f({\mathbf{0}^{2{n}+1}})=0$, implying $[0]\in{\mathbf{CycStr}\left(f\right)}$. By Lemma \[lemma:decomposable-reducible\], there exist FSRs $h$ and $g$ such that $f=h\cascade g$, where $g$ is a subFSR of $f$ satisfying $[0]\in{\mathbf{CycStr}\left(g\right)}$. By Lemma \[lemma:unique-subFSR-01\], $g$ is the LFSR $x_1\oplus x_0$. However, by Lemma \[lemma:unique-factor-01\], $f\neq h\cascade(x_1\oplus x_0)$. Hence, the assumption is absurd and $f$ is indecomposable.
Case (ii): $f_0$ is unsatisfiable. By Lemma \[lemma:f2-f0-satisfiability\], $f_2$ is unsatisfiable. By Algorithm \[alg:fsr-dec\], $f_3(\state{x})=0$ for any $\state{x}\in\BnrSet^{2{n}}$. Then $f$ is exactly the LFSR $p_1$ and $f(x_0,x_1,\dots,x_{2{n}})=(x_{2{n}}\oplus x_{{n}}\oplus x_0)\cascade(x_1\oplus x_0)$. So, $f$ is decomposable.
**PROBLEM**: FSR INDECOMPOSABILITY
[INSTANCE]{}: An FSR $f$ with its feedback logic $f_1$ as a Boolean circuit of size $\sizeof{f_1}$.
[QUESTION]{}: Is $f$ indecomposable?
By Lemmas \[lemma:CKT-SAT-NP-C\], \[lemma:alg-dec-poly-time\] and \[lemma:reduction-indecomposable\], Algorithm \[alg:fsr-dec\] is a polynomial-time Karp reduction from CIRCUIT SATISFIABILITY to FSR INDECOMPOSABILITY. Therefore, we conclude that
\[thm:indecomposability-np-hard\] The FSR INDECOMPOSABILITY problem is [$\mathbf{NP}$]{}-hard.
Conclusion {#sect:conclusion}
==========
Deciding irreducibility/indecomposability of FSRs is meaningful for sophisticated circuit implementation and security analysis of stream ciphers. Here we have proved both the decision problems are [$\mathbf{NP}$]{}-hard. Assuming **P**$\neq$[$\mathbf{NP}$]{}, where **P** is the class of decision problems computed by polynomial-time deterministic Turing machines, it is intractable to find a polynomial-time computable algorithm for either problem.
Furthermore, it is still of theoretical interests to determine the computational complexity of search versions of FSR reducibility/decomposability, i.e., to find a subFSR/factor of a given FSR, where $g$ and $h$ are called factors of $f$ if $f=h\cascade g$. Besides, provided that the input Boolean circuit is satisfiable, Algorithm \[alg:fsr-red\](resp. Algorithm \[alg:fsr-dec\]) constructs an irreducible(resp. indecomposable) FSR. Since it is easy to efficiently find satisfiable Boolean circuits, it remains a question whether Algorithm \[alg:fsr-red\](resp. Algorithm \[alg:fsr-dec\]) can be modified to construct a family of irreducible(resp. indecomposable) FSRs with desirable properties in practice.
Appendices
==========
Appendix: the proof of Statement (i) of Lemma \[lemma:cycle-join\] {#appnd:proof-cycle-join}
------------------------------------------------------------------
Let $F$ denote the state transformation of the FSR $f$.
By Lemma \[lemma:cycle-state-cycle\], it is sufficient to prove the following claim.
*Claim:* For any $\state{u},\state{v}\in \BnrSet^m$, there exists $i\geq0$ satisfying $F^i(\state{u})=\state{v}$ if and only if $\state{u},\state{v}\in\bigcup_{{{\bm{c}}}\in{\mathfrak{C}}}{S_{m}\left({{{\bm{c}}}}\right)}$, where ${\mathfrak{C}}$ is a weakly connected component of $D_g^f$. We prove this claim by induction on the number of arcs in $D_g^f$.
If $D_g^f$ has no arc, then by Eq.(\[eqn:cycle-join-lambda-cnd\]), $f_3({\lfloor{\state{v}}\rfloor_{m-1}})=0$ for any $\state{v}\in\BnrSet^m$. Thus, ${\mathbf{CycStr}\left(g\right)}={\mathbf{CycStr}\left(f\right)}$ and the claim holds.
Now suppose that $D_g^f$ has at least one arc.
Because $D_g^f$ is acyclic, there exists a source ${{\bm{c}}}_0\in{\mathbf{CycStr}\left(g\right)}$ with positive outdegree. Denote $V=\set{\state{v}\in{S_{m}\left({{{\bm{c}}}_0}\right)}:f_3({\lfloor{\state{v}}\rfloor_{m-1}})=1,
{\lambda\left({\state{v}}\right)}=1}$. By Eq.(\[eqn:cycle-join-lambda-cnd\]), $\cset{V}=1$ and there is a unique arc leaving ${{\bm{c}}}_0$. Denote $V=\set{\state{v}_0}$. Let ${{\bm{c}}}_1
$ denote the unique successor of ${{\bm{c}}}_0$, and let ${\mathfrak{C}}$ denote the weakly connected component containing ${{\bm{c}}}_0$. We have ${{\bm{c}}}_1\neq{{\bm{c}}}_0$ because $D_g^f$ is acyclic.
Let ${\state{v}_0}=(v_0,v_1,\dots,v_{m-1})$ and $$\begin{aligned}
f_3'(x_1,\dots,x_m)=&f_3(x_1,\dots,x_m)\oplus\prod_{i=1}^{m-1}(x_i\oplus v_i\oplus1);\\
f'(x_0,x_1,\dots,x_m)=&g(x_0,x_1,\dots,x_m)\oplus f_3'(x_1,\dots,x_m).\end{aligned}$$ Define a directed graph $D_g^{f'}$ with the set of vertices ${\mathbf{CycStr}\left(g\right)}$ such that an arc is incident from ${{\bm{a}}}$ to ${{\bm{b}}}$ if and only if $$\set{\state{v}\in{S_{m}\left({{{\bm{a}}}}\right)}:
f_3'({\lfloor{\state{v}}\rfloor_{m-1}})=1,
{\lambda\left(\state{v}\right)}=1,
{\widehat{\state{v}}}\in {S_{m}\left({{{\bm{b}}}}\right)}}\neq\emptyset.$$ See that $f_3'$ differs from $f_3$ only at $(v_1,\dots,v_{m-1})$ with $f_3'(v_1,\dots,v_{m-1})=0$. Then $D_g^{f'}$ is obtained by removing the arc leaving ${{\bm{c}}}_0$ in $D_g^f$. Besides, Eq.(\[eqn:cycle-join-lambda-cnd\]) also holds for $f_3'$.
Denote $F'$ as the state transformation of $f'$. The cycle joining method gives $$\label{eqn:cycle-join-interchange-GG}
F'(\state{v})=
\left\{
\begin{aligned}
F({\widehat{\state{v}}}),&\text{ if } {\state{v}}\in\set{\state{v}_0,{\widehat{\state{v}_0}}}; \\
F(\state{v}),& \text{ otherwise.}
\end{aligned}
\right.$$
By induction, the claim above is assumed to hold for $f'$. We only have to consider states in $\bigcup_{{{\bm{c}}}\in{\mathfrak{C}}}{S_{m}\left({{{\bm{c}}}}\right)}$. In $D_g^{f'}$, ${\mathfrak{C}}\setminus\set{{{\bm{c}}}_0}$ and $\set{{{\bm{c}}}_0}$ are weakly connected components. Denoting $p=\lengthof{{{\bm{c}}}_0}$ and $q=\sum_{{{\bm{c}}}_0\neq{{\bm{c}}}\in{\mathfrak{C}}}\lengthof{{{\bm{c}}}}$, and using Lemma \[lemma:cycle-state-cycle\], we have $$\label{eqn:cycle-join-subcycles}
\left\{
\begin{aligned}
&\set{F'^i(F({\state{v}_0})):0\leq i< q}=
\bigcup_{{{\bm{c}}}_0\neq{{\bm{c}}}\in{\mathfrak{C}}}{S_{m}\left({{{\bm{c}}}}\right)};\\
&\set{F'^i(F({\widehat{\state{v}_0}})):0\leq i< p}={S_{m}\left({{{\bm{c}}}_0}\right)};\\
&F'^{q-1}(F({\state{v}_0}))={\widehat{\state{v}_0}};\\
&F'^{p-1}(F({\widehat{\state{v}_0}}))={\state{v}_0}.
\end{aligned}
\right.$$ By Eqs. (\[eqn:cycle-join-interchange-GG\]) and (\[eqn:cycle-join-subcycles\]), $F^{p+q}({\state{v}_0})=\state{v}_0$ and $$\set{F^i({\state{v}_0}):0\leq i< p+q}=
\bigcup_{{{\bm{c}}}\in{\mathfrak{C}}}{S_{m}\left({{{\bm{c}}}}\right)}.$$ Thus, the claim also holds for $f$.
The proof of this claim is complete by induction.
Appendix: The operation $\min$ {#appnd:min}
------------------------------
The operation $\min$ outputs the minimum of two integers.
Let $\min_{m}$ denote the operation computing the minimum of two $m$-bit nonnegative integers. Recall that a vector $\state{v}=(v_0,v_1,\dots,v_{m-1})$ is identified as the integer $\sum_{i=0}^{m-1}v_i2^i$. For $m=1$, we have $\min_1(x_{0},y_{0})=x_0\myAND y_0$. For $m\geq 2$, $\state{x}=(x_0,x_1,\dots,x_{m-1})$ and $\state{y}=(y_0,y_1,\dots,y_{m-1})$, we have $$\begin{aligned}
\min{_m}(\state{x},\state{y})= &
(x_{m-1}\oplus y_{m-1}\oplus1)
\times(\min{}_{m-1}({\lceil{\state{x}}\rceil_{m-1}},{\lceil{\state{y}}\rceil_{m-1}})
\parallel x_{m-1})\\
&\oplus (((x_{m-1}\oplus y_{m-1}) \myAND (x_{m-1}\oplus1))\times \state{x})\\
&\oplus (((x_{m-1}\oplus y_{m-1}) \myAND (y_{m-1}\oplus1))\times \state{y}),\end{aligned}$$ and thereby give a recursive description of $\min_m$ in Figure \[fig:max\], where $z=\min{_{m-1}}({\lceil{\state{x}}\rceil_{m-1}},{\lceil{\state{y}}\rceil_{m-1}})
\parallel x_{m-1}$.
(43.5,24) (25.75,8)
Here the multiplying operation $\times$ has a one-bit input $a$ and an $m$-bit input $\state{w}=(w_0,w_1,\dots,w_{m-1})$, and outputs $(a\myAND w_0,a\myAND w_1,\dots,a\myAND w_{m-1})$. Thus, the multiplying operation $\times$ costs $m$ gates. By Figure \[fig:max\], we have $\sizeof{\min_m}=12+13m+\sizeof{\min_{m-1}}$ for any $m\geq2$, and hence $\sizeof{\min_{m}} = (13m^2+37m-44)/2$.
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[^1]: The product FSR of $f$ and $g$ is denoted by $f.g$ in [@GD70], while by $f\cascade g$ in [@MST79]. We follow the latter in order to avoid ambiguity with periods or conventional multiplication.
[^2]: The fan-in of a gate is the number of bits fed into it.
[^3]: Some theorists adopt the term propositional directed acyclic graph(PDAG), and a PDAG is essentially the same as a Boolean circuit.
[^4]: Let $D$ be a directed graph with its set of vertices $V$. An undirected graph $H$ is obtained by taking each arc of $D$ as an edge of $H$. The weakly connected component(s) is(are) the connected component(s) of $H$. Formally, define a binary relation $$R=\set{(a,b)\in V\times V:\text{ there is an arc incident from }
a \text{ to }b \text{ or there is an arc incident from }
b \text{ to }a},$$ and then a weakly connected component of $D$ is an equivalence class w.r.t. the equivalence closure of $R$.
|
---
abstract: 'We give a complete rigorous proof of the full asymptotic expansion of the partition function of the dimer model on a square lattice on a torus for general weights $z_h,z_v$ of the dimer model and arbitrary dimensions of the lattice $m,n$. We assume that $m$ is even and we show that the asymptotic expansion depends on the parity of $n$. We review and extend the results of Ivashkevich, Izmailian, and Hu [@IIH] on the full asymptotic expansion of the partition function of the dimer model, and we give a rigorous estimate of the error term in the asymptotic expansion of the partition function.'
title: 'Dimer Model: Full Asymptotic Expansion of the Partition Function'
---
Pavel Bleher[^1], Brad Elwood[^2], Dražen Petrović[^3]
Indiana University-Purdue University Indianapolis
**
[^4]
Introduction
============
Dimer Model on a Square Lattice
-------------------------------
We consider the dimer model on a square lattice $\Gamma_{m,n}=(V_{m,n},E_{m,n})$ on the torus ${{\mathbb Z}}_m\times {{\mathbb Z}}_n={{\mathbb Z}}^2/(m{{\mathbb Z}}\times n{{\mathbb Z}})$ (periodic boundary conditions), where $V_{m,n}$ and $E_{m,n}$ are the sets of vertices and edges of $\Gamma_{m,n},$ respectively. A *dimer* on $\Gamma_{m,n}$ is a set of two neighboring vertices $\langle x,y\rangle$ connected by an edge. A *dimer configuration* ${\sigma}$ on ${\Gamma}_{m,n}$ is a set of dimers ${\sigma}=\{\langle x_i,y_i\rangle,\;i=1,\ldots,\frac{mn}{2}\}$ which cover $V_{m,n}$ without overlapping. An example of a dimer configuration is shown in Fig. \[F8\]. An obvious necessary condition for a configuration to exist is that at least one of $m,n$ is even, and so we assume that $m$ is even, $m=2m_0$.
![Example of a dimer configuration on a square $6\times 6$ lattice on the torus.[]{data-label="F8"}](Example_Configuration.png)
To define a weight of a dimer configuration, we split the full set of dimers in a configuration ${\sigma}$ into two classes: horizontal and vertical, with respective weights $z_h, z_v>0.$ If we denote the total number of horizontal and vertical dimers in ${\sigma}$ by $N_h({\sigma})$ and $N_v({\sigma}),$ respectively, then the *dimer configuration weight* is $$\label{int1}
w({\sigma})=\prod_{i=1}^{\frac{mn}{2}}w(x_i,y_i)=z_h^{N_h({\sigma})} z_v^{N_v({\sigma})},$$ where $w(x_i,y_i)$ denotes the weight of the dimer $\langle x_i,y_i\rangle\in{\sigma}$. We denote by $\Sigma_{m,n}$ the set of all dimer configurations on ${\Gamma}_{m,n}$. The *partition function* of the dimer model is given by $$\label{int2}
Z=\sum_{{\sigma}\in\Sigma_{m,n}}w({\sigma}).$$ Notice that if all the weights are set equal to one, then $Z$ simply counts the number of dimer configurations, or perfect matchings, on $\Gamma_{m,n}$.
Our goal is to evaluate the full asymptotic series expansion of the partition function $Z$ as $m,n\to\infty$. The free energy of the dimer model on the square lattice was obtained in the papers of Kasteleyn [@Kas1] and Temperley and Fisher [@TempFish]. Our work is based on the Kasteleyn’s expression of the partition function $Z$ on a torus as a linear combination of 4 Pfaffians developed in the works [@Kas1], [@Kas2], [@Kas3] (see also the works of Galluccio and Loebl [@GalluLoe], Tesler [@Tes], and Cimasoni and Reshetikhin [@CimResh]). The constant term in the asymptotic of the partition function was obtained by Ferdinand [@Ferdin] (see also the work of Kenyon, Sun and Wilson [@KSW]).
The asymptotic expansion of the partition function on a torus was developed by Ivashkevich, Izmailian, and Hu [@IIH] and our calculations use their ideas. Ivashkevich, Izmailian, and Hu considered the case when $z_h=z_v$ and $n$ is even. In the present work we extend their calculations to arbitrary weights $z_h,z_v$ and to odd $n$. It is worth noticing that the asymptotic expansions for even and odd values of $n$ are different. We give a complete rigorous proof of the asymptotic expansion of the partition function, with an estimate of the error term. The asymptotic expansion of the partition function is expressed in terms of the classical Jacobi theta functions, Dedekind eta function, and Kronecker double series. The work [@IIH] has been further extended by Izmailian, Oganesyan, and Hu [@IOH] to the dimer model on a square lattice with various boundary conditions for both even and odd $n$. Our result for the dimer model on a torus coincides with the one in [@IOH] for even $n$, and for odd $n$ it coincides except for the value of the elliptic nome in formula below. The difference in the value of the elliptic nome for even and odd $n$ is explained after formula in Section \[a3o\] below.
It follows from , that the partition function $Z$ is a homogeneous polynomial of the variables $z_h,z_v$, and it can be written as $$\label{zeta1}
Z(z_h,z_v)=\sum_{\sigma\in \Sigma_{m,n}} z_h^{N_h({\sigma})}z_v^{N_v({\sigma})}
=z_h^{\frac{mn}{2}}Z(1,\zeta),$$ where $$\label{zeta2}
\zeta=\frac{z_v}{z_h}>0,$$ so without loss of generality we may assume that $$\label{zeta3}
z_h=1,\quad z_v=\zeta,$$ and we will evaluate the full asymptotic series expansion of the partition function $Z(1,{\zeta})$ as $m,n\to\infty$.
To formulate our main result we have to introduce and remind some special functions and operators.
Function $g(x)$ {#Fun_g}
---------------
Introduce the function $$\label{eq2.1}
g(x)=\ln\left({\zeta}\sin(\pi x)+\sqrt{1+{\zeta}^2\sin^2(\pi x)}\right),$$ where ${\zeta}>0$ is defined in . Observe that $g(x)$ has the following properties:
1. $g(-x)= -g(x),$
2. $g(x+1)=-g(x),$
3. $g(x)$ is real analytic on $[0,1]$ and $$\label{eq2.2}
g\left(x\right)= \sum_{p=0}^\infty g_{2p+1}x^{2p+1},$$ where $$\label{eq2.3}
g_1=\pi{\zeta},\quad g_3=-\frac{\pi^3 {\zeta}({\zeta}^2+1)}{6}\,,\quad g_5=\frac{\pi^5{\zeta}({\zeta}^2+1)(9{\zeta}^2+1)}{120}\,,\;\ldots.$$
4. $g(x)\ge C_0x$ on the segment $0\le x\le \frac{1}{2}\,$, with some $C_0>0$.
The constant $C_0$ in the latter inequality can depend on ${\zeta}$. In what follows we assume that ${\zeta}$ is fixed and we do not indicate the dependence of various constants $C_k$ on ${\zeta}$. Unless otherwise is stated, the constants $C_k$ can be different in different inequalities.
Observe that since $g(x)$ is analytic at $x=0$, we have that $$\label{eq2.3a}
\left|g_{2p+1}\right|\le C \xi^p,$$ with some $C,\xi>0$.
Differential Operator ${\Delta}_p$
----------------------------------
Let $\mathcal S_p$ be the set of collections of positive integers\
$(p_1,\ldots,p_r;q_1,\ldots,q_r)$, $1\le r\le p$, such that $$\label{eq2.4}
\begin{aligned}
\mathcal S_p=\left\{
(p_1,\ldots,p_r;q_1,\ldots,q_r)\;\big|\;
0<p_1<\ldots<p_r;\; p_1q_1+\ldots+p_r q_r=p\right\}.
\end{aligned}$$ Introduce the differential operator $$\label{eq2.5}
\begin{aligned}
{\Delta}_p=\sum_{\mathcal S_p}
\frac{(g_{2p_1+1})^{q_1}\ldots (g_{2p_r+1})^{q_r}}{q_1!\ldots q_r!}\,\frac{d^q}{d{\lambda}^q}\,,\quad
q=q_1+\ldots+q_r-1\,.
\end{aligned}$$ Observe that $$\label{eq2.6}
\Delta_1=g_3, \quad
\Delta_2=\frac{g^2_3}{2}\frac{d}{d\lambda}+g_5,\quad\Delta_3=\frac{g_3^3}{3!}\frac{d^2}{d\lambda^2}+g_3g_5\frac{d}{d\lambda}+g_7,\quad \ldots\ .$$
Kronecker’s Double Series
-------------------------
The Kronecker double series of order $p$ with parameters ${\alpha},{\beta}$ is defined as $$\label{eq2.7a}
\begin{aligned}
K^{{\alpha},{\beta}}_p(\tau)=-\frac{p!}{(-2\pi i)^p}\sum_{(j,k)\not=(0,0)}\frac{e(j{\alpha}+k{\beta})}{(k+\tau j)^p}\,,
\end{aligned}$$ where $$\label{eq2.7b}
\begin{aligned}
e(x)=e^{-2\pi ix}.
\end{aligned}$$ We will use the following Kronecker double series with parameters $({\alpha},{\beta})=(\frac{1}{2},\frac{1}{2}), (0,\frac{1}{2}), (\frac{1}{2},0)$, respectively: $$\label{eq2.7}
\begin{aligned}
K^{\frac{1}{2},\frac{1}{2}}_p(\tau)=&-\frac{p!}{(-2\pi i)^p}\sum_{(j,k)\not=(0,0)}\frac{(-1)^{j+k}}{(k+\tau j)^p},\\
K^{0,\frac{1}{2}}_p(\tau)=&-\frac{p!}{(-2\pi i)^p}\sum_{(j,k)\not=(0,0)}\frac{(-1)^{k}}{(k+\tau j)^p},\\
K^{\frac{1}{2},0}_p(\tau)=&-\frac{p!}{(-2\pi i)^p}\sum_{(j,k)\not=(0,0)}\frac{(-1)^{j}}{(k+\tau j)^p}\,.
\end{aligned}$$ We will use it for $\tau$ pure imaginary and $p\ge 4$. Then the double series are absolutely convergent.
Dedekind Eta Function
---------------------
The Dedekind eta function is defined as $$\label{eq2.8}
\begin{aligned}
\eta=\eta(\tau)=e^{\frac{\pi i\tau}{12}}\prod_{k=1}^\infty \left(1-e^{2\pi i\tau k}\right)
=q^{\frac{1}{12}}\prod_{k=1}^\infty \left(1-q^{2k}\right) ,
\end{aligned}$$ where $$\label{eq2.9}
q=e^{\pi i\tau}$$ is the elliptic nome.
Jacobi Theta Functions
----------------------
There are four Jacobi theta functions: $$\label{eq2.10}
\begin{aligned}
\theta_1(z,q)&=2\sum_{k=0}^{\infty}(-1)^k q^{\left(k+\frac{1}{2}\right)^2} \sin\big((2k+1)z\big), \\
\theta_2(z,q)&=2\sum_{k=0}^{\infty}q^{\left(k+\frac{1}{2}\right)^2}\cos\big((2k+1)z\big), \\
\theta_3(z,q)&=1+2\sum_{k=1}^{\infty}q^{k^2}\cos(2k z), \\
\theta_4(z,q)&=1+2\sum_{k=1}^{\infty}(-1)^k q^{k^2} \cos(2k z),
\end{aligned}$$ where $q=e^{\pi i\tau}$ is elliptic nome.
We have the following identities (see, e.g., [@Weber]): $$\label{dj}
\begin{aligned}
&\theta_2=\theta_2(0,q)=\frac{2\eta^2(2\tau)}{\eta(\tau)}\,,\\
&\theta_3=\theta_3(0,q)=\frac{\eta^5(\tau)}{\eta^2(2\tau)\eta^2(\frac{\tau}{2})}\,,\\
&\theta_4=\theta_4(0,q)=\frac{\eta^2(\frac{\tau}{2})}{\eta(\tau)}\,.
\end{aligned}$$
Also (see, e.g., [@IIH]), $$\label{KtoT12}
\begin{aligned}
K_4^{0,\frac{1}{2}}(\tau)&=\frac{1}{30}\left(\frac{7}{8}\theta_2^8-\theta_3^4\theta_4^4\right)\,,\\
K_4^{\frac{1}{2},0}(\tau)&=\frac{1}{30}\left(\frac{7}{8}\theta_4^8-\theta_2^4\theta_3^4\right)\,,\\
K_4^{\frac{1}{2},\frac{1}{2}}(\tau)&=\frac{1}{30}\left(\frac{7}{8}\theta_3^8+\theta_2^4\theta_4^4\right)\,.\\
\end{aligned}$$
Main Result: Full Asymptotic Expansion of the Dimer Model Partition Function
============================================================================
Pfaffians
---------
We would like to evaluate the asymptotic expansion of the dimer model partition function $Z$ on the square lattice, ${\Gamma}_{m,n}$, of dimensions $m\times n$, with periodic boundary conditions where $m,n\to\infty$ under the assumption that there exist positive constants $C_2>C_1$ such that $$\label{fbc1.9}
C_1\le\frac{m}{n}\le C_2.$$ As shown by Kasteleyn [@Kas1; @Kas2; @Kas3], the partition function $Z$ can be written in terms of four Pfaffians as $$\label{tbc1}
Z=\frac{1}{2}\left(-{{\operatorname{Pf}\,}}A_1+{{\operatorname{Pf}\,}}A_2+{{\operatorname{Pf}\,}}A_3+{{\operatorname{Pf}\,}}A_4\right),$$ where $A_1, A_2, A_3, A_4$ are the antisymmetric Kasteleyn matrices with periodic-periodic, periodic-antiperiodic, antiperiodic-periodic, and antiperiodic-antiperiodic boundary conditions, respectively. Their determinants are given by the double product formulae as $$\label{tbc2}
\begin{aligned}
&\det A_i=\prod_{j=0}^{\frac{m}{2}-1}\prod_{k=0}^{n-1} \left[4\left(\sin^2\frac{2\pi(j+{\alpha}_i)}{m}+ {\zeta}^2\sin^2\frac{2\pi(k+{\beta}_i)}{n}
\right)\right],
\end{aligned}$$ with $$\label{tbc3}
\begin{aligned}
({\alpha}_1,{\beta}_1)=(0,0),\quad ({\alpha}_2,{\beta}_2)=(0,1/2),\quad ({\alpha}_3,{\beta}_3)=(1/2,0),
\quad ({\alpha}_4,{\beta}_4)=(1/2,1/2).
\end{aligned}$$ These double product formulae are obtained by diagonalizing the matrices $A_i$ (see [@Kas1; @McCoy; @McCoyWu]). The Pfaffian of a square antisymmetric matrix $A$ is related to its determinant through the classical identity: $$\label{pfdet}
({{\operatorname{Pf}\,}}A)^2=\det A.$$ Observe that $\det A_1=0$ due to the factor $j=k=0$ in , hence $$\label{pfa1}
{{\operatorname{Pf}\,}}A_1=0,$$ and for odd $n$, $\det A_2=0$, due to the factor $j=0$, $k=\frac{n-1}{2}\,$, hence $$\label{pfa2}
{{\operatorname{Pf}\,}}A_2=0, \quad \textrm{if $n$ is odd.}$$ In addition, $$\label{pfa3}
{{\operatorname{Pf}\,}}A_3={{\operatorname{Pf}\,}}A_4, \quad \textrm{if $n$ is odd}$$ (see [@BEP]). As shown in [@KSW],[@BEP], $$\begin{aligned}
&{{\operatorname{Pf}\,}}A_2>0,\quad \textrm{if $n$ is even},\\
&{{\operatorname{Pf}\,}}A_3>0,\quad {{\operatorname{Pf}\,}}A_4>0\quad \textrm{for all $n$},
\end{aligned}$$ hence from we obtain that $$\label{pfai}
\begin{aligned}
&{{\operatorname{Pf}\,}}A_i=\prod_{j=0}^{\frac{m}{2}-1}\prod_{k=0}^{n-1} \left[4\left(\sin^2\frac{2\pi(j+{\alpha}_i)}{m}+ {\zeta}^2\sin^2\frac{2\pi(k+{\beta}_i)}{n}
\right)\right]^{1/2}.
\end{aligned}$$ Combining with , , , we obtain that $$\label{tbc1a}
\begin{aligned}
Z&=\frac{1}{2}\left({{\operatorname{Pf}\,}}A_2+{{\operatorname{Pf}\,}}A_3+{{\operatorname{Pf}\,}}A_4\right),\quad \textrm{if $n$ is even},\\
Z&={{\operatorname{Pf}\,}}A_3,\quad \textrm{if $n$ is odd}.
\end{aligned}$$
Main Result
-----------
Before stating the main theorem, let us introduce some additional notations. Denote $$\label{ntT4}
S=mn,\qquad \nu=\frac{m}{n}\,.$$ We set $$\label{ntT5}
\tau =\begin{dcases} i{\zeta}\nu ,& \text{if $n$ is even,}\\
\frac{i {\zeta}\nu}{2},& \text{if $n$ is odd,}\end{dcases}$$ so that the elliptic nome is equal to $$\label{ntT6}
q=e^{\pi i\tau}=\begin{dcases}e^{-\pi\zeta \nu},& \text{if $n$ is even,}\\
e^{\frac{-\pi {\zeta}\nu}{2}},& \text{if $n$ is odd.}\end{dcases}$$ For brevity we also denote $$\label{ntT7}
\eta=\eta(\tau),\qquad \theta_{k}=\theta_{k}(0,q), \quad k=2,3,4,$$ where $\eta(\tau)$ is the Dedekind eta function, and $\theta_k(z,q)$ are the Jacobi theta functions. The main result is the following asymptotic expansion of the partition function $Z$ in powers of $S^{-1}$, derived by Ivashkevich et al. in [@IIH] in the case ${\zeta}=1$ and $n$ is even. We give a complete rigorous proof of the asymptotic expansion for any ${\zeta}>0$ and for $n$ both even and odd.
\[main\_thmT\_TBC\] If $n$ is even, then as $ m,n\to\infty$ under condition , we have that $$\label{tcT,1.2}
\begin{aligned}
&Z= e^{SF}\left(C^{(2)}e^{R^{(2)}}+C^{(3)}e^{R^{(3)}}+C^{(4)}e^{R^{(4)}}\right),
\end{aligned}$$ where $$\label{tcT,1.3a}
\begin{aligned}
&F=\frac{1}{\pi}\int\limits\limits_0^{\zeta}\frac{\arctan x}{x}\,dx,
\end{aligned}$$ $$\label{tcT,1.3b}
\begin{aligned}
&C^{(2)}= \frac{\theta_4^2}{2\eta^2},\quad
C^{(3)}=\frac{\theta_2^2}{2\eta^2},\quad
C^{(4)}=\frac{\theta_3^2}{2\eta^2},
\end{aligned}$$ and $R^{(j)}$, $j=2,3,4$, admit the asymptotic expansions $$\label{tcT,1.3c}
\begin{aligned}
R^{(j)}\sim\sum_{p=1}^{\infty}\frac{R_{p}^{(j)}}{S^p}\,,\quad j=2,3,4,
\end{aligned}$$ with $$\label{tcT,1.3d}
\begin{aligned}
&R_{p}^{(j)}=-\frac{2^{2p+1}\nu^{p+1}}{p+1}\,{\Delta}_p \left[K_{2p+2}^{{\beta}_j,{\alpha}_j}
\left(\frac{i \nu{\lambda}}{\pi}\right)\right]\bigg|_{{\lambda}=\pi{\zeta}}\,,
\end{aligned}$$ where ${\alpha}_j,{\beta}_j$ are defined in . In particular, by and , $$\label{tcT,1.3e}
\begin{aligned}
&R_{1}^{(2)}=-\frac{2\nu^2g_3}{15}\bigg(\frac{7}{8}\theta_4^8-\theta_2^4\theta_3^4\bigg)\,,
\quad
R_{1}^{(3)}=-\frac{2\nu^2g_3}{15}\bigg(\frac{7}{8}\theta_2^8-\theta_3^4\theta_4^4\bigg)\,,\\
&R_{1}^{(4)}=-\frac{2\nu^2g_3}{15}\bigg(\frac{7}{8}\theta_3^8+\theta_2^4\theta_4^4\bigg)\,.
\end{aligned}$$ Furthermore, if $n$ is odd, then as $ m,n\to\infty$ under condition , we have that $$\label{tcT,1.2a}
\begin{aligned}
&Z= Ce^{SF+R},
\end{aligned}$$ where $F$ is given in , $$\label{tcT,1.3f}
\begin{aligned}
C=\frac{\theta_2}{\eta},
\end{aligned}$$ and $R$ admits the asymptotic expansions $$\label{tcT,1.3g}
\begin{aligned}
R\sim\sum_{p=1}^{\infty}\frac{R_{p}}{S^p}\,,
\end{aligned}$$ with $$\label{tcT,1.3h}
\begin{aligned}
R_{p}=-\frac{\nu^{p+1}}{p+1}\,{\Delta}_p \left[K_{2p+2}^{0,\frac{1}{2}}\left(\frac{i \nu{\lambda}}{2\pi}\right)\right]\bigg|_{{\lambda}=\pi{\zeta}}\,.
\end{aligned}$$ By and , $$\label{tcT,1.3i}
\begin{aligned}
&R_1=-\frac{\nu^2g_3}{60}\bigg(\frac{7}{8}\theta_2^8-\theta_3^4\theta_3^4\bigg)\,.
\end{aligned}$$
As noticed by Kasteleyn [@Kas1], the free energy $F$ in can be expressed in terms of the Euler dilogarithm function $$\label{dilog1}
{\rm L}_2(z)=-\int_0^z\frac{\ln(1-s)\,ds}{s}$$ as $$\label{dilog2}
F({\zeta})=(2i)^{-1}\big[{\rm L}_2(i{\zeta})-{\rm L}_2(-i{\zeta})\big]\,.$$
The proof of Theorem \[main\_thmT\_TBC\] will be given in Sections \[a2e\]–\[a3o\].
Asymptotic behavior of ${{\operatorname{Pf}\,}}A_2$ for even $n$ {#a2e}
================================================================
Since $\sin^2(x+\pi)=\sin^2x$, we can rewrite ${{\operatorname{Pf}\,}}A_2$ in for even $n$ as $$\label{eq8.2}
\begin{aligned}
{{\operatorname{Pf}\,}}A_2=\prod_{j=0}^{\frac{m}{2}-1}\prod_{k=0}^{\frac{n}{2}-1}
\left[4\left(\sin^2\frac{2j\pi}{m}+ {\zeta}^2\sin^2\frac{(2k+1)\pi}{n}\right)\right].
\end{aligned}$$ Using the Chebyshev type identity (see e.g. [@Kas1]), $$\label{eq9.2}
\prod_{j=0}^{\frac{m}{2}-1} \left[4\left( u^2+ \sin^2\frac{2j\pi}{m}\right)\right]=
\left[\left(u+\sqrt{1+u^2}\right)^{\frac{m}{2}}-\left(-u+\sqrt{1+u^2}\right)^{\frac{m}{2}}\right]^2\,,$$ equation is reduced to $$\label{eq9.3}
\begin{aligned}
&{{\operatorname{Pf}\,}}A_2=\prod_{k=0}^{\frac{n}{2}-1}
\left[\left(u_{k}+\sqrt{1+u_{k}^2}\right)^{\frac{m}{2}}-\left(-u_{k}+\sqrt{1+u_{k}^2}\right)^{\frac{m}{2}}\right]^2\,,
\end{aligned}$$ where $$\label{eq9.4}
u_{k}={\zeta}\sin(\pi x_{k})\ge0, \quad x_{k}=\frac{2k+1}{n}\,.$$ Observe that $$\label{eq9.5}
\left(u_{k}+\sqrt{1+u_{k}^2}\right)\left(-u_{k}+\sqrt{1+u_{k}^2}\right)=1\,,$$ hence $$\label{eq9.6}
{{\operatorname{Pf}\,}}A_2=B^{(2)}_{m,n}C^{(2)}_{m,n}\,,$$ where $$\label{eq9.7}
\begin{aligned}
&B^{(2)}_{m,n}=\prod_{k=0}^{\frac{n}{2}-1} \left(u_{k}+\sqrt{1+u_{k}^2}\right)^m,\\
&C^{(2)}_{m,n}=\prod_{k=0}^{\frac{n}{2}-1} \left[1-\frac{1}{\left(u_{k}+\sqrt{1+u_{k}^2}\right)^m}\right]^2\,.
\end{aligned}$$ Respectively, $$\label{eq9.8}
\ln ({{\operatorname{Pf}\,}}A_2)=G^{(2)}_{m,n}+H^{(2)}_{m,n}\,,$$ with $$\label{eq9.9}
\begin{aligned}
G^{(2)}_{m,n}&=m\sum_{k=0}^{\frac{n}{2}-1} \ln\left(u_{k}+\sqrt{1+u_{k}^2}\right)
=m\sum_{k=0}^{\frac{n}{2}-1} g\left(\frac{2k+1}{n}\right),\\
H^{(2)}_{m,n}&=2\sum_{k=0}^{\frac{n}{2}-1} \ln\left[1-\frac{1}{\left(u_{k}+\sqrt{1+u_{k}^2}\right)^m}\right].\\
\end{aligned}$$ The function $g(x)$, defined in , is real analytic, and we will evaluate an asymptotic series expansion of $G^{(2)}_{m,n}$ for large $n$ by using an Euler–Maclaurin type formula and the Bernoulli polynomials $B_k(x)$ (see [@AS] or Appendix \[appD\]).
Evaluation of $G_{m,n}^{(2)}$
-----------------------------
\[lemG\] As $n,m\to\infty$ under condition , we have that $G^{(2)}_{m,n}$ admits the following asymptotic expansion: $$\label{Lem_G2.1}
G^{(2)}_{m,n}\sim SF+\frac{{\gamma}}{6}
-m\sum_{p=1}^\infty\frac{B_{2p+2}\left(\frac{1}{2}\right)g_{2p+1}}{(p+1)\left(\frac{n}{2}\right)^{2p+1}}\,,\quad {\gamma}=\pi\nu{\zeta}.$$
From we have that $$\label{eq9.10}
G_{m,n}^{(2)}=m\,G_{n}^{(2)},\quad G_{n}^{(2)}=\sum_{k=0}^{\frac{n}{2}-1} g\left(\frac{2k+1}{n}\right)\,.$$ Using the Euler-Maclaurin formula , we obtain that $G_n^{(2)}$ is expanded in the asymptotic series in powers of $\frac{1}{n}$ as $$\label{eq9.11}
\begin{aligned}
G_{n}^{(2)}&\sim \frac{n}{2}\int\limits\limits_0^1 g(x)\,dx
+\sum_{p=1}^\infty\frac{B_{p}\left(\frac{1}{2}\right)}{\left(\frac{n}{2}\right)^{p-1}p!}[g^{(p-1)}(1)-g^{(p-1)}(0)].
\end{aligned}$$ From and the equation $g(x+1)=-g(x)$ we obtain that $$\label{eq9.12}
\begin{aligned}
&g^{(2p)}(0)=g^{(2p)}(1)=0,\\
&g^{(2p+1)}(0)=(2p+1)! g_{2p+1}, \quad
g^{(2p+1)}(1)=-(2p+1)!g_{2p+1}.
\end{aligned}$$ Now, becomes $$\label{eq9.13}
\begin{aligned}
G_{n}^{(2)}&\sim \frac{n}{2}\int\limits\limits_0^1 g(x)\,dx
+\sum_{p=1}^\infty\frac{B_{2p}\left(\frac{1}{2}\right)}{\left(\frac{n}{2}\right)^{2p-1}(2p)!}[g^{(2p-1)}(1)-g^{(2p-1)}(0)]\\
&\sim \frac{n}{2}\int\limits\limits_0^1 g(x)\,dx
-\sum_{p=1}^\infty\frac{2B_{2p}\left(\frac{1}{2}\right)(2p-1)!g_{2p-1}}{\left(\frac{n}{2}\right)^{2p-1}(2p)!}\\
&\sim \frac{n}{2}\int\limits\limits_0^1 g(x)\,dx
-\sum_{p=0}^\infty\frac{B_{2p+2}\left(\frac{1}{2}\right)g_{2p+1}}{(p+1)\left(\frac{n}{2}\right)^{2p+1}}.
\end{aligned}$$ Substituting into , we obtain that $$\label{eq9.14}
\begin{aligned}
G_{m,n}^{(2)}&\sim \frac{mn}{2}\int\limits\limits_0^1 g(x)\,dx
-m\sum_{p=0}^\infty\frac{B_{2p+2}\left(\frac{1}{2}\right)g_{2p+1}}{(p+1)\left(\frac{n}{2}\right)^{2p+1}}\,.\\
\end{aligned}$$ Since $B_2(\frac{1}{2})=-\frac{1}{12}$ and $g_1=\pi{\zeta}$, we obtain that $$\label{eq9.15}
\begin{aligned}
G_{m,n}^{(2)}\sim SF+\frac{{\gamma}}{6}
-m\sum_{p=1}^\infty\frac{B_{2p+2}\left(\frac{1}{2}\right)g_{2p+1}}{(p+1)\left(\frac{n}{2}\right)^{2p+1}}\,,\quad {\gamma}=\pi\nu{\zeta}.
\end{aligned}$$ where $$\label{eq9.16}
\begin{aligned}
F=\frac{1}{2}\int\limits_0^1\ln\left({\zeta}\sin(\pi x)+\sqrt{1+{\zeta}^2\sin^2(\pi x)}\right)\,dx\,.
\end{aligned}$$ As shown by Kasteleyn [@Kas1], $$\label{eq9.17}
\begin{aligned}
\frac{1}{2}\int\limits_0^1\ln\left({\zeta}\sin(\pi x)+\sqrt{1+{\zeta}^2\sin^2(\pi x)}\right)\,dx=
\frac{1}{\pi}\int\limits\limits_0^{\zeta}\frac{\arctan x}{x}\,dx,
\end{aligned}$$ hence Lemma \[lemG\] follows.
Next, we evaluate $H^{(2)}_{m,n}$ in .
Evaluation of $H_{m,n}^{(2)}$
-----------------------------
\[lem3.2\] As $n,m\to\infty$ under condition , we have that $H^{(2)}_{m,n}$ admits the following asymptotic expansion: $$\label{Lem_H2.1}
H^{(2)}_{m,n}\sim A^{(2)}+B^{(2)},$$ with $$\label{Lem_H2.2}
\begin{aligned}
&A^{(2)}=4\sum_{k=0}^{\infty}\ln\left(1-e^{- {\gamma}\left(2k+1\right)}\right),\quad {\gamma}=\pi\nu{\zeta},\\
&B^{(2)}=m\sum_{p=1}^\infty\frac{B_{2p+2}\left(\frac{1}{2}\right)\,g_{2p+1}}{(p+1)\left(\frac{n}{2}\right)^{2p+1}}\,+\sum_{p=1}^\infty \frac{R^{(2)}_p}{S^p}\,,
\end{aligned}$$ where $$\label{Lem_H2.3}
R_p^{(2)}=-\frac{2^{2p+1}\nu^{p+1}}{p+1}\,{\Delta}_p
\left[K_{2p+2}^{\frac{1}{2},0}
\left(\frac{i \nu{\lambda}}{\pi}\right)\right]\bigg|_{{\lambda}=\pi{\zeta}}\,.$$
From , , and we have that $$\label{eq9.18}
H_{m,n}^{(2)}=2\sum_{k=0}^{\frac{n}{2}-1} \ln\left[1-e^{-m g (x_{k})}\right],\quad x_k=\frac{2k+1}{n}\,.$$ Since $g(x)\ge C_0x$ on the segment $[0,\frac{1}{2}]$, for some $C_0>0$, we have that $$\label{eq9.19}
e^{-mg(x_{k})}\le e^{-C_0 \nu (2k+1)}\,,\quad \nu=\frac{m}{n}\,,$$ hence the sum in is estimated from above by a geometric series, and for any $L>0$ there is $R>0$ such that $$\label{eq9.21}
H_{m,n}^{(2)}=4\sum_{k=0}^{R\ln n} \ln\left[1-e^{-mg (x_{k})}\right]+\mathcal O(n^{-L}),$$ so that in our calculations we can restrict $k$ to $k\le R\ln n$.\
Following [@IIH], let us expand the logarithm in into the Taylor series $$\label{eq9.22}
H_{m,n}^{(2)}=-4\sum_{k=0}^{R\ln n} \sum_{j=1}^\infty \frac{e^{-mjg (x_{k})}}{j}+\mathcal O(n^{-L}).$$ Observe that $$\label{eq9.22a}
mjg (x_{k})\ge C_0(2k+1)j,$$ hence the series in $j$ converges exponentially and for any $L>0$ there is $R>0$ such that $$\label{eq9.22b}
H_{m,n}^{(2)}=-4\sum_{k=0}^{R\ln n} \sum_{j=1}^{R\ln n} \frac{e^{-mjg (x_{k})}}{j}+\mathcal O(n^{-L}).$$ Expanding now $g(x)$ into power series , we obtain that $$\label{eq9.23}
\begin{aligned}
e^{-mjg(x_{k})}&=e^{-mj \pi{\zeta}x_{k}} \exp\left[-mj\left(\sum_{p=1}^\infty g_{2p+1} x^{2p+1}_{k}\right)\right].
\end{aligned}$$ Since $S=mn$ and $\nu=\frac{m}{n}$, we have that $n^{2p}=\frac{S^p}{\nu^p}\,$. Hence, $$\label{eq9.24}
mx_{k}^{2p+1}=m\left(\frac{2k+1}{n}\right)^{2p+1}=\frac{\left(2k+1\right)^{2p+1}\nu^{p+1}}{S^p}\,$$ and $$\label{eq9.25}
\begin{aligned}
e^{-mjg(x_{k})}
&=e^{-(2k+1)j {\gamma}} \exp\left[-j\sum_{p=1}^\infty \frac{ g_{2p+1}\left(2k+1\right)^{2p+1}\nu^{p+1}}{S^p}\right], \quad {\gamma}=\pi\nu{\zeta}.
\end{aligned}$$ Denote $$\label{eq9.26}
a_p=- \left(2k+1\right)j\nu\, g_{2p+1},\quad x=\frac{\left(2k+1\right)^2\nu}{S}\,.$$ Then formula simplifies to $$\label{eq9.27}
\begin{aligned}
e^{-mjg(x_{k})}
&=e^{-(2k+1)j {\gamma}} \exp\left(\sum_{p=1}^\infty a_p x^p\right).
\end{aligned}$$ Substituting this expression into , we obtain that $$\label{eq9.22b0}
H_{m,n}^{(2)}=-4\sum_{k=0}^{R\ln n} \sum_{j=1}^{R\ln n}
\frac{e^{-(2k+1)j{\gamma}} }{j}\,
\exp\left(\sum_{p=1}^\infty a_p x^p\right)
+\mathcal O(n^{-L}).$$ Expanding the exponent into the Taylor series, we obtain that $$\label{eq9.28}
\begin{aligned}
\exp\left(\sum_{p=1}^\infty a_p x^p\right)
=1+\sum_{p=1}^\infty b_p x^p,
\end{aligned}$$ with $$\label{eq9.29}
\begin{aligned}
b_p= \sum_{\mathcal S_p}
\frac{(a_{p_1})^{q_1}\ldots (a_{p_r})^{q_r}}{q_1!\ldots q_r!}\,,
\end{aligned}$$ where $\mathcal S_p$ is defined in (see [@IIH] and Appendix \[appA\]). Thus, $$\label{eq9.22b1}
H_{m,n}^{(2)}=-4\sum_{k=0}^{R\ln n} \sum_{j=1}^{R\ln n}
\frac{e^{-(2k+1)j{\gamma}} }{j}\,
\left(1+\sum_{p=1}^\infty b_p x^p\right)
+\mathcal O(n^{-L}),$$ or $$\label{eq9.33}
H_{m,n}^{(2)}= A^{(2)}_n+B^{(2)}_n+\mathcal O(n^{-L}),$$ where $$\label{eq9.33a}
\begin{aligned}
&A^{(2)}_n=-4\sum_{k=0}^{R\ln n} \sum_{j=1}^{R\ln n}
\frac{e^{-(2k+1)j {\gamma}}}{j}\,,\\
&B^{(2)}_n=-4\sum_{k=0}^{R\ln n} \sum_{j=1}^{R\ln n} \sum_{p=1}^\infty
\frac{e^{-(2k+1)j{\gamma}} }{j}\,
b_p x^p,\quad x=\frac{\left(2k+1\right)^2\nu}{S}\,.
\end{aligned}$$ We have that $$\label{eq9.33b}
A^{(2)}_n=-4\sum_{k=0}^\infty \sum_{j=1}^\infty
\frac{e^{-(2k+1)j {\gamma}}}{j}+\mathcal O(n^{-L})=
4\sum_{k=0}^{\infty} \ln(1-e^{-(2k+1){\gamma}})+\mathcal O(n^{-L}).$$ We write now $B_n^{(2)}$ as $$\label{eq9.33c}
\begin{aligned}
B^{(2)}_n&=B^{(2)}_{n,K}+R^{(2)}_{n,K},\quad
B^{(2)}_{n,K}=
-4\sum_{k=0}^{R\ln n} \sum_{j=1}^{R\ln n} \sum_{p=1}^{K-1}
\frac{e^{-(2k+1)j{\gamma}} }{j}\,b_p x^p,\\
R^{(2)}_{n,K}&=-4\sum_{k=0}^{R\ln n} \sum_{j=1}^{R\ln n} \sum_{p=K}^{\infty}
\frac{e^{-(2k+1)j{\gamma}} }{j}\,b_p x^p,
\end{aligned}$$ and we would like to estimate the error term $R^{(2)}_{n,K}$. To that end we will prove the following lemma:
\[error\_estimate\] [(Error term estimate)]{} Fix any ${\varepsilon}>0$. Then as $S\to \infty$, $$\label{sl1}
\begin{aligned}
R^{(2)}_{n,K}=\mathcal O(S^{-K(1-{\varepsilon})})\,.
\end{aligned}$$
Remind that $S=mn=\nu n^2$, where $C_1\le \nu\le C_2$, hence $S\to\infty$ implies that $n\to\infty$.
Let us estimate $b_p$. From and we have that $$\label{sl6}
|a_p|=(2k+1)j\nu|g_{2p+1}|\le C_1 (2k+1)j \xi^p,$$ hence $$\label{sl7}
\left| \sum_{p=1}^\infty a_p z^p\right|
\le C_2 (2k+1)j|z|,\quad |z|\le (2\xi)^{-1}, \quad z\in{{\mathbb C}}.$$ This implies that $$\label{sl8}
\left| \sum_{p=1}^\infty a_p z^p\right|
\le C_2,\quad \textrm{if}\quad |z|\le \min\big\{(2\xi)^{-1},[(2k+1)j]^{-1}\big\}, \quad z\in{{\mathbb C}},$$ hence $$\label{sl9}
\left|\exp\left( \sum_{p=1}^\infty a_p z^p\right)\right|
\le C_3=e^{C_2} ,\quad \textrm{if}\quad |z|\le \min\big\{(2\xi)^{-1},[(2k+1)j]^{-1}\big\}, \quad z\in{{\mathbb C}}.$$ By the Cauchy integral formula, $$\label{sl10}
\frac{f^{(p)}(0)}{p!}=\frac{1}{2\pi i}\oint_{|z|=\rho}\frac{f(z)\,dz}{z^{p+1}}\,,$$ applied to $f(z)=\exp\left( \sum_{p=1}^\infty a_p z^p\right)$ and $\rho=\min\big\{(2\xi)^{-1},[(2k+1)j]^{-1}\big\}$, it follows that $$\label{sl11}
|b_p|\le C_3 \big[(2k+1)j\big]^p \quad \textrm{if}\quad (2k+1)j\ge \xi,$$ and $$\label{sl12}
|b_p|\le C_3 \xi^p \quad \textrm{if}\quad (2k+1)j\le \xi.$$ Using these estimates of $b_p$, we will now prove .
As $n\to\infty$, we may assume that $R\ln n>\xi$, and we partition $R^{(2)}_{n,K}$ as follows: $$\label{sl14}
\begin{aligned}
R^{(2)}_{n,K}&=
\sum_{j,k:\,(2k+1)j\le \xi}
\left(\sum_{p=K}^\infty
\frac{e^{-(2k+1)j{\gamma}} }{j}\,
|b_p| x^p\right)\\
&+\sum_{j,k:\,j,k\le R\ln n;\;(2k+1)j> \xi }
\left(\sum_{p=K}^\infty \frac{e^{-(2k+1)j{\gamma}} }{j}\,|b_p| x^p \right),
\quad x=\frac{\left(2k+1\right)^2\nu}{S}\,.
\end{aligned}$$ In the first term there are only finitely many possible values of $j$ and $k$, and by , $$\label{sl15}
\begin{aligned}
\sum_{j,k:\,(2k+1)j\le \xi}
\left(\sum_{p=K}^\infty
\frac{e^{-(2k+1)j{\gamma}} }{j}\,
|b_p| x^p\right)\le \sum_{p=K}^\infty (C_4S^{-1})^p
=\mathcal O(S^{-K(1-{\varepsilon})})\,.
\end{aligned}$$ Consider now the second term in . Using estimate , we obtain that $$\label{sl18}
\begin{aligned}
\sum_{j,k:\,j,k\le R\ln n;\;(2k+1)j> \xi }
\left(\sum_{p=K}^\infty
\frac{e^{-(2k+1)j{\gamma}} }{j}\,
|b_p| x^p \right)\le \sum_{p=K}^\infty \frac{c_p}{S^p}\,,
\end{aligned}$$ with $$\label{sl19}
\begin{aligned}
0<c_p\le
C_6\sum_{k=0}^{R\ln n}
\sum_{j=1}^{R\ln n}
e^{-(2k+1)j{\gamma}}\big[(2k+1)^3j\nu\big]^p\le \big[C_7 \nu(R\ln n)^4\big]^p,
\end{aligned}$$ hence $$\label{sl20}
\begin{aligned}
\sum_{p=K}^\infty \frac{c_p}{S^p}\le \sum_{p=K}^\infty
\left[\frac{C_7 (R\ln n)^4}{n^2}\right]^{p}=\mathcal O(S^{-K(1-{\varepsilon})})\,.
\end{aligned}$$ Thus, $$\label{sl21}
\begin{aligned}
\sum_{j,k:\,j,k\le R\ln n;\;(2k+1)j> \xi }
\left(\sum_{p=K}^\infty
\frac{e^{-(2k+1)j{\gamma}} }{j}\,
|b_p| x^p \right)=\mathcal O(S^{-K(1-{\varepsilon})})\,,
\end{aligned}$$ and is proved.
Next, we would like to replace $R\ln n$ in $B^{(2)}_{n,K}$ in by $\infty$. Denote $$\label{sl22}
\begin{aligned}
B^{(2)}(p)=
-4\sum_{k=0}^{\infty} \sum_{j=1}^{\infty}
\frac{e^{-(2k+1)j{\gamma}} }{j}\,b_p x^p,\quad
B_n^{(2)}(p)=
-4\sum_{k=0}^{R\ln n} \sum_{j=1}^{R\ln n}
\frac{e^{-(2k+1)j{\gamma}} }{j}\,b_p x^p.
\end{aligned}$$ Then using estimate of $b_p$, we obtain that $$\label{sl23}
\begin{aligned}
|B^{(2)}(p)-B^{(2)}_{n}(p)|&=
4\left|\sum_{j,k:\;j>0,\;k\ge 0;\;\max\{j,k\}>R\ln n} \frac{e^{-(2k+1)j{\gamma}} }{j}\,
b_p x^p\right|\\
&\le
C_0 S^{-p}\left|\sum_{j,k:\;j>0,\;k\ge 0;\;\max\{j,k\}>R\ln n} \frac{e^{-(2k+1)j{\gamma}} }{j}\,
[(2k+1)^3j\nu]^p \right|.
\end{aligned}$$ We have that $$\label{sl24}
\begin{aligned}
\sum_{k=R\ln n}^\infty k^{3p} e^{-k{\gamma}}\le C_1(p) n^{-R{\gamma}/2},
\end{aligned}$$ hence from we obtain that $$\label{sl25}
\begin{aligned}
|B^{(2)}(p)-B^{(2)}_{n}(p)|\le C_2(p) n^{-R{\gamma}/2}.
\end{aligned}$$
From Lemma \[error\_estimate\], , and we obtain an asymptotic expansion of $H_{m,n}^{(2)}$ in powers of $S^{-1}$ as $$\label{eq9.34b}
H_{m,n}^{(2)}\sim A^{(2)}+B^{(2)},$$ with $$\label{eq9.34c}
\begin{aligned}
&A^{(2)}=4\sum_{k=0}^{\infty} \ln(1-e^{-(2k+1){\gamma}})\,,
\quad B^{(2)}=\sum_{p=1}^\infty d_p S^{-p},
\end{aligned}$$ where $$\label{eq9.34e}
\begin{aligned}
d_p=-4\nu^p\sum_{k=0}^{\infty} \sum_{j=1}^{\infty}
\frac{e^{-(2k+1)j{\gamma}} b_p(2k+1)^{2p}}{j}\,.
\end{aligned}$$ Here $b_p$ is given by equation , and it satisfies estimate , which shows that the series over $k,j$ in the latter formula is convergent. We can transform $d_p$ as follows.
Substituting expression for $a_{p_1},\ldots,a_{p_r}\,$ into , we obtain that $$\label{eq9.30}
\begin{aligned}
b_p=\sum_{\mathcal S_p}
\frac{(g_{2p_1+1})^{q_1}\ldots (g_{2p_r+1})^{q_r}\left[-\left(2k+1\right)j\nu \right]^{q_1+\ldots+q_r}}{q_1!\ldots q_r!}\,.
\end{aligned}$$ We can simplify the latter expression using operator $\Delta_p$ in . Namely, we have that $$\label{eq9.31a}
\begin{aligned}
{\Delta}_p \left[e^{- \left(2k+1\right)j\nu{\lambda}}\right]\Big|_{{\lambda}=\pi{\zeta}}
&=e^{-(2k+1)j {\gamma}}\\
&\times \sum_{\mathcal S_p}
\frac{(g_{2p_1+1})^{q_1}\ldots (g_{2p_r+1})^{q_r}\left[-(2k+1)j\nu \right]^{q_1+\ldots+q_r-1}}{q_1!\ldots q_r!}\,,
\end{aligned}$$ hence $$\label{eq9.32}
\begin{aligned}
b_p=e^{ \left(2k+1\right)j{\gamma}}\left[-(2k+1)j\nu \right]
{\Delta}_p \left[e^{- \left(2k+1\right)j\nu{\lambda}}\right]\Big|_{{\lambda}=\pi{\zeta}}\,.
\end{aligned}$$ Returning back to formula , we obtain that $$\label{eq9.34d}
\begin{aligned}
d_p=4\nu^{p+1}\sum_{k=0}^{\infty} \sum_{j=1}^{\infty}
(2k+1)^{2p+1}\,{\Delta}_p \left[e^{- \left(2k+1\right)j\nu{\lambda}}\right]\Big|_{{\lambda}=\pi{\zeta}}\,.
\end{aligned}$$ To relate $d_p$ to the Kronecker double series, introduce the function $$\label{eq9.35}
\begin{aligned}
F^{(2)}_p({\lambda})=\sum_{k=0}^{\infty}
\sum_{j=1}^\infty \left(k+\frac{1}{2}\right)^{2p+1} e^{-2\nu j {\lambda}\left(k+\frac{1}{2}\right)}\,.
\end{aligned}$$ Then $$\label{eq9.36}
\begin{aligned}
d_p=2^{2p+3}\nu^{p+1} {\Delta}_p \big[F^{(2)}_p({\lambda})\big]\big|_{{\lambda}=\pi{\zeta}}\,.
\end{aligned}$$ The function $F^{(2)}_p({\lambda})$ can be expressed in terms of the Kronecker double series of a complex argument. More precisely, from equation in Appendix \[appF\] we have that $$\label{eq9.37}
F^{(2)}_p({\lambda})=\frac{B_{2p+2}\left(\frac{1}{2}\right)-K_{2p+2}^{\frac{1}{2},0}\left(\frac{i\nu{\lambda}}{\pi}\right)}{4(p+1)}.$$ Furthermore, since the free term in the operator ${\Delta}_p$ in is equal to $g_{2p+1}$, we obtain that $$\label{eq9.38}
{\Delta}_p F^{(2)}_p({\lambda})=\frac{B_{2p+2}\left(\frac{1}{2}\right)}{4(p+1)}g_{2p+1}-\frac{{\Delta}_p K_{2p+2}^{\frac{1}{2},0}\left(\frac{i\nu{\lambda}}{\pi}\right)}{4(p+1)},$$ and therefore, $$\label{eq9.39}
\begin{aligned}
B^{(2)}&=\sum_{p=1}^\infty \frac{2^{2p+1}\nu^{p+1}B_{2p+2}\left(\frac{1}{2}\right)}{S^p(p+1)}g_{2p+1}-
\sum_{p=1}^\infty \frac{2^{2p+1}\nu^{p+1}}{S^p(p+1)}
{\Delta}_p
\left[K_{2p+2}^{\frac{1}{2},0}
\left(\frac{i \nu{\lambda}}{\pi}\right)\right]\bigg|_{{\lambda}=\pi{\zeta}}\\
&=m\sum_{p=1}^\infty\frac{B_{2p+2}\left(\frac{1}{2}\right)}{(p+1)\left(\frac{n}{2}\right)^{2p+1}}g_{2p+1}+\sum_{p=1}^\infty \frac{R^{(2)}_p}{S^p}\,,
\end{aligned}$$ (recall that $S=\nu n^2$), and this completes the proof of Lemma \[lem3.2\].
Evaluation of $ \ln ({{\operatorname{Pf}\,}}A_2)$
-------------------------------------------------
\
To evaluate $\ln ({{\operatorname{Pf}\,}}A_2)$, substitute and into to obtain $$\label{eq9.41}
\begin{aligned}
\ln ({{\operatorname{Pf}\,}}A_2)&\sim\bigg(SF+\frac{{\gamma}}{6}
-m\sum_{p=1}^\infty\frac{B_{2p+2}\left(\frac{1}{2}\right)}{(p+1)\left(\frac{n}{2}\right)^{2p+1}}g_{2p+1}\bigg)\\
&+\bigg(4\sum_{k=0}^{\infty} \ln(1-e^{-(2k+1){\gamma}})+m\sum_{p=1}^\infty\frac{B_{2p+2}\left(\frac{1}{2}\right)}{(p+1)\left(\frac{n}{2}\right)^{2p+1}}g_{2p+1}+\sum_{p=1}^\infty \frac{R^{(2)}_p}{S^p}\bigg)\\
&= SF+\frac{{\gamma}}{6}+\sum_{k=1}^{\infty}\ln\left(1-e^{- \left(2k-1\right){\gamma}}\right)^{4}+\sum_{p=1}^\infty \frac{R^{(2)}_p}{S^p}\,,
\quad {\gamma}=\pi\nu{\zeta}.
\end{aligned}$$ Note that the series containing the Bernoulli polynomials $B_{2p+2}(\frac{1}{2})$ cancel out. If we let the elliptic nome be equal to $$\label{eq9.42}
q=e^{-{\gamma}}\,,$$ then by using and , we obtain that $$\label{eq9.43}
\begin{aligned}
e^\frac{{\gamma}}{6}\prod_{k=1}^\infty(1-e^{-(2k-1){\gamma}})^4&=q^{-1/6}\prod_{k=1}^{\infty}(1-q^{2k-1})^4=\left[q^{-1/24}\prod_{k=1}^{\infty}\frac{(1-q^{k})}{(1-q^{2k})}\right]^4\\
&=\left[\frac{q^{1/24}\prod_{k=1}^{\infty}(1-q^{k})}{q^{1/12}\prod_{k=1}^{\infty}(1-q^{2k})}\right]^4
=\left[\frac{\eta\left(\frac{\tau}{2}\right)}{\eta(\tau)}\right]^4=\frac{\theta_4^2}{\eta^2}\,.
\end{aligned}$$ Therefore, equation implies that $$\label{eq9.44}
{{\operatorname{Pf}\,}}A_2=e^{SF}\frac{\theta_4^2}{\eta^2}\,e^{R^{(2)}},
\quad R^{(2)}\sim
\sum_{p=1}^\infty \frac{R^{(2)}_p}{S^p}\,.$$
Asymptotic expansions of ${{\operatorname{Pf}\,}}A_3$ and ${{\operatorname{Pf}\,}}A_4$ for even $n$ {#a3e}
===================================================================================================
The asymptotic expansions of ${{\operatorname{Pf}\,}}A_3$ and ${{\operatorname{Pf}\,}}A_4$ for even $n$ can be obtained in the same way as the one of ${{\operatorname{Pf}\,}}A_2$. Let us briefly discuss them.
From formula we have that $$\label{pfa3a}
\begin{aligned}
&{{\operatorname{Pf}\,}}A_3=\prod_{j=0}^{\frac{m}{2}-1}\prod_{k=0}^{n-1} \left[4\left(\sin^2\frac{2\pi(j+\frac{1}{2})}{m}+ {\zeta}^2\sin^2\frac{2\pi k}{n}
\right)\right]^{1/2}.
\end{aligned}$$ Using that $\sin^2(x+\pi)=\sin^2x$, we can rewrite the latter formula for even $n$ as $$\label{pfa3b}
\begin{aligned}
&{{\operatorname{Pf}\,}}A_3=\prod_{j=0}^{\frac{m}{2}-1}\prod_{k=0}^{\frac{n}{2}-1} \left[4\left(\sin^2\frac{2\pi(j+\frac{1}{2})}{m}+ {\zeta}^2\sin^2\frac{2\pi k}{n}
\right)\right].
\end{aligned}$$ Using the Chebyshev type identity (see [@Kas1]), $$\label{eq3.2}
\prod_{j=0}^{\frac{m}{2}-1} \left[4\left( u^2+ \sin^2\frac{(2j+1)\pi}{m}\right)\right]=
\left[\left(u+\sqrt{1+u^2}\right)^{\frac{m}{2}}+\left(-u+\sqrt{1+u^2}\right)^{\frac{m}{2}}\right]^2\,,$$ we obtain that $$\label{eq3.3}
\begin{aligned}
&{{\operatorname{Pf}\,}}A_3=\prod_{k=0}^{\frac{n}{2}-1}
\left[\left(u_{k}+\sqrt{1+u_{k}^2}\right)^{\frac{m}{2}}+\left(-u_{k}+\sqrt{1+u_{k}^2}\right)^{\frac{m}{2}}\right]^2\,.
\end{aligned}$$ where $$\label{eq3.4}
u_{k}={\zeta}\sin(\pi x_{k})\ge0, \quad x_{k}=\frac{2k}{n}\,,$$ which implies that $$\label{eq3.8}
\ln ({{\operatorname{Pf}\,}}A_3)=G^{(3)}_{m,n}+H^{(3)}_{m,n}\,,$$ where $$\label{eq3.9}
\begin{aligned}
&G^{(3)}_{m,n}=m\sum_{k=0}^{\frac{n}{2}-1} \ln\left(u_{k}+\sqrt{1+u_{k}^2}\right)
=m\sum_{k=0}^{\frac{n}{2}-1} g\left(\frac{2k}{n}\right),\\
&H^{(3)}_{m,n}=2\sum_{k=0}^{\frac{n}{2}-1} \ln\left[1+\frac{1}{\left(u_{k}+\sqrt{1+u_{k}^2}\right)^m}\right].\\
\end{aligned}$$ Using the Euler–Maclaurin formula, we obtain the following asymptotic expansion: $$\label{Lem_G2.2}
G^{(3)}_{m,n}\sim SF-\frac{{\gamma}}{3}
-m\sum_{p=1}^\infty\frac{B_{2p+2}\left(0\right)g_{2p+1}}{(p+1)\left(\frac{n}{2}\right)^{2p+1}}\,, \quad {\gamma}=\pi\nu{\zeta}.$$ Next, we obtain an asymptotic expansion of $H^{(3)}_{m,n}$: $$\label{Lem_H3.1}
H^{(3)}_{m,n}\sim A^{(3)}+B^{(3)},$$ with $$\label{Lem_H3.2}
\begin{aligned}
&A^{(3)}=4\sum_{k=1}^{\infty}\ln\left(1+e^{-2 k{\gamma}}\right)+2\ln2,
\\
&B^{(3)}=m\sum_{p=1}^\infty\frac{B_{2p+2}\left(0\right)}{(p+1)\left(\frac{n}{2}\right)^{2p+1}}g_{2p+1}+\sum_{p=1}^\infty \frac{R^{(3)}_p}{S^p}\,,
\end{aligned}$$ where $$\label{Lem_H3.3}
R_p^{(3)}=-\frac{2^{2p+1}\nu^{p+1}}{p+1}\,{\Delta}_p
\left[K_{2p+2}^{0,\frac{1}{2}}
\left(\frac{i \nu{\lambda}}{\pi}\right)\right]\bigg|_{{\lambda}=\pi{\zeta}}\,.$$ Substituting and into we obtain that $$\label{eq3.41}
\begin{aligned}
\ln ({{\operatorname{Pf}\,}}A_3)&\sim
SF-\frac{{\gamma}}{3}+4\sum_{k=1}^{\infty}\ln\left(1+e^{-2 k{\gamma}}\right)+2\ln2+\sum_{p=1}^\infty \frac{R^{(3)}_p}{S^p}\,.
\end{aligned}$$ Let $q=e^{-{\gamma}}$. Since $$\label{eq3.43}
\begin{aligned}
4e^{-\frac{{\gamma}}{3}}\prod_{k=1}^\infty \left(1+e^{-2 k{\gamma}}\right)^4&=
4q^{1/3}\prod_{k=1}^{\infty}(1+q^{2k})^4=4\left[q^{1/12}\prod_{k=1}^{\infty}\frac{(1-q^{4k})}{(1-q^{2k})}\right]^4\\
&=4\left[\frac{q^{1/6}\prod_{k=1}^{\infty}(1-q^{4k})}{q^{1/12}\prod_{k=1}^{\infty}(1-q^{2k})}\right]^4=4\left[\frac{\eta(2 \tau)}{\eta(\tau)}\right]^4=\frac{\theta_2^2}{\eta^2}\,,
\end{aligned}$$ we obtain that $$\label{eq3.44}
{{\operatorname{Pf}\,}}A_3=e^{SF}\frac{\theta_2^2}{\eta^2}\,e^{R^{(3)}},
\quad R^{(3)}\sim
\sum_{p=1}^\infty \frac{R^{(3)}_p}{S^p}\,.$$
Let us turn to ${{\operatorname{Pf}\,}}A_4$. Since $\sin^2(x+\pi)=\sin^2x$, we can rewrite ${{\operatorname{Pf}\,}}A_4$ in for even $n$ as $$\label{eq5.2}
\begin{aligned}
{{\operatorname{Pf}\,}}A_4&=\prod_{j=0}^{\frac{m}{2}-1}\prod_{k=0}^{\frac{n}{2}-1} \left[4\left(\sin^2\frac{(2j+1)\pi}{m}+{\zeta}^2\sin^2\frac{(2k+1)\pi}{n}\right)\right]\,.
\end{aligned}$$ Using identity , we obtain that $$\label{eq6.3}
\begin{aligned}
&{{\operatorname{Pf}\,}}A_4=\prod_{k=0}^{\frac{n}{2}-1}
\left[\left(u_{k}+\sqrt{1+u_{k}^2}\right)^{\frac{m}{2}}+\left(-u_{k}+\sqrt{1+u_{k}^2}\right)^{\frac{m}{2}}\right]^2\,,
\end{aligned}$$ where $$\label{eq6.4}
u_{k}={\zeta}\sin(\pi x_{k})\ge0, \quad x_{k}=\frac{2k+1}{n}\,,$$ which implies that $$\label{eq6.8}
\ln ({{\operatorname{Pf}\,}}A_4)=G^{(4)}_{m,n}+H^{(4)}_{m,n}\,,$$ where $$\label{eq6.9}
\begin{aligned}
&G^{(4)}_{m,n}=m\sum_{k=0}^{\frac{n}{2}-1} \ln\left(u_{k}+\sqrt{1+u_{k}^2}\right)
=m\sum_{k=0}^{\frac{n}{2}-1} g\left(\frac{2k+1}{n}\right),\\
&H^{(4)}_{m,n}=2\sum_{k=0}^{\frac{n}{2}-1} \ln\left[1+\frac{1}{\left(u_{k}+\sqrt{1+u_{k}^2}\right)^m}\right].\\
\end{aligned}$$ Using the Euler–Maclaurin formula, we obtain the following asymptotic expansion: $$\label{Lem_G4.2}
G^{(4)}_{m,n}\sim SF+\frac{{\gamma}}{6}
-m\sum_{p=1}^\infty\frac{B_{2p+2}\left(\frac{1}{2}\right)g_{2p+1}}{(p+1)\left(\frac{n}{2}\right)^{2p+1}}\,, \quad {\gamma}=\pi\nu{\zeta},$$ and then, similar to Lemma \[lem3.2\], we obtain that $$\label{Lem_H4.1}
H^{(4)}_{m,n}\sim A^{(4)}+B^{(4)},$$ with $$\label{Lem_H4.2}
\begin{aligned}
&A^{(4)}=4\sum_{k=0}^{\infty}\ln\left(1+e^{-{\gamma}(2k+1)}\right),\\
&B^{(4)}=m\sum_{p=1}^\infty\frac{B_{2p+2}\left(\frac{1}{2}\right)}{(p+1)\left(\frac{n}{2}\right)^{2p+1}}g_{2p+1}+\sum_{p=1}^\infty \frac{R^{(4)}_p}{S^p}\,,
\end{aligned}$$ where $$\label{Lem_H4.3}
R_p^{(4)}=-\frac{2^{2p+1}\nu^{p+1}}{p+1}\,{\Delta}_p
\left[K_{2p+2}^{\frac{1}{2},\frac{1}{2}}
\left(\frac{i \nu{\lambda}}{\pi}\right)\right]\bigg|_{{\lambda}=\pi{\zeta}}\,.$$ Substituting and into , we obtain that $$\label{eq4}
{{\operatorname{Pf}\,}}A_4=e^{SF}\frac{\theta_3^2}{\eta^2}\,e^{R^{(4)}},
\quad R^{(4)}\sim
\sum_{p=1}^\infty \frac{R^{(4)}_p}{S^p}\,.$$ Substituting equations , , and into , we obtain the asymptotic formula for $Z$, , for even $n$.
Asymptotic behavior of ${{\operatorname{Pf}\,}}A_3$ for odd $n$ {#a3o}
===============================================================
From equation we have that $$\label{tbc2.2}
\begin{aligned}
{{\operatorname{Pf}\,}}A_3=\prod_{j=0}^{\frac{m}{2}-1}\prod_{k=0}^{n-1} \left[4\left(\sin^2\frac{(2j+1)\pi}{m}+{\zeta}^2\sin^2\frac{2\pi k}{n}\right)\right]^{1/2}\,.
\end{aligned}$$ For odd $n$, using the identity $\sin^2(x+\pi)=\sin^2x$, we can rewrite the latter formula as $$\label{odd1}
\begin{aligned}
{{\operatorname{Pf}\,}}A_3&=\prod_{j=0}^{\frac{m}{2}-1}\prod_{k=0}^{n-1} \left[4\left(\sin^2\frac{(2j+1)\pi}{m}+{\zeta}^2\sin^2\frac{\pi k}{n}\right)\right]^{1/2}\,.
\end{aligned}$$ Indeed, if we take $0\le k\le \frac{n-1}{2}\,$ in , then we obtain factors with $\sin^2\frac{2\pi k}{n}$, while if we take $k=\frac{n+1}{2}+k'\,,$ $0\le k'\le \frac{n-3}{2}\,,$ then we obtain factors with $$\sin^2\frac{2\pi}{n}\left(\frac{n+1}{2}+k'\right) =\sin^2\frac{\pi(2k'+1) }{n}\,.$$ Combining these two cases, we obtain .
From , using identity , we obtain that $$\label{eq4.2}
\begin{aligned}
{{\operatorname{Pf}\,}}A_3=\prod_{k=0}^{n-1}
\left[\left(u_{k}+\sqrt{1+u_{k}^2}\right)^{\frac{m}{2}}+\left(-u_{k}+\sqrt{1+u_{k}^2}\right)^{\frac{m}{2}}\right],
\end{aligned}$$ where $$\label{eq4.21}
u_{k}={\zeta}\sin(\pi x_{k})\ge0, \quad x_{k}=\frac{k}{n}\,,$$ which implies that $$\label{eq4.3}
\ln ({{\operatorname{Pf}\,}}A_3)=G^{(3)}_{m,n}+H^{(3)}_{m,n}\,,$$ where $$\label{eq4.4}
\begin{aligned}
&G^{(3)}_{m,n}=\frac{m}{2}\,\sum_{k=0}^{n-1} \ln\left(u_{k}+\sqrt{1+u_{k}^2}\right)
=\frac{m}{2}\,\sum_{k=0}^{n-1} g\left(\frac{k}{n}\right),\\
&H^{(3)}_{m,n}=\sum_{k=0}^{n-1} \ln\left[1+\frac{1}{\left(u_{k}+\sqrt{1+u_{k}^2}\right)^m}\right].\\
\end{aligned}$$ Formulae - are similar to - for even $n$, but $x_k=\frac{2k}{n}$ for even $n$, while $x_k=\frac{k}{n}$ for odd $n$. This leads to the difference in elliptic nome for even and odd $n$.
Using the Euler–Maclaurin formula, we obtain that $$\label{odd2}
G^{(3)}_{m,n}\sim SF-\frac{{\gamma}}{12}
-\frac{m}{2}\sum_{p=1}^\infty\frac{B_{2p+2}\left(0\right)}{(p+1)n^{2p+1}}g_{2p+1}\,, \quad {\gamma}=\pi\nu{\zeta},$$ and similar to the even case, we obtain the asymptotic expansion of $H^{(3)}_{m,n}$ as $$\label{odd3}
H^{(3)}_{m,n}\sim A^{(3)}+B^{(3)},$$ with $$\label{odd4}
\begin{aligned}
&A^{(3)}=2\sum_{k=1}^\infty \ln(1+e^{-{\gamma}k}),\\
&B^{(3)}=\frac{m}{2}\sum_{p=1}^\infty \frac{B_{2p+2}(0)g_{2p+1}}{(p+1)n^{2p+1}}+\sum_{p=1}^\infty\frac{R_p^{(3)}}{S^p}\,,
\end{aligned}$$ where $$\label{odd5}
R_p^{(3)}=-\frac{\nu^{p+1}}{p+1}\,{\Delta}_p
\left[K_{2p+2}^{0,\frac{1}{2}}
\left(\frac{i \nu{\lambda}}{2\pi}\right)\right]\bigg|_{{\lambda}=\pi{\zeta}}\,.$$
Combining , , and , we obtain that $$\label{eq4.17}
\begin{aligned}
\ln({{\operatorname{Pf}\,}}A_3)&\sim SF-\frac{{\gamma}}{12}+2\sum_{k=1}^\infty \ln(1+e^{-{\gamma}k})+\sum_{p=1}^\infty\frac{R_p^{(3)}}{S^p}+\ln 2.
\end{aligned}$$ Let now $$\label{eq4.18}
q=e^{\pi i\tau}=e^{-\frac{{\gamma}}{2}}\,,$$ then $$\label{eq4.19}
\begin{aligned}
2e^{-\frac{{\gamma}}{12}}\prod_{k=1}^\infty \left(1+e^{-{\gamma}k}\right)^2&=2q^{1/6}\prod_{k=1}^{\infty}(1+q^{2k})^2
=2\left[\prod_{k=1}^{\infty}\frac{q^{1/6}(1-q^{4k})}{q^{1/12}(1-q^{2k})}\right]^2\\
&=2\left[\frac{\eta(2\tau)}{\eta(\tau)}\right]^2=\frac{\theta_2}{\eta}\,,
\end{aligned}$$ hence $$\label{eq4.20}
\begin{aligned}
\ln Z=\ln({{\operatorname{Pf}\,}}A_3)&\sim SF+\ln \frac{\theta_2}{\eta}
+\sum_{p=1}^\infty\frac{R_p^{(3)}}{S^p}\,.
\end{aligned}$$ This finishes the proof of Theorem \[main\_thmT\_TBC\].
Exponent of a Taylor Series {#appA}
===========================
We have that $$\label{exp1}
\begin{aligned}
\exp\left(\sum_{p=1}^\infty a_px^p\right)=1+\sum_{p=1}^\infty b_px^p,
\end{aligned}$$ where $$\label{exp2}
\begin{aligned}
b_p= \sum_{\mathcal S_p}
\frac{(a_{p_1})^{q_1}\ldots (a_{p_r})^{q_r}}{q_1!\ldots q_r!}\,,
\end{aligned}$$ and $\mathcal S_p$ is the set of collections of positive integers $(p_1,\ldots,p_r;q_1,\ldots,q_r)$, $1\le r\le p$, such that $$\label{exp3}
\begin{aligned}
\mathcal S_p=\left\{
(p_1,\ldots,p_r;q_1,\ldots,q_r)\;\big|\;
0<p_1<\ldots<p_r;\; p_1q_1+\ldots+p_r q_r=p\right\}.
\end{aligned}$$ The series in are understood as formal ones.
Expanding the exponent into the Taylor series, we obtain that $$\label{exp4}
\begin{aligned}
\exp\left(\sum_{p=1}^\infty a_px^p\right)=1+\sum_{k=1}^\infty \frac{1}{k!}\left(\sum_{p=1}^\infty a_p x^p\right)^k.
\end{aligned}$$ By the multinomial formula, $$\label{exp5}
\begin{aligned}
\frac{1}{k!}\left(\sum_{p=1}^\infty a_p x^p\right)^k
=\sum_{0<p_1<\ldots<p_r,\; q_1>0,\ldots, q_r>0:\; q_1+\ldots+q_r=k}
\frac{(a_{p_1}x^{p_1})^{q_1}\ldots (a_{p_r}x^{p_r})^{q_r}}{q_1!\ldots q_r!},
\end{aligned}$$ hence $$\label{exp6}
\begin{aligned}
\exp\left(\sum_{p=1}^\infty a_p x^p\right)&=1+\sum_{k=1}^\infty
\sum_{0<p_1<\ldots<p_r,\; q_1>0,\ldots, q_r>0:\; q_1+\ldots+q_r=k}
\frac{(a_{p_1}x^{p_1})^{q_1}\ldots (a_{p_r}x^{p_r})^{q_r}}{q_1!\ldots q_r!}\\
&=1+\sum_{r=1}^\infty
\sum_{0<p_1<\ldots<p_r,\; q_1>0,\ldots, q_r>0}
\frac{(a_{p_1}x^{p_1})^{q_1}\ldots (a_{p_r}x^{p_r})^{q_r}}{q_1!\ldots q_r!}.
\end{aligned}$$ Combining terms with $ p_1q_1+\ldots+p_r q_r=p$, we obtain formulae , .
Bernoulli’s Polynomials {#appC}
=======================
Bernoulli’s polynomials are defined recursively by the equations, $$\label{bp1}
\begin{aligned}
B_k'(x)=kB_{k-1}(x),\quad \int\limits_0^1 B_k(x)\,dx=0,\quad k=1,2,\ldots;\quad B_0(x)=1.
\end{aligned}$$ In particular, $$\label{bp1a}
\begin{aligned}
B_1(x)=x-\frac{1}{2}\,,\quad B_2(x)=x^2-x+\frac{1}{6}\,,\quad
B_3(x)=x^3-\frac{3x^2}{2}+\frac{x}{2}\,.
\end{aligned}$$ The Bernoulli periodic functions $\widehat B_k(x)$ are defined by the periodicity condition $\widehat B_k(x+1)=\widehat B_k(x)$ and by the condition $\widehat B_k(x)=B_k(x)$ for $0\le x\le 1$. Their Fourier series is equal to $$\label{bp2}
\begin{aligned}
\widehat B_k(x)=-\frac{k!}{(-2\pi i)^k} \sum_{\ell\not=0} \frac{e^{-2\pi i \ell x}}{\ell^k}\,.
\end{aligned}$$ For $k\ge 2$ the Fourier series is absolutely convergent, and for $k=1$ it converges in $L^2[0,1]$. The generating function of the Bernoulli polynomials is $$\label{bp7}
\begin{aligned}
G({\lambda}; x):=\frac{{\lambda}\, e^{{\lambda}x}}{e^{{\lambda}}-1}=\sum_{k=0}^\infty \frac{{\lambda}^k B_k(x)}{k!}\,.
\end{aligned}$$ Substituting into , we obtain that for $0\le x\le 1$ and $|{\lambda}|<2\pi$, $$\label{bp12}
\begin{aligned}
\frac{{\lambda}\, e^{{\lambda}x}}{e^{{\lambda}}-1}&-1-{\lambda}\left(x-\frac{1}{2}\right)=\sum_{k=2}^\infty \frac{{\lambda}^k B_k(x)}{k!}
=-\sum_{k=2}^\infty \sum_{\ell\not=0}\frac{{\lambda}^k }{(-2\pi i)^k}\, \frac{e^{-2\pi i \ell x}}{\ell^k}\\
&=-\sum_{\ell\not=0}e^{-2\pi i \ell x}\sum_{k=2}^\infty \left(\frac{{\lambda}}{-2\pi i \ell}\right)^k
=-\sum_{\ell\not=0}e^{-2\pi i \ell x}\left(\frac{{\lambda}}{-2\pi i \ell}\right)^2\frac{1}{1+\frac{{\lambda}}{2\pi i \ell}}\\
&=\frac{{\lambda}^2}{4\pi^2}\sum_{\ell\not=0}\frac{e^{-2\pi i \ell x}}{\ell\left(\ell+\frac{{\lambda}}{2\pi i }\right)}\,.
\end{aligned}$$ Taking ${\lambda}=2\pi i z$, where $|z|<1$, $x={\alpha}$, and $\ell=k$, we obtain that $$\label{bp13}
\begin{aligned}
\frac{2\pi i z e^{2\pi i z {\alpha}}}{e^{2\pi i z}-1}
=1+2\pi iz\left({\alpha}-\frac{1}{2}\right)-z^2\sum_{k\not=0}\frac{e^{-2\pi i k{\alpha}}}{k\left(k+z\right)}\,,
\end{aligned}$$ or $$\label{bp14}
\begin{aligned}
\frac{ e^{2\pi i z {\alpha}}}{e^{2\pi i z}-1}
=\frac{1}{2\pi i z}+\left({\alpha}-\frac{1}{2}\right)-\frac{z}{2\pi i}\sum_{k\not=0}\frac{e^{-2\pi i k{\alpha}}}{k\left(k+z\right)}\,,
\end{aligned}$$ We can rewrite the latter equation as $$\label{bp15}
\begin{aligned}
\frac{ e^{2\pi i z {\alpha}}}{e^{2\pi i z}-1}
=\frac{1}{2\pi i z}+\left({\alpha}-\frac{1}{2}\right)-\frac{1}{2\pi i}\sum_{k\not=0}e^{-2\pi i k{\alpha}}\left(\frac{1}{k}-\frac{1}{k+z}\right)\,,
\end{aligned}$$
The Euler–Maclaurin Formula {#appD}
===========================
Let $f(x)$ be an analytic function on the interval $[a,b]$. We partition the interval $[a,b]$ into $N$ equal intervals of the length $$\label{em1}
h=\frac{b-a}{N}\,.$$ Let $$\label{em2}
x_k=a+kh+{\alpha}h,\quad k=0,1,\ldots, N-1,$$ where $0\le {\alpha}\le 1$. Then the Euler–Maclaurin formula with a remainder is $$\label{em3}
\begin{aligned}
\sum_{k=0}^{N-1} f(x_k)
&=\frac{1}{h}\int\limits_a^b f(x)\,dx
+\sum_{p=1}^\ell\frac{B_{p}({\alpha})h^{p-1}}{p!}[f^{(p-1)}(b)-f^{(p-1)}(a)]+R_{\ell}({\alpha}),
\end{aligned}$$ where $B_p({\alpha})$ is the Bernoulli polynomial and the remainder $R_{\ell}({\alpha})$ can be written as $$\label{em4}
\begin{aligned}
R_{\ell}({\alpha})= \frac{h^{\ell}}{\ell !}\int\limits_0^1 \widehat B_{\ell}({\alpha}-\tau)\left[ \sum_{k=0}^{N-1} f^{(\ell)}(a+kh+\tau h)\right]\,d\tau,
\end{aligned}$$ where $\widehat B_{\ell}(x)$ is the periodic Bernoulli function.
Thus, the Euler-Maclaurin formula gives an asymptotic series, $$\label{em5}
\begin{aligned}
\sum_{k=0}^{N-1} f(x_k)
&\sim \frac{1}{h}\int\limits_a^b f(x)\,dx
+\sum_{p=1}^\infty\frac{B_{p}({\alpha})h^{p-1}}{p!}[f^{(p-1)}(b)-f^{(p-1)}(a)].
\end{aligned}$$ In general, since both $B_p(x)$ and $f^{(p)}(x)$ grow like $p!$, the series on the right in diverges.
Kronecker’s Double Series of Pure Imaginary Argument {#appF}
====================================================
A classical reference to the Kronecker double series is the book of Weil [@Weil]. In this Appendix we review and specify some results of Ivashkevich, Izmailian, and Hu [@IIH]. Let us consider the Kronecker double series with parameters $(\alpha,\beta)=(\frac{1}{2},0)$ as defined in with argument $\tau=2ir{\zeta}$ and $({\alpha},{\beta})=(\frac{1}{2},\frac{1}{2}),(0,\frac{1}{2})$ as defined in with argument $\tau=ir{\zeta}$. Observe that in all cases, if $p$ is odd, the terms $(j,k)$ and $(-j,-k)$ cancel each other. Hence $K^{\frac{1}{2},0}_{2p-1}(\tau)=K^{\frac{1}{2},\frac{1}{2}}_{2p-1}(\tau)=K^{0,\frac{1}{2}}_{2p-1}(\tau)=0$ for $p=1,2,\ldots\ $. Therefore, we will take $p$ to be even. Let us first consider the case $(\alpha,\beta)=(\frac{1}{2},0)$.\
Case $(\alpha,\beta)=(\frac{1}{2},0)$
-------------------------------------
From , we have that $$\label{kr1}
\begin{aligned}
K_{2p}^{\frac{1}{2},0}(\tau)=\frac{(-1)^{p+1} (2p)!}{(2\pi )^{2p}}\sum_{(j,k)\not=(0,0)}\frac{(-1)^{k}}{(k+\tau j)^{2p}}\,.
\end{aligned}$$ Separating terms with $j=0$, we obtain that $$\label{kr2}
\begin{aligned}
K_{2p}^{\frac{1}{2},0}(\tau)=\frac{(-1)^{p+1} (2p)!}{(2\pi )^{2p}}\sum_{k\not=0}\frac{(-1)^k}{k^{2p}}
+\frac{(-1)^{p+1} (2p)!}{(2\pi )^{2p}}\sum_{j\not=0}\sum_{k=-\infty}^\infty\frac{(-1)^{k}}{(k+\tau j)^{2p}}\,.
\end{aligned}$$ The first term is just the Fourier series for the Bernoulli polynomial $B_{2p}\left(x\right)$ evaluated at $x=\frac{1}{2}$. Let us transform the second term. Since the terms $(j,k)$ and $(-j,-k)$ give the same contribution, we can write that $$\label{kr3}
\begin{aligned}
K_{2p}^{\frac{1}{2},0}(\tau)=B_{2p}\left(\frac{1}{2}\right)
+\frac{2(-1)^{p+1} (2p)!}{(2\pi )^{2p}}\sum_{j=1}^\infty\sum_{k=-\infty}^\infty\frac{(-1)^{k}}{(k+\tau j)^{2p}}\,.
\end{aligned}$$ When $z=iy$, $y>0$, and ${\alpha}=1/2$, identity reads $$\label{kr4}
\begin{aligned}
\frac{ e^{-\pi y }}{1-e^{-2\pi y}}
=\frac{1}{2\pi y}+\frac{1}{2\pi i}\sum_{k\not=0}(-1)^k\left(\frac{1}{k}-\frac{1}{k+iy}\right)\,.
\end{aligned}$$ Expanding the left hand side into the geometric series, we obtain that $$\label{kr5}
\begin{aligned}
\sum_{k=0}^\infty e^{-2\pi y(k+\frac{1}{2})}
=\frac{1}{2\pi y}+\frac{1}{2\pi i}\sum_{k\not=0}(-1)^k\left(\frac{1}{k}-\frac{1}{k+iy}\right)\,.
\end{aligned}$$ Differentiating this identity $(2p-1)$ times with respect to $y$, we obtain that $$\label{kr6}
\begin{aligned}
\sum_{k=0}^\infty \left[-2\pi\left(k+\frac{1}{2}\right)\right]^{2p-1}e^{-2\pi y(k+\frac{1}{2})}
=-\frac{1}{2\pi i}\sum_{k=-\infty}^\infty \frac{(-i)^{2p-1}(2p-1)!(-1)^k}{(k+iy)^{2p}}\,,
\end{aligned}$$ or equivalently, $$\label{kr7}
\begin{aligned}
\frac{(-1)^p(2p)!}{(2\pi)^{2p}}\sum_{k=-\infty}^\infty \frac{(-1)^k}{(k+iy)^{2p}}
=2p \sum_{k=0}^\infty \left(k+\frac{1}{2}\right)^{2p-1}e^{-2\pi y(k+\frac{1}{2})}.
\end{aligned}$$ Using this formula in with $y=r{\zeta}j$, we obtain that $$\label{kr8}
\begin{aligned}
K_{2p}^{\frac{1}{2},0}(\tau)=B_{2p}\left(\frac{1}{2}\right)
-4p \sum_{j=1}^\infty \sum_{k=0}^\infty \left(k+\frac{1}{2}\right)^{2p-1}e^{-2r{\lambda}j(k+\frac{1}{2})}\,.
\end{aligned}$$ Thus, we have the following proposition:
We have that $$\label{kr9}
\begin{aligned}
\sum_{j=1}^\infty \sum_{k=0}^\infty \left(k+\frac{1}{2}\right)^{2p-1}e^{-2r{\lambda}j(k+\frac{1}{2})}
=\frac{B_{2p}\left(\frac{1}{2}\right)-K_{2p}^{\frac{1}{2},0}(\tau)}{4p}\,.
\end{aligned}$$
Applying it for $K_{2p+2}^{\frac{1}{2},0}(\tau)$, we obtain that $$\label{kr10}
\begin{aligned}
\sum_{j=1}^\infty \sum_{k=0}^\infty \left(k+\frac{1}{2}\right)^{2p+1}e^{-2r{\lambda}j(k+\frac{1}{2})}
=\frac{B_{2p+2}\left(\frac{1}{2}\right)-K_{2p+2}^{\frac{1}{2},0}(\tau)}{4(p+1)}\,.
\end{aligned}$$
Case $(\alpha,\beta)=(\frac{1}{2},\frac{1}{2})$
-----------------------------------------------
From , we have that $$\label{kr11}
\begin{aligned}
K_{2p}^{\frac{1}{2},\frac{1}{2}}(\tau)=\frac{(-1)^{p+1} (2p)!}{(2\pi )^{2p}}\sum_{(j,k)\not=(0,0)}\frac{(-1)^{k+j}}{(k+\tau j)^{2p}}\,.
\end{aligned}$$ Separating terms with $j=0$, we obtain that $$\label{kr12}
\begin{aligned}
K_{2p}^{\frac{1}{2},\frac{1}{2}}(\tau)=\frac{(-1)^{p+1} (2p)!}{(2\pi )^{2p}}\sum_{k\not=0}\frac{(-1)^k}{k^{2p}}
+\frac{(-1)^{p+1} (2p)!}{(2\pi )^{2p}}\sum_{j\not=0}\sum_{k=-\infty}^\infty\frac{(-1)^{k+j}}{(k+\tau j)^{2p}}\,.
\end{aligned}$$ The first term is just the Fourier series for the Bernoulli polynomial $B_{2p}\left(x\right)$ evaluated at $x=\frac{1}{2}$. Let us transform the second term. Since the terms $(j,k)$ and $(-j,-k)$ give the same contribution, we can write that $$\label{kr13}
\begin{aligned}
K_{2p}^{\frac{1}{2},\frac{1}{2}}(\tau)=B_{2p}\Big(\frac{1}{2}\Big)
+\frac{2(-1)^{p+1} (2p)!}{(2\pi )^{2p}}\sum_{j=1}^\infty\sum_{k=-\infty}^\infty\frac{(-1)^{k+j}}{(k+\tau j)^{2p}}\,.
\end{aligned}$$ Now from identity , with $y=r{\zeta}j$, we obtain that $$\label{kr14}
\begin{aligned}
K_{2p}^{\frac{1}{2},\frac{1}{2}}(\tau)=B_{2p}\Big(\frac{1}{2}\Big)
-4p \sum_{j=1}^\infty \sum_{k=0}^\infty (-1)^j\left(k+\frac{1}{2}\right)^{2p-1}e^{-2r{\lambda}j(k+\frac{1}{2})}\,.
\end{aligned}$$ Thus, we have the following proposition:
We have that $$\label{kr15}
\begin{aligned}
\sum_{j=1}^\infty \sum_{k=0}^\infty (-1)^j\left(k+\frac{1}{2}\right)^{2p-1}e^{-2r{\lambda}j(k+\frac{1}{2})}
=\frac{B_{2p}\left(\frac{1}{2}\right)-K_{2p}^{\frac{1}{2},\frac{1}{2}}(\tau)}{4p}\,.
\end{aligned}$$
Applying it for $K_{2p+2}(\tau)$, we obtain that $$\label{kr16}
\begin{aligned}
\sum_{j=1}^\infty \sum_{k=0}^\infty (-1)^j\left(k+\frac{1}{2}\right)^{2p+1}e^{-2r{\lambda}j(k+\frac{1}{2})}
=\frac{B_{2p+2}\left(\frac{1}{2}\right)-K_{2p+2}^{\frac{1}{2},\frac{1}{2}}(\tau)}{4(p+1)}\,.
\end{aligned}$$
Case $(\alpha,\beta)=(0,\frac{1}{2})$
-------------------------------------
From , we have that $$\label{kr17}
\begin{aligned}
K_{2p}^{0,\frac{1}{2}}(\tau)=\frac{(-1)^{p+1} (2p)!}{(2\pi )^{2p}}\sum_{(j,k)\not=(0,0)}\frac{(-1)^{j}}{(k+\tau j)^{2p}}\,.
\end{aligned}$$ Separating terms with $j=0$, we obtain that $$\label{kr18}
\begin{aligned}
K_{2p}^{0,\frac{1}{2}}(\tau)=\frac{(-1)^{p+1} (2p)!}{(2\pi )^{2p}}\sum_{k\not=0}\frac{1}{k^{2p}}
+\frac{(-1)^{p+1} (2p)!}{(2\pi )^{2p}}\sum_{j\not=0}\sum_{k=-\infty}^\infty\frac{(-1)^{j}}{(k+\tau j)^{2p}}\,.
\end{aligned}$$ The first term is just the Fourier series for the Bernoulli polynomial $B_{2p}\left(x\right)$ evaluated at $x=0$. Let us transform the second term. Since the terms $(j,k)$ and $(-j,-k)$ give the same contribution, we can write that $$\label{kr19}
\begin{aligned}
K_{2p}^{0,\frac{1}{2}}(\tau)=B_{2p}\left(0\right)
+\frac{2(-1)^{p+1} (2p)!}{(2\pi )^{2p}}\sum_{j=1}^\infty\sum_{k=-\infty}^\infty\frac{(-1)^{j}}{(k+\tau j)^{2p}}\,.
\end{aligned}$$ When $z=iy$, $y>0$, and ${\alpha}=0$, identity reads $$\label{kr20}
\begin{aligned}
\frac{1}{1-e^{-2\pi y}}
=\frac{1}{2\pi y}+\frac{1}{2}+\frac{1}{2\pi i}\sum_{k\not=0}\left(\frac{1}{k}-\frac{1}{k+iy}\right)\,.
\end{aligned}$$ Expanding the left hand side into the geometric series, we obtain that $$\label{kr21}
\begin{aligned}
\sum_{k=0}^\infty e^{-2\pi yk}
=\frac{1}{2\pi y}+\frac{1}{2}+\frac{1}{2\pi i}\sum_{k\not=0}\left(\frac{1}{k}-\frac{1}{k+iy}\right)\,.
\end{aligned}$$ Differentiating this identity $(2p-1)$ times with respect to $y$, we obtain that $$\label{kr22}
\begin{aligned}
\sum_{k=0}^\infty \left[-2\pi k\right]^{2p-1}e^{-2\pi yk}
=-\frac{1}{2\pi i}\sum_{k=-\infty}^\infty \frac{(-i)^{2p-1}(2p-1)!}{(k+iy)^{2p}}\,,
\end{aligned}$$ or equivalently, $$\label{kr23}
\begin{aligned}
\frac{(-1)^p(2p)!}{(2\pi)^{2p}}\sum_{k=-\infty}^\infty \frac{1}{(k+iy)^{2p}}
=2p \sum_{k=0}^\infty k^{2p-1}e^{-2\pi yk}.
\end{aligned}$$ Using this formula in with $y=r{\zeta}j$, we obtain that $$\label{kr24}
\begin{aligned}
K_{2p}^{0,\frac{1}{2}}(\tau)=B_{2p}\left(0\right)
-4p \sum_{j=1}^\infty \sum_{k=0}^\infty (-1)^jk^{2p-1}e^{-2{\lambda}r jk}\,.
\end{aligned}$$ Thus, we have the following proposition:
We have that $$\label{kr25}
\begin{aligned}
\sum_{j=1}^\infty \sum_{k=0}^\infty (-1)^jk^{2p-1}e^{-2{\lambda}r jk}
=\frac{B_{2p}\left(0\right)-K_{2p}^{0,\frac{1}{2}}(\tau)}{4p}\,.
\end{aligned}$$
Applying it for $K_{2p+2}^{0,\frac{1}{2}}(\tau)$, we obtain that $$\label{kr26}
\begin{aligned}
\sum_{j=1}^\infty \sum_{k=0}^\infty (-1)^jk^{2p+1}e^{-2{\lambda}r jk}
=\frac{B_{2p+2}\left(0\right)-K_{2p+2}^{0,\frac{1}{2}}(\tau)}{4(p+1)}\,.
\end{aligned}$$
[99]{}
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[^1]: pbleher@iupui.edu
[^2]: bradelwood@gmail.com
[^3]: petrovic.drazen@gmail.com
[^4]: The first author is supported in part by the National Science Foundation (NSF) Grant DMS-1565602.
|
---
abstract: 'Pupil mapping is a promising and unconventional new method for high contrast imaging being considered for terrestrial exoplanet searches. It employs two (or more) specially designed aspheric mirrors to create a high-contrast amplitude profile across the telescope pupil that does not appreciably attenuate amplitude. As such, it reaps significant benefits in light collecting efficiency and inner working angle, both critical parameters for terrestrial planet detection. While much has been published on various aspects of pupil mapping systems, the problem of sensitivity to wavefront aberrations remains an open question. In this paper, we present an efficient method for computing the sensitivity of a pupil mapped system to Zernike aberrations. We then use this method to study the sensitivity of a particular pupil mapping system and compare it to the concentric-ring shaped pupil coronagraph. In particular, we quantify how contrast and inner working angle degrade with increasing Zernike order and rms amplitude. These results have obvious ramifications for the stability requirements and overall design of a planet-finding observatory.'
author:
- Ruslan Belikov
- 'N. Jeremy Kasdin'
- 'Robert J. Vanderbei'
bibliography:
- '../lib/refs.bib'
title: 'Diffraction-Based Sensitivity Analysis of Apodized Pupil Mapping Systems'
---
Introduction {#sec:Introduction}
============
The impressive discoveries of large extrasolar planets over the past decade have inspired widespread interest in finding and directly imaging Earth-like planets in the habitable zones of nearby stars. In fact, NASA has plans to launch two space telescopes to accomplish this, the [*Terrestrial Planet Finder Coronagraph (TPF-C)*]{} and the [*Terrestrial Planet Finder Interferometer (TPF-I)*]{}, while the European Space Agency is planning a similar multi-satellite mission called Darwin. These missions are currently in the concept study phase. In addition, numerous ground-based searches are proceeding using both coronagraphic and interferometric approaches.
Direct imaging of Earth-like extrasolar planets in the habitable zones of Sun-like stars poses an extremely challenging problem in high-contrast imaging. Such a star will shine $10^{10}$ times more brightly than the planet. And, if we assume that the star-planet system is $10$ parsecs from us, the maximum separation between the star and the planet will be roughly $0.1$ arcseconds.
[[**[Design Concepts for TPF-C.]{}**]{}]{} For TPF-C, for example, the current baseline design involves a traditional [*Lyot coronagraph*]{} consisting of a modern 8th-order occulting mask (see, e.g., @KCG04) attached to the back end of a Ritchey-Chretien telescope having an $8$m by $3.5$m elliptical primary mirror. Alternative innovative back-end designs still being considered include [*shaped pupils*]{} (see, e.g., @KVSL02 and @VKS04), a [*visible nuller*]{} (see, e.g., @SLSWL04) and [*pupil mapping*]{} (see, e.g., @Guy03 where this technique is called [*phase-induced amplitude apodization*]{} or [*PIAA*]{}). By pupil mapping we mean a system of two lenses, or mirrors, that takes a flat input field at the entrance pupil and produce an output field that is amplitude modified but still flat in phase (at least for on-axis sources).
[[**[The Pupil Mapping Concept.]{}**]{}]{} The pupil mapping concept has received considerable attention recently because of its high throughput and small effective inner working angle (IWA). These benefits could potentially permit more observations over the mission lifetime, or conversely, a smaller and cheaper overall telescope. As a result, there have been numerous studies over the past few years to examine the performance of pupil mapping systems. In particular, @Guy03 [@TV03; @VT04; @GPGMRW05] derived expressions for the optical surfaces using ray optics. However, these analyses made no attempt to provide a complete diffraction through a pupil mapping system. More recently, @Van05 provided a detailed diffraction analysis. Unfortunately, this analysis showed that a pupil mapping system, in its simplest and most elegant form, cannot achieve the required $10^{-10}$ contrast; the diffraction effects from the pupil mapping systems themselves are so detrimental that contrast is limited to $10^{-5}$. In [@GPGMRW05] and @PGRMWBG06, a [*hybrid pupil mapping*]{} system was proposed that combines the pupil mapping mirrors with a modest apodization of oversized entrance and exit pupils. This combination does indeed achieve the needed high-contrast point spread function (PSF). In this paper, we call such systems [*apodized pupil mapping*]{} systems.
A second problem that must be addressed is the fact that a simple two-mirror (or two-lens) pupil mapping system introduces non-constant angular magnification for off-axis sources (such as a planet). In fact, the off-axis magnification for light passing through a small area of the exit pupil is directly proportional to the amplitude amplification in that small area. For systems in which the exit amplitude amplification is constant, the magnification is also constant. But, for high-contrast imaging, we are interested in amplitude profiles that are far from constant. Hence, off-axis sources do not form images in a formal sense (the “images” are very distorted.) @Guy03 proposed an elegant solution to this problem. He suggested using this system merely as a mechanism for concentrating (on-axis) starlight in an image plane. He then proposed that an occulter be placed in the image plane to remove the starlight. All other light, such as the distorted off-axis planet light, would be allowed to pass through the image plane. On the back side would be a second, identical pupil mapping system (with the apodizers removed), that would “umap” the off-axis beam and thus remove the distortions introduced by the first system (except for some beam walk—see @VT04).
[[**[Sensitivity Analysis.]{}**]{}]{} What remains to be answered is how apodized pupil mapping behaves in the presence of optical aberrations. It is essential that contrast be maintained during an observation, which might take hours during which the wavefront will undoubtedly suffer aberration due to the small dynamic perturbations of the primary mirror. An understanding of this sensitivity is critical to the design of TPF-C or any other observatory. In @Green04, a detailed sensitivity analysis is given for shaped pupils and various Lyot coronagraphs (including the $8$th-order image plane mask introduced in @KCG04). Both of these design approaches achieve the needed sensitivity for a realizable mission. So far, however, no comparable study has been done for apodized pupil mapping. One obstacle to such a study is the considerable computing power required to do a full 2-D diffraction simulation.
[[**[Aberrations Given by Zernike Polynomials.]{}**]{}]{} In this paper, we present an efficient method for computing the effects of wavefront aberrations on apodized pupil mapping. We begin with a brief review of the design of apodized pupil mapping systems in Section \[sec:Review of Pupil Mapping and Apodization\]. We then present in Section \[sec:Diffraction Analysis\] a semi-analytical approach to computing the PSF of systems such as pupil-mapping and concentric rings in the presence of aberrations represented by Zernike polynomials. For such aberrations, it is possible to integrate analytically the integral over azimuthal angle, thereby reducing the computational problem from a double integral to a single one, eliminating the need for massive computing power.
In Section \[sec:Simulations\], we present the sensitivity results for an apodized pupil mapping system and a concentric ring shaped pupil coronagraph, and compare the results.
Review of Pupil Mapping and Apodization {#sec:Review of Pupil Mapping and Apodization}
=======================================
In this section, we review the apodized pupil mapping approach and introduce the specific system that we study in subsequent sections. It should be noted that this apodized pupil mapping design may not be the best possible. Rather, it is merely an example of such a system that achieves high contrast. Other examples can be found in the recent paper by @PGRMWBG06. Our aim in this paper is not to identify the best such system. Instead, our aim is to develop tools for carrying a full diffraction analysis of any apodized pupil mapping system.
Pupil Mapping via Ray Optics {#subsec:Pupil Mapping via Ray Optics}
----------------------------
We begin by summarizing the ray-optics description of pure pupil mapping. An on-axis ray entering the first pupil at radius $r$ from the center is to be mapped to radius ${\tilde{r}}= {\tilde{R}}(r)$ at the exit pupil (see Figure \[fig:1\]). Optical elements at the two pupils ensure that the exit ray is parallel to the entering ray. The function ${\tilde{R}}(r)$ is assumed to be positive and increasing or, sometimes, negative and decreasing. In either case, the function has an inverse that allows us to recapture $r$ as a function of ${\tilde{r}}$: $r = R({\tilde{r}})$. The purpose of pupil mapping is to create nontrivial amplitude profiles. An amplitude profile function $A({\tilde{r}})$ specifies the ratio between the output amplitude at ${\tilde{r}}$ to the input amplitude at $r$ (in a pure pupil-mapping system the input amplitude is constant). @VT04 showed that for any desired amplitude profile $A({\tilde{r}})$ there is a pupil mapping function $R({\tilde{r}})$ that achieves it (in a ray-optics sense). Specifically, the pupil mapping is given by $$\label{1}
R({\tilde{r}}) = \pm \sqrt{\int_0^{{\tilde{r}}} 2 A^2(s) s ds} .$$ Furthermore, if we consider the case of a pair of lenses that are planar on their outward-facing surfaces, then the inward-facing surface profiles, $h(r)$ and ${\tilde{h}}({\tilde{r}})$, that are required to obtain the desired pupil mapping are given by the solutions to the following ordinary differential equations: $$\label{2}
\frac{\partial h}{\partial r}(r)
= \frac{r-{\tilde{R}}(r)}{ |n-1| \sqrt{z^2
+ \frac{n+1}{n-1}(r-{\tilde{R}}(r))^2} },
\qquad
h(0) = z,$$ and $$\label{3}
\frac{\partial {\tilde{h}}}{\partial {\tilde{r}}}({\tilde{r}})
= \frac{R({\tilde{r}})-{\tilde{r}}}{ |n-1| \sqrt{\phantom{\tilde{|}}z^2
+ \frac{n+1}{n-1}(R({\tilde{r}})-{\tilde{r}})^2} },
\qquad
{\tilde{h}}(0) = 0.$$ Here, $n \ne 1$ is the refractive index and $z$ is the distance separating the centers ($r=0$, ${\tilde{r}}= 0$) of the two lenses.
Let $S(r,{\tilde{r}})$ denote the distance between a point on the first lens surface $r$ units from the center and the corresponding point on the second lens surface ${\tilde{r}}$ units from its center. Up to an additive constant, the optical path length of a ray that exits at radius ${\tilde{r}}$ after entering at radius $r = R({\tilde{r}})$ is given by $$\label{4}
Q_0({\tilde{r}}) = S(R({\tilde{r}}),{\tilde{r}}) + |n|({\tilde{h}}({\tilde{r}})-h(R({\tilde{r}}))).$$ @VT04 showed that, for an on-axis source, $Q_0({\tilde{r}})$ is constant and equal to $-(n-1)|z|$.[^1]
High-Contrast Amplitude Profiles {#subsec:High-Contrast Amplitude Profiles}
--------------------------------
If we assume that a collimated beam with amplitude profile $A({\tilde{r}})$ such as one obtains as the output of a pupil mapping system is passed into an ideal imaging system with focal length $f$, the electric field $E(\rho)$ at the image plane is given by the Fourier transform of $A({\tilde{r}})$: $$\label{5}
E(\xi,\eta)
=
\frac{E_0}{\lambda i f}
e^{\pi i \frac{\xi^2 + \eta^2}{\lambda f}}
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}
e^{-2 \pi i \frac{{\tilde{x}}\xi + {\tilde{y}}\eta}{\lambda f}}
A(\sqrt{{\tilde{x}}^2+{\tilde{y}}^2}) d{\tilde{y}}d{\tilde{x}}.$$ Here, $E_0$ is the input amplitude which, unless otherwise noted, we take to be unity. Since the optics are azimuthally symmetric, it is convenient to use polar coordinates. The amplitude profile $A$ is a function of ${\tilde{r}}= \sqrt{{\tilde{x}}^2+{\tilde{y}}^2}$ and the image-plane electric field depends only on image-plane radius $\rho = \sqrt{\xi^2 + \eta^2}$: $$\begin{aligned}
E(\rho)
& = &
\frac{1}{\lambda i f}
e^{\pi i \frac{\xi^2 + \eta^2}{\lambda f}}
\int_0^{\infty}\int_0^{2 \pi}
e^{-2 \pi i \frac{{\tilde{r}}\rho}{\lambda f} \cos(\theta - \phi)}
A({\tilde{r}}) {\tilde{r}}d\theta d{\tilde{r}}\label{6} \\
& = &
\frac{2 \pi}{\lambda i f}
e^{\pi i \frac{\xi^2 + \eta^2}{\lambda f}}
\int_0^{\infty} J_0\left(-2 \pi \frac{{\tilde{r}}\rho}{\lambda f}\right)
A({\tilde{r}}) {\tilde{r}}d{\tilde{r}}. \label{7}\end{aligned}$$ The point-spread function (PSF) is the square of the electric field: $$\label{8}
\mbox{Psf}(\rho) = |E(\rho)|^2 .$$ For the purpose of terrestrial planet finding, it is important to construct an amplitude profile for which the PSF at small nonzero angles is ten orders of magnitude reduced from its value at zero. A paper by @VSK03 explains how these functions are computed as solutions to certain optimization problems. The high-contrast amplitude profile used in the rest of this paper is shown in Figure \[fig:2\].
Apodized Pupil Mapping Systems {#sec:hybrid}
------------------------------
\[subsec:Apodized Pupil Mapping Systems\]
@Van05 showed that pure pupil mapping systems designed for contrast of $10^{-10}$ actually achieve much less than this due to harmful diffraction effects that are not captured by the simple ray tracing analysis outlined in the previous section. For most systems of practical real-world interest (i.e., systems with apertures of a few inches and designed for visible light), contrast is limited to about $10^{-5}$. @Van05 considered certain hybrid designs that improve on this level of performance but none of the hybrid designs presented there completely overcame this diffraction-induced contrast degradation.
In this section, we describe an apodized pupil mapping system that is somewhat more complicated than the designs presented in @Van05. This hybrid design, based on ideas proposed by Olivier Guyon and Eugene Pluzhnik (see @PGRMWBG06), involves three additional components. They are
1. a preapodizer $A_0$ to soften the edge of the first lens/mirror so as to minimize diffraction effects caused by hard edges,
2. a postapodizer to smooth out low spatial frequency ripples produced by diffraction effects induced by the pupil mapping system itself, and
3. a backend phase shifter to smooth out low spatial frequency ripples in phase.
Note that the backend phase shifter can be built into the second lens/mirror. There are several choices for the preapodizer. For this paper, we choose Eqs. (3) and (4) in @PGRMWBG06 for our pre-apodizer: $$A_0(r) = \frac{A(r)(1 + \beta)}{A(r) + \beta A_{\mathrm{max}}},$$ where $A_{\mathrm{max}}$ denotes the maximum value of $A(r)$ and $\beta$ is a scalar parameter, which we take to be $0.1$. It is easy to see that
- $A(r)/A_{\mathrm{max}} \le A_0(r) \le 1$ for all $r$,
- $A_0(r)$ approaches $1$ as $A(r)$ approaches $A_{\mathrm{max}}$, and
- $A_0(r)$ approaches $0$ as $A(r)$ approaches $0$.
Incorporating a post-apodizer introduces a degree of freedom that is lacking in a pure pupil mapping system. Namely, it is possible to design the pupil mapping system based on an arbitrary amplitude profile and then convert this profile to a high-contrast profile via an appropriate choice of backend apodizer. We have found that a simple Gaussian amplitude profile that approximately matches a high-contrast profile works very well. Specifically, we used $${A_{\mathrm{pupmap}}}({\tilde{r}}) = 3.35 e^{-22({\tilde{r}}/{\tilde{a}})^2},$$ where ${\tilde{a}}$ denotes the radius of the second lens/mirror.
The backend apodization is computed by taking the actual output amplitude profile as computed by a careful diffraction analysis, smoothing it by convolution with a Gaussian distribution, and then apodizing according to the ratio of the desired high-contrast amplitude profile $A({\tilde{r}})$ divided by the smoothed output profile. Of course, since a true apodization can never intensify a beam, this ratio must be further scaled down so that it is nowhere greater than unity. The Gaussian convolution kernel we used has mean zero and standard deviation ${\tilde{a}}/\sqrt{100,000}$.
The backend phase modification is computed by a similar smoothing operation applied to the output phase profile. Of course, the smoothed output phase profile (measured in radians) must be converted to a surface profile (having units of length). This conversion requires us to assume a certain specific wavelength. As a consequence, the resulting design is correct only at one wavelength. The ability of the system to achieve high contrast degrades as one moves away from the design wavelength.
Star Occulter and Reversed System {#subsec:Star Occulter and Reversed System}
---------------------------------
It is important to note that the PSFs in Figure \[fig:2\] correspond to a bright on-axis source (i.e., a star). Off-axis sources, such as faint planets, undergo two effects in a pupil mapping system that differ from the response of a conventional imaging system: an effective magnification and a distortion. These are explained in detail in @VT04 and @TV03. The magnification, in particular, is due to an overall narrowing of the exit pupil as compared to the entrance pupil. It is this magnification that provides pupil mapped systems their smaller effective inner working angle. The techniques in Section \[sec:Diffraction Analysis\] will allow us to compute the exact off-axis diffraction pattern of an apodized pupil mapped coronagraph and thus to see these effects.
While the effective magnification of a pupil mapping system results in an inner working angle advantage of about a factor of two, it does not produce high-quaity diffraction limited images of off-axis sources because of the distortion inherent in the system. @Guy03 proposed the following solution to this problem. He suggested using this system merely as a mechanism for concentrating (on-axis) starlight in an image plane. He then proposed that an occulter be placed in the image plane to remove the starlight. All other light, such as the distorted off-axis planet light, would be allowed to pass through the image plane. On the back side would be a second, identical pupil mapping system (with the apodizers removed), that would “umap” the off-axis beam and thus remove the distortions introduced by the first system (except for some beam walk—see @VT04). A schematic of the full system (without the occulter) is shown in Figure \[fig:3\]. Note that we have spaced the lenses one focal length from the flat sides of the two lenses. As noted in [@VT04], such a spacing guarantees that these two flat surfaces form a conjugate pair of pupils.
Diffraction Analysis {#sec:Diffraction Analysis}
====================
In [@Van05], it was shown that a simple Fresnel analysis is inadequate for validating the high-contrast imaging capabilities we seek. Hence, a more accurate approximation was presented. In this section, we give a similar but slightly different approximation that is just as effective for studying pupil mapping but is better suited to the full system we wish to analyze.
Propagation of General Wavefronts {#subsec:Propagation of General Wavefronts}
---------------------------------
The goal of this section is to derive an integral that describes how to propagate a scalar electric field from one plane perpendicular to the direction of propagation to another parallel plane positioned downstream of the first. We assume that the electric field passes through a lens at the first plane, then propagates through free space until reaching a second lens at the second plane through which it passes. In order to cover the apodized pupil mapping case discussed in the previous section, we allow both the entrance and exit fields to be apodized.
Suppose that the input field at the first plane is ${E_{\mathrm{in}}}(x,y)$. Then the electric field at a particular point on the second plane can be well-approximated by superimposing the phase-shifted waves from each point across the entrance pupil (this is the well-known Huygens-Fresnel principle—see, e.g., Section 8.2 in [@BW99]). If we assume that the two lenses are given by radial “height” functions $h(r)$ and ${\tilde{h}}({\tilde{r}})$, then we can write the exit field as $$\label{9}
{E_{\mathrm{out}}}({\tilde{x}}, {\tilde{y}})
=
{A_{\mathrm{out}}}({\tilde{r}})
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}
\frac{1}{\lambda i Q({\tilde{x}},{\tilde{y}},x,y)}
e^{ 2 \pi i Q({\tilde{x}},{\tilde{y}},x,y) /\lambda }
{A_{\mathrm{in}}}(r) {E_{\mathrm{in}}}(x,y)
dydx,$$ where $$\label{20}
Q({\tilde{x}},{\tilde{y}},x,y) =
\sqrt{(x-{\tilde{x}})^2+(y-{\tilde{y}})^2+(h(r)-{\tilde{h}}({\tilde{r}}))^2}
+ n(Z - h(r) + {\tilde{h}}({\tilde{r}}))$$ is the optical path length, $Z$ is the distance between the planar lens surfaces, ${A_{\mathrm{in}}}(r)$ denotes the input amplitude apodization at radius $r$, ${A_{\mathrm{out}}}({\tilde{r}})$ denotes the output amplitude apodization at radius ${\tilde{r}}$, and where, of course, we have used $r$ and ${\tilde{r}}$ as shorthands for the radii in the entrance and exit planes, respectively.
As before, it is convenient to work in polar coordinates: $$\label{10}
{E_{\mathrm{out}}}({\tilde{r}},{\tilde{\theta}})
=
{A_{\mathrm{out}}}({\tilde{r}})
\int_0^{\infty}\int_0^{2 \pi}
\frac{1}{\lambda i Q({\tilde{r}},r,\theta-{\tilde{\theta}})}
e^{ 2 \pi i Q({\tilde{r}},r,\theta-{\tilde{\theta}})/\lambda)}
{A_{\mathrm{in}}}(r) {E_{\mathrm{in}}}(r,\theta)
r d\theta dr,$$ where $$\label{21}
Q({\tilde{r}},r,\theta) =
\sqrt{r^2-2r{\tilde{r}}\cos\theta+{\tilde{r}}^2+(h(r)-{\tilde{h}}({\tilde{r}}))^2}
+ n(Z - h(r) + {\tilde{h}}({\tilde{r}})) .$$ For numerical tractability, it is essential to make approximations so that the integral over $\theta$ can be carried out analytically, thereby reducing the double integral to a single one. To this end, we need to make an appropriate approximation to the square root term: $$\label{13}
S = \sqrt{r^2-2r{\tilde{r}}\cos\theta+{\tilde{r}}^2+(h(r)-{\tilde{h}}({\tilde{r}}))^2} .$$
A simple crude approximation is adequate for the $1/Q({\tilde{r}},r,\theta-{\tilde{\theta}})$ amplitude-reduction factor in [Eq. (\[10\])]{}. We approximate this factor by the constant $1/Z$.
The $Q({\tilde{r}},r,\theta-{\tilde{\theta}})$ appearing in the exponential must, on the other hand, be treated with care. The classical Fresnel approximation is to replace $S$ by the first two terms in a Taylor series expansion of the square root function about $(h(r)-{\tilde{h}}({\tilde{r}}))^2$. As we already mentioned, this approximation is too crude. It is critically important that the integrand be exactly correct when the pair $(r,{\tilde{r}})$ correspond to rays of ray optics. Here is a method that does this. First, we add and subtract $S({\tilde{r}},r,0)$ from $Q({\tilde{r}},r,\theta)$ in [Eq. (\[10\])]{} to get $$\begin{aligned}
Q({\tilde{r}},r,\theta-{\tilde{\theta}})
& = &
S({\tilde{r}},r,\theta-{\tilde{\theta}})-S({\tilde{r}},r,0)+S({\tilde{r}},r,0)
+ |n|\left({\tilde{h}}({\tilde{r}})-h(r)\right)
\nonumber \\
& = &
\frac{S({\tilde{r}},r,\theta-{\tilde{\theta}})^2-S({\tilde{r}},r,0)^2}{
S({\tilde{r}},r,\theta-{\tilde{\theta}})+S({\tilde{r}},r,0)}
+S({\tilde{r}},r,0) + |n|\left({\tilde{h}}({\tilde{r}})-h(r)\right)
\nonumber \\
& = &
\frac{r {\tilde{r}}- r {\tilde{r}}\cos(\theta-{\tilde{\theta}})}{
(S({\tilde{r}},r,\theta-{\tilde{\theta}})+S({\tilde{r}},r,0))/2}
+S({\tilde{r}},r,0) + |n|\left({\tilde{h}}({\tilde{r}})-h(r)\right) .
\label{141}\end{aligned}$$ So far, these calculations are exact. The only approximation we now make is to replace $S({\tilde{r}},r,\theta-{\tilde{\theta}})$ in the denominator of [Eq. (\[141\])]{} with $S({\tilde{r}},r,0)$ so that the denominator becomes just $S({\tilde{r}},r,0)$. Putting this all together, we get a new approximation, which we refer to as the [*S-Huygens*]{} approximation: $$\begin{aligned}
{E_{\mathrm{out}}}({\tilde{r}},{\tilde{\theta}})
& \approx &
\frac{{A_{\mathrm{out}}}({\tilde{r}})}{\lambda i Z}
\int_0^{\infty} K(r,{\tilde{r}})
\int_0^{2 \pi}
e^{2 \pi i \left( -\frac{{\tilde{r}}r \cos(\theta - {\tilde{\theta}})}{S({\tilde{r}},r,0)}
\right) /\lambda} {E_{\mathrm{in}}}(r,\theta) d\theta
\; {A_{\mathrm{in}}}(r) r dr , \label{160}\end{aligned}$$ where $$K(r,{\tilde{r}})
=
e^{ 2 \pi i \left(
\frac{ r {\tilde{r}}}{S({\tilde{r}},r,0)} + S({\tilde{r}},r,0) + |n|({\tilde{h}}({\tilde{r}}) - h(r))
\right) /\lambda}$$ (note that we have dropped an $exp(2 \pi i n Z / \lambda)$ factor since this factor is just a constant unit complex number which would disappear anyway at the end when we compute intensities).
The only reason for making approximations to the Huygens-Fresnel integral is to simplify the dependence on $\theta$ so that the integral over this variable can be carried out analytically. For example, if we now assume that the input field ${E_{\mathrm{in}}}(r,\theta)$ does not depend on $\theta$, then the inner integral can be evaluated explicitly and we get $$\begin{aligned}
{E_{\mathrm{out}}}({\tilde{r}},{\tilde{\theta}})
& \approx &
\frac{2 \pi {A_{\mathrm{out}}}({\tilde{r}})}{\lambda i Z}
\int_0^{\infty} K(r,{\tilde{r}})
J_0\left( \frac{2 \pi {\tilde{r}}r}{\lambda S({\tilde{r}},r,0)} \right)
{E_{\mathrm{in}}}(r) \; {A_{\mathrm{in}}}(r) r dr , \label{161}\end{aligned}$$ Removing the dependency on $\theta$ greatly simplifies computations because we only need to compute a 1D integral instead of 2D. In the next subsection we will show how to achieve similar reductions in cases where the dependence of ${E_{\mathrm{in}}}$ on $\theta$ takes a simple form.
Figure \[fig:4\] shows plots characterizing the performance of an apodized pupil mapping system analyzed using the techniques described in this section. The specifications for this system are as follows. The designed-for wavelength is $632.8$nm. The optical elements are assumed to be mirrors separated by $0.375$m. The system is an on-axis system and we therefore make the non-physical assumption that the mirrors don’t obstruct the beam. That is, the mirrors are invisible except when they are needed. The mirrors take as input a $0.025$m on-axis beam and produce a $0.025$m pupil-remapped exit beam. The second mirror is oversized by a factor of two; that is, its diameter is $0.050$m. The postapodizer ensures that only the central half contributes to the exit beam. The first mirror is also oversized appropriately as shown in the upper-right subplot of Figure \[fig:4\]. After the second mirror, the exit beam is brought to a focus. The focal length is $2.5$m. The lower-right subplot in Figure \[fig:4\] shows the ideal PSF (in black) together with the achieved PSF at three wavelengths: at $70\%$ (green), $100\%$ (blue), and $130\%$ (red) of the design wavelength. At the design wavelength, the achieved PSF matches the ideal PSF almost exactly. Note that there is minor degradation at the other two wavelengths mostly at low spatial frequencies.
We end this section by pointing out that the S-Huygens approximation given by is the basis for all subsequent analysis in this paper. It can be used to compute the propagation between every pair of consecutive components in apodized pupil mapping and concentric ring systems. It should be noted that the approximation does not reduce to the standard Fresnel or Fourier approximations even when considering such simple scenarios as free-space propagation of a plane wave or propagation from a pupil plane to an image plane. Even for these elementary situations, the S-Huygens approximation is superior to the usual textbook approximations.
Propagation of Azimuthal Harmonics {#subsec:Propagation of Azimuthal Harmonics}
----------------------------------
In this section, we assume that $E(r,\theta) = E(r) e^{i n \theta}$ for some integer $n$. We refer to such a field as an [*$n$th-order azimuthal harmonic*]{}. We will show that an $n$th-order azimuthal harmonic will remain an $n$th-order azimuthal harmonic after propagating from the input plane to the output plane described in the previous section. Only the radial component $E(r)$ changes, which enables the reduction of the computation from 2D to 1D. Arbitrary fields can also be propagated, by decomposing them into azimuthal harmonics and propagating each azimuthal harmonic separately. Computation is thus greatly simplified even for arbitrary fields, especially for the case of fields which can be described by only a few azimuthal harmonics to a high precision, such as Zernike aberrations, which we consider in subsection \[subsec:Zernike\]. This improvement in computation efficiency is important, because a full 2D diffraction simulation of an apodized pupil mapping system with the precision of greater than $10^{10}$ typically overwhelms the memory of a mainstream computer. By reducing the computation from 2D to 1D, however, the entire apodized pupil mapping system can be simulated with negligible memory requirements and takes only minutes.
Suppose that the input field in an optical system described by is an $n$th-order azimuthal harmonic ${E_{\mathrm{in}}}(r,\theta) =
{E_{\mathrm{i}}}(r) e^{i n \theta}$ for some integer $n$. Then the output field is also an $n$th-order azimuthal harmonic ${E_{\mathrm{out}}}({\tilde{r}},{\tilde{\theta}}) =
{E_{\mathrm{o}}}({\tilde{r}}) e^{i n {\tilde{\theta}}}$ with radial part given by $${E_{\mathrm{o}}}({\tilde{r}})
=
\frac{2 \pi i^{n-1} {A_{\mathrm{out}}}({\tilde{r}})}{\lambda Z}
\int_0^{\infty} K(r,{\tilde{r}}) {E_{\mathrm{i}}}(r)
J_n\left(\frac{2 \pi r {\tilde{r}}}{\lambda S({\tilde{r}},r,0)}\right) \; {A_{\mathrm{in}}}(r) r dr .$$
We start by substituting the azimuthal harmonic form of ${E_{\mathrm{in}}}$ into and regrouping factors to get $$\begin{aligned}
{E_{\mathrm{out}}}({\tilde{r}},{\tilde{\theta}})
& = &
\frac{{A_{\mathrm{out}}}({\tilde{r}})}{\lambda i Z}
\int_0^{\infty} K(r,{\tilde{r}})
\int_0^{2 \pi}
e^{2 \pi i \left( -\frac{{\tilde{r}}r \cos(\theta - {\tilde{\theta}})}{S({\tilde{r}},r,0)}
\right) /\lambda} {E_{\mathrm{i}}}(r)e^{i n \theta} d\theta
\; {A_{\mathrm{in}}}(r) r dr \nonumber \\
& = &
\frac{{A_{\mathrm{out}}}({\tilde{r}})}{\lambda i Z}
e^{i n {\tilde{\theta}}}
\int_0^{\infty} K(r,{\tilde{r}}) {E_{\mathrm{i}}}(r)
\int_0^{2 \pi}
e^{2 \pi i \left( -\frac{{\tilde{r}}r \cos(\theta - {\tilde{\theta}})}{S({\tilde{r}},r,0)}
\right) /\lambda}
e^{i n (\theta-{\tilde{\theta}})}
d\theta
\; {A_{\mathrm{in}}}(r) r dr . \nonumber \\\end{aligned}$$ The result then follows from an explicit integration over the $\theta$ variable: $$\begin{aligned}
{E_{\mathrm{out}}}({\tilde{r}},{\tilde{\theta}})
& = &
\frac{2 \pi i^{n-1} {A_{\mathrm{out}}}({\tilde{r}})}{\lambda Z}
e^{i n {\tilde{\theta}}}
\int_0^{\infty} K(r,{\tilde{r}}) {E_{\mathrm{i}}}(r)
J_n\left(\frac{2 \pi r {\tilde{r}}}{\lambda S({\tilde{r}},r,0)}\right)
\; {A_{\mathrm{in}}}(r) r dr \nonumber\end{aligned}$$
Decomposition of Zernike Aberrations into Azimuthal Harmonics {#subsec:Zernike}
-------------------------------------------------------------
The theorem shows that the full 2D propagation of azimuthal harmonics can be computed efficiently by evaluating a 1D integral. However, suppose that the input field is not an azimuthal harmonic, but something more familiar, such as a $(l,m)$-th Zernike aberration:
$$\label{alpha}
{E_{\mathrm{in}}}(r,\theta) = e^{i\epsilon Z_l^m(r/a) \cos(m \theta)} ,$$
where $\epsilon$ is a small number. ($\epsilon/2\pi$ and $\epsilon/\pi$ are the peak-to-valley phase variations across the aperture of radius $a$ for $m=0$ and $m\neq0$, respectively.)
Recall that the definition of the $n$th-order Bessel function is $$\label{Jn}
J_n(x) = \frac{1}{2 \pi i^n}
\int_0^{2 \pi} e^{i x \cos \theta} e^{i n \theta} d \theta .$$ From this definition we see that $i^k J_n(x)$ are simply the Fourier coefficients of $e^{i x \cos(\theta)}$. Hence, the Fourier series for the complex exponential is given simply by the so-called [*Jacobi-Anger expansion*]{} $$\label{jacobi-anger}
e^{i x \cos \theta} = \sum_{k= -\infty}^{\infty} i^k J_k(x) e^{i k \theta}.$$ The Zernike aberration can be decomposed into azimuthal harmonics using the Jacobi-Anger expansion: $$\begin{aligned}
\label{Zernike}
e^{i\epsilon Z_l^m(r/a) \cos(m \theta)}
& = &
\sum_{k= -\infty}^{\infty} i^k J_k(\epsilon Z_l^m(r/a)) e^{i km
\theta}\\
& = &
J_0(\epsilon Z_l^m(r/a))+\sum_{k= 1}^{\infty} i^k J_k(\epsilon Z_l^m(r/a)) e^{i km \theta}\end{aligned}$$
Note that $$|J_k(x)| \approx \frac{1}{k!} \left(\frac{x}{2}\right)^k$$ for $0 \le x \ll 1$. Hence, if we assume that $\epsilon \sim
10^{-3}$, then the $k$’th term is of the order $10^{-3k}$. The field amplitude in the high-contrast region of the PSF will be dominated by the $k=1$ term and be on the order of $10^{-3}$. If we drop terms of $k=3$ and above, we are introducing an error on the order of $10^{-9}$ in amplitude. The error in intensity will be dominated by a cross-product of the $k=3$ and the $k=1$ term, or $10^{-12}$ across the dark region. So, in this case, Zernike aberrations can be more than adequately modeled using just 3 azimuthal harmonic terms. For $\epsilon \sim 10^{-2}$, the number of terms goes up to 5 for an error tolerance of $10^{-12}$. In practice, even this small number of terms was actually found to be overly conservative.
In order to compute the full 2D response for a given Zernike aberration, we simply decompose it into a few azimuthal harmonics, propagate them separately, and sum the results at the end. This method could also be applied to any arbitrary field.
Simulations {#sec:Simulations}
===========
The entire 4-mirror apodized pupil mapping system can be modeled as the following sequence of 7 steps:
1. Propagate an input wavefront from the front (flat) surface of the first pupil mapping lens to the back (flat) surface of the second pupil mapping lens as described in Section \[subsec:Apodized Pupil Mapping Systems\].
2. Propagate forward a distance $f$.
3. Propagate through a positive lens with focal length $f$ to a focal plane $f$ units downstream.
4. Multiply by star occulter.
5. Propagate through free-space a distance $f$ then through a positive lens to recollimate the beam.
6. Propagate forward a distance $f$.
7. Propagate backwards through a pupil mapping system having the same parameters as the first one.
A similar analysis can be carried out for a concentric ring shaped pupil system, or even a pure apodization system, as follows:
1. Choose ${A_{\mathrm{in}}}$ to represent either the concentric ring binary mask or some other azimuthally symmetric apodization.
2. Choose $h$ as appropriate for a focusing lens and let ${\tilde{h}}\equiv 0$.
3. Propagate through this system a distance $f$ to the image plane.
4. Multiply by star occulter.
The theorem can be applied to every propagation step, so that an azimuthal harmonic will remain an azimuthal harmonic throughout the entire system. Hence, our computation strategy was to decompose the input field into azimuthal harmonics, propagate each one separately through the entire system by repeated applications of the theorem, and sum them at the very end.
Figure \[fig:5\] shows a cross section plot of the PSF as it appears at first focus and second focus in our apodized pupil mapping system (the first focus plot is indistinguishable from the case of ideal apodization or concentric ring shaped pupils). There are two plots for second focus: one with the occulter in place and one without it. Note that without the occulter, the PSF matches almost perfectly the usual Airy pattern. With the occulter, the on-axis light is suppressed by ten orders of magnitude.
The electric field for a planet is just a slightly tilted and much fainter field than the field associated with the star. Hence, the methods presented here (specifically using the $(1,1)$-Zernike, i.e. tilt) can be used to generate planet images. Some such scenarios are shown in Figure \[fig:6\]. The first row shows how an off-axis source, i.e. planet, looks at the first focus. As discussed earlier, at this focal plane off-axis sources do not form good images. This is clearly evident in this figure. The second row shows the planet as it appears at the second image plane, which is downstream from the reversed pupil mapping system. In this case, the off-axis source is mostly restored and the images begin to look like standard Airy patterns as the angle increases from about $2 \lambda/D$ outward. Figure \[fig:7\] shows corresponding cross sectional plots for the apodized pupil mapping system at the second focus. The third row in Figure \[fig:6\] shows how a planet would appear at a focal plane of a concentric ring shaped pupil system.
Figure \[fig:8\] shows how the off-axis source is attenuated as a function of the angle from optical axis, for the case of our apodized pupil mapping system (at second focus) and the concentric ring coronagraph. For the case of apodized pupil mapping, the $50\%$ point occurs at about $2.5 \lambda/D$.
Figures \[fig:9\] and \[fig:10\] show the distortions/leakage from an on-axis source in the presence of various Zernike aberrations, for apodized pupil mapping and the concentric ring shaped pupil systems, respectively. The Zernike aberrations are assumed to be $1/100$th wave rms.
Figure \[fig:11\] shows the corresponding cross-section sensitivity plots for both the apodized pupil mapping system and the concentric ring shaped pupil system. From this plot it is easy to see both the tighter inner working angle of apodized pupil mapping systems as well as their increased sensitivity to wavefront errors. Finally, Figure \[fig:12\] demonstrates contrast degradation measured at three angles, $2$, $4$, and $8 \lambda/D$, as a function of severity of the Zernike wavefront error. The rms error is expressed in waves.
Conclusions {#sec:Conclusions}
===========
We have presented an efficient method for calculating the diffraction of aberrations through optical systems such as apodized pupil mapping and shaped pupil coronagraphs. We presented an example for both systems and computed their off-axis responses and aberration sensitivities. Figures \[fig:11\] and \[fig:12\] show that our particular apodized pupil mapping system is more sensitive to low order aberrations than the concentric ring masks. That is, contrast and IWA degrade more rapidly with increasing rms level of the aberrations. Thus, for a particular telescope, our pupil mapping system will achieve better throughput and inner working angle, but suffer greater aberration sensitivity.
We note that there is a spectrum of apodized pupil mapping systems, out of which we selected but one example. The two extremes, pure apodization and pure pupil mapping, both have serious drawbacks. On the one end, pure apodization loses almost an order of magnitude in throughput and suffers from an unpleasantly large IWA. At the other extreme, pure pupil mapping fails to achieve the required high contrast due to diffraction effects. There are several points along this spectrum that are superior to the end points. We have focused on just one such point, which is similar to the design suggested by [@GPGMRW05]. We leave it to future work to determine if this is the best design point. For example, clearly one can improve the aberration sensitivity by relaxing the inner working angle and throughput requirements. Such analysis is beyond the scope of this paper, but we have provided here the tools to analyze the sensitivity of these kinds of designs.
[**Acknowledgements.**]{} This research was partially performed for the Jet Propulsion Laboratory, California Institute of Technology, sponsored by the National Aeronautics and Space Administration as part of the TPF architecture studies and also under JPL subcontract number 1260535. The third author also received support from the ONR (N00014-05-1-0206).
[^1]: For a pair of mirrors, put $n=-1$. In that case, $z<0$ as the first mirror is “below” the second.
|
---
abstract: 'A theory of feedback controlled heat transport in quantum systems is presented. It is based on modelling heat engines as driven multipartite systems subject to projective quantum measurements and measurement-conditioned unitary evolutions. The theory unifies various results presented in the previous literature. Feedback control breaks time reversal invariance. This in turn results in the fluctuation relation not being obeyed. Its restoration occurs by an appropriate accounting of the information gain and information use via measurements and feedback. We further illustrate an experimental proposal for the realisation of a Maxwell demon using superconducting circuits and single photon on-chip calorimetry. A two level qubit acts as a trapdoor which, conditioned on its state is coupled to either a hot resistor or a cold one. The feedback mechanism alters the temperatures felt by the qubit and can result in an effective inversion of temperature gradient, where heat flows from cold to hot thanks to information gain and use.'
address:
- 'NEST, Scuola Normale Superiore & Istituto Nanoscienze-CNR, I-56126 Pisa, Italy'
- 'Low Temperature Laboratory, Department of Applied Physics, Aalto University School of Science, 00076 AALTO, Finland'
- 'ICTP, Strada Costiera 11, Trieste 34151, Italy'
- 'NEST, Scuola Normale Superiore & Istituto Nanoscienze-CNR, I-56126 Pisa, Italy'
author:
- Michele Campisi
- Jukka Pekola
- Rosario Fazio
title: 'Feedback controlled heat transport in quantum devices: Theory and solid state experimental proposal'
---
Introduction
============
In a famous thought experiment Maxwell envisioned a method for apparently defying the second law of thermodynamics by means of a feedback control mechanism [@Maruyama09RMP81]. Maxwell’s idea is based on a malicious demon, an intelligent being that is able to observe the microscopic dynamics of a system, and acts on it so as to steer it toward defying the second law. In one of Maxwell’s original concepts, the system is a container with two chambers, containing respectively a hot gas and a cold gas. The two chambers are separated by a wall presenting a trap-door which the demon can open and close at will. The demon observes the erratic motion of the gas particle and when sees a particle of the cold chamber approach the trap-door with sufficiently high velocity, she/he swiftly opens the door as to let the particle go through and closes it immediately afterwards. In this way, particle after particle, heat flows from the cold chamber to the hot chamber in contradiction with the second law.
Advance in nanotechnology has made the possibility of bringing Maxwell demons and similar devices from the realm of thought experiments to the realm of real experiments [@Toyabe10NP6; @Koski14PRL113; @Koski14PNAS111; @Berut12Nature483]. Both theoretical and experimental studies so far have focused mainly on situations where feedback control is operated as a measurement-conditioned driving on some working substance (classical or quantum) coupled to a single temperature, so as to withdraw energy from the latter in contradiction with the second law as formulated by Kelvin. Interesting realistic proposals have appeared in Refs. [@Strasberg13PRL110; @Bergli13PRE88]. Situations where heat flows between different temperature reservoirs is controlled, however have not been addressed so far, neither theoretically nor experimentally. The main motivation of the present work is that of filling that gap. In the following we shall present the general theory of feedback controlled heat transport in quantum devices, and shall describe a possible experimental realisation thereof.
The theory presented here builds on previous works concerning fluctuation relations in presence of measurements without feedback [@Campisi10PRL105; @Campisi11PRE83] and with feedback [@Morikuni11JSP143], combined with an inclusive approach where quantum heat engines are seen as mechanically driven multipartite systems starting in a multi-temperature initial state [@Jarzynski99JSM96; @Campisi14JPA47; @Campisi15NJP17; @Campisi16JPA49a]. Reference [@Morikuni11JSP143] reported on the theory of a one-measurement based feedback control on a quantum working-substance prepared by contact with a single bath. That formalism is here extended to the case of many heat baths, and also repeated measurements, to allow for the study of continuous feedback control of heat flow in a multi reservoir scenario. Previous work concerning repeated measurements appeared in Refs [@Horowitz10PRE82] for classical systems in contact with a single bath. Fluctuation relations need to be modified by a mutual information term, which we shall explicitely provide.
Our experimental proposal is based on the fast developing advancements in experimental solid state low temperature techniques: in particular the calorimetric measurement scheme that has been put forward by one of us and co-workers [@Pekola10PRL105; @Gasparinetti15PRAPP3]. As proven by some recent theoretical proposals [@Campisi15NJP17; @Karimi16PRB94] the method opens up a new avenue for the practical management of heat and work on a chip by means of superconducting devices, particularly superconducting qubits. Here we illustrate the possible implementation of very simple feedback controlled heat transport where the trapdoor is realised by a superconducting qubit whose coupling with two resistors at different temperatures is controlled based on the outcomes of continuous calorimetric monitoring of the resistors themselves.
![Feedback controlled heat transport. A bi-partite system starting in a two temperature Gibbs state is observed by a Demon, who measures an observable $A$. Depending on the outcome $a_j$ of the measurement the demon applies a quantum gate $U_j$ to the bi-partite system with the aim of beating the second law. Each partition is composed of a heat reservoir and possibly one part of a working substance. The whole system evolves with unitaries interrupted by projections.[]{data-label="fig:1"}](Fig1.pdf){width="\linewidth"}
Theory
======
Following [@Campisi16JPA49a] we model a generic heat transport/heat engine scenario as a driven multi-partite system starting in the factorised state, see Figure \[fig:1\] $$\begin{aligned}
\rho_0 = \bigotimes_{l} \frac{e^{-\beta_l H_l}}{Z_l} \label{eq:eq-initial1}\end{aligned}$$ where $H_i$ is the Hamiltonian of each partition including a heat bath and possibly a portion of the working substance, and $Z_i$ is the corresponding partition function [@Campisi16JPA49a]. Let the total Hamiltonian be $$\begin{aligned}
H(t) = \sum_l H_l+ V(t)\end{aligned}$$ where $V(t)$ is an interaction term that is switched on for the time interval $t \in [0,\tau]$ over which the system is monitored. We assume that at times $t_1<t_2< \dots t_K$ some observable $A$ is measured thus causing the wave function describing the compound to collapse onto the subspace spanned by the eigenvectors belonging to the measured eigenvalue $a_j$. Following [@Morikuni11JSP143] we shall assume that there can be a measurement error where the eigenvalue $a_{k}$ is recorded instead of the actual eigenvalue $a_j$. This is assumed to happen with probability $\varepsilon [k|j]$. The choice of the interaction $V(t)$ in the interval $(t_{i},t_{i+1})$ is dictated by the sequence of recorded eigenvalues, or more simply the recorded sequence $\{k_1,k_2,... k_{i}\}=\mathbf{k}_i$, that is for $t_i<t<t_{i+1}$ $V(t)=V_{\mathbf{k}_i}(t)$. The corresponding unitary operator describing the evolution in the time span $t_i<t<t_{i+1}$ is $U_{\mathbf{k}_i}= \overleftarrow{\exp}\left[\int_{t_i}^{t_{i+1}}ds\, H_{\mathbf{k}_i}(s) \right]$ where $\overleftarrow{\exp}$ denotes time ordered exponential and $H_{\mathbf{k}_i}(t)= \sum_l H_l+ V_{\mathbf{k}_i(t)}$. We shall denote the un-conditioned evolution operator from time $t=0$ to the time of the first measurement $t= t_1$ as $U_0$. Note that the sequence of recorded labels $\mathbf{k}_j$ generally differs from the sequence of labels $\{j_1,j_2,... j_{i}\}=\mathbf{j}_i$ specifying in which subspace the system state was actually projected at the measurement times $t_1, t_2, \dots t_i$. As customary in the context of the fluctuation theorem we shall assume that besides the intermediate measurements of $A$, all $H_l$’s are measured at times $t=0,t=\tau$ giving the eigenvalues $E_n^l$, $E_m^l$ respectively.
The quantity of primary interest is the probability $p(m,\mathbf{k},\mathbf{j},n)$ that $n$ is obtained in the first energy measurement, the sequence $\mathbf{j}$ is realised, the sequence $\mathbf{k}$ is recorded and $m$ is obtained in the final energy measurement. Here we have introduced the simplified notations $\mathbf{j}=\mathbf{j}_K$, $\mathbf{k}=\mathbf{k}_K$. The explicit expression of $p(m,\mathbf{k},\mathbf{j},n)$ is: $$\begin{aligned}
\label{eq:pnjkm}
p(m,\mathbf{k},\mathbf{j},n) &=
\Tr\, P_m A_{\mathbf{k},\mathbf{j}} U_0 P_n U_0^\dagger A_{\mathbf{k},\mathbf{j}}^\dagger P_m p^0_n \\
A_{\mathbf{k},\mathbf{j}}&=\overleftarrow{\Pi}_i\left(U_{\mathbf{k}_i}\pi_{j_i}\sqrt{\varepsilon[k_i | j_i]}\right)\end{aligned}$$ where $p^0_n= \Pi_l e^{- \beta_l E_n^l}/Z_l$ denotes the probability of obtaining the eigenvalue $E_n= \sum_l E_n^l$ in the first measurement; $P_n$ denotes the corresponding projector; $\pi_j$ denotes the projector onto the subspace belonging to the eigenvalue $a_j$ of $A$; the symbol $\overleftarrow{\Pi}_i$ denotes $i$-ordered product, that is, $\overleftarrow{\Pi}_i(U_{\mathbf{k}_i}\pi_{j_i}\varepsilon[k_i | j_i])= U_{\mathbf{k}_K}\pi_{j_K}\sqrt{\varepsilon[j_K | k_K]} \cdots U_{\mathbf{k}_2}\pi_{j_2}\sqrt{\varepsilon[j_2 | k_2]} U_{\mathbf{k}_1}\pi_{j_1}\sqrt{\varepsilon[j_1| k_1]}$.
Let $\Delta E_l = E_m^l-E_n^l$ be the energy change in the partition $l$ observed in a single realisation of the feedback driven protocol. Using the cyclic property of the trace and completeness $\sum P_n=\mathbb{1}$, we obtain the following: $$\begin{aligned}
\label{eq:FT1}
\langle e^{-\sum_l \beta_l \Delta E_l} \rangle = \gamma = \sum_{\mathbf{j},\mathbf{k}} \Tr A_{\mathbf{k},\mathbf{j}}^\dagger \rho_0 A_{\mathbf{k},\mathbf{j}}\end{aligned}$$ The proof is reported in the appendix. This relation extends the result presented in Ref. [@Morikuni11JSP143] to the case of multipartite system with initial multi temperature state, and to repeated measurements.[^1] The quantity $\Tr A_{\mathbf{k},\mathbf{j}}^\dagger \rho_0 A_{\mathbf{k},\mathbf{j}}$ represents the probability that the sequences $\mathbf{j}^\dagger= \{j_K, ...j_2,j_1\}$, $\mathbf{k}^\dagger= \{k_K, ... k_2,k_1\}$, are realised under the backward evolution specified by the adjoint Kraus operators $A_{\mathbf{k},\mathbf{j}}^\dagger$. The total probability $\gamma$ does not generally add to one. The reason for that is that the $i$-th evolution $U_{\mathbf{k}_i}^\dagger$ occurs before the the $i$-th eigenvalue $j_i$ is realised in the backward map. The feedback loop is evidently not time-reversal symmetric, and such lack of reversibility breaks the fluctuation theorem $\langle e^{\sum_l \beta_l \Delta E_l} \rangle = 1$ which in fact is a manifestation of time-reversal symmetry [@Campisi11RMP83]. This is reflected by the fact that the quantum channel specified by the Kraus operators $A_{\mathbf{k},\mathbf{j}}$ is generally not *unital*.[^2] The adjoint of a non-unital quantum channel is not trace preserving. In the case of feedback control the quantum channel $\sum_{\mathbf{j},\mathbf{k}} A_{\mathbf{k},\mathbf{j}}\rho_0 A_{\mathbf{k},\mathbf{j}}^\dagger$ is generally not unital, as a consequence its adjoint is generally not trace preserving, hence we have generally $\gamma \neq 1$. Lack of unitality generally reflects lack of time-reversal symmetry. Examples are thermalisation maps, namely maps that have a thermal state (not the identity) as a fixed point. Physically these are realised by means of weak contact of a system with a thermal bath, leading to irreversible dynamics. Likewise feedback control breaks the symmetry. This observation reveals some analogy between feedback control and dissipative dynamics.
Before proceeding let us comment briefly on the origin of lack of unitality in feedback controlled systems, in order to gain insight in the issue. For simplicity let us consider the case of a single measurement $K=1$. Let us begin by noticing that the quantum channel specified by the $A_{k,j}$ is trace preserving. We have $\Tr \sum_{k,j} A_{k,j} \rho A_{k,j}^\dagger = \sum_{k,j} \varepsilon[k|j] \Tr \, U_{k}\pi_j \rho \pi_j U_k ^\dagger = \sum_{k,j} \varepsilon[k|j] \Tr\, \pi_j \rho \pi_j = \sum_{j} \Tr\, \pi_j \rho \pi_j =\Tr \, \rho$, where we have used the cyclic property of the trace, unitarity $U_{k}^\dagger U_k=\mathbb{1}$, idempotence $\pi_j \pi_j = \pi_j$, normalisation $\sum_{k} \varepsilon[k|j] = 1$, and completeness $\sum \pi_j = \mathbb{1}$. Let us now turn to unitality. We have $\sum_{k,j} A_{k,j}A_{k,j}^\dagger = \sum_{k,j} \varepsilon[k|j] U_{k}\pi_j U_k^\dagger$. If the evolution $U_k$ did not dependent on $k$, that is $U_k=\bar{U}$ was chosen regardless of the recorded value $k$ (e.g, $\bar{U}$ is pre-specified or is completely random), one could perform the sum over $k$ using $\sum_k \varepsilon[k|j]=1$ and then use $\sum \pi_j = \mathbb{1}$ to conclude the map is unital. Feedback, implying explicit dependence on $k$ of $U_k$ breaks unitality. Unitality would occur also in the case when $\varepsilon[k|j]$ does not depend on $j$, meaning the measurement outcome $k$ is completely random and has no correlation with the actual state $j$. In sum if the feedback control measurement is off, either because one decides not to use the information gathered in the measurement, or because the measurement gathers no information in the first place, unitality is recovered, and the fluctuation theorem is restored. This result is in agreement with the established fact that projective measurements without feedback control do not alter the validity of the fluctuation theorem [@Campisi10PRL105; @Campisi09PRE80; @Watanabe14PRE89]. Here we have further learned that noise, i.e. choosing the $U$’s between the measurements completely randomly, also does not affect the integral fluctuation relation.
Let us now turn to thermodynamics. Using Jensen’s inequality, Eq. (\[eq:FT1\]) implies: $$\begin{aligned}
\label{eq:2ndLaw1}
\sum_l \beta_l \langle \Delta E_l \rangle \geq- \ln{\gamma}\end{aligned}$$ In the case when the map is unital it is $\gamma=1$, and the second law of thermodynamics is recovered [@Campisi16JPA49a]. When $\gamma>1$ the condition $\sum_l \beta_l \langle \Delta E_l \rangle <0 $ is not forbidden, and the apparent violation of the second law becomes possible. This occurs with a proper “demonic” design of the feedback control. When $\gamma<1$ instead the second law is more strictly enforced by means of an “angelic” intervention.
As shown in Refs. [@Morikuni11JSP143; @Sagawa10PRL104] in the case of a single measurement (in either classical or quantum systems) the fluctuation relation can be restored if an information theoretic term, in the form of a mutual information, is added to the exponent in the exponential average. Ref. [@Horowitz10PRE82] reports the extension to the case of repeated measurements in the classical scenario. All these results are for a single-temperature initial state. In the present set-up we find as well an information theoretic correction term (see the appendix for a proof): $$\begin{aligned}
\label{eq:FT2}
\langle e^{-\sum_l \beta_l \Delta E_l- J_{\mathbf{k},\mathbf{j}}} \rangle =1\end{aligned}$$ where $J_{\mathbf{k},\mathbf{j}}$ is defined by the following set of equations: $$\begin{aligned}
J_{\mathbf{k},\mathbf{j}} &= \ln \frac{p(\mathbf{k},\mathbf{j})}{p(\mathbf{j}:\mathbf{k})p(\mathbf{k})}\\
p(\mathbf{k},\mathbf{j}) &= \sum_{n,m} p(m,\mathbf{k},\mathbf{j},n)\\
p(\mathbf{k})&= \sum_{n,,\mathbf{j},m} p(m,\mathbf{k},\mathbf{j},n)= \sum_\mathbf{j} p(\mathbf{k},\mathbf{j})\\
p(\mathbf{j}:\mathbf{k}) &= \frac{p(\mathbf{k},\mathbf{j})}{\Pi_i \varepsilon[k_i | j_i]} \label{eq:pj|k}\end{aligned}$$ The symbol $p(\mathbf{k},\mathbf{j}) $ represents the joint probability that the sequence $\mathbf{j}$ is realised and the sequence $\mathbf{k}$ is recorded, while $p(\mathbf{k})$ is the probability that $\mathbf{k}$ is recorded. The symbol $p(\mathbf{j}:\mathbf{k})$ stands for the probability that the sequence $\mathbf{j}$ is realised, conditioned on $\mathbf{k}$ being the record. More explicitely $$\begin{aligned}
p(\mathbf{j}:\mathbf{k}) &= \frac{p(\mathbf{k},\mathbf{j})}{\Pi_i \varepsilon[k_i | j_i]}
= \sum_{n,m}\Tr\, P_m B_{\mathbf{k},\mathbf{j}} U_0 P_n U_0^\dagger B_{\mathbf{k},\mathbf{j}}^\dagger P_m p^0_n \\
B_{\mathbf{k},\mathbf{j}}&=\overleftarrow{\Pi}_i\left(U_{\mathbf{k}_i}\pi_{j_i}\right) = \frac{A_{\mathbf{k},\mathbf{j}}}{\Pi_i \sqrt{\varepsilon[k_i | j_i]}}\, . \label{eq:Bkj}\end{aligned}$$ The operators $B_{\mathbf{k},\mathbf{j}}$ differ from the operators $A_{\mathbf{k},\mathbf{j}}$ by the term containing the conditional probability $\varepsilon[k_i | j_i]$. Note that the Bayes rule does not apply here, i.e. generally it is $p(\mathbf{k},\mathbf{j}) \neq p(\mathbf{j}:\mathbf{k})p(\mathbf{k})$. The reason is that $\mathbf{j}$ and $\mathbf{k}$ are concatenated with each other. An outcome $j_i$ influences the record $k_i$, which in turn influences the next outcome $j_{i+1}$ and so on. The quantity $J_{\mathbf{k},\mathbf{j}}$ measures the degree of such mutual influence, or correlation between the two sequences $\mathbf{j}$ and $\mathbf{k}$ [^3]. In absence of feedback, namely when there is no correlation between the two sequences, $J_{\mathbf{k},\mathbf{j}}$ is null and the standard relation is recovered. Note that given a feedback rule, generally $\langle J_{\mathbf{k},\mathbf{j}} \rangle $ would grow with the length $K$ of the sequences, i.e. the number of measurements. It is accordingly expected that $\langle J_{\mathbf{k},\mathbf{j}} \rangle \propto K$ in the large $K$ regime.
With Jensen’s inequality Eq. (\[eq:FT2\]) implies $$\begin{aligned}
\label{eq:2ndLaw2}
\sum_l \beta_l \langle \Delta E_l \rangle \geq - \langle J_{\mathbf{k},\mathbf{j}} \rangle\end{aligned}$$ We thus have found two bounds to $\sum_l \beta_l \langle \Delta E_l \rangle$.
By looking directly at the $\sum_l \beta_l \langle \Delta E_l \rangle$ as in Ref. [@Campisi16JPA49a] we have found a third bound whose interpretation is most direct and straightforward. Let $$\begin{aligned}
\rho_\tau &= \sum_{n,\mathbf{j},\mathbf{k},m} P_m A_{\mathbf{k},\mathbf{j}} U_0 P_n \rho_0 P_n U_0^\dagger A_{\mathbf{k},\mathbf{j}}^\dagger P_m = \sum_{\mathbf{j},\mathbf{k}} A_{\mathbf{k},\mathbf{j}} U_0 \rho_0 U_0^\dagger A_{\mathbf{k},\mathbf{j}}^\dagger\end{aligned}$$ be the system density matrix at time $\tau$. In the second equality we have used completeness $\sum_m P_m=\mathbb{1}$ and the fact that the initial state has no coherences in the energy eigenbasis $\sum_n P_n \rho_0 P_n = \rho_0$. Simple manipulations, similar to those employed in Ref. [@Campisi16JPA49a] lead to the following salient result $$\begin{aligned}
\sum_l \beta_l \langle \Delta E_l \rangle = \sum_i D[\rho^l_\tau || \rho_0^l] + I[\rho_\tau] + \Delta \mathcal H\end{aligned}$$ where $$\begin{aligned}
D[\rho^l_\tau || \rho_0^l] &= \Tr \rho^l_\tau \ln \rho^l_\tau - \Tr \rho^l_\tau \ln \rho^l_0 \label{eq:D}\\
I[\rho_\tau] &= -\sum_l \Tr \rho_\tau^l \ln \rho_\tau^l + \Tr \rho_\tau \ln \rho_\tau \label{eq:I}\\
\Delta \mathcal H &= -\Tr \rho_\tau \ln \rho_\tau + \Tr \rho_0 \ln \rho_0 \label{eq:DH}\end{aligned}$$ denote the Kullback Leibler divergence between the final state $\rho_\tau$ and the initial state $\rho_0$, Eq. (\[eq:D\]); the total amount of correlations (mutual information) that builds up among the partitions as a consequence of their interaction during the time span $[0,\tau]$, Eq. (\[eq:I\]); and the total change in von-Neumann entropy of the whole compound, Eq. (\[eq:DH\]). Here $\rho_t^l= \Tr'_l \rho_t$ is the reduced state of partition $l$ at time $t$ ($\Tr'_l$ denotes trace over all partitions but the $l$-th). The mutual information $I$ among the partitions of the system (measuring all correlations, quantal and classical), which develops generally due to their interaction $V(t)$ (and can also occur in absence of measurements and feedback [@Campisi16JPA49a]), should not be confused with the classical mutual information $J_{\mathbf{k},\mathbf{j}}$ between the realisation sequence $\mathbf{j}$ and the record sequence $\mathbf{k}$ caused by the feedback mechanism.
Both the Kullback Leibler divergence $D[\rho^i_\tau || \rho_0^i]$ and the mutual information $I[\rho_t]$ are non negative quantities. We thus arrive at the central inequality: $$\begin{aligned}
\sum_l \beta_l \langle \Delta E_l \rangle \geq \Delta \mathcal H
\label{eq:2ndLaw3}\end{aligned}$$ In the standard no measurement case, $\rho_\tau$ is linked to $\rho_0$ via a unitary map, hence $\Delta \mathcal H =0$ and one recovers the result of Ref. [@Campisi16JPA49a], namely $\sum_i \beta_i \langle \Delta E_i \rangle = \sum_i D[\rho^i_\tau || \rho_0^i] + I[\rho_\tau]$, and the second law in its standard form. Note that when there are measurements, but no feedback, the $\rho_\tau$ is linked to $\rho_0$ via a *unital* map, implying $\gamma = 0$, $\langle J_{\mathbf{k},\mathbf{j}} \rangle= 0$, and $\Delta \mathcal H \geq 0$ hence $\sum_l \beta_l \langle \Delta E_l \rangle \geq \Delta \mathcal H \geq0$, meaning that, as is already known [@Campisi10PRL105; @Campisi09PRE80; @Watanabe14PRE89] the second law is not altered by the mere application of projective measurements that interrupt an otherwise unitary dynamics. However Eq. (\[eq:2ndLaw3\]) clearly indicates that there is a dissipation term associated with quantum-mechanical measurements, which is not present in the classical case. In sum through Eq. (\[eq:2ndLaw3\]) we see that there is a thermodynamic cost associated to quantum measurements.
Combining Eqs. (\[eq:2ndLaw1\],\[eq:2ndLaw2\],\[eq:2ndLaw3\]) the second law of thermodynamics, in presence of feedback control takes the form $$\begin{aligned}
\sum_l \beta_l \langle \Delta E_l \rangle \geq \max[-\ln \gamma,-\langle J_{\mathbf{k},\mathbf{j}} \rangle ,\Delta \mathcal H ]
\label{eq:2ndLaw4}\end{aligned}$$
Illustrative example
====================
To exemplify the theory above we consider a prototypical model of quantum heat engine whose working substance is made of two qubits [@Campisi15NJP17; @Campisi16JPA49a; @Quan07PRE76]. Their Hamiltonian reads $$\begin{aligned}
H= H_1 + H_2 = \frac{\hbar \omega}{2}\sigma_z^1 + \frac{\hbar \omega}{2}\sigma_z^2\end{aligned}$$ where $\sigma_z^i$ denote Pauli operators. We assume the two qubits have same level spacing $\hbar\omega$ and are initially in the state: $$\begin{aligned}
\rho_0 = \frac{e^{-\beta_1 H_1}}{Z_1} \otimes \frac{e^{-\beta_2 H_2}}{Z_2}\end{aligned}$$ with $Z_i$ their partition functions. At $t=0$ the $\sigma_z^i$’s are measured collapsing the two qubits in the state $|k\rangle = |k'\rangle |k''\rangle$, with $k',k''= \pm, \pm$. We assume classical error in the measurement of each qubit $\epsilon[+|+]=\epsilon[-|-]=q$, $\epsilon[-|+]=\epsilon[+|-]=1-q$ for some $q \in [0,1]$. Accordingly the eigenvalues $j=j',j''$ are recorded with probability $\varepsilon[k|j]= \epsilon[k'|j']\epsilon[k''|j'']$. If the states $|+,+\rangle , |-,+\rangle, |-,-\rangle$, are recorded we do nothing: $U_{+,+}= U_{-,+}= U_{-,-}=\mathbb{1}$; else, i.e., if $k=|-,+\rangle $ we apply a swap operation, $U_{-,+}=U_{SWAP}$, that maps $|-,+\rangle$ into $|+,-\rangle$. The system is now in a joint eigenstate $|m\rangle= U_k |j\rangle$ of the two qubits Hamiltonian $H$, hence the final measurement of the $\sigma_z^i$ is irrelevant. At the end of the process each qubit is allowed to relax to thermal equilibrium with their respective thermal baths of inverse temperatures $\beta_i$ so as to re-establish the initial state $\rho_0$. Accordingly the average energies $\langle \Delta E_i \rangle $ acquired by each qubit during the process equals the average heats that they release in the baths in the thermal relaxation step. Due to the feedback mechanism energy may be withdrawn from the cold bath and released in the hot one. Note that, due to the fact that the two qubits have same level spacing the SWAP operation does not alter their total energy. Namely there is no energy injection by the Demon: to steer the energy flow he only uses information. The set-up is illustrated in Fig. \[fig:2\] panel a).
![Panel a). Scheme of a two-qubit feedback controlled refrigerator. Two qubits are prepared each in thermal equilibrium with a thermal bath. When the Demon sees the cold qubit in the excited state and the hot qubit in ground state, he swaps them. He then lets them thermalise each with its own bath and starts over. He thus transfers heat from the cold bath to the hot bath without investing energy. Panels b,c). $\sum \beta_l \langle \Delta E_l \rangle$, $-\ln \gamma$, $-\langle J_{k,j} \rangle$, $\Delta \mathcal H$ as a function of the error probability $q$ for $\beta_1=\hbar\omega$, and two different values of $\beta_2$.[]{data-label="fig:2"}](Fig2.pdf){width="\linewidth"}
The relevant probability chain is a bit simpler than in the general case because the first energy measurement is itself here also the first feedback measurement. It reads $
p(m,k,j) = \Tr P_m U_k P_j U_k^\dagger P_m p_j^0 \varepsilon[k|j] = \Tr P_m A_{k,j} A_{k,j}^\dagger P_m p_j^0$ with $
A_{k,j} = \sqrt{\varepsilon[k|j]} U_k P_j $. For $\gamma$ we have $\gamma= \sum_{j,k} {\varepsilon[k|j]} \Tr P_j U_k^\dagger \rho_0 U_k P_j $. The final state is $\rho_f= \sum_{j,k} {\varepsilon[k|j]} U_k P_j \rho_0 P_j U_k^\dagger$. The probability $p(j:k)$ that the outcome $j$ is realised conditioned on $k$ being recorded is simply the marginal probability $p(j)$ that $j$ is realised because the record $k$ comes chronologically after the realisation of $j$ and hence cannot have any influence on it. The quantity $J_{k,j}$ boils down then to the logarithm of the ratio $p(j,k)/p(j)p(k)$ [@Morikuni11JSP143] hence its expectation is the non-negative mutual information between $j$ and $k$: $\langle J_{k,j} \rangle=\sum_{j,k} p(j,k) [\ln p(j,k)/p(j)p(k)]$.
Panels b,c) of Fig. \[fig:2\] show $\sum \beta_l \langle \Delta E_l \rangle, -\ln \gamma, -\langle J_{k,j}\rangle , \Delta \mathcal H$ for two choices of $\beta_2$ and same $\beta_1$, as a function of the error probability $q$. In accordance with Eq. (\[eq:2ndLaw4\]) we see that $\sum \beta_l \langle \Delta E_l \rangle$ is bounded from below by $-\ln \gamma, -\langle J_{k,j}\rangle$ and $\Delta \mathcal H$. Independent of all other parameters the refrigerator cannot work in the region $q<1/2$ where $j$ and $k$ are anti-correlated, while it may only work if $q>1/2$. This is captured by $-\ln \gamma$ being positive in the region $0<q<1/2$ and negative for $1/2<q<1$. At $q=1/2$ outcome and recording are fully uncorrelated, which restores unitality as discussed above and implies $\ln \gamma=0$. Regarding $\Delta \mathcal H$, while it tends to be closer to $\sum \beta_l \langle \Delta E_l \rangle$ in the operation region ($q>1/2$), it greatly departs from it in the non-operation region, where it can even get negative values. Notably in both panels there is a value of $q$ for which the bound is saturated by $\Delta \mathcal H$. Regarding $-\langle J_{k,j}\rangle$ we note it is everywhere non-positive as expected. Furthermore it is symmetric with respect to $q\rightarrow 1-q$. This reflects the fact that the mutual information does not distinguish between correlation and anti-correlation. The maximum $-\langle J_{k,j}\rangle =0$ is attained at $q=1/2$ where $j,k$ are uncorrelated, and the standard fluctuation relation is recovered (i.e., $\gamma=1$). In both panels we see that $\Delta \mathcal H > -\langle J_{k,j}\rangle$. Whether this a generic bound is yet to be understood. We note that while at $q=1/2$ both $-\ln \gamma$ and $-\langle J_{k,j}\rangle$ are null, $\Delta \mathcal H$ is non-negative, reflecting the fact that in absence of feedback there is nonetheless an entropic cost associated to measurements, as discussed above. Such cost can be counterbalanced in presence of feedback (note that $\Delta H$ may be negative for $q \neq 1/2$). Confronting now the two panels, we see that the higher the thermal gradient $\beta_2-\beta_1$, the larger is the point $q$ where the engine starts operating, i.e. where $\sum \beta_l \langle \Delta E_l \rangle$ turns from positive into negative: As intuition suggests the more the gradient the better must your measurement be. This feature is captured also by $\Delta \mathcal H$ but not by $-\ln \gamma, -\langle J_{k,j}\rangle$. Also the smaller the gradient the more the shape of the function $\sum \beta_l \langle \Delta E_l \rangle$ resembles that of $-\ln \gamma$, with the shift between the two being approximately the value of $\Delta \mathcal H$ at $q=1/2$: that is $\sum \beta_l \langle \Delta E_l \rangle \simeq -\ln \gamma + \Delta \mathcal H|_{q=1/2}$.
Experimental proposal
=====================
![Set up of the proposed experiment. A superconducting qubit (black rectangle) embodies a Maxwell demon trap door, and two resistors embedded in RLC circuits embody the two chambers of different temperatures. Qubit and RLC circuits are inductively coupled. Calorimetric monitoring of photons entering and exiting each resistor is applied, allowing to both measure heat exchanged by each resistor, and monitoring the state of the qubit at any time. When the qubit is up a feedback algorithm drives the resonance frequency of the cold RLC circuit out of tune with the qubit frequency, while keeping the hot RLC in tune with it (and vice versa) so that an overall heat current flows from cold to hot. The resonance frequencies of the RLC circuits are controlled by tuning their non-linear inductive elements, i.e., SQUIDs, via application of external magnetic flux $\Phi_i$. []{data-label="fig:3"}](Fig3.pdf){width="\linewidth"}
The general theory developed above allows for a joint information theoretic and thermodynamic analysis of feedback controlled dynamics in the broad scenario where a demon can influence not only the amount of work being provided by the outside as in previous works [@Toyabe10NP6; @Koski14PRL113; @Koski14PNAS111], but also the heat flow between the various parts of a compound system, e.g. the heat flow between various heat baths.
The progress of solid state technology on the other hand allows to realise such feedback controlled heat transport mechanisms in real devices. The example illustrated above can be experimentally realised by introducing a feedback mechanism in the two-superconducting qubits scheme illustrated in in Ref. [@Campisi15NJP17]. Below we illustrate a design that is of more immediate realisation. It is a based on a single qubit and it does not involve any qubit-operation, but only manipulations of qubit-bath couplings. The proposal that we put forward here is based on two ingredients that enable unique capabilities allowing for the implementation of a Maxwell demon based on a most simple concept. The two ingredients are a two-level-system acting as quantum trap door and the calorimetric measurement scheme developed in Refs. [@Pekola10PRL105; @Gasparinetti15PRAPP3].
The one qubit set-up is illustrated in Fig. \[fig:3\]. The two-level system is embodied by a superconducting qubit of level spacing $\hbar \omega$. The two chambers are embodied by two resistors being kept at different temperatures. Qubit and resistors con exchange energy (i.e. heat) in the form of photons of energy $\hbar \omega$ associated to the TLS absorbing/emitting one photon from/to one of the two baths. The resistors are embedded into an RLC loop of tunable resonance frequency. This results into a tuneable TLS/resistor coupling. When an RLC circuit is far detuned from $\omega$, the qubit is effectively decoupled from the resistor, while maximal coupling occurs when it is in tune with the qubit. The resonance frequency can be tuned by using a SQUID as a non-linear and tuneable inductor, its inductance being governed by a controllable threading magnetic flux.
When a photon enters/exits one of the two resistors, its electronic temperature undergoes a positive/negative jump followed by a fast decay. Two calorimeters [@Pekola10PRL105; @Gasparinetti15PRAPP3] continuously monitor the two resistors, and count how many photons enter/exit them. This allows for a directional full counting statistics of heat. Most remarkably it also allows to infer the state of the TLS at each time. If an absorption (in either resistor) is observed, it means the TLS jumped down, hence it was up before the absorption was detected, and is down afterwards. This allows to experimentally access the quantum state trajectory of the TLS.
The feedback concept is extremely simple: as soon as a jump-down is observed, turn on the interaction with the cold resistor and turn off the interaction with the hot resistor. Vice-versa for the observation of a jump up. This results in a net flow of heat from the cold resistor to the hot one. Based on the above general analysis the apparent violation of the second law is understood in terms of lack of time-reversal symmetry of feedback control, leading to an overall non-unital dynamics of resistors plus TLS. In a practical realisation one is realistically not able to fully turn off the interactions. Furthermore there will be some delay time $\delta$ between measurement being performed and feedback being realised, giving rise effectively to possible error $\varepsilon[k_i|j_i]$ between measured state $k_i$ and actual state $j_i$ of the qubit.
Modelling
=========
In the following we model the dynamics of the proposed experiment. We model the evolution of the two level system via a standard Lindblad master equation $$\dot \rho = -i [H_S,\rho] + \mathcal L_L \rho + \mathcal L_R \rho$$ where $H_S= -E_0 (\Delta \sigma_x + q \sigma_z)=$ is the two level system Hamiltonian expressed in terms of the Pauli matrices $\sigma_\alpha$, and $\mathcal{L}_l$ are Lindblad operators $$\begin{aligned}
\mathcal L_l \rho &= \Gamma_l^\downarrow D[\sigma] \rho + \Gamma_l^\uparrow D[\sigma^\dagger]\rho\end{aligned}$$ expressed in terms of and the super-operator $D[O]\rho = O\rho O^\dagger -\frac{1}{2} O^\dagger O \rho- \frac{1}{2}\rho O^\dagger O $ and the rising and lowering spin operators $\sigma^\dagger,\sigma$ of the Hamiltonian $H_S$, defined via $\sigma^\dagger |-\rangle = |+\rangle, \sigma^\dagger |+\rangle = 0, \sigma |+\rangle= |-\rangle, \sigma |-\rangle=0$, where $|- (+)\rangle$ is the ground (excited) state of $H_S$. Here $l=L,R$ denote either the left or the right reservoir. The rates $\Gamma_l^{\downarrow\uparrow}$ for jump down/up in the $l$’th resistor are given by $$\begin{aligned}
\label{eq:rates}
\Gamma_l^{\downarrow\uparrow} &= \frac{E_0^2 M_l^2}{\hbar^2 \Phi_0} \frac{\Delta^2}{(q^2+\Delta^2)} S_{I,l}(\pm \omega)\end{aligned}$$ where $
S_{I,l}(\omega) = S_{V,l}(\omega) [R_l^2(1+Q_l^2[\omega/\omega_{LC,l}-\omega_{LC,l}/\omega ]^2)]^{-1}
$ is the current noise spectrum expressed in terms of the voltage noise spectrum $
S_{V,l}(\omega) =2 R_l \hbar \omega (1-e^{-\beta_l \hbar \omega})^{-1}
$, $Q_l=\sqrt{L_l / C_l}/R_l $ is the quality factor and $\omega_{LC,l}=1/\sqrt{L_l C_l}$ the resonance frequency of resonator $l$, expressed in terms of its resistance, inductance and capacitance $R_l,L_l,C_l$. By increasing $L_j$ the rates $\Gamma_l^{\downarrow\uparrow} $, can be quenched, namely the interaction between the TLS and the $l$-th resistor can be turned off. The symbol $M_l$ stands for the mutual inductance between the qubit and the $l$-th resistor and $\Phi_0$ is the flux quantum. Note that the rates are detailed balanced: $$\begin{aligned}
\Gamma_l^{\downarrow} = e^{\beta_l \hbar \omega} \Gamma_l^{\uparrow}\end{aligned}$$ The study of heat and work fluctuations requires the study of the dynamics to be performed at the level of single quantum-jump trajectories [@Campisi15NJP17; @Hekking13PRL111], resulting from the unravelling of the master equation. This is here achieved by means of the Monte Carlo wave function (MCWF) method [@Molmer93JOSAB; @BreuerPetruccioneBOOK]. In the specific case under study of a two level system subject to dissipation terms leading to full wave function collapse in either state $|-\rangle$ or $|+\rangle$, this results in a classical dichotomous Poisson process with rates $\Gamma_l^{\downarrow\uparrow}$ [@Campisi15NJP17].
The basis of our numerical experiment is the generation of such dichotomous Poisson random trajectories. We chose the right reservoir as the cold one and the left as the hot one. The TLS is assumed to be initially in equilibrium with the left bath. We produce a large sample of trajectories and build the normalised historgram $h(N_R)$ of the number $N_R$ of photons entering the right reservoir. Since the heat $Q_R$ entering the right reservoir is given as $Q_R=\hbar N_R \omega$, the statistics $h(N_R)$ is the heat statistics. In absence of feedback it satisfies the fluctuation relation $$\begin{aligned}
\label{eq:heatFT}
\frac{h(N_R)}{h(-N_R)}= e^{-\Delta \beta \hbar\omega N_R}, \qquad \text{no feedback}\end{aligned}$$ The feedback is introduced as follows. At each moment in time we distinguish between the actual state of the system $j=\pm$ and the knowledge $k=\pm$ we have about it. The latter does not necessarily coincide with the former because we allow for some delay-time $\delta$ between a jump occurring in the TLS and our knowledge of the state of the qubit being updated accordingly. The delay time thus effectively introduces an error probability $\varepsilon[\pm|\pm]$ between the actual state and the knowledge about the state, at each time. At each time, conditioned on the knowledge $k$ of the state we use either one set of rates favouring the interaction with either the cold or hot bath. More explicitly, let $\Gamma^{\downarrow\uparrow|\pm}_{l}$ be the rate for jump down (up) in $l$-th bath conditioned on TLS being measured to be in state $\pm$. In accordance with Eq. (\[eq:rates\]) we use the following rates $$\begin{aligned}
\Gamma^{\downarrow|+}_{L}= \frac{A}{1-e^{-\beta_L \hbar \omega}} \qquad \Gamma^{\uparrow|+}_{L}= \Gamma^{\downarrow|+}_{L} e^{-\beta_L \hbar \omega} \\
\Gamma^{\downarrow|+}_{R}= \frac{B}{1-e^{-\beta_R \hbar \omega}} \qquad \Gamma^{\uparrow|+}_{R}= \Gamma^{\downarrow|+}_{R}e^{-\beta_R \hbar \omega}\\
\Gamma^{\downarrow|-}_{L}= \frac{B}{1-e^{-\beta_L \hbar \omega}} \qquad \Gamma^{\uparrow|-}_{L}= \Gamma^{\downarrow|-}_{L} e^{-\beta_L \hbar \omega} \\
\Gamma^{\downarrow|-}_{R}= \frac{A}{1-e^{-\beta_R \hbar \omega}} \qquad \Gamma^{\uparrow|-}_{R}= \Gamma^{\downarrow|-}_{R}e^{-\beta_R \hbar \omega}\end{aligned}$$ where $A,B$ are determined by the circuitry parameters, and can be tuned via external fluxes $\Phi_i$. With $B<A$, this means that energy exchange with the right (cold) bath is larger when the TLS is believed to be down, so that it becomes more likely that energy flows out of the cold reservoir. Similarly energy exchange with the left (hot) bath is larger when the TLS is believed to be up, so that it becomes more likely that energy flows in the hot reservoir. Overall this results in an effect that contrasts the natural flow from hot to cold. The largest effect can be achieved when turning off the unwanted interaction completely, namely when $B=0$. Having in mind a realistic set-up here we keep the ratio $A/B$ finite, meaning partial turning-off is considered.
![Left: typical histogram $h(N_R)$. Right: $\ln h(N_R)/h(-N_R)$ as a function of $N_R$. Straight dashed line is $\Delta \beta \hbar \omega N_R$. Straight solid line is $-\Delta \beta_\text{eff} \hbar \omega N_R$. Here $k_B T_L = 1.1,k_B T_R=1$. These thermal energies are expressed in units of $-\hbar \omega$. $\omega$ also fixes the time unit. Delay time is $\delta=0.5\times 10^3$ in those time units. It is $A=1 \times10^{-3}$, $B=0.5 \times 10^{-3}$ corresponding to the largest rate timescale $\bar t =1.2642 \times 10^3$. The simulation time is $20 \bar t$. The statistics is built on a sample of $5 \times 10^6$ trajectories. []{data-label="fig:4"}](Fig4.pdf){width="\linewidth"}
Because of the feedback the fluctuation relation (\[eq:heatFT\]) is not obeyed. However it can be proved (see appendix) that, due to the feedback mechanism, the TLS feels the effective temperature gradient $$\begin{aligned}
\label{eq:DeltaTeffective}
\Delta \beta^\text{eff}=\beta^\text{eff}_L- \beta^\text{eff}_R = \Delta \beta + \frac{2}{\hbar\omega}\ln \frac{\varepsilon[+|+] A+ {\varepsilon[-|+] B}}{{\varepsilon[+|-] A+\varepsilon[-|-]B}} \end{aligned}$$ we thus see that by tuning the ratio $A/B$ the effective temperature gradient can be manipulated and if the errors associated to the measurement is not too big, it can even be inverted as compared to the original thermal gradient $\Delta \beta$. So the overall effect of the demon is to change the “temperatures felt” by the TLS. Accordingly the following fluctuation relation $$\begin{aligned}
\label{eq:heatFTeffective}
\frac{h(N_R)}{h(-N_R)}= e^{-\Delta \beta_\text{eff} \hbar\omega N_R}\end{aligned}$$ is obeyed by the histogram $h(N_R)$. This immediately allows to interpret the quantity $$\begin{aligned}
J_\text{exp}=-\frac{2 Q_R}{\hbar\omega}\ln \frac{\varepsilon[+|+] A+ {\varepsilon[-|+] B}}{{\varepsilon[+|-] A+\varepsilon[-|-]B}} \end{aligned}$$ via Eq. (\[eq:FT2\]) as the mutual information encoded in a trajectory along which a heat $Q_R$ is exchanged with the $R$ bath. Note that when $A=B$, the feedback has no effect and accordingly $J_\text{exp}=0$. Likewise if $\varepsilon[+|+]=\varepsilon[+|-]$ (hence $\varepsilon[-|+]=\varepsilon[-|-]$) meaning no correlation between state and knowledge thereof, feedback control does not work and again $J_\text{exp}=0$. Most importantly the experimental mutual information $J_\text{exp}$ is proportional to the heat exchanged. This allows for accessing a fluctuating information theoretic quantity by means of a thermodynamic measurements in a realistic experimental scenario.
Figure \[fig:4\] shows typical histograms $h(N_R)$ for realistic parameters. We also plotted the quantity $\ln h(N_R)/h(-N_R)$ finding a good agreement with the theoretical prediction $-\Delta \beta_\text{eff} \hbar\omega N_R$. The effective conditional probabilities $\varepsilon[k|j]$ were obtained by recording for each trajectory the total time when state was $j$ and knowledge was $k$, and averaging their value over the whole ensemble of trajectories. The observed deviation is a consequence of the fact that error here is not introduced in the form of an outcome being missed (as assumed in deriving Eq. (\[eq:heatFTeffective\])), but rather being reported with some delay. With the histogram $h(N_R)$ we computed $\sum \beta_l \langle \Delta E_l\rangle = \hbar \omega \Delta \beta \langle N_R \rangle = -0.0862$, $-\ln \gamma = -\ln \langle e^{\Delta \beta \hbar \omega N_R } \rangle=-0.1205$, $-\langle J_\text{exp}\rangle= -0.2873 $, for the chosen parameters. The computed values are in agreement with the prediction of Eq. (\[eq:2ndLaw4\]). The proposed experiment does not allow to measure $\Delta \mathcal H$, which would require accessing the full system+baths density matrix.
Energy spent by the Demon
-------------------------
What is the energy cost incurred by the demon to open/close the trap-door? To roughly estimate that we model the LCR circuit as a classical harmonic oscillator (LC circuit) in contact with a heat bath (the resistor) at temperature $T$. To open/close the door towards one of the two reservoirs, the demon switches the LC frequency from $\omega_i$ to another frequency $\omega_f$ so as to put it in/off resonance with the qubit. If the operation is carried in a quasi static manner, the work done is equal to the free energy change: $W= k_B T \ln (\omega_f/\omega_i)$. The operation would in this case be reversible, and the work lost when opening the door will be retrieved when opening it. The overall cost of a open/close cycle would be null in this limiting case. The other limiting case is when the switch is infinitely fast. The overall cost of a single open/close cycle in this case would be non-negative in accordance with the second law of thermodynamics, and amounts to $W= k_B T (\omega_f/\omega_i- \omega_i/\omega_f)^2/2$. The overall work incurred in a repeated feedback operation is proportional to the number of open/close cycles, which in turn is proportional to the net number of energy quanta being transported, namely the total heat transported. Interestingly we note that the faster the open/close operation, the more effective is the feedback mechanism, the more energy needs to be invested.
Conclusions
===========
We have developed a general quantum theory of repeated feedback control in a multiple heat reservoir scenario. The main effect of feedback control is that it induces a generally non-unital dynamics of the full reservoirs+system compound. As a consequence the standard bound set by the second law od thermodynamics on the dissipation quantifier $\sum_l \beta_l \langle \Delta E_l \rangle$ is shifted and may become negative. We have illustrated an experimental proposal where a single superconducting qubit plays the role of a trap-door that is subject to feedback control. The envisaged method for simultaneously measuring the qubit state and the heat exchanged by each reservoir is single photon calorimetry.
Acknowledgements {#acknowledgements .unnumbered}
================
This research was supported by a Marie Curie Intra European Fellowship within the 7th European Community Framework Programme through the project NeQuFlux grant n. 623085 (M.C.), by Unicredit Bank (M.C.), by the Academy of Finland contract no. 272218 (J.P.), and by the COST action MP1209 “Thermodynamics in the quantum regime”.
Derivation of Eq. (\[eq:FT1\])
==============================
$$\begin{aligned}
\langle e^{-\sum_l \beta_l \Delta E_l} \rangle &= \sum_{n,\mathbf{j},\mathbf{k},m} p(m,\mathbf{k},\mathbf{j},n) e^{-\sum \beta_l E_m^l} e^{\sum \beta_l E_n^l} \nonumber \\
&= \sum_{n,\mathbf{j},\mathbf{k},m} \Tr\, P_m A_{\mathbf{k},\mathbf{j}} U_0 P_n U_0^\dagger A_{\mathbf{k},\mathbf{j}}^\dagger P_m \frac{ e^{-\sum \beta_l E_n^l}}{Z} e^{-\sum \beta_l E_m^l} e^{\sum \beta_l E_n^l} \nonumber \\
&= \sum_{\mathbf{j},\mathbf{k},m} \Tr\, P_m A_{\mathbf{k},\mathbf{j}} A_{\mathbf{k},\mathbf{j}}^\dagger P_m \frac{ e^{-\sum \beta_l E_m^l}}{Z} \nonumber \\
&= \sum_{\mathbf{j},\mathbf{k}} \Tr\, A_{\mathbf{k},\mathbf{j}}^\dagger \rho_0 A_{\mathbf{k},\mathbf{j}} \nonumber
\end{aligned}$$
Eq. (\[eq:pnjkm\]) and $\rho_0= \sum _n P_n e^{-\sum \beta_l E_n^l}/Z$ have been used to obtain the second line. Completeness $\sum_n P_n= \mathbb{1}$ and unitarity $U_0 U_0^\dagger = \mathbb{1}$ led to the third line. Fourth line follows from the cyclical property of the trace, idempotence $P_mP_m= P_m$ and $\rho_0= \sum _m P_m { e^{-\sum \beta_l E_m^l}}/{Z}$.
Derivation of Eq. (\[eq:FT2\])
==============================
Using Eq. (\[eq:pj|k\]), the exponentiated fluctuating mutual information can be conveniently expressed as $$\begin{aligned}
e^{-J_{\mathbf{k},\mathbf{j}}} = \frac{p(\mathbf{j}:\mathbf{k})p(\mathbf{k})}{p(\mathbf{k},\mathbf{j})}
= \frac{p(\mathbf{k})}{\Pi_i \varepsilon[k_i | j_i]}\end{aligned}$$ hence $$\begin{aligned}
\langle e^{-\sum_l \beta_l \Delta E_l- J_{\mathbf{k},\mathbf{j}}} \rangle &= \sum_{n,\mathbf{j},\mathbf{k},m} p(m,\mathbf{k},\mathbf{j},n) e^{-\sum \beta_l E_m^l} e^{\sum \beta_l E_n^l} \frac{p(\mathbf{k})}{\Pi_i \varepsilon[k_i | j_i]}\\
&= \sum_{n,\mathbf{j},\mathbf{k},m} \Tr\, P_m B_{\mathbf{k},\mathbf{j}} U_0 P_n U_0^\dagger B_{\mathbf{k},\mathbf{j}}^\dagger P_m \frac{ e^{-\sum \beta_l E_n^l}}{\Pi_l Z_l} e^{-\sum \beta_l E_m^l} e^{\sum \beta_l E_n^l} p(\mathbf{k})\nonumber \\
&= \sum_{\mathbf{j},\mathbf{k},m} \Tr\, P_m B_{\mathbf{k},\mathbf{j}} B_{\mathbf{k},\mathbf{j}}^\dagger P_m \frac{ e^{-\sum \beta_l E_m^l}}{\Pi_l Z_l}p(\mathbf{k}) \nonumber \\
&= \sum_{\mathbf{k},m} \Tr\, P_m \frac{ e^{-\sum \beta_l E_m^l}}{\Pi_l Z_l}p(\mathbf{k}) \nonumber \\
&= \Tr \rho_0 \sum_\mathbf{k} p(\mathbf{k})\\
&= 1\end{aligned}$$ Eq. (\[eq:pnjkm\]), $\rho_0= \sum _n P_n e^{-\sum \beta_l E_n^l}/\Pi_l Z_l$ and Eq. (\[eq:Bkj\]) have been used to obtain the second line. Completeness $\sum_n P_n= \mathbb{1}$ and unitarity $U_0 U_0^\dagger = \mathbb{1}$ led to the third line. The fourth line follows from $\sum_{\mathbf{j}} B_{\mathbf{k},\mathbf{j}} B_{\mathbf{k},\mathbf{j}}^\dagger = \mathbb{1}$ which follows by expanding the $i$-ordered products, apply idempotence $\pi_{j}\pi_{j}=\pi_{j}$, completeness $\sum_{j} \pi_{j}= \mathbb{1}$, and unitarity $U_j U_j^\dagger = \mathbb{1}$. Cyclical property of the trace, idempotence $P_mP_m= P_m$ and $\rho_0= \sum _m P_m { e^{-\sum \beta_l E_m^l}}/{\Pi_l Z_l}$ lead to the fifth line. The final result is a consequence of normalisation of $\rho_0$ and of $p(\mathbf{k})$.
Derivation of Eq. (\[eq:DeltaTeffective\])
==========================================
Under the operation of the demon the TLS experiences effective temperatures of the baths that differ from their actual value. To fix ideas, let us for the moment, assume no delay time and no error in the measurement. The qubit is effectively subject to the following effective rates $\Gamma^{\downarrow,\text{eff}}_{s} = \Gamma^{\downarrow|+}_{s}, \Gamma^{\uparrow,\text{eff}}_{s} = \Gamma^{\uparrow|-}_{s}$. Accordingly, the detailed balance temperatures are shifted: $$\begin{aligned}
e^{\beta^\text{eff}_L \hbar \omega} &= \Gamma^{\downarrow,\text{eff}}_{L}/ \Gamma^{\uparrow,\text{eff}}_{L} = (A/B) e^{\beta_L\hbar \omega}\\
e^{\beta^\text{eff}_R \hbar \omega} &= \Gamma^{\downarrow,\text{eff}}_{R}/ \Gamma^{\uparrow,\text{eff}}_{R} = (B/A) e^{\beta_R\hbar \omega}\end{aligned}$$ where we used the explicit expressions Eq. (\[eq:rates\]). This implies the effective temperatures $$\begin{aligned}
\beta^\text{eff}_L &= \frac{1}{\hbar \omega}\ln (A/B) +\beta_L\\
\beta^\text{eff}_R &= \frac{1}{\hbar \omega}\ln (B/A) +\beta_R
\end{aligned}$$
Let us now introduce the errors $\varepsilon[\pm|\pm]$ related to the measurement. The stochastic process describing the dynamics of the TLS is still Poissonian with one rate occurring in case of right measurement and one rate occurring in the other case. The idea is that monitoring is continuous, or better, occurring with a sampling time interval $dt$, which we assume short compared to all rates $\Gamma_{R,L}^{\uparrow,\downarrow|\pm}$. Let us imagine the system is in state $j=+$. There is a probability $\varepsilon[+|+]$ the observation is $k=+$ and a probability $\varepsilon[-|+]$ the observation is $k=-$. Thus the probability to undergo a jump down in the $s$ reservoir in the interval $dt$ is $$\begin{aligned}
p(dt) &= \varepsilon[+|+] e^{-\Gamma^{\downarrow|+}_{s}dt} + \varepsilon[-|+] e^{-\Gamma^{\downarrow|-}_{s}dt} \\
& \simeq \varepsilon[+|+](1-\Gamma^{\downarrow|+}_{s}dt) + \varepsilon[-|+](1-\Gamma^{\downarrow|-}_{s}dt) \\
& = 1- (\varepsilon[+|+]\Gamma^{\downarrow|+}_{s}+\varepsilon[-|+]\Gamma^{\downarrow|-}_{s})dt \\
& = e^{-(\varepsilon[+|+]\Gamma^{\downarrow|+}_{s}+\varepsilon[-|+]\Gamma^{\downarrow|-}_{s})dt}\end{aligned}$$ Similarly for the jump up. Overall the TLS experience the new rates $$\begin{aligned}
\Gamma^{\downarrow}_{s,\text{eff}} &= \varepsilon[+|+]\Gamma^{\downarrow|+}_{s}+ \varepsilon[-|+]\Gamma^{\downarrow|-}_{s}
\\
\Gamma^{\uparrow}_{s,\text{eff}} &= \varepsilon[+|-] \Gamma^{\uparrow|+}_{s} +\varepsilon[-|-] \Gamma^{\uparrow|-}_{s}\end{aligned}$$ Accordingly $$\begin{aligned}
e^{\beta^\text{eff}_s\hbar \omega} &= \frac{\Gamma^{\downarrow}_{s,\text{eff}} }{\Gamma^{\uparrow}_{s,\text{eff}} } =
\frac{\varepsilon[+|+]\Gamma^{\downarrow|+}_{s}+ \varepsilon[-|+]\Gamma^{\downarrow|-}_{s}}{\varepsilon[+|-] \Gamma^{\uparrow|+}_{s} +\varepsilon[-|-] \Gamma^{\uparrow|-}_{s}}\\
\beta_s^\text{eff} &= \frac{1}{\hbar \omega} \ln \frac{\varepsilon[+|+]\Gamma^{\downarrow|+}_{s}+ \varepsilon[-|+]\Gamma^{\downarrow|-}_{s}}{\varepsilon[+|-] \Gamma^{\uparrow|+}_{s} +\varepsilon[-|-] \Gamma^{\uparrow|-}_{s}}
\end{aligned}$$ Plugging in the explicit expressions we get $$\begin{aligned}
\beta_L^\text{eff} & = \beta_L + \frac{1}{\hbar \omega} \ln
\frac{\varepsilon[+|+]A+ \varepsilon[-|+]B}{\varepsilon[+|-]A +\varepsilon[-|-] B}\\
\beta_R^\text{eff} &= \beta_R + \frac{1}{\hbar \omega} \ln \frac{\varepsilon[+|-]A +\varepsilon[-|-] B}{\varepsilon[+|+]A+ \varepsilon[-|+]B}\end{aligned}$$ Hence Eq. (\[eq:DeltaTeffective\]).
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[^1]: For simplicity we restricted to the case of cyclic $H(t)$. The extension to non-cyclic case is straightforward.
[^2]: We recall that a quantum channel specified by Kraus operators $M_i$, $\rho \rightarrow \sum_i M_i \rho M_i^\dagger$ that is trace preserving $\sum M_i^\dagger M_i = \mathbb{1}$, is unital when it maps the identity into itself $\sum M_i M_i^\dagger = \mathbb{1}$.
[^3]: Eq. (\[eq:FT2\]) is reminiscent of a similar relation reported by Vedral [@Vedral12JPA45], see Eq. (8) there. The two relations fundamentally differ in various respects. Notably in the meaning of the mutual information term. In our case measuring the correlation between outcomes and their records, in the case of Ref. [@Vedral12JPA45] measuring the correlation between the measurements themselves
|
---
abstract: 'An accurate determination of the Hubble constant remains a puzzle in observational cosmology. The possibility of a new physics has emerged with a significant tension between the current expansion rate of our Universe measured from the cosmic microwave background by the Planck satellite and from local methods. In this paper, new tight estimates on this parameter are obtained by considering two data sets from galaxy distribution observations: galaxy cluster gas mass fractions and baryon acoustic oscillation measurements. By considering the flat and non-flat $\Lambda$CDM models, we obtain, respectively: $H_0=65.89^{+1.54}_{-1.50}$ km s$^{-1}$ Mpc$^{-1}$ and $H_0=64.31^{+ 4.50}_{- 4.45}$ km s$^{-1}$ Mpc$^{-1}$ at $2\sigma$ c.l. in full agreement with the Planck satellite results. Our results also support a negative value for the deceleration parameter at least in 3$\sigma$ c.l..'
author:
- 'R. F. L. Holanda$^{1}$'
- 'G. Pordeus-da-Silva$^{1,2}$'
- 'S. H. Pereira$^{3}$'
bibliography:
- 'referenciasBibTex.bib'
title: A low Hubble Constant from galaxy distribution observations
---
Introduction
============
During the past decades, the efforts of observational cosmology have been mainly focused on a precise determination of the parameters that describe the evolution of the Universe. Undoubtedly, one of the most important quantities to understand the cosmic history is the current expansion rate $H_0$, which is fundamental to answer important questions concerning different phases of cosmic evolution, as a precise determination of the cosmic densities, the mechanism behind the primordial inflation as well as the current cosmic acceleration (see [@suyu2012hubble] for a broad discussion).
Nowadays, the most reliable measurements of the Hubble constant are obtained from distance measurements of galaxies in the local Universe using Cepheid variables and Type Ia Supernovae (SNe Ia), which furnishes $H_0=74.03\pm 1.42$ km s$^{-1}$ Mpc$^{-1}$ [@Riess:2019cxk]. The value of $H_0$ can also be estimated from a cosmological model fit to the cosmic microwave background (CMB) radiation anisotropies. By assuming the flat $\Lambda$CDM model, the $H_0$ estimate is $H_0=67.36\pm 0.54$ km s$^{-1}$ Mpc$^{-1}$ [@Aghanim:2018eyx][^1]. These two $H_0$ values are discrepant by $\simeq 4.4\sigma$, which gives rise to the so-called $H_0$-tension problem[^2].
For this reason, new models beyond the standard cosmological one (the flat $\Lambda$) that could alleviate this tension become appealing. Some extensions of the $\Lambda$CDM model that allow to reduce the $H_0$ tension are: the existence of a new relativistic particle [@D_Eramo_2018], small spatial curvature effects [@Bolejko_2018], evolving dark energy models [@M_rtsell_2018], among others [@Riess:2019cxk]. Then, new methods to estimate $H_0$ are welcome in order to bring some light on this puzzle. Precise measurements of the cosmic expansion rate $H(z)$ are important to provide more restrictive constraints on cosmological parameters as well as new insights into some fundamental questions that range from the mechanism behind the primordial inflation and current cosmic acceleration to neutrino physics (see. e.g., [@suyu2012hubble] for a broad discussion).
The Hubble constant has also been estimated from galaxy cluster systems by using their angular diameter distances obtained from the Sunyaev-Zel’dovich effect (SZE) plus X-ray observations. For instance, @Reese_2002 used 18 angular diameter distances of galaxy clusters with redshifts ranging from $z =0.14$ up to $z=0.78$ and obtained $H_0= 60 \pm 4$ km s$^{-1}$ Mpc$^{-1}$ (only statistical errors) for an $\{\Omega_\text{m}=0.3$, $\Omega_{\Lambda}=0.7\}$ cosmology. @Bonamente_2006 considered 38 angular diameter distances of galaxy clusters in the redshift range $0.14 \leq z \leq 0.89$ and obtained $H_0= 76.9 \pm 4$ km s$^{-1}$ Mpc$^{-1}$ (only statistical errors) also for an $\{\Omega_\text{m}=0.3$, $\Omega_{\Lambda}=0.7\}$ cosmology. In both cases, it was assumed a spherical morphology to describe the clusters. Without fixing cosmological parameters, the authors of the Ref.@Holanda_2012 estimated $H_0$ by using a sample of angular diameter distances of 25 galaxy clusters (described by an elliptical density profile) jointly with baryon acoustic oscillations (BAO) and the CMB Shift Parameter signature. [The $H_0$ value obtained in the framework of $\Lambda$CDM model with arbitrary curvature was $H_0 = 74^{+8.0}_{-7.0}$ km s$^{-1}$ Mpc$^{-1}$ at 2$\sigma$ c.l.. By considering a flat $w$CDM model with a constant equation of state parameter, they obtained $H_0 = 72^{+10}_{-9.0}$ km s$^{-1}$ Mpc$^{-1}$ at 2$\sigma$ c.l.. In both cases were considered the statistical and systematic errors. As one may see, due to large error bars, the results found are in agreement with the current Riess et al. local estimate [@Riess:2019cxk] and with the Planck satellite estimate within 2$\sigma$.]{} It is worth to comment that the constraints on the Hubble constant via X-ray surface brightness and SZE observations of the galaxy clusters depend on the validity of the cosmic distance duality relation (CDDR): $D_L(1+z)^{-2}/D_A=1$, where $D_L$ is the luminosity distance and $D_A$ is the angular diameter distance [@Uzan_2004; @HOLANDA_2012b].
In this paper, we obtain new and tight estimates on the Hubble constant by combining two data sets from galaxy distribution observations in redshifts: 40 cluster galaxy gas mass fractions (GMF) and 11 baryon acoustic oscillation (BAO) measurements. The $H_0$ estimates are performed in two scenarios: flat and non-flat $\Lambda$CDM model. This last one is motivated by recent discussions in the literature concerning a possible cosmological curvature tension, with the Planck CMB spectra preferring a positive curvature at more than 99% c.l. [@Di_Valentino_2019; @George; @Di_Valentino2]. We show that the combination of these two independent data sets provides an interesting method to constrain the Hubble constant. For both models, tight estimates are found and our results support low Hubble constant values in agreement with the Planck results. Our results also indicate a universe in accelerated expansion in more than $3\sigma$ c.l..
This paper is organized as follows: Section \[Section2\] presents the two cosmological models and data sets used. Section \[Section3\] presents the main results and analysis and Section \[Section4\] finishes with conclusions.
Cosmological Models and data sets. {#Section2}
==================================
In order to estimate the Hubble constant, we consider two cosmological scenarios: the flat and non-flat $\Lambda$CDM models, where both consider the cosmic dynamics dominated by a cold dark matter (CDM) component and cosmological constant ($\Lambda$), usually related to the constant vacuum energy density with negative pressure. By considering a constant equation of state for dark energy, $p_{\Lambda}=-\rho_{\Lambda}$, and the Universe described by a homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker geometry, we obtain from the Einstein equation the following expression for the Hubble parameter: $$H(z)=H_{0}\sqrt{\Omega_{\text{m,}0}(1+z)^{3}+\Omega_{\text{k,}0}(1+z)^{2}+\Omega_\Lambda} \,,
\label{Hz}$$ where $H_0$ is the current Hubble constant, generally expressed in terms of the dimensionless parameter $h\equiv H_{0}/(100$ km s$^{-1}$ Mpc$^{-1})$, $\Omega_{\text{m,}0}$, $\Omega_\Lambda$ and $\Omega_{\text{k,}0}$ are the current dimensionless parameter of matter density (baryons + dark matter), dark energy density and curvature density ($\Omega_{\text{k,0}}\equiv 1-\Omega_{\text{m,}0}-\Omega_\Lambda$), respectively. Note that if $\Omega_{\text{k,0}}=0$ the flat $\Lambda$CDM model is recovered.
Data sets and $\protect\chi^{2}$ function
-----------------------------------------
In this section, we present the data sets used in the statistical analyses and their respective $\chi^{2}$ function.
### Sample I: Baryonic Acoustic Oscillation data
This sample is composed of 11 measures obtained by 7 different surveys presented in Table \[tabBAO\]. The relevant physical quantities for the BAO data are the angular diameter distance[^3]: $$\begin{gathered}
D_{\text{A}}(z) = \frac{1}{(1+z)}\times \\
\left\{ \begin{array}{lll} \frac{H_{0}^{-1}}{\sqrt{|-\Omega_{\text{k,0}}|}}\text{sin}\left(\frac{\sqrt{|-\Omega_{\text{k,0}}|}}{H_{0}^{-1}}\int_{0}^{z}\frac{dz'}{H(z')} \right) & \text{if} & \Omega_{\text{k,0}}<0 \\
\int_{0}^{z}\frac{dz'}{H(z')} & \text{if} & \Omega_{\text{k,0}}= 0 \\
\frac{H_{0}^{-1}}{\sqrt{|-\Omega_{\text{k,0}}|}}\text{sinh}\left(\frac{\sqrt{|-\Omega_{\text{k,0}}|}}{H_{0}^{-1}}\int_{0}^{z}\frac{dz'}{H(z')} \right) & \text{if} & \Omega_{\text{k,0}}>0
\end{array} \right . \ ,
\label{DA z}\end{gathered}$$ the spherically-averaged distance: $$D_{\text{V}}(z)=\left[(1+z)^{2}D^{2}_{\text{A}}(z)\frac{z}{H(z)} \right]^{1/3}$$ and the sound horizon at the drag epoch [@Eisenstein_1998]: $$r_{\text{s}}(z_{\text{d}})=\frac{2}{3k_{\text{eq}}}\sqrt{\frac{6}{R(z_{\text{eq}})}}\text{ln}\left[\frac{\sqrt{1+R(z_{\text{d}})}+\sqrt{R(z_{\text{d}})+R(z_{\text{d}})}}{1+\sqrt{R(z_{\text{eq}})}}\right] ,$$ where $z_{\text{d}}$ is the drag epoch redshift, $z_{\text{eq}}$ is the equality redshift, $k_{\text{eq}}$ is the scale of the particle horizon at the equality epoch and [@Eisenstein_1998] $$R(z)\equiv \frac{3\rho_{\text{b}}}{4\rho_{\gamma}}=31.5(\Omega_{\text{b}}h^{2})\left(\frac{T_{\text{CMB}}}{2.7 \ \text{K}}\right)^{-4}\left(\frac{z}{10^3}\right)^{-1}$$ is the ratio of the baryon to photon momentum density. Here we assume $\Omega_{\text{b}}h^{2}=0.0226$ [@Cooke_2016], $T_{\text{CMB}}=2.72548$ [@Fixsen_2009] and for $z_{\text{eq}}$, $k_{\text{eq}}$ and $z_{\text{d}}$, we use the fit obeyed by @Eisenstein_1998.
Survey Set I $z$ $d_z (z)$ $\sigma_{d_z}$ –
--------------- ------- ------------------- ------------------------ ----------------- --
6dFGS 0.106 0.336 0.015 –
SDSS-LRG 0.35 0.1126 0.0022 –
Survey Set II $z$ $ \mathcal{D}(z)$ $\sigma_{\mathcal{D}}$ $r_s^{\rm fid}$
BOSS-MGS 0.15 664 25 148.69
BOSS-LOWZ 0.32 1264 25 149.28
BOSS-CMASS 0.57 2056 20 149.28
BOSS-DR12 0.38 1477 16 147.78
BOSS-DR12 0.51 1877 19 147.78
BOSS-DR12 0.61 2140 22 147.78
WiggleZ 0.44 1716 83 148.6
WiggleZ 0.60 2221 101 148.6
WiggleZ 0.73 2516 86 148.6
: BAO data set consisting of 1 measurement of the survey 6dFGS [@Beutler:2011hx], 1 of SDSS-LRG [@Padmanabhan:2012hf], 1 of BOSS-MGS [@Ross:2014qpa], 1 of BOSS-LOWZ [@Anderson:2014], 1 of BOSS-CMASS [@Anderson:2014], 3 of BOSS-DR12 [@Alam:2017hwk] and 3 of WiggleZ [@Kazin:2014qga]. The BAO variable $\mathcal{D}(z)$, $\sigma_{\mathcal{D}}$ and $r_s^{\rm fid}$ have units of Mpc, while $d_z(z)$ and $\sigma_{d_z}$ are dimensionless.
\[tabBAO\]
For survey set I, the BAO quantity is given by: $$d_z(z)=\frac{r_{\text{s}}(z_{\text{d}})}{D_{\text{V}}(z)} \,, \label{dz}$$ with a $\chi^{2}$ function given by: $$\chi^{2}_{\text{BAO,I}}=\sum_{i=1}^{2} \left[\frac{d_{z}^{\text{th}}(z_{i})-d_{z, \ i}^{\text{ob}}}{\sigma_{d_{z , \ i}^{\text{ob}}}}\right]^2 \,.$$
On the other hand, the BAO quantity for survey set II is given by: $$\mathcal{D}(z)=\frac{D_{\text{V}}(z)}{r_{\text{s}}(z_{\text{d}})} r_s^{\rm fid} \,, \label{alpha}$$ and, in this specific case, the $\chi^{2}$ function is: $$\begin{aligned}
\chi^{2}_{\text{BAO,II}} = &\sum_{i=1}^{6} \left[\frac{\mathcal{D}^{\text{th}}(z_{i})-\mathcal{D}_{i}^{\text{ob}}}{\sigma_{\mathcal{D}_{i}^{\text{ob}}}}\right]^2 \nonumber \\
& + \left[\vec{\mathcal{D}}^{\text{th}}-\vec{\mathcal{D}}^{\text{ob}}\right]^{\text{T}}\text{\textbf{C}}^{-1}_{\text{WiggleZ}}\left[\vec{\mathcal{D}}^{\text{th}}-\vec{\mathcal{D}}^{\text{ob}}\right] \,, \end{aligned}$$ where $\text{\textbf{C}}^{-1}_{\text{WiggleZ}}$ is the inverse covariance matrix, whose explicit form is [@Kazin:2014qga]: $$10^{-4}\left(
\begin{array}{rrr} 2.17898878 & -1.11633321 & 0.46982851 \\
-1.11633321 & 1.70712004 & -0.71847155 \\
0.46982851 & -0.71847155 & 1.65283175 \end{array} \right).$$ Unlike the others, the data points of the WiggleZ survey are correlated.
### Sample II: Galaxy Cluster Gas Mass Fractions
The gas mass fractions (GMF) considered in this work corresponds to 40 Chandra observations from massive and dynamically relaxed galaxy clusters in redshift range $0.078 \leq z \leq 1.063$ from the @Mantz:2014xba (see Figure \[fig:GMF\]). These authors incorporated a robust gravitational lensing calibration of the X-ray mass estimates. The measurements of the gas mass fractions were performed in spherical shells at radii near $r_{2500}$[^4], rather than integrated at all radii ($< r_{2500}$). This approach significantly reduces systematic uncertainties compared to previous works that also estimated galaxy cluster gas mass fractions.
![Measurements of $f_{\text{gas}}\left( z\right)$ used in our analysis. Details on this sample are presented in Table 2 of the @Mantz:2014xba.[]{data-label="fig:GMF"}](FigDadosGMF){width="\columnwidth"}
The gas mass fraction quantity for a cluster is given by [@Mantz:2014xba]: $$f_{\text{gas}}^{\text{X-ray}}\left( z\right) =A(z)K(z)\gamma (z) \frac{\Omega _{\text{b}}(z)}{\Omega _{\text{m}}(z)} \left[ \frac{D_{A}^{\text{fid}}\left( z\right) }{D_{A}\left( z\right) }\right] ^{\frac{3}{2}} \ , \label{Eq fgas}$$ where $$A(z)=\left[\frac{H(z)D_{A}(z)}{H^{\text{fid}}(z)D^{\text{fid}}_{A}(z)} \right]^{\eta}$$ stands for the angular correction factor ($\eta=0.442\pm 0.035$), $\Omega_{\rm{m}}(z)$ is the total mass density parameter, which corresponds to the sum of the baryonic mass density parameter, $\Omega_{\rm{b}}(z)$, and the dark matter density parameter, $\Omega_{\rm{c}}(z)$. The term in brackets corrects the angular diameter distance $D_A(z)$ from the fiducial model used in the observations, $D_A^\mathrm{fid}(z)$, which makes these measurements model-independent. The parameters $\gamma(z)$ and $K(z)$ correspond, respectively, to the depletion factor, i.e., the rate by which the hot gas fraction measured in a galaxy cluster is depleted with respect to the baryon fraction universal mean and to the bias of X-ray hydrostatic masses due to both astrophysical and instrumental sources. We adopt the value of $\gamma=0.848 \pm 0.085$ in our analysis, which was obtained from hydrodynamical simulations [@Planelles_2013] (see also a detailed discussion in section 4.2 in the Ref. @Mantz:2014xba). The $\gamma$ parameter has also been estimated via observational data (SNe Ia, gas mass fraction, Hubble parameter) with values in full agreement with those from hydrodynamical simulations (see @Holanda_2017 and @Zheng_2019). Finally, for the parameter $K(z)$, we have used the value reported by @Applegate_2016 in which [*Chandra*]{} hydrostatic masses to relaxed clusters were calibrated with accurate weak lensing measurements from the Weighing the Giants project. The $K(z)$ parameter was estimated to be $K=0.96 \pm 0.09 \pm 0.09$ (1$\sigma$ statistical plus systematic errors) and no significant trends with mass, redshift or the morphological indicators were verified.
Observe that by assuming $\omega _{\text{b,}0}\equiv \Omega _{\text{b,}0}h^{2}=$ $0.0226\pm 0.00034$ [@Cooke_2016], we can rewrite equation (\[Eq fgas\]) as $$f_{\text{gas}}^{\text{X-ray}}\left( z\right) =
\frac{K \ \gamma \ \omega _{\text{b,}0}}{\Omega _{\text{m,}0}h^2} \left[\frac{H(z)D_{A}(z)}{H^{\text{fid}}(z)D^{\text{fid}}_{A}(z)} \right]^{\eta}\left[ \frac{D_{A}^{\text{fid}}\left( z\right) }{D_{A}\left(z\right) }\right] ^{\frac{3}{2}} .$$ Therefore, for this sample, the $\chi ^{2}$ function is given by, $$\chi_{\text{GMF}}^{2}=\sum_{i=1}^{40}\frac{\left[ f_{\text{gas}}^{\text{th}}(z_i)-f_{\text{gas}, \ i}^{\text{ob}}\right] ^{2}}{\sigma _{\text{tot}, \ i}^{2}}\text{ ,}$$ with a total uncertainty given by[^5]$$\begin{gathered}
\sigma _{\text{tot}, \ i}^{2} = \sigma^{2}_{f_{\text{gas}}^{\text{ob}}, \ i} + \left[f_{\text{gas}}^{\text{th}}(z_i) \right]^{2} \left\{ \left( \frac{\sigma_{K} }{K}\right)^{2} + \left( \frac{\sigma_{\gamma }}{\gamma }\right)^{2} \right. \\
\left. + \ln^2 \left[\frac{H(z_i)D_{A}(z_i)}{H^{\text{fid}}(z_i)D^{\text{fid}}_{A}(z_i)}\right]\sigma_{\eta}^{2} \right\} \ ,\end{gathered}$$ where, $\sigma _{K}=0.127$, $\sigma _{\gamma }=0.085$, and $\sigma_\eta =0.035$.
Results and Discussions {#Section3}
=======================
The statistical analysis is performed by the construction of the $\chi^{2}$ function, $$\chi^{2}_{\text{tot.}}(\{\alpha\})=\chi^{2}_{\text{BAO,I}}+\chi^{2}_{\text{BAO,II}}+\chi^{2}_{\text{GMF}} \ .$$ From this function, we are able to construct the likelihood distribution function, $\mathcal{L}(\{\alpha\})=Be^{-\frac{1}{2}\chi^{2}_{\text{tot.}}(\{\alpha\})}$, where $B$ is the normalization factor and $\{\alpha\}$ is the set of free parameters of the cosmological model in question, that is, {$\Omega_{\text{m,}0},h$} and {$\Omega_{\text{m,}0},\Omega_{\Lambda},h$} for the flat and non-flat $\Lambda$CDM models, respectively.
Flat $\protect\Lambda$CDM model
-------------------------------
Figure \[fig:LCDM\] shows the contours and likelihoods for the $\Omega_{\text{m,}0}$ and $h$ parameters obtained in the context of the flat $\Lambda$CDM model. The contours delimited by dotted green lines correspond to the analysis using only GMF, the ones delimited by the dashed pink lines correspond to the analysis using only BAO, and the ones delimited by solid blue lines are referring to the joint analysis GMF + BAO. As one may see, the GMF sample alone does not restrict the value of parameter $h$ (or equivalently $H_0$) but provides tight restrictions to the value of parameter $\Omega_{\text{m,}0}$. From the joint analysis GMF + BAO, we obtain from the $\Omega_{\text{m,}0}-h$ plane (with two free parameters): $h=0.659^{+ 0.012 + 0.020}_{- 0.011 - 0.018}$ and $\Omega_{\text{m,}0} = 0.311^{+ 0.016 + 0.026}_{- 0.015 - 0.025}$ at $1\sigma$ and $2\sigma$ c.l..
By marginalizing over the parameter $\Omega_{\text{m,}0}$, we obtain the likelihood function for the $h$ parameter (see Figure \[fig:h\]), with: $h=0.659^{+ 0.008 + 0.015}_{- 0.007 - 0.015}$ at $1\sigma$ and $2\sigma$ c.l.. On the other hand, by marginalizing over the parameter $h$, we obtain the likelihood function of the $\Omega_{\text{m,}0}$ parameter as $\Omega_{\text{m,}0} = 0.311^{+ 0.010 + 0.021}_{- 0.010 - 0.020}$ at $1\sigma$ and $2\sigma$ c.l..
![Contours and likelihoods of parameters $\Omega_{\text{m,}0}$ and $h$ for the flat $\Lambda$CDM model. The contours delimited by dotted green, dashed pink, and solid blue lines correspond to the analysis using only GMF, BAO and the joint analysis GMF + BAO, respectively. Regions with darker and lighter colors delimit the $1$- and $2\sigma$ c.l. regions, respectively.[]{data-label="fig:LCDM"}](Flat_LCDM_3PlotFgas){width="\columnwidth"}
Figure \[fig:h\] (left) shows the likelihood of $h$ parameter for the flat (solid blue line) $\Lambda$CDM model and also the $1\sigma$ c.l. regions estimate of the $h$ parameter made by @Aghanim:2018eyx in a flat background model and @Riess:2019cxk, cosmological model independent. As one may see, our estimate is in agreement with that one from the CMB anisotropies (within $2\sigma$ c.l.) and it is strongly discrepant with the estimate made by @Riess:2019cxk. Being more specific, our estimate of $H_0$ in a flat $\Lambda$CDM model presents a discrepancy of $5.0\sigma$ with that performed by @Riess:2019cxk. A discrepancy also occurs if we compare our estimate with the most recent estimate of $H_0$ obtained by SH0ES Collaboration, i.e., $H_{0}=73.5 \pm 1.4 \text{ km s}^{-1}\text{Mpc}^{-1} $ [@Reid_2019]. On the other hand, our estimate is in agreement with several other estimates of $H_0$ that used samples with intermediate redshifts in a flat universe [@Busti_2014; @Abbott_2018; @Yu_2018; @Chen_2017; @Holanda_2014; @da_Silva_2018].
Non-flat $\protect\Lambda $CDM model
------------------------------------
Figure \[fig:NonFlat\] shows the contours and likelihoods for the $\Omega_{\text{m,}0}$, $\Omega_{\Lambda}$, and $h$ parameters obtained in the context of the non-flat $\Lambda$CDM model. Similar to the case of the previous section, the contours delimited by solid red lines correspond to the joint analysis BAO + GMF. For this analysis, by marginalizing over the parameter $\Omega_{\Lambda}$, we obtain from the $\Omega_{\text{m,}0}-h$ plane (with two free parameters), the intervals: $h=0.644^{+ 0.035 + 0.057}_{- 0.034 - 0.056}$, and $\Omega_{\text{m,}0} = 0.305^{+ 0.024 + 0.042}_{- 0.022 - 0.034}$ at $1\sigma$ and $2\sigma$ c.l..
Similarly, by marginalizing over the parameter $\Omega_{\text{m,}0}$, we obtain from the $\Omega_{\Lambda}-h$ plane (with two free parameters) the values: $h=0.645^{+ 0.034 +0.057}_{- 0.034 - 0.056}$ and $\Omega_{\Lambda} = 0.660^{+ 0.146 + 0.227}_{- 0.179 - 0.312}$ at $1\sigma$ and $2\sigma$ c.l..
Finally, by marginalizing on the parameter $h$, we obtain from the $\Omega_{\text{m,}0}-\Omega_{\Lambda}$ plane (with two free parameters) the intervals: $\Omega_{\text{m,}0}=0.305^{+ 0.023 + 0.040}_{- 0.022 - 0.035}$ and $\Omega_{\Lambda} = 0.661^{+ 0.152 + 0.232}_{- 0.170 - 0.301}$ at $1\sigma$ and $2\sigma$ c.l..
On the other hand, in order to obtain the likelihood function for the $h$ parameter, we marginalize over $\Omega_{\text{m,}0}$ and $\Omega_{\Lambda}$ parameters (see Figure \[fig:h\]). From this likelihood, the following estimate is found: $h=0.643^{+ 0.023 + 0.045}_{- 0.022 - 0.045}$ at $1\sigma$ and $2\sigma$ c.l.. Similarly, by marginalizing over the $h$ and $\Omega_{\Lambda}$ parameters, we obtain the likelihood for the parameter $\Omega_{\text{m,}0}$ with the following intervals: $\Omega_{\text{m,}0} = 0.305^{+ 0.016 + 0.031}_{- 0.014 - 0.029}$ at $1\sigma$ and $2\sigma$ c.l.. Finally, by marginalizing over the $h$ and $\Omega_{\text{m,}0}$ parameters, we obtain the likelihood for the parameter $\Omega_{\Lambda}$ as $\Omega_{\Lambda} = 0.663^{+ 0.094 + 0.204}_{- 0.120 - 0.230}$ at $1\sigma$ and $2\sigma$ c.l..
Figure \[fig:h\] (right) shows the likelihood of $h$ parameter for the non-flat (dashed red line) $\Lambda$CDM model, together with $1\sigma$ c.l. regions of the estimate of the $h$ parameter obtained from CMB anisotropies @Aghanim:2018eyx in a non-flat background and that one from the Ref. @Riess:2019cxk (local method). From the figure, it is evident that our estimate is in full agreement (within $1\sigma$ c.l.) with that one of $H_0$ from the CMB anisotropies @Aghanim:2018eyx and strongly discrepant with the local estimate made by @Riess:2019cxk, a discrepancy of $3.6\sigma$ c.l..
Table \[Tab:parametros\] shows a synthesis of the results presented in the last two subsections. More specifically, we show the estimates of the free parameters of both cosmological models at $2\sigma$ c.l. obtained from their respective likelihoods. For the $H_0$ parameter, we have seen that the estimate for the flat case is compatible within 2$\sigma$ with the Hubble constant measurement from the Planck results and it is in full disagreement with the local estimate. The estimates obtained here are considerably tighter than those ones from the Ref.@Holanda_2012, where angular diameter distances of galaxy clusters plus BAO and shift parameter were used. The $\Omega_{\text{m},0}$ parameter for the flat model is also compatible with the estimate ($\Omega_{\text{m,}0} = 0.3158\pm 0.0073$) from @Aghanim:2018eyx within $1\sigma$ c.l.. Also for the non-flat case, the estimates for $\Omega_{\text{m,}0}$ and $\Omega_{\Lambda}$ are in good agreement to @Aghanim:2018eyx at $2\sigma$ c.l..
### Deceleration parameter and curvature density parameter
Using the uncertainties propagation and the values estimated for $\Omega_{\text{m,}0}$ and $\Omega_{\Lambda}$ from their likelihoods, we can estimate the current value of the deceleration parameter by using $q_0 = \Omega_{\text{m,}0}/2 -\Omega_{\Lambda}$. We obtain $q_0 = -0.533 \pm 0.046$ and $q_0 = -0.483 \pm 0.340 $ both at $3\sigma$ c.l. for the flat and non-flat $\Lambda$CDM model, respectively. Moreover, similarly, we estimate the following value for the current curvature density parameter at $1\sigma$ c.l.: $\Omega_{\text{k},0}=0.056 \pm 0.108$. These results indicate an accelerating expansion of the universe in more than $3\sigma$ c.l., and $\Omega_{\text{k,}0}$ compatible with a spatially flat curvature within $1\sigma$ c.l..
Flat $\Lambda$CDM model Non-flat $\Lambda$CDM model
--------------------------------------------------------- ----------------------------- -----------------------------
$H_{0}$ $\left[\text{km s}^{-1}\text{Mpc}^{-1} \right]$ $65.9^{+ 1.54}_{- 1.50}$ $64.3^{+4.50}_{-4.45}$
$\Omega_{\text{m,}0}$ $0.311^{+ 0.021}_{- 0.020}$ $0.305^{+ 0.031}_{- 0.029}$
$\Omega_{\Lambda}$ — $0.663^{+ 0.204}_{- 0.230}$
$\chi^{2}_{\text{min}} $ $25.91$ $ 25.90$
: A summary of the constraints in $2\sigma$ c.l. on the set of free parameters of the flat and non-flat $\Lambda$CDM model.[]{data-label="Tab:parametros"}
Conclusions {#Section4}
===========
The recent $H_0$ tension between the local and global methods for the Hubble constant value has motivated the search for new tests and data analyses that could alleviate (or solve) the discrepancy in the $H_0$ value estimated by different methods.
In this paper, we obtained new and tight estimates on the Hubble constant by combining 40 galaxy cluster gas mass fraction measurements with 11 baryon acoustic oscillation data in the frameworks of flat and non-flat $\Lambda$CDM models. The data sets are in the following range of redshift $0.078\leq z \leq 1.023 $. For both cosmological models, the gas mass fraction sample alone did not restrict the value of $H_0$, but put restrictive limits on the $\Omega_M$ parameter. However, from the joint analysis with the 11 BAO data, the restriction on the possible $H_0$ values was notable. By considering the flat and non-flat $\Lambda$CDM model, we obtained, respectively: $H_0=65.89^{+1.54}_{-1.50}$ km s$^{-1}$ Mpc$^{-1}$ and $H_0=64.31^{+ 4.50}_{- 4.45}$ km s$^{-1}$ Mpc$^{-1}$ at $2\sigma$ c.l. For both cases, the estimates indicated a low value of $H_0$ as those ones obtained by the Planck satellite results from the CMB anisotropies observations. For the flat model, the agreement is within $2\sigma$ c.l., while for the non-flat model the concordance is within $1\sigma$ c.l. (see Figure \[fig:h\]).
As a final analysis, an estimate for the current deceleration parameter and for the current curvature density parameter were obtained. The results pointed to a Universe in accelerated expansion in more than $3\sigma$ c.l.. We obtained $q_0 = -0.533 \pm 0.046$ and $q_0 = -0.483 \pm 0.340 $ both in $3\sigma$ c.l. for the flat and non-flat $\Lambda$CDM models, respectively. For the non-flat model, although the best fit suggested a positive curvature, our analysis is compatible with a spatially flat curvature within $1\sigma$ c.l..
In the coming years, the eROSITA [@merloni2012erosita] mission will make an all-sky X-ray mapping of thousand of galaxy clusters and will provide accurate information on gas mass fraction measurements, which will turn the analysis proposed here even more robust.
RFLH thanks financial support from Conselho Nacional de Desenvolvimento Cientıfico e Tecnologico (CNPq) (No.428755/2018-6 and 305930/2017-6). SHP would like to thank CNPq for financial support, No.303583/2018-5.
[^1]: Another recent estimate of $H_0$ has been reported by the H0LiCOW collaboration based on lensing time-delays observations, $H_0 = 71.9^{+2.4}_{-3.0}$ km s$^{-1}$ Mpc$^{-1}$, which is in moderate tension with Planck. However, when combined with clustering data, a value of $H_0 = 66.98 \pm 1.18$ km s$^{-1}$ Mpc$^{-1}$ is obtained [@rusu2019h0licow].
[^2]: We recommend @Freedman:2017yms for an overview and history, as well as @Verde_2019 for the current state of this intriguing problem.
[^3]: The Robustness of baryon acoustic oscillations constraints in models beyond the flat $\Lambda$CDM model have been verified and discussed in details, for instance, in the works of @bao1 and @bao2.
[^4]: This radii is that one within which the mean cluster density is 2500 times the critical density of the Universe at the cluster’s redshift.
[^5]: Note that, as in the BAO analysis, here we overlook the uncertainty regarding the $\Omega _{\text{b,}0}h^{2}$, as we consider it insignificant.
|
---
abstract: 'We investigate geometrical properties of the random $K$-satisfiability problem using the notion of $x$-satisfiability: a formula is $x$-satisfiable is there exist two SAT assignments differing in $Nx$ variables. We show the existence of a sharp threshold for this property as a function of the clause density. For large enough $K$, we prove that there exists a region of clause density, below the satisfiability threshold, where the landscape of Hamming distances between SAT assignments experiences a gap: pairs of SAT-assignments exist at small $x$, and around $x=\frac{1}{2}$, but they do not exist at intermediate values of $x$. This result is consistent with the clustering scenario which is at the heart of the recent heuristic analysis of satisfiability using statistical physics analysis (the cavity method), and its algorithmic counterpart (the survey propagation algorithm). Our method uses elementary probabilistic arguments (first and second moment methods), and might be useful in other problems of computational and physical interest where similar phenomena appear.'
address:
- |
LATP, UMR 6632 CNRS et Université de Provence\
13453 Marseille CEDEX, France
- |
LPTMS, UMR 8626 CNRS et Université Paris-Sud\
91405 Orsay CEDEX, France
- |
LPTMS, UMR 8626 CNRS et Université Paris-Sud\
91405 Orsay CEDEX, France
- |
Physics Department, Politecnico di Torino\
Corso Duca degli Abruzzi 24, 10129 Torino, Italy
author:
- Hervé Daudé
- Marc Mézard
- Thierry Mora
- Riccardo Zecchina
title: |
Pairs of SAT Assignment\
in Random Boolean Formulæ
---
satisfiability, clustering 75.10.Nr ,75.40.-s ,75.40.Mg
Introduction and outline
========================
Consider a string of Boolean variables —or equivalently a string of *spins*— of size $N$: $\vec\sigma=\{\sigma_i\}\in\{-1,1\}^{N}$. Call a $K$-clause a disjunction binding $K$ of these Boolean variables in such a way that one of their $2^K$ joint assignments is set to [false]{}, and all the others to [true]{}. A formula in a conjunctive normal form (CNF) is a conjunction of such clauses. The satisfiability problem is stated as: does there exist a truth assignment $\vec\sigma$ that satisfies this formula? A CNF formula is said to be *satisfiable* (SAT) if this is the case, and *unsatisfiable* (UNSAT) otherwise.
The satisfiability problem is often viewed as the canonical constraint satisfaction problem (CSP). It is the first problem to have been shown NP-complete [@cook], i.e. at least as hard as any problem for which a solution can be checked in polynomial time.
The $P\neq NP$ conjecture states that no general polynomial-time algorithm exists that can decide whether a formula is SAT or UNSAT. However formulas which are encountered in practice can often be solved easily. In order to understand properties of some typical families of formulas, one introduces a probability measure on the set of instances. In the random $K$-SAT problem, one generates a random $K$-CNF formula $F_K(N,M)$ as a conjunction of $M=N\alpha$ $K$-clauses, each of them being uniformly drawn from the $2^K\binom{N}{K}$ possibilities. In the recent years the random $K$-satisfiability problem has attracted much interest in computer science and in statistical physics. Its most striking feature is certainly its sharp threshold.
Throughout this paper, ‘with high probability’ (w.h.p.) means with a probability which goes to one as $N \to \infty$.
\[stc\] For all $K\geq 2$, there exists $\alpha_c(K)$ such that:
- if $\alpha<\alpha_c(K)$, $F_K(N,N\alpha)$ is satisfiable w.h.p.
- if $\alpha>\alpha_c(K)$, $F_K(N,N\alpha)$ is unsatisfiable w.h.p.
The random $K$-SAT problem, for $N$ large and $\alpha$ close to $\alpha_c(K)$, provides instances of very hard CNF formulas that can be used as benchmarks for algorithms. For such hard ensembles, the study of the typical complexity could be crucial for the understanding of the usual ‘worst-case’ complexity.
Although Conjecture \[stc\] remains unproved, Friedgut established the existence of a non-uniform sharp threshold [@Friedgut].
\[fried1\] For each $K\geq 2$, there exists a sequence $\alpha_N(K)$ such that for all $\epsilon>0$: $$\lim_{N\to\infty}\mathbf{P}(F_K(N,N\alpha)\textrm{ is satisfiable})=
\left \{ \begin{array}{ll}1 & \textrm{if } \alpha=(1-\epsilon)\alpha_N(K) \\ 0 & \textrm{if } \alpha=(1+\epsilon)\alpha_N(K).\end{array} \right.$$
A lot of efforts have been devoted to finding tight bounds for the threshold. The best upper bounds so far were derived using first moment methods [@kirousis; @dubois], and the best lower bounds were obtained by second moment methods [@achliomoore; @achlioperes]. Using these bounds, it was shown that $\alpha_c(K)=2^K \ln(2)-O(K)$ as $K\to\infty$.
On the other hand, powerful, self-consistent, but non-rigorous tools from statistical physics were used to predict specific values of $\alpha_c(K)$, as well as heuristical asymptotic expansions for large $K$ [@MZ; @MPZ; @MMZ-RSA]. The *cavity method* [@Cavity], which provides these results, relies on several unproven assumptions motivated by spin-glass theory, the most important of which is the partition of the space of SAT-assignments into many *states* or *clusters* far away from each other (with Hamming distance greater than $cN$ as $N\to\infty$), in the so-called hard-SAT phase.
So far, the existence of such a clustering phase has been shown rigorously in the simpler case of the random XORSAT problem [@XORSAT-CDMM; @XORSAT-MRZ; @XORSAT-DM] in compliance with the prediction of the cavity method, but its existence is predicted in many other problems, such as $q$-colorability [@mulet; @braunstein] or the Multi-Index Matching Problem [@martinmezardrivoire]. At the heuristic level, clustering is an important phenomenon, often held responsible for entrapping local search algorithm into non-optimal metastable states [@montanarisemerjian]. It is also a limiting feature for the belief propagation iterative decoding algorithms in Low Density Parity Check Codes [@montanari; @FLMR].
In this paper we provide a rigorous analysis of some geometrical properties of the space of SAT-assignments in the random $K$-SAT problem. This study complements the results of [@MMZ_prl], and its results are consistent with the clustering scenario. A new characterizing feature of CNF formulas, the ‘$x$-satisfiability’, is proposed, which carries information about the spectrum of distances between SAT-assignments. The $x$-satisfiability property is studied thoroughly using first and second moment methods previously developed for the satisfiability threshold.
The Hamming distance between two assignments $(\vec \sigma,\vec \tau)$ is defined by $$d_{\vec\sigma\vec\tau}=\frac{N}{2}-\frac{1}{2}\sum_{i=1}^{N}\sigma_i\tau_i \ .$$ (Throughout the paper the term ‘distance’ will always refer to the Hamming distance.) Given a random formula $F_K(N,N\alpha)$, we define a ‘SAT-$x$-pair’ as a pair of assignments $(\vec \sigma,\vec \tau)\in\{-1,1\}^{2N}$, which both satisfy $F$, and which are at a fixed distance specified by $x$ as follows: $$\label{eq:condepsilon}
d_{\vec\sigma\vec\tau} \in [Nx-\epsilon(N),Nx+\epsilon(N)].$$ Here $x$ is the proportion of distinct values between the two configurations, which we keep fixed as $N$ and $d$ go to infinity. The resolution $\epsilon(N)$ has to be $\geq 1$ and sub-extensive: $\lim_{N\rightarrow\infty}\epsilon(N)/N=0$, but its precise form is unimportant for our large $N$ analysis. For example we can choose $\epsilon(N)=\sqrt{N}$.
A CNF formula is $x$-satisfiable if it possesses a SAT-$x$-pair.
Note that for $x=0$, $x$-satisfiability is equivalent to satisfiability, while for $x=1$, it is equivalent to Not-All-Equal satisfiability, where each clause must contain at least one satisfied literal and at least one unsatisfied litteral [@gareyjohnson].
The clustering property found heuristically in [@MPZ; @MZ] suggests the following:
\[cluster\] For all $K\geq K_0$, there exist $\alpha_1(K)$, $\alpha_2(K)$, with $\alpha_1(K)<\alpha_2(K)$, such that: for all $\alpha\in(\alpha_1(K),\alpha_2(K))$, there exist $x_1(K,\alpha)<x_2(K,\alpha)<x_3(K,\alpha)$ such that:
- for all $x\in [0,x_1(K,\alpha)]\cup[x_2(K,\alpha),x_3(K,\alpha)]$, a random formula $F_K(N,N\alpha)$ is $x$-satisfiable w.h.p.
- for all $x \in [x_1(K,\alpha),x_2(K,\alpha)]\cup [x_3(K,\alpha),1]$, a random formula $F_K(N,N\alpha)$ is $x$-unsatisfiable w.h.p.
Let us give a geometrical interpretation of this conjecture. The space of SAT-assignments is partioned into non-empty regions whose diameter is smaller than $x_1$; the distance between any two of these regions is at least $x_2$, while $x_3$ is the maximum distance between any pair of SAT-assignments. This interpretation is compatible with the notion of clusters used in the statistical physics approach. It should also be mentioned that in a contribution posterior to this work [@AchlioptasRicci06], the number of regions was shown to be exponential in the size of the problem, further supporting the statistical mechanics picture.
Conjecture \[cluster\] can be rephrased in a slightly different way, which decomposes it into two steps. The first step is to state the *Satisfiability Threshold Conjecture* for pairs:
\[sharpxsat\] For all $K\geq 2$ and for all $x$, $0<x<1$, there exists an $\alpha_c(K,x)$ such that:
- if $\alpha<\alpha_c(x)$, $F_K(N,N\alpha)$ is $x$-satisfiable w.h.p.
- if $\alpha>\alpha_c(x)$, $F_K(N,N\alpha)$ is $x$-unsatisfiable w.h.p.
\[x-stconj\]
The second step conjectures that for $K$ large enough, as a function of $x$, the function $\alpha_c(K,x)$ is non monotonic and has two maxima: a local maximum at a value $x_M(K)<1$, and a global maximum at $x=0$.
In this paper we prove the equivalent of Friedgut’s theorem:
\[fried2\] For each $K\geq 3$ and $x$, $0<x<1$, there exists a sequence $\alpha_N(K,x)$ such that for all $\epsilon>0$: $$\lim_{N\to\infty}\mathbf{P}(F_K(N,N\alpha)\textrm{ is }x\textrm{-satisfiable})=
\left \{ \begin{array}{ll}1 & \textrm{if } \alpha=(1-\epsilon)\alpha_N(K,x), \\ 0 & \textrm{if } \alpha=(1+\epsilon)\alpha_N(K,x),\end{array} \right.$$
and we obtain two functions, $\alpha_{LB}(K,x)$ and $\alpha_{UB}(K,x)$, such that:
- For $\alpha>\alpha_{UB}(K,x)$, a random $K$-CNF $F_K(N,N\alpha)$ is $x$-unsatisfiable w.h.p.
- For $\alpha<\alpha_{LB}(K,x)$, a random $K$-CNF $F_K(N,N\alpha)$ is $x$-satisfiable w.h.p.
The two functions $\alpha_{LB}(K,x)$ and $\alpha_{UB}(K,x)$ are lower and upper bounds for $\alpha_N(K,x)$ as $N$ tends to infinity. Numerical computations of these bounds indicate that $\alpha_N(K,x)$ is non monotonic as a function of $x$ for $K\geq 8$, as illustrated in Fig. \[alpha8\]. More precisely, we prove
\[nonmonotonic\] For all $\epsilon>0$, there exists $K_0$ such that for all $K\geq K_0$, $$\begin{aligned}
\min_{x\in\left(0,\frac{1}{2}\right)} \alpha_{UB}(K,x)& \leq & (1+\epsilon) \frac{2^{K}\ln 2}{2},\label{ineq1}\\
\alpha_{LB}(K,0) & \geq & (1-\epsilon) 2^{K}\ln 2,\label{ineq2}\\
\alpha_{LB}(K,1/2) & \geq & (1-\epsilon) 2^{K}\ln 2.\label{ineq3}\end{aligned}$$
This in turn shows that, for $K$ large enough and in some well chosen interval of $\alpha$ below the satisfiability threshold $\alpha_c\sim 2^K\ln 2$, SAT-$x$-pairs exist for $x$ close to zero and for $x=\frac{1}{2}$, but they do not exist in the intermediate $x$ region. Note that Eq. was established by [@achlioperes].
In section \[first\] we establish rigorous and explicit upper bounds using the first-moment method. The existence of a gap interval is proven in a certain range of $\alpha$, and bounds on this interval are found, which imply Eq. in Theorem \[nonmonotonic\]. Section \[second\] derives the lower bound, using a weighted second-moment method, as developed recently in [@achliomoore; @achlioperes], and presents numerical results. In section \[largek\] we discuss the behavior of the lower bound for large $K$. The case of $x=\frac{1}{2}$ is treated rigorously, and Eq. in Theorem \[nonmonotonic\] is proven. Other values of $x$ are treated at the heuristic level. Section \[prooffriedgut\] presents a proof of Theorem \[fried2\]. We discuss our results in section \[conclu\].
Upper bound: the first moment method {#first}
====================================
The first moment method relies on Markov’s inequality:
\[lemmafirst\] Let $X$ be a non-negative random variable. Then $$\mathbf{P}(X\geq 1)\leq \mathbf{E}(X)\ .
\label{lemfirst}$$
We take $X$ to be the number of pairs of SAT-assignments at fixed distance: $$\label{Z1}
Z(x,F)=\sum_{\vec \sigma,\vec \tau}\delta\left(d_{\vec\sigma\vec\tau}\in[Nx+\epsilon(N),Nx-\epsilon(N)]\right)
\;\delta\left[\vec\sigma,\,\vec\tau \in S(F)\right],$$ where $F=F_K(N,N\alpha)$ is a random $K$-CNF formula, and $S(F)$ is the set of SAT-assignments to this formula. Throughout this paper $\delta(A)$ is an indicator function, equal to $1$ if the statement $A$ is true, equal to $0$ otherwise. The expectation $\mathbf{E} $ is over the set of random $K$-CNF formulas. Since $Z(x,F)\geq 1$ is equivalent to ‘$F$ is $x$-satisfiable’, (\[lemfirst\]) gives an upper bound for the probability of $x$-satisfiability.
The expected value of the double sum can be rewritten as: $$\mathbf{E}(Z)=2^N \sum_{d\in[Nx+\epsilon(N),Nx-\epsilon(N)]\cap \mathbb{N}}\binom{N}{d}\mathbf{E}\left[\delta\left(\vec\sigma,\vec\tau\in S(F)\right)\right].$$ where $\vec\sigma$ and $\vec\tau$ are any two assignments with Hamming distance $d$. We have $\delta\left(\vec\sigma,\vec\tau\in S(F)\right)=\prod_c \delta\left(\vec\sigma,\vec\tau\in S(c)\right)$, where $c$ denotes one of the $M$ clauses. All clauses are drawn independently, so that we have: $$\mathbf{E}(Z)\leq (2\epsilon(N)+1)2^N \max_{d\in[Nx+\epsilon(N),Nx-\epsilon(N)]\cap \mathbb{N}}\left\{\binom{N}{d}\left({\mathbf{E}\left[\delta\left(\vec\sigma,\vec\tau\in S(c)\right)\right]}\right)^{M}\right\},$$ where we have bounded the sum by the maximal term times the number of terms. $\mathbf{E}\left[\delta\left(\vec\sigma,\vec\tau\in S(c)\right)\right]$ can easily be calculated and its value is: $1-2^{1-K}+2^{-K}(1-x)^{K}+o(1)$. Indeed there are only two realizations of the clause among $2^K$ that do not satisfy $c$ unless the two configurations overlap exactly on the domain of $c$.
Considering the normalized logarithm of this quantity, $$F(x,\alpha)=\lim_{N\to\infty}\frac{1}{N}\ln \mathbf{E}(Z)=\ln 2+H_2(x)+\alpha\ln\left(1-2^{1-K}+2^{-K}(1-x)^{K}\right),$$ where $H_2(x)=-x\ln x-(1-x)\ln(1-x)$ is the two-state entropy function, one can deduce an upper bound for $\alpha_N(K,x)$. Indeed, $F(x,\alpha)<0$ implies $\lim_{N\to\infty}\mathbf{P}(Z(x,F)\geq 1)=0$. Therefore:
\[thUB\] For each $K$ and $0<x<1$, and for all $\alpha$ such that $$\label{alUB}
\alpha>\alpha_{UB}(K,x)=-\frac{\ln 2+H_2(x)}{\ln(1-2^{1-K}+2^{-K}(1-x)^{K})},$$ a random formula $F_K(N,N\alpha)$ is $x$-unsatisfiable w.h.p.
We observe numerically that a ‘gap’ ($x_1,x_2$ and $\alpha$ such that $x_1<x<x_2\Longrightarrow F(x,\alpha)< 0$) appears for $K\geq 6$. More generally, the following results holds, which implies Eq. in Theorem \[nonmonotonic\]:
\[thgap\] Let $\epsilon\in (0,1)$, and $\{y_K\}_{K\in \mathbb{N}}$ be a sequence verifying $Ky_K\to\infty$ and $y_K=o(1)$. Denote by $H_2^{-1}(u)$ the smallest root to $H_2(x)=u$, with $u\in[0,\ln 2]$.
There exists $K_0$ such that for all $K\geq K_0$, $\alpha\in[(1+\epsilon)2^{K-1}\ln 2,\alpha_N(K))$ and $x\in [y_K,H_2^{-1}(\alpha 2^{1-K}-\ln 2-\epsilon)]\cup[1-H_2^{-1}(\alpha 2^{1-K}-\ln 2-\epsilon),1]$, $F_K(N,N\alpha)$ is $x$-unsatisfiable w.h.p.
[*Proof.* ]{}Clearly $(1+\epsilon)2^{K-1}\ln(2)<\alpha_N(K)$ since $\alpha_N(K)=2^K \ln(2)-O_K(K)$ [@achlioperes]. Observe that $(1-y_K)^K =o(1)$. Then for all $\delta>0$, there exists $K_1$ such that for all $K\geq K_1$, $x>y_K$: $$\alpha_{UB}(x)<(1+\delta)2^{K-1}(\ln 2+H_2(x)).$$ Inverting this inequality yields the theorem. $\Box$
The choice (\[Z1\]) of $X$, although it is the simplest one, is not optimal. The first moment method only requires the condition $X\geq 1$ to be equivalent to the $x$-satisfiability, and better choices of $X$ exist which allow to improve the bound. Techniques similar to the one introduced separately by Dubois and Boufkhad [@dubois] on the one hand, and Kirousis, Kranakis and Krizanc [@kirousis] on the other hand, can be used to obtain two tighter bounds. Quantitatively, it turns out that these more elaborate bounds provide only very little improvement over the simple bound (see Fig. \[upper\]). For the sake of completeness, we give without proof the simplest of these bounds:
\[thref1\]The unique positive solution of the equation $$\begin{aligned}
H_2(x)&+\alpha\ln\left(1-2^{1-K}+2^{-K}(1-x)^{K}\right)\nonumber\\
&+(1-x)\ln\left[2-\exp\left(-K\alpha\frac{2^{1-K}-2^{-K}(1-x)^{K-1}}{1-2^{1-K}+2^{-K}(1-x)^{K}}\right)\right]\nonumber\\
&+x\ln\left[2-\exp\left(-K\alpha\frac{2^{1-K}-2^{1-K}(1-x)^{K-1}}{1-2^{1-K}+2^{-K}(1-x)^{K}}\right)\right]=0\end{aligned}$$ is an upper bound for $\alpha_N(K,x)$. For $x=0$ we recover the expression of [@kirousis].
The proof closely follows that of [@kirousis] and presents no notable difficulty. We also derived a tighter bound based on the technique used in [@dubois], gaining only a small improvement over the bound of Theorem \[thref1\] (less than $.001\%$).
Lower bound: the second moment method {#second}
=====================================
The second moment method uses the following consequence of Chebyshev’s inequality:
\[lemmasecond\] If $X$ is a non-negative random variable, one has: $$\mathbf{P}(X>0)\geq \frac{\mathbf{E}(X)^2}{\mathbf{E}(X^2)}.$$
It is well known that the simplest choice of $X$ as the number of SAT-assignments (in our case the number of SAT-$x$-pairs) is bound to fail. The intuitive reason [@achliomoore; @achlioperes] is that this naive choice favors pairs of SAT-assignments with a great number of satisfying litterals. It turns out that such assignments are highly correlated, since they tend to agree with each other, and this causes the failure of the second-moment method. In order to deal with [*balanced*]{} (with approximately half of literals satisfied) and uncorrelated pairs of assignments, one must consider a weighted sum of all SAT-assignments. Following [@achliomoore; @achlioperes], we define: $$Z(x,F)=\sum_{\vec\sigma,\vec\tau}\delta\left(d_{\vec\sigma\vec\tau}=\lfloor Nx\rfloor\right)W(\vec\sigma,\vec\tau,F),$$ where $\lfloor Nx \rfloor$ denotes the integer part of $Nx$. Note that the condition $d_{\vec\sigma\vec\tau}=\lfloor Nx\rfloor$ is stronger than Eq. . The weights $W(\vec\sigma,\vec\tau,F)$ are decomposed according to each clause: $$\begin{aligned}
W(\vec\sigma,\vec\tau,F)&=&\prod_c W(\vec\sigma,\vec\tau,c),\\
\textrm{with}\quad W(\vec\sigma,\vec\tau,c)&=&W(\vec u,\vec v),\end{aligned}$$ where $\vec u,\vec v$ are $K$-component vectors such that: $u_i=1$ if the $i^\textrm{th}$ litteral of $c$ is satisfied under $\vec\sigma$, and $u_i=-1$ otherwise (here we assume that the variables connected to $c$ are arbitrarily ordered). $\vec v$ is defined in the same way with respect to $\vec \tau$. In order to have the equivalence between $Z>0$ and the existence of pairs of SAT-assignments, we impose the following condition on the weights: $$W(\vec u,\vec v)=\left\{\begin{array}{ll} 0&\textrm{if}\quad\vec u=(-1,\ldots,-1)\quad\textrm{or}\quad\vec v=(-1,\ldots,-1),\\
>0 &\textrm{otherwise}.\end{array}\right.$$
Let us now compute the first and second moments of $Z$:
$$\mathbf{E}(Z)=2^N\binom{N}{\lfloor Nx\rfloor}f_1(x)^M,$$
where $$\begin{aligned}
f_1(x)&=&\mathbf{E}[W(\vec\sigma,\vec\tau,c)]\\
&=&2^{-K}\sum_{\vec u,\vec v}W(\vec u,\vec v)(1-x)^{|\overrightarrow{u\cdot v}|}x^{K-|\overrightarrow{u\cdot v}|}.\end{aligned}$$ Here $|\vec u|$ is the number of indices $i$ such that $u_i=+1$, and $\overrightarrow{u\cdot v}$ denotes the vector $(u_1 v_1,\ldots,u_K v_K)$.
Writing the second moment is a little more cumbersome:
$$\label{E2}
\mathbf{E}(Z^2)=2^N\sum_{\mathbf{a}\in V_N\cap \{0,1/N,2/N,\ldots,1\}^8}\frac{N!}{\prod_{i=0}^{7}(Na_i)!}f_2(\mathbf{a})^M,$$
where $$\begin{aligned}
f_2(\mathbf{a})&=&\mathbf{E}[W(\vec\sigma,\vec\tau,c)W(\vec\sigma,\vec\tau,c)]\nonumber\\
&=&2^{-K}\sum_{\vec u,\vec v,\vec u',\vec v'}W(\vec u,\vec v)W(\vec u',\vec v')\prod_{i=1}^{K}a_0^{\delta(u_i=v_i=u'_i=v'_i)}a_1^{\delta(u_i=v_i=u'_i\ne v'_i)}\nonumber\\
&&a_2^{\delta(u_i=v_i=v'_i\ne u'_i)}a_3^{\delta((u_i=v_i)\ne(u'_i=v'_i))}a_4^{\delta(u_i=u'_i=v'_i\ne v_i)}\nonumber\\
&&a_5^{\delta((u_i=u'_i)\ne(v_i=v'_i))}a_6^{\delta((u_i=v'_i)\ne(u'_i=v_i))}a_7^{\delta(u'_i=v'_i=u_i\ne u_i)}\end{aligned}$$ $\mathbf{a}$ is a 8-component vector giving the proportion of each type of quadruplets $(\tau_i,\sigma_i,\tau'_i,\sigma'_i)$ —$\vec\tau$ being arbitrarily (but without losing generality) fixed to $(1,\ldots,1)$— as described in the following table:
$a_0$ $a_1$ $a_2$ $a_3$ $a_4$ $a_5$ $a_6$ $a_7$
------------- ------- ------- ------- ------- ------- ------- ------- -------
$\tau_i$ + + + + + + + +
$\sigma_i$ + + + + $-$ $-$ $-$ $-$
$\tau'_i$ + + $-$ $-$ + + $-$ $-$
$\sigma'_i$ + $-$ + $-$ + $-$ + $-$
The set $V_N\subset [0,1]^8$ is a simplex specified by: $$\label{simplexN}
\left\{\begin{array}{l}\lfloor N(a_4+a_5+a_6+a_7)\rfloor=\lfloor Nx\rfloor \\
\lfloor N(a_1+a_2+a_5+a_6)\rfloor=\lfloor Nx\rfloor\\
\sum_{i=0}^7 a_i=1\end{array}\right.$$
These three conditions correspond to the normalization of the proportions and to the enforcement of the conditions $d_{\vec\sigma\vec\tau}=\lfloor Nx\rfloor$, $d_{\vec\sigma'\vec\tau'}= \lfloor Nx\rfloor$. When $N\to\infty$, $V=\bigcap_{N\in\mathbb{N}} V_N$ defines a five-dimensional simplex described by the three hyperplanes: $$\label{simplex}
\left\{\begin{array}{l}a_4+a_5+a_6+a_7=x \\
a_1+a_2+a_5+a_6=x\\
\sum_{i=0}^7 a_i=1\end{array}\right.$$
In order to yield an asymptotic estimate of $\mathbf{E}(Z^2)$ we first use the following lemma, which results from a simple approximation of integrals by sums:
Let $\psi(\mathbf{a})$ be a real, positive, continuous function of $\mathbf{a}$, and let $V_N$, $V$ be defined as previously. Then there exists a constant $C_0$ depending on $x$ such that for sufficiently large $N$: $$\label{E3}
\sum_{\mathbf{a}\in V_N\cap \{1/N,2/N,\ldots,1\}^8}\frac{N!}{\prod_{i=0}^{7}(Na_i)!}\psi(\mathbf{a})^N
\leq C_0 N^{3/2} \int_V{\,\mathrm{d}}\mathbf{a}\ e^{N[H_8(\mathbf{a})+\ln \psi(\mathbf{a})]},$$ where $H_8(\mathbf{a})=-\sum_{i=1}^8 a_i\ln a_i$.
A standard Laplace method used on Eq. (\[E3\]) with $\psi=2(f_2)^\alpha$ yields:
For each $K,x$, define: $$\label{defphi}
\Phi(\mathbf{a})=H_8(\mathbf{a})-\ln 2-2H_2(x)+\alpha\ln f_2(\mathbf{a})-2\alpha\ln f_1(x).$$ and let $\mathbf{a}_0\in V$ be the global maximum of $\Phi$ restricted to $V$. Suppose that $\partial_{\mathbf{a}}^2 \Phi(\mathbf{a}_0)$ is definite negative. Then there exists a constant $C_1$ such that, for $N$ sufficiently large, $$\label{cond}
\frac{\mathbf{E}(Z)^2}{\mathbf{E}(Z^2)}\geq C_1\exp(-N\Phi(\mathbf{a}_0)).$$
Obviously $\Phi(\mathbf{a}_0) \ge 0 $ in general. In order to use Lemma \[lemmasecond\], one must find the weights $W(\vec u,\vec v)$ in such a way that $\max_{\mathbf{a}\in V}\Phi(\mathbf{a})= 0$. We first notice that, at the particular point $\mathbf{a}^*$ where the two pairs are uncorrelated with each other, $$a_0^*=a_3^*=\frac{(1-x)^2}{2},\quad a_1^*=a_2^*=a_4^*=a_7^*=\frac{x(1-x)}{2},\quad
a_5^*=a_6^*=\frac{x^2}{2},$$ we have the following properties:
- $H_8(\mathbf{a}^*)=\ln 2+2H_2(x)$,
- $\partial_{\mathbf{a}}H_8(\mathbf{a}^*)=0,$ $\partial_{\mathbf{a}}^2 H_8(\mathbf{a}^*)$ definite negative,
- $f_1(x)^2=f_2(\mathbf{a}^*)$ and hence $\Phi(\mathbf{a}^*)=0$.
(Note that the derivatives $\partial_{\mathbf{a}}$ are taken in the simplex $V$). So the weights must be chosen in such a way that $\mathbf{a}^*$ be the global maximum of $\Phi$. A necessary condition is that $\mathbf{a}^*$ be a local maximum, which entails $\partial_{\mathbf{a}}f_2(\mathbf{a}^*)=0$.
Using the fact that the number of common values between four vectors $\vec u,\vec v,\vec u',\vec v'\in\{-1,1\}^K$ can be written as: $$\frac{1}{8}\left(K+\vec u\cdot\vec v+\vec u\cdot\vec u'+\vec u\cdot\vec v'+\vec v\cdot\vec u'+\vec v\cdot\vec v'+ \vec u'\cdot\vec v'+\overrightarrow{u\cdot v}\cdot\overrightarrow{u'\cdot v'}\right)$$ we deduce from $\partial_{\mathbf{a}}f_2(\mathbf{a}^*)=0$ the condition: $$\label{eqnln1}
\sum_{\vec u,\vec v}W(\vec u,\vec v)\left\{\begin{array}{l}\vec u\\ \vec v\end{array}\right. (1-x)^{|\overrightarrow{u\cdot v}|}x^{K-|\overrightarrow{u\cdot v}|}=0,$$ $$\begin{aligned}
0&=&K(2x-1)^2{\left[\sum_{\vec u,\vec v}W(\vec u,\vec v) (1-x)^{|\overrightarrow{u\cdot v}|}x^{K-|\overrightarrow{u\cdot v}|} \right]}^2\nonumber\\
&&+{\left[\sum_{\vec u,\vec v}W(\vec u,\vec v) \overrightarrow{u\cdot v}\, (1-x)^{|\overrightarrow{u\cdot v}|}x^{K-|\overrightarrow{u\cdot v}|}\right]}^2\nonumber\\
&&+2(2x-1)\left[\sum_{\vec u,\vec v}W(\vec u,\vec v) \vec u\cdot\vec v\, (1-x)^{|\overrightarrow{u\cdot v}|}x^{K-|\overrightarrow{u\cdot v}|}\right]\nonumber\\
&&\times\left[\sum_{\vec u,\vec v}W(\vec u,\vec v) (1-x)^{|\overrightarrow{u\cdot v}|}x^{K-|\overrightarrow{u\cdot v}|}\right].\label{eqnln2}\end{aligned}$$
If we suppose that $W$ is invariant under simultaneous and identical permutations of the $u_i$ or of the $v_i$ (which we must, since the ordering of the variables by the label $i$ is arbitrary), the $K$ components of all vectorial quantities in Eqs. (\[eqnln1\]), (\[eqnln2\]) should be equal. Then we obtain equivalently: $$\begin{aligned}
\label{eqnln1p}
\sum_{\vec u,\vec v}W(\vec u,\vec v)(2|\vec u|-K)\, (1-x)^{|\overrightarrow{u\cdot v}|}x^{K-|\overrightarrow{u\cdot v}|}=0 \quad \textrm{and}\quad \vec u\leftrightarrow \vec v,\\
\label{eqnln2p}
\sum_{\vec u,\vec v}W(\vec u,\vec v)(K(2x-1)+\vec u\cdot\vec v) (1-x)^{|\overrightarrow{u\cdot v}|}x^{K-|\overrightarrow{u\cdot v}|}=0,\end{aligned}$$
We choose the following simple form for $W(\vec u,\vec v)$: $$W(\vec u,\vec v)=\left\{\begin{array}{ll} 0&\textrm{if}\quad\vec u=(-1,\ldots,-1)\quad\textrm{or}\quad\vec v=(-1,\ldots,-1),\\
\lambda^{|\vec u|+|\vec v|}\nu^{|\overrightarrow{u\cdot v}|} &\textrm{otherwise}.\end{array}\right.$$ Although this choice is certainly not optimal, it turns out particularly tractable. Eqs. and simplify to: $$\label{lambdanu}
\begin{split}
{[\nu (1-x)]}^{K-1}=&(\lambda^2+1-2\lambda\nu){\left(2\lambda x+\nu (1-x)(1+\lambda^2)\right)}^{K-1}\\
{\left(\nu (1-x)+\lambda x\right)}^{K-1}=&(1-\lambda\nu){\left(2\lambda x+\nu (1-x)(1+\lambda^2)\right)}^{K-1}.
\end{split}$$ We found numerically a unique solution $\lambda>0, \nu>0$ to these equations for any value of $K \ge 2$ that we checked.
Fixing $(\lambda,\nu)$ to a solution of , we seek the largest value of $\alpha$ such that the local maximum $\mathbf{a}^*$ is a global maximum, i.e. such that there exists no $\mathbf{a}\in V$ with $\Phi(\mathbf{a})>0$. To proceed one needs analytical expressions for $f_1(x)$ and $f_2(\mathbf{a})$. $f_1$ simply reads: $$\begin{aligned}
f_1(x)&=&2^{-K}{\left((1-x)\nu(1+\lambda^2) +2x\lambda\right)}^K-2\cdot
2^{-K}{\left(x\lambda+(1-x)\nu\right)}^K\nonumber\\
&&+2^{-K}((1-x)\nu)^K.\end{aligned}$$ $f_2$ is calculated by Sylvester’s formula, but its expression is long and requires preliminar notations. We index the $16$ possibilities for $(u_i,v_i,u_i',v_i')$ by a number $r \in\{0,\ldots,15\}$ defined as: $$r=8 \frac{1-u_i}{2}+ 4 \frac{1-v_i}{2} + 2 \frac{1-u_i'}{2} + \frac{1-v_i'}{2} \ .$$ For each index $r$, define $$\begin{aligned}
l(r)&=\delta(u_i=1)+ \delta(v_i=1)+\delta(u_i'=1)+ \delta(v_i'=1), \\
n(r)&=\delta(u_i v_i=1)+ \delta(u_i' v_i'=1),\end{aligned}$$ and $$\begin{aligned}
z_r&=&{\lambda}^{l(r)} {\nu}^{n(r)} \times\left\{\begin{array}{ll}a_r&\textrm{if}\quad r\leq 7\\a_{15-r}&\textrm{if}\quad r \ge 8\end{array}\right. \ .\end{aligned}$$ Also define the four following subsets of $\{0,\ldots,15\}$: $A_0$ is the set of indices $r$ corresponding to quadruplets of the form $(-1,v_i,u_i',v_i')$. $A_0=\{r\in\{0,\ldots,15\}\,|\,u_i=-1\}$. Similarly, $A_1=\{r\,|\,v_i=-1\}$, $A_2=\{r\,|\,u'_i=-1\}$ and $A_3=\{r\,|\,v'_i=-1\}$.
Then $f_2$ is given by: $$\begin{aligned}
2^K f_2(\mathbf{a})&=&{\left(\sum_{j=0}^{15}z_j\right)}^{K}-\sum_{k=0}^3{\left(\sum_{j\in A_k}z_j\right)}^{K}+\sum_{0\leq k<k'\leq 3}{\left(\sum_{j\in A_k\cap A_{k'}}z_j\right)}^{K}\nonumber\\
&&-\sum_{0\leq k<k'<k''\leq 3}{\left(\sum_{j\in A_k\cap A_{k'}\cap A_{k''}}z_j\right)}^{K}+{\left(\sum_{j\in A_0\cap A_1\cap A_2 \cap A_3}z_j\right)}^{K}.\end{aligned}$$
We can now state our lower-bound result:
\[th\_lb\] Let $\alpha_+\in (0,+\infty]$ be the smallest $\alpha$ such that $\partial^2_\mathbf{a} \Phi(\mathbf{a}^*)$ is not definite negative. For each $K$ and $x\in (0,1)$, and for all $\alpha\leq\alpha_{LB}(K,x)$, with $$\label{alpha2}
\alpha_{LB}(K,x)=\min\left[\alpha_+,\inf_{\mathbf{a}\in V_+}\frac{\ln
2+2H_2(x)-H_8(\mathbf{a})}{\ln f_2(\mathbf{a})-2\ln f_1(x)}
\right],$$ where $V_+=\{\mathbf{a}\in V\ |\ f_2(\mathbf{a})>f_1^2\left(1/2\right)\}$, and where $(\lambda,\nu)$ is chosen to be a positive solution of (\[lambdanu\]), the probability that a random formula $F_K(N,N\alpha)$ is $x$-satisfiable is bounded away from $0$ as $N \to\infty$.
This is a straightforward consequence of the expression (\[defphi\]) of $\Phi(\mathbf{a})$.
Theorem \[fried2\] and Lemma \[th\_lb\] immediately imply:
For all $\alpha< \alpha_{LB}(K,x)$ defined in Lemma \[th\_lb\], a random $K$-CNF formula $F_K(N,N\alpha)$ is $x$-satisfiable w.h.p.
We devised several numerical strategies to evaluate $\alpha_{LB}(K,x)$. The implementation of Powell’s method on each point of a grid of size $\mathcal{N}^5$ ($\mathcal{N}=10,15,20$) on $V$ turned out to be the most efficient and reliable. The results are given by Fig. \[alpha8\] for $K=8$, the smallest $K$ such that the picture given by Conjecture \[cluster\] is confirmed. We found a clustering phenomenon for all the values of $K \ge 8$ that we checked. In the following we shall provide a rigorous estimate of $\alpha_{LB}\left(K,\frac{1}{2}\right)$ at large $K$.
Large $K$ analysis {#largek}
==================
Asymptotics for $x=\frac{1}{2}$
-------------------------------
The main result of this section is contained in the following theorem, which implies Eq. in Theorem \[nonmonotonic\]:
\[theolb\] The large $K$ asymptotics of $\alpha_{LB}(K,x)$ at $x=1/2$ is given by: $$\alpha_{LB}(K,1/2)\sim 2^K\ln 2.$$
The proof primarily relies on the following results:
\[lemlambda\] Let $\nu=1$ and $\lambda$ be the unique positive root of: $$\label{eqlambda}
(1-\lambda){(1+\lambda)}^{K-1}-1=0.$$ Then $(\lambda,\nu)$ is solution to (\[lambdanu\]) with $x=\frac{1}{2}$ and one has, at large $K$: $$\lambda-1 \sim -2^{1-K}.$$
\[hardlemma1\] Let $x=\frac{1}{2}$. There exist $K_0>0$, $C_1>0$ and $C_2>0$ such that for all $K\geq K_0$, and for all $\mathbf{a}\in V$ s.t. $|{\mathbf{a}}-{\mathbf{a}}^*|<1/8$, $$\label{2ndmajo}
\left\vert \ln f_2(\mathbf{a})-2\ln f_1(1/2) \right\vert \leq K^2C_1{\vert\mathbf{a}-\mathbf{a}^*\vert}^2 2^{-2K} + C_2 {\vert\mathbf{a}-\mathbf{a}^*\vert}^3 2^{-K}$$
\[hardlemma2\] Let $x=\frac{1}{2}$. There exist $K_0>0$, $C_0>0$ such that for $K\geq K_0$, for all $\mathbf{a}\in V$, $$\label{1stmajo}
\begin{split}
\left|\ln f_2(\mathbf{a})-2\ln f_1(1/2)\right|\leq 2^{-K}\left[(a_0+a_1+a_4+a_5)^K +(a_0+a_2+a_4+a_6)^K\right. \\
\left. +(a_0+a_1+a_6+a_7)^K +(a_0+a_2+a_5+a_7)^K\right] + C_0 K2^{-2K}
\end{split}$$
The proofs of these lemmas are defered to sections \[proofhardlemma1\] and \[proofhardlemma2\].
Proof of Theorem \[theolb\]
---------------------------
We first show that $\partial^2_{\mathbf{a}}\Phi(\mathbf{a}^*)$ is definite negative for all $\alpha<2^K$, when $K$ is sufficiently large. Indeed $\partial^2_{\mathbf{a}}H_8(\mathbf{a}^*)$ is definite negative and its largest eigenvalue is $-4$. Using Lemma \[hardlemma1\], for $\mathbf{a}\in V$ close enough to $\mathbf{a}^*$: $$\Phi({\mathbf{a}})\leq -2|{\mathbf{a}}-{\mathbf{a}}^*|^{2}+\alpha C_1{\vert\mathbf{a}
-\mathbf{a}^*\vert}^{2} K^2 2^{-2K} + \alpha C_2 {\vert\mathbf{a}-\mathbf{a}^*\vert}^{3} 2^{-K}.$$ Therefore $$\Phi({\mathbf{a}})\leq -|{\mathbf{a}}-{\mathbf{a}}^*|^{2}\quad\textrm{ for }K\textrm{ large enough, }|{\mathbf{a}}-{\mathbf{a}}^*|<\frac{1}{2C_2}\textrm{ and }\alpha<2^K.$$
Using Theorem \[th\_lb\], we need to find the minimum, for $a\in V_+$, of $$G(K,{{\mathbf{a}}})\equiv \frac{3\ln 2-H_8(\mathbf{a})}{\ln f_2(\mathbf{a})-2\ln f_1(1/2)}.$$ We shall show that $$\inf_{{\mathbf{a}}\in V_{+}} G(K,{\mathbf{a}})\sim 2^K\ln 2.\label{equivG}$$ We divide this task in two parts. The first part states that there exists $R>0$ and $K_1$ such that for all $K\geq K_1$, and for all ${\mathbf{a}}\in V_+$ such that $|{\mathbf{a}}-{\mathbf{a}}^*|<R$, $G(K,{\mathbf{a}})>2^K$. This is a consequence of Lemma \[hardlemma1\]; using the fact that $3\ln 2-H_8({\mathbf{a}})\geq |{\mathbf{a}}-{\mathbf{a}}^*|^{2}$ for ${\mathbf{a}}$ close enough to ${\mathbf{a}}^*$, one obtains: $$G(K,{\mathbf{a}})\geq \frac{2^K}{C_1 K^2 2^{-K}+C_2 |{\mathbf{a}}-{\mathbf{a}}^*|}$$ which, for $K$ large enough and ${\mathbf{a}}$ close enough to ${\mathbf{a}}^*$, is greater than $2^K$.
The second part deals with the case where ${\mathbf{a}}$ is far from ${\mathbf{a}}^*$, i.e. $|{\mathbf{a}}-{\mathbf{a}}^*|>R$. First we put a bound on the numerator of $G({\mathbf{a}})$: there exists a constant $C_3>0$ such that for all ${\mathbf{a}}\in V$ s.t. $|{\mathbf{a}}-{\mathbf{a}}^*|>R$, one has $3\ln 2-H_8({\mathbf{a}})>C_3$.
Looking at Eq. , it is clear that, in order to minimize $G(K,{\mathbf{a}})$, ${\mathbf{a}}$ should be ‘close’ to at least one the four hyperplanes defined by $$\begin{split}
a_0+a_1+a_4+a_5=1, &\qquad
a_0+a_2+a_4+a_6=1,\\
a_0+a_1+a_6+a_7=1, &\qquad
a_0+a_2+a_5+a_7=1.
\end{split}$$ More precisely, we say for instance that ${\mathbf{a}}$ is *close to* the first hyperplane defined above iff $$a_0+a_1+a_4+a_5>1-K^{-1/2}$$ Now suppose that ${\mathbf{a}}$ is *not* close to that hyperplane. Then the corresponding term goes to $0$: $$(a_0+a_1+a_4+a_5)^K\leq \left(1-K^{-1/2}\right)^{K}\sim \exp(-\sqrt{K})\quad\textrm{as }K\to \infty.$$
We classify all possible cases according to the number of hyperplanes ${\mathbf{a}}\in V_+$ is close to:
- $\mathbf{a}$ is close to none of the hyperplanes. Then $$G(K,{\mathbf{a}})\geq \frac{2^K C_3}{4\exp(-\sqrt{K})+C_0K2^{-K}}>2^K\qquad\textrm{for }K\textrm{ large enough.}$$
- $\mathbf{a}$ is close to one hyperplane only, e.g. the first hyperplane $a_0+a_1+a_4+a_5=1$ (the other hyperplanes are treated equivalently). As $\sum_{i=0}^{7}a_i=0$, one has $$a_2<K^{-1/2},\quad a_3<K^{-1/2},\quad a_6<K^{-1/2},\quad a_7<K^{-1/2}.$$ This implies $H_{8}({\mathbf{a}})<2\ln 2 + 2\ln K/\sqrt{K}$, and we get: $$G(K,{\mathbf{a}})\geq \frac{2^K [\ln 2-2\ln K/\sqrt{K}]}{1+C_0K 2^{-K}+3\e^{-\sqrt{K}}}\geq 2^K (\ln 2) \left[1-3\ln K/\sqrt{K}\right]$$ for sufficiently large $K$.
- ${\mathbf{a}}$ is close to two hyperplanes. It is easy to check that these hyperplanes must be either the first and the fourth ones, or the second and the third ones. In the first case we have $a_0+a_5>1-3/\sqrt{K}$ and in the second case $a_0+a_6>1-3/\sqrt{K}$. Both cases imply: $H_{8}({\mathbf{a}})<\ln 2 + 3\ln K/\sqrt{K}$. One thus obtains: $$G(K,{\mathbf{a}})\geq \frac{2^K [2\ln 2-3\ln K/\sqrt{K}]}{2+C_0 K2^{-K}+2\e^{-\sqrt{K}}}\geq 2^K (\ln 2) \left[1-3\ln K/\sqrt{K}\right].$$
- One can check that ${\mathbf{a}}$ cannot be close to more than two hyperplanes.
To sum up, we have proved that for $K$ large enough, for all ${\mathbf{a}}\in V_+$, $$G(K,{\mathbf{a}})\geq 2^K (\ln 2) \left[1-3\ln K/\sqrt{K}\right],$$ Clearly, $\alpha_{LB}(K,1/2)=\inf_{{\mathbf{a}}\in V_+} G(K,{\mathbf{a}})<\alpha_{UB}(K,1/2)$. Since from Theorem \[thUB\] we know that $\alpha_{UB}(K,1/2)\sim 2^K\ln 2$, this proves Eq. .
Proof of Lemma \[hardlemma1\] {#proofhardlemma1}
-----------------------------
Let $x=\frac{1}{2}$ and choose $\nu=1$ and $\lambda$ the unique positive root of Eq. . Let $\epsilon_i=a_i-1/8$, and ${ \mbox{\boldmath$\epsilon$}}=(\epsilon_0,\ldots,\epsilon_7)$. We expand $f_2({\mathbf{a}})$ in series of ${ \mbox{\boldmath$\epsilon$}}$. The zeroth order term is $f_2(1/8,\ldots,1/8)=f_1^2(1/2)$. The first order term vanishes. We thus get: $$\label{f2dev}
f_2({\mathbf{a}})=f_1^{2}(1/2)+B_0-B_1+B_2-B_3+B_4,$$ with [$$\begin{aligned}
B_0&=&\sum_{q=2}^K\binom{K}{q}{\left(\frac{1}{2}\sum_{i=0}^7 p_i(\lambda)\epsilon_i\right)}^q {\left[\frac{1+\lambda}{2}\right]}^{4(K-q)},\\
B_1&=& 2^{-K}\sum_{a=1}^4\sum_{q=2}^K \binom{K}{q} {\left[\sum_{i=0}^7 \left(\lambda^{\ell_{ai}}-1\right)\epsilon_i\right]}^q {\left[\frac{1+\lambda}{2}\right]}^{3(K-q)},\\
B_2&=& 2^{-2K}\sum_{a=1}^6\sum_{q=2}^K \binom{K}{q} [2r_a(\lambda,{ \mbox{\boldmath$\epsilon$}})]^q {\left[\frac{1+\lambda}{2}\right]}^{2(K-q)},\\
B_3&=& 2^{-3K}\sum_{a=1}^4\sum_{q=2}^K \binom{K}{q} [4s_a(\lambda,{ \mbox{\boldmath$\epsilon$}})]^q {\left[\frac{1+\lambda}{2}\right]}^{K-q},\\
B_4&=& 2^{-4K} \sum_{k=2}^K (8\epsilon_0)^q.\end{aligned}$$]{} In $B_0$, $p_i(\lambda)=\lambda^{l(i)}+\lambda^{l(15-i)}-2-4(\lambda-1)$. We have used the fact that $\sum_{i=0}^7 \epsilon_i=0$. Using $l(i)+l(15-i)=4$, one obtains $|p_i(\lambda)|\leq 11(\lambda-1)^{2}\leq 11 \cdot 2^{4-2K}$, since $|\lambda-1|\leq 2^{2-K}$ for $K$ large enough, by virtue of Lemma \[lemlambda\].
In $B_1$, we have used again $\sum_{i=0}^7 \epsilon_i=0$. $\ell_{ai}$ is either $l(i)$ or $l(15-i)$, depending on $a$. In both cases $|\lambda^{\ell_{ai}}-1|\leq 4|\lambda-1|\leq 2^{4-K}$. In $B_2$ and $B_3$, the expressions of $r_a(\lambda,{ \mbox{\boldmath$\epsilon$}})$ and $s_{a}(\lambda,{ \mbox{\boldmath$\epsilon$}})$ are given by: $$\begin{split}
r_1=\epsilon_0+\lambda(\epsilon_1+\epsilon_2)+\lambda^{2}\epsilon_3, &\qquad r_2=\epsilon_0+\lambda(\epsilon_1+\epsilon_4)+\lambda^{2}\epsilon_5,\\
r_3=\epsilon_0+\lambda(\epsilon_2+\epsilon_4)+\lambda^{2}\epsilon_6, &\qquad r_4
=\epsilon_0+\lambda(\epsilon_1+\epsilon_7)+\lambda^{2}\epsilon_6,\\
r_5=\epsilon_0+\lambda(\epsilon_2+\epsilon_7)+\lambda^{2}\epsilon_5, &\qquad r_6
=\epsilon_0+\lambda(\epsilon_4+\epsilon_7)+\lambda^{2}\epsilon_3,
\end{split}$$ $$s_1=\epsilon_0+\lambda \epsilon_1,\qquad
s_2=\epsilon_0+\lambda \epsilon_2, \qquad
s_3=\epsilon_0+\lambda \epsilon_4,\qquad
s_4=\epsilon_0+\lambda \epsilon_7.\qquad$$
In order to prove Lemma \[hardlemma1\] we will use the following fact:
Let $y$ be a real variable such that $|y|\leq 1$. Then $$\left|\sum_{k=2}^K \binom{K}{k} y^k\right|\leq \frac{K(K-1)}{2}y^{2}+2^K |y|^{3}.$$
One has $|2r_{a}|\leq 8|{ \mbox{\boldmath$\epsilon$}}|$, $|4s_{a}|\leq 8|{ \mbox{\boldmath$\epsilon$}}|$, and $|8\epsilon_0|\leq 8|{ \mbox{\boldmath$\epsilon$}}|$. Therefore, for $|{ \mbox{\boldmath$\epsilon$}}|<1/8$, one can write: $$\begin{aligned}
|B_0|&\leq & \frac{K(K-1)}{2}(11\cdot 2^6)^{2} 2^{-4K} |{ \mbox{\boldmath$\epsilon$}}|^{2} + (11\cdot 2^6)^{3} 2^{-5K} |{ \mbox{\boldmath$\epsilon$}}|^{3}\label{majo0}\\
|B_1|&\leq & 4\frac{K(K-1)}{2}2^{14} 2^{-3K} |{ \mbox{\boldmath$\epsilon$}}|^{2} + 2^{21} 2^{-3K} |{ \mbox{\boldmath$\epsilon$}}|^{3}\label{majo1}\\
|B_i|&\leq & \binom{4}{i}\frac{K(K-1)}{2} 2^6 2^{-iK} |{ \mbox{\boldmath$\epsilon$}}|^{2} + 2^9 2^{-(i-1)K} |{ \mbox{\boldmath$\epsilon$}}|^{3}\quad\textrm{for }2\leq i\leq 4.\end{aligned}$$
Observe that $$\label{asympf}
f_1(1/2)={\left[{\left(\frac{1+\lambda}{2}\right)}^K-2^{-K}\right]}^{2}=1+O(K2^{-K})$$ and that for $K$ large enough, $$\left|\ln\frac{f_2({\mathbf{a}})}{f_1^2(1/2)}\right|\leq \frac{2}{f_1(1/2)^2}\sum_{i=0}^{4}|B_i|,$$ which proves Lemma \[hardlemma1\].
Proof of Lemma \[hardlemma2\] {#proofhardlemma2}
-----------------------------
Note that the bounds on $B_0$ and $B_1$ , remain valid for any ${ \mbox{\boldmath$\epsilon$}}$. Therefore $B_0=O(2^{-2K})$ and $B_1=O(2^{-2K})$ uniformly. We bound $B_3$ by observing that: $$\begin{split}
B_3=&2^{-K}\left[(a_0+\lambda a_1)^K+(a_0+\lambda a_2)^K+(a_0+\lambda a_4)^K+(a_0+\lambda a_7)^K\right]\\
&-2^{-3K}\sum_{a=1}^4{\left[\frac{1+\lambda}{2}\right]}^K\left[1+K \left(\frac{8s_a(\lambda,{ \mbox{\boldmath$\epsilon$}})}{1+\lambda}\right)\right].
\end{split}$$ Since $(a_0+\lambda a_1)\leq a_0+a_1\leq 1/2$ and likewise for the three other terms, one has $B_3=O(2^{-2K})$ uniformly in ${\mathbf{a}}$. A similar argument yields $B_4=O(2^{-2K})$. There remains $B_2$, which we write as: $$\begin{split}
B_2=&2^{-K}\sum_{0\leq k<k'\leq 3}{\left(\sum_{j\in A_k\cap A_{k'}}z_j\right)}^{K}\\
&-2^{-2K}\sum_{a=1}^6{\left[\frac{1+\lambda}{2}\right]}^{2K}\left[1+K \left(\frac{8r_a(\lambda,{ \mbox{\boldmath$\epsilon$}})}{(1+\lambda)^2}\right)\right]
\end{split}$$ The second term of the sum is $O(K2^{-2K})$. The first term is made of six contributions. Two of them, namely $2^{-K}(a_0+\lambda(a_1+a_2)+\lambda^2 a_3)$ and $2^{-K}(a_0+\lambda(a_4+a_7)+\lambda^2 a_3)$, are $O(2^{-2K})$, because of the condition on distances. Among the four remaining contributions, we show how to deal with one of them, the others being handled similarly. This contribution can be written as: $$(a_0+\lambda(a_1+a_4)+\lambda^2 a_5)^K=(a_0+a_1+a_4+a_5)^K {\left(1+\frac{(\lambda-1)(a_1+a_4)+(\lambda^2-1)a_5}{a_0+a_1+a_4+a_5}\right)}^K.$$ We distinguish two cases. Either $a_0+a_1+a_4+a_5\leq 1/2$, and we get trivially: $$(a_0+\lambda(a_1+a_4)+\lambda^2 a_5)^K-(a_0+a_1+a_4+a_5)^K=O(2^{-K}),$$ since both terms are $O(2^{-K})$; or $a_0+a_1+a_4+a_5\geq 1/2$, and then: $$\begin{split}
\left|(a_0+\lambda(a_1+a_4)+\lambda^2 a_5)^K-(a_0+a_1+a_4+a_5)^K\right|\leq\\
\qquad\left|{\left(1+\frac{(\lambda-1)(a_1+a_4)+(\lambda^2-1)a_5}{a_0+a_1+a_4+a_5}\right)}^K-1\right|=O(K2^{-K}).
\end{split}$$ Using again Eq. finishes the proof of Lemma \[hardlemma2\].$\Box$
Heuristics for arbitrary $x$
----------------------------
For arbitrary $x$, the function to minimize in (\[alpha2\]) is hard to study analytically. Here we present what we believe to be the correct asymptotic expansion of $\alpha_{LB}(K,x)$ at large $K$. Hopefully this temptative analysis could be used as a starting point towards a rigorous analytical treatment for any $x$.
A careful look at the numerics suggests the following Ansatz on the position of the global maximum, at large $K$: $$\label{ans}
\begin{split}
&a_0=1-x+o(1),\quad a_6=x+o(1)\\
&a_i=o(1)\quad\text{for }i\neq 0,6.
\end{split}$$ A second, symmetric, maximum also exists around $a_0=1-x$, $a_5=x$. Plugging this locus into Eq. leads to the following conjecture:
For all $x\in(0,1]$, the asymptotics of $\alpha_{LB}(x)$ is given by: $$\lim_{K\to \infty}2^{-K}\alpha_{LB}(K,x)=\frac{\ln 2+H(x)}{2},$$ and the limit is uniform on any closed sub-interval of $(0,1]$.
This conjecture is consistent with both our numerical simulations and our result at $x=\frac{1}{2}$.
Proof of Theorem \[fried2\] {#prooffriedgut}
===========================
Starting with the sharpness criterion for monotone properties of the hypercube given by E. Friedgut and J. Bourgain, we will prove Theorem \[fried2\] by using techniques and tools developped by N. Creignou and H. Daudé for proving the sharpness of monotone properties in random CSPs.\
First we make precise some notations for this study on random $K$-CNF formula over $N$ Boolean variables $ \{x_1, \ldots, x_ N\} .$ A $K$-clause $C$ is given in disjunctive form: $C=x_1^{\varepsilon_1}\vee \ldots \vee x_K^{\varepsilon_K}$ where $\varepsilon_i \in \{0,1\}$ ($x_i^0$ is the positive literal $x_i$ and $x_i^1$ is the negative one $\overline {x_i}$). A $K$-CNF formula $F$ is a finite conjunction of $K$-clauses, $\Omega(F)$ will denote the set of distinct variables occurring in $F$, $\Omega(F)\subset \{x_1, \ldots, x_ N\} $. In this Boolean framework, $S(F)$ the set of satisfying assignments to $F$, becomes a subset of $\{0,1\}^N$.\
Now, let us recall how a slight change of our probability measure on formul[æ]{} gives a convenient product probability space for studying $x$-satisfiability.
$x$-unxatisfiability as a monotone property
-------------------------------------------
In our case the number of clauses in a random formula $F_K(N, N\alpha)$ is fixed to $M=N\alpha$. We define another kind of random formula $G_K(N, N\alpha)$ by allowing each of the ${{\mathcal{N}}} = 2^K {N\choose K}$ possible clauses to be present with probability $p= \alpha N / {{\mathcal{N}}}$. Then, assigning $1$ to each clause if it is present and $0$ otherwise, the hypercube $\{0,1\}^{{\mathcal{N}}}$ stands for the set of all possible formul[æ]{}, endowed with the so-called product measure $\mu _p$, where $p$ is the probability for $1$, and $1-p$ for $0$.
More generally, let ${\mathcal{N}}$ be a positive integer, a property $Y\subset\{0,1\}^{{\mathcal{N}}}$ is called monotone if , for any $y, y' \in \{0,1\}^{{\mathcal{N}}}$, $y\leq y'$ and $y \in Y$ implies $y' \in Y$. In that case $\mu_p( y \in Y )$ is an increasing function of $p\in [0,1]$ where $$\mu_p(y_1,\cdots, y_{{\mathcal{N}}})=p^{\vert y \vert } \cdot (1-p)^{{{\mathcal{N}}}- \vert y \vert} \hbox { where } \vert y \vert = {\sharp \{1\leq i \leq {{\mathcal{N}}} \ / \ y_i =1\}}.$$ For any non trivial $Y$ we can define for every $\beta \in ]0,1[$ the unique $p_{\beta}\in ]0,1[$ such that: $$\mu_{p_{\beta}}(y \in Y ) = \beta.$$
In our case $Y$ will be the property of being $x$-unsatisfiable. If we put: $$\label{subsetD}
{{\mathcal{D}}} =\Bigl \{ (\vec{\sigma}, \vec{\tau}) \in \{0,1\} ^N \times \{0,1\} ^N \quad s. t.\quad d_{\vec{\sigma} \vec{\tau}} \in [Nx - \varepsilon(N), Nx + \varepsilon(N) ] \Bigr \}$$ then $x$-unsatisfiability can be read: $$F \in Y \Longleftrightarrow S(F) \times S(F) \quad \cap \quad {{\mathcal{D}}} = \emptyset.$$ Observe that the number of clauses in $G_K(N, N\alpha)$ is distributed as a binomial law $\mathrm{Bin} ( {{\mathcal{N}}}, p=\alpha N / {{\mathcal{N}}})$ peaked around its expected value $p\cdot {{\mathcal{N}}}= \alpha N $. Therefore, from well known results on monotone property of the hypercube, [@JansonLR-99 page 21 and Corollary 1.16 page 19], our Theorem \[fried2\] is equivalent to the following result, which establishes the sharpness of the monotone property $Y$ under $\mu_p$.
\[sharpmodel2\] For each $K\geq 3$ and $x, 0<x<1$, there exists a sequence $\alpha_N(K,x)$ such that for all $\eta >0$: $$\lim_{N\to \infty} \mu_p( F \mbox { is } x-unsatisfiable) = \left\{ \begin{array} {rl} 1 \mbox { if } p\cdot {{\mathcal{N}}}=(1-\eta ) \alpha_N(K,x)N, \\ 0 \mbox{ if } p\cdot {{\mathcal{N}}}=(1+\eta ) \alpha_N(K,x)N. \end{array} \right.$$
This theorem will be proved using general results on monotone properties of the hypercube. We state these results below without proof.
General tools
-------------
The main tool used to prove the existence of a sharp threshold will be a sharpness criterion stemming from Bourgain’s result [@Friedgut] and from a remark by Friedgut on the possibility to strengthen his criterion [@Friedgut-05 Remark following Theorem 2.2]. Thus, a slight strengthening of Bourgain’s proof in the appendix of [@Friedgut] combined with an observation made in [@CreignouD-03 Theorem 2.3, page 130] gives the following sharpness criterion:
\[sharpcriterion\]
Let $Y_{{\mathcal{N}}}\subset \{0,1\}^{{{\mathcal{N}}}}$ be a sequence of monotone properties, then $Y$ has a sharp threshold as soon as there exists a sequence $T_{{\mathcal{N}}}$ with $T_{{\mathcal{N}}}\supset Y_{{\mathcal{N}}}$ such that for any $\beta \in ]0,1[$ and every $D\geq 1$ the three following conditions are satisfied: $$\label{cond0}
p_{\beta}=o(1),$$ $$\label{cond1}
\mu_{p_{\beta}} (y \ \mathrm {s.t.} \ \exists\, z \in T, \ z\subset y, \ \vert z \vert \leq D ) = o(1),$$ $$\label{cond2}
\forall \, z_0 \notin T, \, \vert z_0 \vert \leq D \quad
\mu_{p_{\beta}} (y \in Y ,\ y\setminus z_0 \notin Y \quad \vert \quad y \supset z_0 \ ) = o(1).$$
We end this subsection by recalling two general results on monotone properties defined on finite sets, established in [@CreignouD-04].
\[perco1\]
Let $U=\{1,\ldots, {{\mathcal{N}}}\}$ be partitioned into two sets $U'$ and $U''$ with $\#U'={{\mathcal{N}}}', \#U''={{\mathcal{N}}}''$ and ${{\mathcal{N}}}={{\mathcal{N}}}'+{{\mathcal{N}}}''$. For any $u\subset U$ let us denote $u'=u\cap U'$ and $u''= u\cap U''. $ Let $Y\subset \{0,1\}^{{{\mathcal{N}}}}$ be a monotone property. For any element $u$, let ${{\mathcal{A}}}(u)$ be the set of elements from $U'$ that are essential for property $Y$ at $u$: $ {{\mathcal{A}}}(u) = \left\{ i\in U' \hbox { s.t. } u \cup \{i\} \in Y
\right\}.$ Then, for any $a>0$ the following holds $$\mu_{p} (u\in Y, u''\not\in Y) \le \frac{1}{(1-p)^{{{\mathcal{N}}}'}}\cdot\mu_{p}
(u\not\in Y, \#{{\mathcal{A}}}(u)\ge a) + \frac{a\cdot p}{(1-p)^{{{\mathcal{N}}}'}} .$$
For the second result we consider a sequence of monotone properties $Y_{{\mathcal{N}}}\subset \{0,1\}^{{{\mathcal{N}}}}$. For any fixed $u\in \{0,1\}^{{{\mathcal{N}}}}$, ${{\mathcal{B}}}_j (u)$ will be the set of collections of $j$ elements such that one can reach property $Y$ from $u$ by adding this collection, thus $\#{{\mathcal{B}}}_j (u)\le {{{\mathcal{N}}} \choose j}$.
\[perco2\] Let $Y_{{\mathcal{N}}}\subset \{0,1\}^{{{\mathcal{N}}}}$ be a sequence of monotone properties. For any integer $j\geq 1$, for any $b>0$ and as soon as ${{\mathcal{N}}}\cdot p$ tends to infinity, the following estimate holds $$\mu_{p} \Bigl (u\not\in Y, \#{{\mathcal{B}}}_j (u)\ge b\cdot {{{\mathcal{N}}} \choose
j}\Bigr) =o(1),$$ $${{\mathcal{B}}}_j (u)=\left\{
\{i_1,\ldots, i_j\}, 1\leq i_1<\ldots< i_j\leq {{\mathcal{N}}}, \hbox{ such that } u\cup \{ i_1,\ldots,
i_j\}\in Y\right\}.$$
Proof of Theorem \[sharpmodel2\] (main steps)
---------------------------------------------
As usual, the first two conditions $\eqref{cond0}$ and $\eqref{cond1}$ are easy to verify for the $x$-unsatisfiability property. For the first one we have: $$\mu_{p}(F \textrm{ is }x\textrm{-satisfiable})\leq \mu_{p}(F \textrm{ is satisfiable} )\leq 2^N (1-p)^{ N \choose K }.$$ This shows that $\displaystyle p_{\beta} \leq {N \ln (2)-\ln(1-\beta) \over {N\choose K}}$ , thus for $x$-unsatisfiability we get: $$\label{cond0OK}
\forall \beta \in ]0,1[ \quad p_{\beta}(N)= O(N^{1-K}).$$
For the second condition, let $H(F)$ be the $K$-uniform hypergraph associated to a formula $F$: its vertices are the $\Omega(F)$ variables occurring in $F$, each index set of a clause $C$ in $F$ corresponds to an hyperedge. Let us recall, see [@KL-02], that a $K$-uniform connected hypergraph with $v$ vertices and $w$ edges is called a *hypertree* when $(K-1) w-v=-1$; it is said to be *unicyclic* when $(K-1) w-v=0$, and *complex* when $(K-1) w-v \geq 1$. Let $T$ be the set of formul[æ]{} $F$ such that $H(F)$ has at least one complex component. We will rule out $\eqref{cond1}$ (and also $\eqref{cond2}$) by using the following result on non complex formul[æ]{}, the proof of which is deferred to the next subsection:
\[simple\] Let $K\geq 3$. If $G$ is a $K$-CNF-formula on $v$ variables whose associated hypergraph is an hypertree or unicyclic then for all integer $d\in\{0, \ldots, v\} $ there exits $(\vec \sigma,\vec \tau) \in S(G) \times S(G) $ such that $d_{\vec \sigma \vec \tau}=d.$
In particular, this result shows that any $x$-unsatisfiable formula has at least one complex component, i. e. $T\supset Y. $ Then observe that there is $O(N^{(K-1) s -1})$ distinct complex components of size $s$ with $N$ vertices. Thus we get for all $p:$ $\displaystyle \mu_{p} (F \ \mathrm {s.t.} \ \exists\, G \in T, \ G \subset F, \ \vert G \vert \leq D ) \leq \sum_{s\leq D} O(N^{(K-1)s -1}) \cdot p^s, $ and $\eqref{cond1}$ follows from $\eqref{cond0OK}$\
In order to prove $\eqref{cond2}$, let us introduce some tools inspired of [@CreignouD-04].
For each positive integer $t$ and $\Delta=(\Delta_1,\ldots ,\Delta_t)\in \{0,1\}^t$, a $\Delta$-assignment is an assignment for which the $t$ first values of the variables are equal to $\Delta_1,\ldots ,\Delta_t$. Then $S_{\Delta}(F)$ will denote the set of satisfying $\Delta$-assignments to $F$: $S_{\Delta}(F) \subset S(F) \subset \{0,1\}^N$.
For any pair of $t$-tuples $(\Delta, \Delta ' ) \in \{0,1\}^t \times \{0,1\}^t$ we define $Y^{\Delta, \Delta'}$: $$F \in Y^{\Delta, \Delta'} \Longleftrightarrow S_{\Delta}(F) \times S_{\Delta '}(F) \quad \cap \quad {{\mathcal{D}}}_x = \emptyset.$$ Observe that $Y^{\Delta, \Delta'}$ is a monotone property containing $Y$.
Now we come back to $\eqref{cond2}$ with $F_0 \notin T$, so that the hypergraph associated to the booster formula $F_0$ has no complex components. $S(F_0) \not= \emptyset$ and w.l.o.g. we can suppose that $\Omega(F_0)=\{1,\ldots, t\}$. Then, for $ F \in Y$ such that $F \supset F_0$ with $F\setminus F_0 \notin Y$, let $F''$ denote the largest subformula of $F$ such that $\Omega(F'') \cap \{1,\ldots,t\} =\emptyset$. We have the two following claims whose proof is postponed to the next subsection.
\[claim1\] For any $(\Delta, \Delta ' ) \in S(F_0) \times S(F_0), \quad F \setminus F_0 \in Y^{\Delta, \Delta'}$.
\[claim2\] There exits $(\Delta, \Delta ' ) \in S(F_0) \times S(F_0)$ such that $ F '' \notin Y^{\Delta, \Delta'}$.
Thus $\eqref{cond2}$ is proved as soon as for any $ \beta \in]0,1[$ and $(\Delta, \Delta ' ) \in \{0,1\}^{t}\times \{0,1\}^{t}$: $$\label{fromCD2a}
\mu_{p_{\beta}} (\ F\setminus F_0 \in Y^{\Delta, \Delta'}, F '' \notin Y^{\Delta, \Delta'} \quad \vert \quad F \supset F_0 \ ) = o(1).$$ The two first events in the R.H.S. of (\[fromCD2a\]) do not depend on the set of clauses in $F_0$ thus by independence under the product measure and recalling that $Y^{\Delta, \Delta'}$ is a monotone property we are led to prove that: $$\mu_{p_{\beta}} (\ F \in Y^{\Delta, \Delta'}, F '' \notin Y^{\Delta, \Delta'}\ ) = o(1).$$ From $\eqref{cond0OK}$ we know that $p_{\beta}(N)=O(N^{1-K})$. Let ${{\mathcal{N}}}'=\Theta(N^{K-1})$ be the number of clauses having at least one variable in $\{1,\ldots,t\}$, then Lemma \[perco1\], applied to the monotone property $Y^{\Delta, \Delta'}$, shows that the above assertion is true as soon as we are able to prove that for all $\gamma >0$:
$$\label{fromCD2b}
\mu_{p_{\beta}} (\ F \notin Y^{\Delta, \Delta'}, \# {{\mathcal{A}}}_{\Delta, \Delta'}(F) \geq \gamma \cdot N^{K-1} \ ) =o(1).$$
where ${{\mathcal{A}}}_{\Delta, \Delta'}(F)$ is the set of $K$-clauses $C$ on $N$ variables having at least one variable in $ \{x_1, \ldots ,x_t \} $ and such that $F \wedge C \in Y^{\Delta, \Delta'}$.
Then let ${{\mathcal{B}}}_{\Delta, \Delta'}(F)$ be the set of collections of $(K-1)$ $K$-clauses $\{C_1, \ldots , C_{K-1}\}$ such that $F \wedge C_1\wedge \ldots \wedge C_{K-1} \in Y^{\Delta, \Delta'}$. From lemma \[perco1\] we deduce that (\[fromCD2b\]) is true as soon as the following result is proved:
\[comblemma\] For all $ t, K\geq 3, \gamma>0$ and $(\Delta, \Delta ' ) \in \{0,1\}^t \times \{0,1\}^t$, there exits $\theta >0$ such that for all $N$, the following holds: $$\label{fromCD3}
\# {{\mathcal{A}}}_{\Delta, \Delta'}(F) \geq \gamma \cdot N^{K-1} \Longrightarrow \# {{\mathcal{B}}}_{\Delta, \Delta'}(F) \geq \theta \cdot N^{K \cdot (K-1)}.$$
Again the proof of this last result is deferred to the next subsection that furnishes a detailed and complete proof of Theorem \[sharpmodel2\].
Detailed proofs {#Proofs}
---------------
### Lemma \[simple\]
When $G$ has a leaf-clause, that is a clause $C=x_1^{\varepsilon_1}\vee \ldots \vee x_K^{\varepsilon_K}$ having only one variable, say $x_1$, in common with $G \setminus C$, the assertion can be proved by induction on the number of clauses in $G$. Indeed from a pair of satisfying assignments $(\vec \sigma,\vec \tau) \in S(G \setminus C) \times S( G\setminus C)$ with $d_{\vec \sigma \vec \tau}=d $ and a pair of satisfying assignments at distance $d' \in\{0, \ldots, K-1\}$ for $C'=x_2^{\varepsilon_2}\vee \ldots \vee x_K^{\varepsilon_K}$, one gets a pair of satisfying assignments at distance $d+d'$. But $C'$ is a $K-1$-clause, thus for any $d' \in \{0, \ldots, K-1\}$ $C'$ has a pair of satisfying assignments at distance $d'$.\
When any $K$-clause $C_i$ of $G=C_1\wedge \ldots\wedge C_l$ has exactly two variables in common with $G\setminus C_i$ then we can write $C_1=x_1^{\mu_1}\vee x_2^{\nu_2} \vee C'_1, C_2=x_2^{\mu_2}\vee x_3^{\nu_3} \vee C'_2, \ldots, C_l=x_l^{\mu_l}\vee x_1^{\nu_1} \vee C'_l$ where the $C'_j$ are $(K-2)$-clauses. A variable in $C'_j$ occurs exactly once in formula $G$ and the set of variables in these $C'_j$ is equal to $\{ x_{l+1}, \ldots,x_v\}$. In particular this set is disjoint from the set of variables of the $2$-CNF formula $(x_1^{\mu_1}\vee x_2^{\nu_2}) \wedge (x_2^{\mu_2}\vee x_3^{\nu_3}) \wedge \ldots \wedge(x_l^{\mu_l}\vee x_1^{\nu_1})$. First observe that this $2$-CNF cyclic formula has always a satisfying assignment $(\sigma_{1}, \ldots, \sigma_{l})$ and together with any truth value for the $(x_j, j >l)$ it gives a satisfying assignment for $G$. Thus, for $G$, one gets a pair of satisfying assignments at distance $d$ for any $d\leq v-l$. Second, as $\Omega(C'_j) \cap \Omega(C'_k)=\emptyset$ when $j \not = k$ a satisfying assignment $\sigma_{l+1}, \ldots, \sigma_{v}$ can easily be found for $ C'_1\wedge \ldots \wedge C'_l$. Together with any truth values of the $(x_i, i\leq l) $ it gives a satisfying assignment for $G$. Then, from the satisfying assignment $(\sigma_{1}, \ldots, \sigma_{l},1-\sigma_{l+1}, \ldots, 1-\sigma_{v} )$ one gets, for any $d\geq v-l$, a pair of satisfying assignments at distance $d$.
### Claims \[claim1\] and \[claim2\]
Observe that any SAT-$x$-pair $(\vec \sigma,\vec \tau)$ for $F\setminus F_0$ with $(\sigma_1, \ldots, \sigma_t) \in S(F_0)$ and $(\tau_1, \ldots, \tau_t) \in S(F_0)$ is also a SAT-$x$-pair for $F$. This proves the first claim by contradiction.\
For the second claim, $F\setminus F_0 \notin Y$ so there exits a SAT-$x$-pair $(\vec \sigma,\vec \tau) \in S(F\setminus F_0) \times S(F\setminus F_0) $. By construction, the set of satisfying assignment of $F''$ does not depend on the first $t$ coordinates. Let $d_t$ be the Hamming distance between $(\sigma_1,\ldots \sigma_t)$ and $(\tau_1,\ldots \tau_t)$. We know that all components of the hypergraph associated to formula $F_0$ are simple and lemma $\eqref{simple}$ shows that there exits $(\sigma'_1,\ldots \sigma'_t) \in S(F_0)$ and $(\tau'_1,\ldots \tau'_t) \in S(F_0) $ such that $d_{\vec \sigma'\vec \tau'}=d_t$. Hence $(\sigma'_1,\ldots \sigma'_t, \sigma_{t+1}, \ldots, \sigma_N)$ and $(\tau'_1,\ldots \tau'_t, \tau_{t+1}, \ldots, \tau_N)$ form now a SAT-$x$-pair for $F''$, thus proving the second claim.
### Lemma \[comblemma\]
In [@ErdosS-82], Erdös and Simonovits proved that any sufficiently dense uniform hypergraph always contains specific subhypergraphs. In particular they considered a generalization of the complete bipartite graph specified by two integers $h\ge 2$ and $m\ge 1$. Let us denote by $K_{h}(m)$ the $h$-uniform hypergraph with $h\cdot m$ vertices partitioned into $h$ classes $V_1, \cdots, V_h$ with $\#V_i =m$ and whose hyperedges are those $h$-tuples, which have exactly one vertex in each $V_i$. Thus $K_{h}(m)$ has $m^h$ hyperedges, for $h=2$ it is a complete bipartite graph $K(m,m)$.
For proving Lemma \[comblemma\], we need a small variation on a result of Erdös and Simonovits which differs only in that it deals with ordered $h$-tuples as opposed to sets of size $h$. More precisely, let us consider hypergraphs on $n$ vertices, say $\{x_1,\ldots, x_n\}$, we will say that two disjoint subsets of vertices $A$ and $B$ verify $A<B$ if for all $x_i$ in $A$ and all $x_j$ in $B$ we have $i<j$. Let $H$ be an $h$-uniform hypergraph with vertex set $\{x_1,\ldots, x_n\}$, then any $h$-uniform subhypergraph $K_{h}(m)$ with $V_1<\ldots <V_h$ is called an *ordered copy of $K_{h}(m)$* in $H$. Thus, the ordered version of the theorem from Erdös and Simonovits about supersaturated uniform hypergraphs [@ErdosS-82 Corollary 2, page 184] can be stated as follows.
\[ErdosS theorem\][(Ordered Erdös-Simonovits)]{} Given $c>0$ and two integers $h \ge 2$ and $m \ge 1 $, there exist $c'>0$ and $N$ such that for all integers $n\ge N$, if $H$ is a $h$-uniform hypergraph over $n$ vertices having at least $c\cdot {n\choose h} $ hyperedges then $H$ contains at least $c'n^{hm}$ ordered copies of $K_{h}(m)$.
We will also use the following observation made when one consider an assignment of two colours, say $0$ and $1$, to the hyperedges of $K_{h}(m)$. First let’s say that a vertex $s$ is $c$-marked if $s$ belongs to at least one $c$-colored hyperedge. A subset of vertices $S$ is said $c$-marked if any $s$ in $S$ is $c$-marked.
\[marked\] Let $h \ge 2$, $m \ge 1 $, and $V_1, \cdots, V_{h}$ the partition associated to $K_{h}(m)$. Consider an assignment of two colours to the $m^{h}$ hyperedges of $K_{h}(m)$, then at least one of the $V_i$ is marked.
Indeed, suppose that $V_1, \cdots, V_{h}$ are not $c$-marked. Now consider a vertex $s\in V_1$ then $s$ is $(1-c)$ marked else by construction of $K_{h}(m)$, $V_i$ would be $c$-marked for all $i\geq 2$. Hence $V_1$ becomes $(1-c)$-marked.\
Now let us show (\[fromCD3\]), in other words that for any $K$-CNF formula $F$ such that ${{\mathcal{A}}}_{\Delta, \Delta'}(F)$ is dense then ${{\mathcal{B}}}_{\Delta, \Delta'}(F)$ is also dense. For more readability we will restrict our attention to the special case $K=3$, in using the above fact the proof will be easily extendable to any $K\geq 3$. Suppose there exist $\Theta(N^2)$ clauses in ${{\mathcal{A}}}_{\Delta, \Delta'}(F)$ then, by the pigeon hole principle, at least for one of the eight types of clause we can find $\Theta(N^2)$ clauses of this type in ${{\mathcal{A}}}_{\Delta, \Delta'}(F)$. Suppose, for example, that
$$\# \bigl \{ C=\overline {x_{i_1}} \ \vee x_{i_2}\vee \ \overline {x_{i_3}}, \ 1\leq i_1<i_2<i_3\leq N, i_1\leq t, \ F \wedge C \in Y^{\Delta, \Delta'}\bigr \} =\Theta(N^2).$$
From well chosen elements in ${{\mathcal{A}}}_{\Delta, \Delta'}(F)$ we now exhibit an element in ${{\mathcal{B}}}_{\Delta, \Delta'}(F)$. We consider the graph $H(F)$ associated to formula $F$: the set of vertices is $\{1, \ldots, N\}$ and for each $C=\overline {x_{i_1}} \ \vee x_{i_2}\vee \ \overline {x_{i_3}} \in {{\mathcal{A}}}_{\Delta, \Delta'}(F)$ we create an edge $\{i_2,i_3\}$. Let $(\vec \sigma,\vec \tau)$ be a SAT-$x$-pair for $F$, then either $\sigma \notin S(C)$ or $\tau \notin S(C)$. Now, following a fixed ordering on the set of pairs of thruth assignments we put the colour $0$ on the non colored edge $\{i_2,i_3\}$ if $\sigma_{i_2}=0$ and $\sigma_{i_3}=1$ else we put the color $1$, having in this case $\tau_{i_2}=0$ and $\tau_{i_3}=1$. Now, let’s take an ordered copy of $K(3,3)$ in $H(F)$ with partition $A=\{j_1, j_2, j_3 \}$ and $B=\{j_4, j_5, j_6 \}$. From Fact \[marked\] we know that one part, say $A$, is marked. In such a case we have $\sigma_{j_1}=0,\sigma_{j_2}=0, \sigma_{j_3}=0$ (A is $0$-marked) or $\tau_{j_1}=0,\tau_{j_2}=0, \tau_{j_3}=0$ (A is $1$-marked) hence $(\vec \sigma,\vec \tau)$ is no longer a SAT-$x$-pair for $F\wedge (x_{j_1} \ \vee x_{j_2}\vee \ x_{j_3})$. If $B$ is marked then $(\vec \sigma,\vec \tau)$ is no longer a SAT-$x$-pair for $F\wedge ( \overline {x_{j_4}} \ \vee \overline {x_{j_5}}\vee \ \overline {x_{j_6}})$. Thus in any case $\{ ( x_{j_1} \ \vee x_{j_2}\vee \ x_{j_3} ), (\overline {x_{j_4}} \ \vee \overline {x_{j_5}}\vee \ \overline {x_{j_6}})\} \in {{\mathcal{B}}}_{\Delta, \Delta'}(F)$.
By hypothesis $H(F)$ is a dense graph so from Theorem \[ErdosS theorem\] we can find $\Theta(N^6)$ copies of $K(3,3)$ in $H(F)$. The above construction provide $\Theta(N^6)$ elements in $ {{\mathcal{B}}}_{\Delta, \Delta'}(F)$ thus proving that this set is also dense.
A general sharpness result
--------------------------
Note that the above proof does not use any information about the shape of the set ${\mathcal{D}}$ defining the $x$-unsatisfiability in terms of a subset of $\{0,\ldots, N\}$, namely the interval $ [Nx - \varepsilon(N), Nx + \varepsilon(N) ]$ (see $\eqref{subsetD}$). Actually we can consider properties defined by a non empty proper subset of $\{0,\ldots, N\}$ and we have proved the following general result:
\[generalsharp\] Let $J_N$ be a non empty subset of $\{0, \ldots, N \}$ and consider $${{\mathcal{D}}_{J}} =\Bigl \{ (\vec{\sigma}, \vec{\tau}) \in \{0,1\} ^N \times \{0,1\} ^N \quad s. t.\quad d_{\vec{\sigma} \vec{\tau}} \in J_N \Bigr \}.$$ Let $K\geq 3$ and $Y_{J}$ be the set of $K$-CNF formula defined as: $$F \in Y_{J} \Longleftrightarrow S(F) \times S(F) \quad \cap \quad {{\mathcal{D}}_{J}} = \emptyset.$$ Then, $Y_J$ is a monotone property exhibiting a sharp threshlold.
On one hand, any upper bound for the satisfiability threshold, for instance $\eqref{cond0OK}$, is an upper bound for all $Y_J$ threshold. On the other hand, lemma \[simple\] tells us that a non complex formula does not belongs to $Y_J$. Then, from [@KL-02], we know that w.h.p a formula whose ratio between the number of clauses and the number of varibles is less than $1/ K(K-1)$, has no complex component. Thus it provides a lower bound for all $Y_J$ threshold.
Discussion and Conclusion {#conclu}
=========================
We have developed a simple and rigorous probabilistic method which is a first step towards a complete characterization of the clustered hard-SAT phase in the random satisfiability problem. Our result is consistent with the clustering picture and supports the validity of the one-step replica symmetry breaking scheme of the cavity method for $K\geq 8$.
The study of $x$-satisfiability has the advantage that it does not rely on a precise definition of clusters. Indeed, it is important to stress that the “appropriate” definition for clusters may vary according to the problem at hand. The natural choice seems to be the connected components of the space of SAT-assignments, where two adjacent assignments have by definition Hamming distance $1$. However, although this naive definition seems to work well on the satisfiability problem, it raises major difficulties on some other problems. For instance, in $q$-colorability, it is useful to permit color exchanges between two adjacent vertices in addition to single-vertex color changes. In XORSAT, the naive definition is inadequate, since jumps from solution to solution can involve a large, yet finite, Hamming distance due to the hard nature of linear Boolean constraints [@MontanariSemerjian05-2].
On the other hand, the existence of a gap in the $x$-satisfiability property is stronger than the original clustering hypothesis. Clusters are expected to have a typical size, and to be separated by a typical distance. However, even for typical formulas, there exist atypical clusters, the sizes and separations of which may differ from their typical values. Because of this variety of cluster sizes and separations, a large range of distances is available to pairs of SAT-assignments, which our $x$-satisfiability analysis takes into account. What we have shown suggests that, for typical formulas, the maximum size of all clusters is smaller than the minimum distance between two clusters (for a certain range of $\alpha$ and $K\geq 8$). This is a sufficient condition for clustering, but by no means a necessary one. As a matter of fact, our large $K$ analysis conjectures that $\alpha_1(K)$ (the smaller $\alpha$ such that Conjecture \[cluster\] is verified) scales as $2^{K-1} \ln 2$, whereas $\alpha_d(K)$ (where the replica symmetry breaking occurs) and $\alpha_s(K)$ (where the one-step RSB Ansatz is supposed to be valid) scale as $2^K\ln K/K$ [@MMZ-RSA]. According to the physics interpretation, in the range $\alpha_s(K)<a<\alpha_1(K)$, there exist clusters, but they are not detected by the $x$-satisfiability approach. This limitation might account for the failure of our method for small values of $K$ —even though more sophisticated techniques for evaluating the $x$-satisfiability threshold $\alpha_c(K,x)$ might yield some results for $K<8$. Still, the conceptual simplicity of our method makes it a useful tool for proving similar phenomena in other systems of computational or physical interest.
A better understanding of the structure of the space of SAT-assignments could be gained by computing the average configurational entropy of pairs of clusters at fixed distance, which contains details about how intra-cluster sizes and inter-cluster distances are distributed. This would yield the value of the $x$-satisfiability threshold. Such a computation was carried out at a heuristic level within the framework of the cavity method for the random XORSAT problem [@MoraMezard06], and should be extendable to the satisfiability problem or to other CSPs.
This work has been supported in part by the EC through the network MTR 2002-00319 ‘STIPCO’ and the FP6 IST consortium ‘EVERGROW’.
This paper, signed in alphabetic order, is based on previous work by Mora Mézard and Zecchina reported in Sec. 1–4, 6. The proof in Sec. 5 is due to Daudé.
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---
abstract: 'Recent review article by S. Samuel “On the speed of gravity and the Jupiter/Quasar measurement” published in the [*International Journal of Modern Physics D*]{}, [**13**]{} (2004) 1753, provides the reader with a misleading “theory” of the relativistic time delay in general theory of relativity. Furthermore, it misquotes original publications by Kopeikin and Fomalont & Kopeikin related to the measurement of the speed of gravity by VLBI. We summarize the general relativistic principles of the Lorentz-invariant theory of propagation of light in time-dependent gravitational field, derive Lorentz-invariant expression for the relativistic time delay, and finally explain why Samuel’s “theory” is conceptually incorrect and confuses the speed of gravity with the speed of light.'
address: |
Department of Physics and Astronomy, University of Missouri-Columbia,\
Columbia, Missouri 65211, USA\
kopeikins@missouri.edu
author:
- 'Sergei M. Kopeikin'
title: 'Comments on the paper by S. Samuel “On the speed of gravity and the Jupiter/Quasar measurement”'
---
Introduction
============
Exact mathematical solution of the light geodesic equation in time-dependent gravitational field of arbitrary moving bodies has been constructed in a series of our publications [@ksge; @ks; @km]. This solution predicts that a light particle (radio wave) is deflected by the gravitational field of the moving body from its retarded, with respect to observer, position. The retarded position of the light-ray deflecting body originates from the Lienard-Wiechert solution of the linearized Einstein equations and can be found by solving the retarded-time equation which is a null characteristic of the gravitational field. We proposed relativistic VLBI experiment to measure this effect of retardation of gravity by the field of moving Jupiter via observation of light bending from a quasar [@apjl] and successfully completed this experiment in September 2002. The experiment testing observational phase-referencing technique with several reference calibrators is described in [@pr1] and the results of the main experiment of September 2002 are published in Astrophysical Journal [@apj].
Samuel’s paper [@s2] is an attempt to review the results of the experiment by making use of a linear Lorentz transformation of the static Shapiro time delay. This approach is physically insufficient for conceptual analysis of the experiment which requires matching of the first and second order effects in the relativistic theory of the time delay based on the Lienard-Wiechert solution of the Einstein equations. This is the main reason for the erroneous conslusions about the nature of the experiment mispresented in [@s2]. In the present paper we outline the basic equations of the complete Lorentz-invariant theory of the time delay and show mistakes in Samuel’s linearized “theory” [@s2] originating from his first publication [@s1].
Retardation of Gravity and the Lienard-Wiechert Potentials
==========================================================
We denote the barycentric coordinates of the solar system as $x^\alpha=(x^0,x^i)$, where $x^0=ct$, $x^i={\bm x}$, and $c$ is the fundamental speed limit. The metric tensor $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, where $\eta_{\mu\nu}={\rm diag(-1,1,1,1)}$ is the Minkowski metric, and $h_{\alpha\beta}$ describes gravitational field of the solar system in the linearized post-Minkowskian [@dam] approximation[^1]. We impose the harmonic gauge conditions $\partial_\nu h^{\mu\nu}-1/2\partial^\mu h=0$, where $\partial_\nu\equiv\partial/\partial x^\nu$ denotes a partial derivative with respect to coordinate $x^\nu$, and $h\equiv\eta^{\mu\nu}h_{\mu\nu}$.
Linearized Einstein equations in harmonic coordinates assume the following form $$\label{ee}
\left(-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}+\nabla^2\right)h^{\mu\nu}=-\frac{8\pi G}{c^4}\left(T^{\mu\nu}-\frac{1}{2}\eta^{\mu\nu}T\right)\;,$$ where $\nabla\equiv\partial/\partial x^i$, $T^{\mu\nu}$ is the stress-energy tensor of matter generating the gravitational field, and $T\equiv\eta^{\mu\nu}T_{\mu\nu}$. In what follows, we take the stress-energy tensor in the standard form of moving point-like masses with Dirac’s delta-functions as amtter’s support [@ll]. In linear approximation the gravitational field of the solar system bodies moving along their orbits with acceleration can be found from Eq. (\[ee\]) separately for each body. For Jupiter, the perturbation of the metric tensor, $h^{\mu\nu}$, is obtained as the Lienard-Wiechert solution of the Einstein equations (\[ee\]) which yields $$\label{1}
h^{\mu\nu}=-\frac{2GM_J}{c^4}\frac{2u^\mu u^\nu+\eta^{\mu\nu}}{r_\alpha
u^\alpha}\;.$$ Here $M_J$ is the mass of Jupiter, $u^\alpha=\gamma(s)(c, {\bm v}_J(s))$ is its four-velocity with ${\bm v}_J(s)=d{\bm x}(s)/ds$, $\gamma(s)=(1-v_J^2(s)/c^2)^{-1/2}$ and the null-cone distance, $r^\alpha=x^\alpha-x^\alpha_J(s)$, and Jupiter’s worldline $x^\alpha_J(s)=(cs, {\bm x}_J(s))$ are functions of the retarded time $s$, determined as a solution of null cone equation $\eta_{\mu\nu}r^\mu r^\nu=0$, that is $$\label{2}
s=t-\frac{1}{c}|{\bm x}-{\bm x}_J(s)|\;.$$ Eq. (\[2\]) describes the propagation of Jupiter’s gravity field from Jupiter to the point ${\bm x}$ with the fundamental speed $c$ [^2]. The letter $c$ used here is the speed of gravity and should not be confused with the speed of light (the speed of a radio wave from a quasar measured by VLBI) as it has been mistakenly done by Asada [@ass]. This fundamental conclusion of general relativity can be tested experimentally by observing gravitational interaction of light from a background source (quasar, star) with the time-dependent field of moving Jupiter and/or other solar system bodies [@apjl].
Electromagnetic Phase
=====================
Radio interferometry (VLBI) measures the phase $\varphi$ of the wave front coming from a quasar. The phase is determined from the eikonal equation for electromagentic field (the quasar radio wave) in the geometric optics approximation of Maxwell’s equations [@ll] $$\label{eik}
g^{\mu\nu}\partial_\mu\varphi\partial_\nu\varphi=0\;,$$ where in the linear approximation $g^{\mu\nu}=\eta^{\mu\nu}-h^{\mu\nu}$. Solution of the eikonal equation is obtained by iterations after substitution of the Lienard-Wiechert gravitational potentials, Eq. (\[1\]), into Eq. (\[eik\]) [@abgr] $$\label{3}
\varphi=\varphi_0+\frac{\nu}{c}\left[k_\alpha x^\alpha+\frac{2GM_J}{c^2}\left(k_\alpha
u^\alpha\right)\ln\Phi\right]\;,$$ where $\Phi\equiv -k_{\alpha} r^\alpha$, $\varphi_0$ is a constant, $\nu$ is the radio frequency, $k^\alpha=(1,{\bm k})$ is a null vector associated with the propagating radio wave, and the unit vector ${\bm k}$ is directed from the quasar to VLBI station [^3].
Solution for eikonal given by Eq. (\[3\]) describes propagation of a plane electromagentic wave scattered on the time-dependent gravitational potential of a point-like mass (Jupiter) moving with time-dependent velocity. We abandoned the acceleration-dependent terms in Eq. (\[3\]) because of their smallness. Influence of the acceleration-dependent terms has been analysed in [@ks]. We notice that Eq. (\[3\]) is Lorentz-invariant because it consists of the products of four-dimensional vectors remaining invariant with respect to the Lorentz transformations of the both Maxwell and Einstein equations. In particular, the argument of the logarithmic fucntion is $\Phi\equiv -k_{\alpha} r^\alpha=r-{\bm k}\cdot{\bm r}$, where $r=|{\bm r}|=\sqrt{{\bm r}\cdot{\bm r}}$, ${\bm r}={\bm x}-{\bm x}_J(s)$, and depend on the retarded position of Jupiter, ${\bm x}_J(s)$, calculated at the retarded time $s$ defined by Eq. (\[2\]) of the null characteristic of the gravitational field of Jupiter. Thus, precise interferometric measurement of phase $\varphi$ of the electromagnetic wave scattered by the gravitational field of a moving gravitating body (Jupiter) allows to determine the null characteristic of the gravitational field and measure the ultimate speed of propagation of gravity which is expected to be equal to the speed of light in vacuum. This is because the measurable phase $\varphi\sim\ln\Phi(s)$ is a logarithmic function of the retarded time $s$ as evident from Eq. (\[3\]).
It is important to notice that the null characteristic of the gravitational field connecting the point of observation, ${\bm x}$, and the retarded position of Jupiter, ${\bm x}_J(s)$, can not be confused with the null characteristic of the electromagnetic field propagating along the space-time direction defined by the null vector $k^\alpha$ because this vector is not parallel to the null vector $r^\alpha = x^\alpha-x^\alpha(s)$ (see Fig. 1). Unfortunately, the four-dimensional Minkowskian diagram of the experiment shown in Fig. 1, was not understood by Asada [@ass] and Samuel [@s1; @s2] who confused the null characteristic of the quasar radio wave and that of the gravitational field of Jupiter.
The Lorentz-invariant Theory of Time Delay
==========================================
The Lorentz-invariant relativistic time delay equation, generalizing the static Shapiro delay, can be obtained outright from equation (\[3\]). We note that the phase $\varphi$ of electromagnetic wave, emitted at the point $x^\alpha_0=(ct_0,{\bm x}_0)$ and received at the point $x^\alpha=(ct,{\bm x})$, remains constant along the wave’s path [@ll]. Indeed, if $\lambda$ is an affine parameter along the path, one has for the phase’s derivative $$\label{phase}
\frac{d\varphi}{d\lambda}=\frac{\partial\varphi}{\partial x^\alpha}\frac{
dx^\alpha}{d\lambda}=0\;,$$ due to the orthogonality of the light rays and their wave fronts. Eq. (\[phase\]) means that $\varphi\left(x^\alpha(\lambda)\right)={\rm const.}$ in accordance with our assertion. Equating two values of the phase at the event of emission, $x^\alpha_0$, and that of observation, $x^\alpha$, and separating the time and space coordinates one obtains from (\[3\]) $$\label{tde}
t-t_0=\frac{1}{c}{\bm k}\cdot\left({\bm x}-{\bm
x}_0\right)-\frac{2GM_J}{c^3}\frac{1-c^{-1}{\bm k}\cdot{\bm
v}_J}{\sqrt{1-v^2_J/c^2}}\ln\left(r-{\bm k}\cdot{\bm
r}\right)\;,$$ where both the distance ${\bm
r}={\bm x}-{\bm x}_J(s)$, $r=|{\bm x}-{\bm x}_J(s)|$, and the retarded time $s$ are defined by the gravity null-cone equation (\[2\]). The time delay (\[tde\]) of light propagating through the gravitational field of an arbitrary moving body was derived first by Kopeikin and Schäfer [@ks] who solved equations for light geodesics in the retarded Lienard-Wiechert gravitational field of the body. Klioner [@kl] also obtained this expression for the case of a uniformly moving body by making use of the Lorentz transformation of the Shapiro delay (that is, both the Maxwell and Einstein equations) from static to moving frame. Samuel’s paper [@s2] represents, in fact, rather convoluted linearized analogue of Klioner’s painstaking calculation [@kl].
Relationship Between Our Theory and Samuel’s Calculations
=========================================================
Let us approximate Eq. (\[3\]) by introducing two angles $\Theta$ and $\theta$ between vector ${\bm k}$ and unit vectors ${\bm p}={\bm R}/R$ and ${\bm l}={\bm r}/r$ correspondingly, where vectors ${\bm
R}={\bm x}-{\bm x}_J(t)$ and ${\bm r}={\bm x}-{\bm
x}_J(s)$ connect the point of observation ${\bm x}\equiv{\bm x}(t)$ with the present, ${\bm x}_J(t)$, and retarded, ${\bm x}_J(s)$, positions of Jupiter, respectively. By definition $\cos\Theta={\bm k}\cdot{\bm p}$ and $\cos\theta={\bm k}\cdot{\bm l}$. The product $\Phi\equiv -k_\alpha r^\alpha=r(1-\cos\theta)\simeq r\theta^2/2$ for small angles. Hence, the phase variation caused by space-time difference $\delta x^\alpha=(c\delta t, \delta{\bm x})$ between two VLBI antennas is $$\label{5}
\delta\varphi=\frac{\nu}{c}\left(k_\alpha\delta x^\alpha+\frac{4GM_J}{c^2}\frac{\delta\theta}{\theta}\right)\;,$$ where $\delta\theta=-{\bm n}\cdot\delta{\bm x}/r$ with ${\bm n}={\bm
l}\times({\bm k}\times{\bm l})$ as the impact vector of the light ray with respect to the retarded (due to the finite speed of gravity) position of Jupiter ${\bm x}_J(s)$. Undetectable terms of order $v_J/c$ have been neglected. The quantity $\delta
t=t_2-t_1$ is the measurable VLBI time delay and $\delta{\bm x}={\bm x}_2-{\bm x}_1\equiv{\bm B}$ is a baseline between two VLBI stations. Since VLBI stations measure the same wave front, $\delta\varphi=0$. Thus, for sufficiently small angle $\theta$ Eq. (\[5\]) yileds $$\begin{aligned}
\label{6a}\delta t&=&c^{-1}{\bm k}\cdot{\bm B}+\Delta\;,\\\nonumber\\\label{6}
\Delta&=&-\frac{4GM_J}{c^3r}\frac{{\bm n}\cdot{\bm B}}{\theta}\;,\end{aligned}$$ where we neglected small terms of the second order of magnitude.
Eq. (\[6\]) can be also derived from Eq. (\[tde\]) [@cqg]. It exactly coincides with the leading order term in Eq. (3.9) of Samuel’s paper [@s2] and allows to establish the following relationship between Samuel’s notations for the Earth-Jupiter distance $R_{EJ}$ and the “observable angle” $\theta_{obs}$ [^4], and our notations for the null-cone distance $r$ and the angle $\theta$. Specifically, neglecting the Earth finite-size effects in our calculations, one obtains $R_{EJ}=r$ and $\theta_{obs}=\theta$. In what follows, we prefer to keep on using our notations.
Eq. (\[6\]) is the excess time delay caused by the scattering of the radio wave from the quasar on the gravitational Lienard-Wiechert potential of [*moving*]{} Jupiter. Though the terms depending on Jupiter’s velocity do not show up in Eq. (\[6\]) explicitly, they are surely incorporated in it [*implicitly*]{} through the retarded position of Jupiter, ${\bm x}_J(s)$, entering both the distance $r$, the angle $\theta$, and the impact parameter vector ${\bm n}$. Such implicit dependence of the relativistic expressions on velocity of gravitating body is typical for the, so-called, post-Minkowskian approximation scheme of general relativity operating with the retarded Lienard-Wiechert solutions of the Einstein equations [@dam; @bel]. Thus, Eq. (\[6\]) describes dynamical situation since (because of the orbital motion of Jupiter) the retarded quantities, $r$, $\theta$, and ${\bm n}$, are not reduced to their static counterparts after their post-Newtonian expansion around the time of observation $t$.
The post-Newtonian expansion of the retarded position of Jupiter, ${\bm x}_J(s)$, in Eq. (\[6\]) around time of photon’s arrival to observer, $t$, yields $$\label{yu}
{\bm x}_J(s)={\bm x}_J(t)+{\bm v}_J(t)(s-t)+O(s-t)^2\;.$$ The difference $s-t$ is calculated by solving the gravity null-cone Eq. (\[2\]). Substituting this solution to Eq. (\[yu\]) yields $$\label{wa}
{\bm x}_J(s)={\bm x}_J(t)-\frac{1}{c}{\bm v}_J(t)R+O(s-t)^2\;,$$ where $R={\bm R}|$, and ${\bm R}={\bm x}-{\bm x}_J(t)$ is the vector lying on the hypersurface of constant time $t$ and connecting the point of observation, ${\bm x}$, and the present position of Jupiter, ${\bm x}_J(t)$.
The post-Newtonian expansion given in Eq. (\[wa\]) originates from the retarded nature of the Lienard-Wiechert gravitational potentials and, thus, describes the effect of the retardation of gravity [@apjl]. The presence of the retardation of gravity effect through the retarded position of Jupiter in the Lorentz-invariant Eq. (\[3\]) and its small-angle approximation Eq. (\[6\]) reflects the causal property of gravitational field which is a consequence of its finite speed. Causality and Lorentz-invariance of the gravitational field are tightly connected fundamental concepts and the measurement of the causal nature (retardation) of gravity in the Fomalont-Kopeikin experiment is equivalent to the proof that the gravitational field is Lorentz-invariant and vice versa.
Samuel also uses the post-Newtonian expansion of the retarded coordinate of Jupiter in Eq. (5.3) of his paper [@s2]. He believes that the post-Newtonian expansion “arises because the position of Jupiter changes as the quasar signals travel from the Jupiter region to Earth”. Jupiter does move as the quasar signal travels towards observer but the distance traveled by Jupiter is proportional not to the difference between the time $t^*$ of the closest approach of the quasar signal to Jupiter and the time of observation, that is $t^*-t$, but to the difference between the retarded time $s$ and the time of observation $t$, that is $s-t$, as clearly follows from Eq. (\[yu\]). Therefore, the fuzzy concept of “the Jupiter region”, which is repeatedly used by Samuel without rigorous mathematical definition, is, in effect, the retarded position of Jupiter defined by the Lienard-Wiechert solution of the gravity field equations. This consideration makes it evident that the origin of the post-Newtonian expansion in Eqs. (5.3)–(5.6) of Samuel’s paper [@s2] is caused by the speed of gravity which propagates from moving Jupiter towards observer as well as the quasar signal does. This point was emphasized in our papers [@apjl; @abgr]. Samuel overlooked the gravitational physics of the Jupiter-quasar experiment because of his approximate and, hence, insufficient solution of the problem of propagation of light rays in time-dependent gravitational fields. He was able to integrate the light-ray geodesics only for the case of the small-angle approximation ($\theta\ll 1$) when the time of the closest approach, $t^*$, of the quasar signal to Jupiter is comparable with the retarded time $s$ along the null characteristic of Jupiter’s non-stationary gravitational field. It is for this reason that Samuel confused the time $t^*$ and the retarded time $s$ and replaced the concept of the propagation of gravity from the retarded position of Jupiter by the concept of the propagation of light from “the Jupiter region” [^5].
Substitution of the post-Newtonian expansion (\[wa\]) to Eq. (\[6\]) yields $$\label{p1}
\Delta =\Delta_S+\Delta_R\;.$$ Here $$\label{p2}
\Delta_S=-(4GM_J/c^3\Theta)({\bm N}\cdot{\bm B})\;$$ is the static Shapiro time delay caused by Jupiter’s gravitational field at the time of observation, where ${\bm N}={\bm p}\times({\bm k}\times{\bm p})$ is the impact vector of the light ray with respect to the present position of Jupiter ${\bm x}_J(t)$, and $$\label{7}
\Delta_R =\frac{4GM_J}{c^4R\Theta^2}\biggl[2
({\bm N}\cdot{\bm v}_J){\bm N}-({\bm K}\times({\bm v}_J\times{\bm K})\biggr]\;,$$ is the post-Newtonian correction to the Shapiro time delay due to the the finite speed of gravity in the gravity null-cone equation (\[2\]). Eq. (\[7\]) is the same as Eq. (4) from [@apj]. It describes the first post-Newtonian $v_J/c$ correction to $\Delta_S$ and can be detected because of the amplifying factor $\sim 1/\Theta^2$. Samuel’s Eq. (5.6) is an approximate form of our Eq. (\[7\]). We notice that Samuel’s notation for the angle $\theta_1\equiv\Theta$ in our notations. Physical origin of the post-Newtonian correction $\Delta_R$ to the static Shapiro time delay $\Delta_S$ is due to the Lorentz-invariant nature of the gravitational field caused by its finite speed of gravity as follows from Eq. (\[yu\]).
In his first paper [@s1] Samuel incorrectly assumed that the experiment directly compared the radio position of the quasar with the optical position of Jupiter, and that the direction of Jupiter was determined by “sunlight that has been reflected off of Jupiter” (see the second paragraph in section III of Samuel’s paper [@s1] describing figure 1 which is similar with figure 2 of Samuel’s paper [@s2]). This assumption would correspond to direct measurement of the angle $\theta$ and hence no $v_J/c$ terms would be observed since they are not evident in Eq. (\[6\]). This explains why Samuel has erroneously decided “that the $v/c$ effects are too small to have been measured in the recent experiment involving Jupiter and quasar J0842+1845” [@s1]. The experiment, however, monitored the position of the quasar as a function of the atomic time by the arrival of the quasar’s photons at the telescope, while the Jupiter’s position entering the time delay Eq. (\[tde\]) was determined separately by fitting VLBI data for the quasar to a precise JPL ephemeris, evaluated at the same atomic time as the arrival of a photon via standard transformations from ephemeris time to atomic time [@apj]. The result of our fitting procedure was that Jupiter deflects light from its retarded position ${\bm x}_J(s)$ but not from its present position ${\bm x}_J(t)$. Thus, the difference $\Delta-\Delta_S$ was measured and the $v_J/c$ correction $\Delta_R$ was determined within precision of 20% [@apj]. The measurement of the post-Newtonian correction (\[7\]) to the Shapiro time delay (\[p2\]) is direct demonstration that gravity does propagate with the same speed as the speed of light. Samuel’s claim that “Fomalont and Kopeikin’s announcement that the speed of gravity is the speed of light to within 20% has no content” is based on his inability to distinguish between gravitational and electromagnetic effects in the post-Newtonian expansion of the Lorentz-invariant time delay Eq. (\[tde\]) predicting that any moving body deflects light by its gravitational field from the retarded position in accordance with the causal (Lorentz-invariant) nature of gravity.
Discussion and Further Particular Comments
==========================================
On the Ideology of the Experiment
---------------------------------
The paper by Samuel [@s2] is an attempt to protect his misleading calculations [@s1] published in [@s1] which conceptual inconsistency was revealed in [@apj; @cqg; @abgr; @vc]. Unfortunately, it does not provide either new mathematical details or more deep insight to the problem. By making use of a linearized Lorentz transform of a spherically-symmetric gravitational field Samuel succeeds in calculation of the first few terms of the differential Shapiro time delay caused by moving Jupiter that approximates the original result by Kopeikin [@apjl] (see Eq. (\[tde\])) given in terms of the retarded time, $s$, connecting the point of observation, ${\bm x}\equiv{\bm x}(t)$, and the retarded position of a light-ray deflecting body (Jupiter), ${\bm x}_J(s)$. Our derivation of Eq. (\[tde\]) makes it clear that the electromagnetic signal from a quasar is observed at the time, $t$, and Jupiter deflects it at the retarded time, $s=t-r/c$, where $r=|x(t)-x_J(s)|$ is the radial coordinate of Jupiter with respect to observer directed along the null characteristic of the gravitational field defined by the null cone equation $\eta_{\mu\nu}r^\mu r^\nu=0$. Both the retarded coordinate of Jupiter, ${\bm x}_J(s)$, and the retarded time, $s$, originate from the Lienard-Wiechert solution of the linearized Einstein equations which is a hyperbolic (wave-type) D’Alembert equation. Lorentz-invariant theory of the propagation of light rays through the time-dependent field of the gravitational retarded potentials reveals that the light particle (photon) is deflected by moving Jupiter when it is located at the retarded position ${\bm x}_J(s)$ due to the finite speed of gravity. Experimental testing whether Jupiter deflects light from its orbital position taken at the retarded time $s$ due to the finite speed of gravity or at the time of observation $t$, is a direct probe of the numerical value of the speed of gravity which must be equal to the speed of light according to Einstein. This is the key idea of the experiment which has been put forward in our publication [@apjl] and practically tested in September 2002 [@apj]. Unfortunately, the einsteinian gravitational physics of the experiment is greatly misunderstood and conceptually misrepresented both in Samuel’s papers [@s1; @s2] and in [@ass; @will; @car] [^6].
What Was Observed and Tested in the Experiment
----------------------------------------------
Samuel prefers to re-express our original result for time delay (\[tde\]) in terms of the “observable angle” $\theta_{obs}\equiv\theta$ in terms of “the distance of the closest approach $\xi$”. The vertex of this angle is at the point of observation, ${\bm x}(t)$, and it has two legs – one leg is directed in the sky towards the quasar and another one is directed towards retarded position of Jupiter, ${\bm x}_J(s)$. If both Jupiter and the quasar were observed simultaneously in radio or in optics, then, the legs composing the “observable angle” $\theta$ would be formed by the null characteristics of electromagnetic signals emanating correspondingly from the quasar and from Jupiter to the observer. However, Jupiter was not observed by VLBI in the Fomalont-Kopeikin experiment at all, only the quasar was - so that only one leg of “the observable angle” is the null characteristic of electromagentic field. Jupiter affected the propagation of the quasar radio wave by its gravitational field and the effective position of Jupiter in the sky (the second leg of “the observable angle”) was determined in the process of the data analysis of the residual phase of the quasar radio wave. The time delay Eq. (\[tde\]) tells us that the direction in the sky connecting the observer and Jupiter is the null characteristic of the gravitational field of moving Jupiter. Thus, Samuel’s “observable angle” $\theta_{obs}\equiv\theta$ is made of the null lines one of which belongs to the electromagnetic field (radio wave from the quasar) but another one belongs to the gravitational field of moving Jupiter, which deflects the radio signal of the quasar not instantaneously but with the retardation to comply with the Lorentz-invariant symmetry of the Einstein equations [^7].
The goal of our measurement was to distinguish between the two hypothesis:
- Einsteinian gravity. Jupiter deflects light by its gravitational field from its retarded position ${\bm x}_J(s)$ as shown in our time-delay Eq. (\[tde\]);
- Newtonian gravity. Jupiter deflects light by its gravitational field instantaneously and its position in the time delay Eq. (\[tde\]) must be taken at the time of observation, $t$.
In effect, we did not measure any angles in the sky directly during the time of observation as Samuel seems to believe. What was measured is the residual phase $\delta\varphi$ of the quasar radio wave, allowing us to determine the difference between times of arrival of the same front of the quasar radio wave to two VLBI stations [@apj]. This difference depends on the position of Jupiter on its orbit and our goal was to prove that the observed differential time delay is affected by the gravitational field of Jupiter acting from its retarded position, ${\bm x}_J(s)$, but not from its position, ${\bm x}_J(t)$, taken at the time of observation. The difference between the two hypothesis gives rise to the post-Newtonian correction Eq. (\[7\]) to the Shapiro time delay (\[p2\]) which had to be zero if the speed of gravity would be infinite. We proved experimentally [@apj] that the first hypothesis is correct within 20% (that is the post-Newtonian correction $\Delta_R\not=0$) which means that the speed of gravity has the same value as the speed of light.
Samuel misinterprets the physical origin of the observed retardation-of-gravity effect in the time delay of light because of his insufficiently elaborated mathematical solution of the problem of light propagation in the time-dependent gravitational field of a moving body. While our approach starts from the basic principles and solution of the gravity field equations, Samuel’s development deals only with the final product of the theory – the time delay Eq. (\[6a\]), which incorporates effects of the propagation of both gravitational and electromagentic fields. When making Lorentz transformation of the time delay Eq. (\[6a\]) Samuel has missed the point that this transformation transforms not only the electromagnetic but the gravitational field as well. Thus, the experiment effectively measures the discrepancy between the Lorentz-invariant symmetry of the gravitational field with respect to the Lorentz transformation symmetry of the electromagnetic field which could arise due to the possible difference between the speed of gravity and light. The experiment has confirmed the Lorentz invariance of the gravitational field because no difference between the speeds of gravity and light was found within 20% [@apj].
On the Relativistic Jargon and the Symbol $c$
---------------------------------------------
Einstein’s theory of general relativity predicts that an electromagnetic wave is deflected by the gravitational field of moving body (Jupiter) from its retarded position, ${\bm x}_J(s)$, taken at the retarded instant of time $s=t-r/c$, where $c$ is conceptually the fundamental speed of propagation of gravitational field because it enters the Lienard-Wiechert gravitational potentials. For historical reasons, it is customary to call the symbol $c$ as “the speed of light”. We emphasize however that this term is misleading when one considers the Lienard-Wiechert solution of the Einstein gravity field equations since in this case $c$ is the speed of gravity characterizing the degree of opening of the null cone made up of the characteristics of the gravitational field. This interpetation of the symbol $c$ entering the retarded coordinate of a moving gravitating body remains true in the case of the physical process of scattering of electromagnetic wave on the Lienard-Wiechert gravitational potential of this body. This point was misinterpeted by Asada [@ass], Will [@will], and Carlip [@car]. Samuel [@s2; @s1] seems to be also confused with the existing conventional terminology and is about to interpret any physical effect depending on $c$ as associated with propagation of light because $c$ is always “the speed of light” by his definition [^8]. Einstein’s gravity field equations contain “the speed of light” constant $c$ but it does not mean that the measurement of this fundamental constant for gravitational field (as it is done in [@apj]) is reduced (or equivalent) to the measurement of the physical speed of light even if the light is used for such measurement. Various fasets of the symbol $c$ and the pitfalls related to the physical interpetations of various experiments have been discussed recently in [@cqg; @uzan].
On the Point of the Closest Approach
------------------------------------
Samuel’s calculations are also based on the assumption that an electromagnetic signal is effectively deflected at the point of its closest approach to the moving Jupiter [^9]. From the first glance this assumption looks plausible but turns out to be generally incorrect in exact mathematical solution of the problem shown in the present paper as well as in our previous works [@ksge; @ks; @km; @apjl; @abgr; @pla] and the work of Klioner [@kl]. First of all, gravitational field is long-ranged and the process of gravitational light deflection (electromagentic wave scattering) does not take place in one event. Relativistic time delay of light is the integral effect and, strictly speaking, it can be expressed in terms of the impact parameter of the light ray, taken at the time of the closest approach $t^*$, only approximately (see [@ksge; @ks; @km] for more detail). But even in the case of the approximation the point of the closest approach of light to the moving body loses its physical content when the Lorentz transformation is applied to transform the time delay from the static to a moving frame. Advanced calculation (missed in [@s2; @s1]) makes it evident that the integral effect of the light-ray time delay is coming out from the retarded position of the light-ray deflecting body [@ks; @km; @apjl] which is in accordance with the Lienard-Wiechert solution of the Einstein gravity field equations describing propagation of the gravitational field [^10].
Recently Klioner [@kl] has calculated the deflection and the time delay of light in the gravitational field of a uniformly moving body by making use of the Lorentz transformation technique. In contrast to the linearized Samuel’s “theory” [@s2; @s1], Klioner’s calculation takes into account all velocity-dependent terms in the Lorentz transformation which significantly supersedes Samuel’s consideration [@s2; @s1]. Klioner’s paper [@kl] delivers an independent proof that Jupiter must deflect light from its retarded position ${\bm x}_J(s)$ in accordance with the Lienard-Wiechert solution of the gravity field equations [@ks; @km; @apjl; @cqg]. The effect of the retardation of gravity in the case of a uniformly moving body is equivalent to the property of the gravitational field to be Lorentz-invariant and to propagate with the fundamental speed $c$ which, at the same time, is the parameter of the Lorentz transformation of the Einstein gravity field equations. Our experiment has tested this property of gravity by observing that Jupiter’s gravitational field deflects the quasar radio wave out of Jupiter’s retarded position lying on the null characteristic of the gravitational field which connects the VLBI station and Jupiter and is given by the retarded time Eq. (\[2\]).
Different mathematical technique [@ksge; @ks; @km; @pla] also demonstrates that the point of the closest approach of photon to the moving body plays no physical role because the time of the closest approach, $t*$, drops out of the final equation for the light bending and/or relativistic time delay. What matters is the retarded time $s=t-r/c$ of the Lienard-Wiechert solution of Einstein’s equations, where $c$ is the speed of propagation of gravity from the moving body to the point of observation. The time of the closest approach, $t*$, approximates the retarded time $s=t-r/c$, if the impact parameter of the light ray to the moving body is small enough (see [@ksge; @apjl] for more detail). This mathematical approximation is physically possible because the gravitational field propagates with the same speed as light in general theory of relativity. In fact, Samuel [@s1; @s2] unknowingly used this approximation to replace the retarded time $s$ with the time of the closest approach $t^*$ which has led him to the confusion of the propagation of gravity effect with that of the propagation of the quasar signal. His statement that the "Shapiro time delay difference is due to relatively [*short-distance effects*]{} is correct but these [*short-distance effects*]{} occur near the retarded position of Jupiter ${\bm x}_J(s)$ taken at the retarded time $s$ while the time of the closest approach $t^*$ is an approximation which, if it is used improperly, simply misinterpets the gravitational physics of the Jupiter-quasar experiment.
On the Minkowski Diagram of the Jupiter-Quasar Experiment
---------------------------------------------------------
Samuel’s graphic analysis of the Fomalont-Kopeikin experiment is shown in figures 1 and 2 of the paper [@s2]. Unfortunately they do not grasp the relativistic spirit of the Jupiter-quasar experiment. The Minkowski diagram is much more adequate for picking up the idea of the measurement of the speed of gravity with VLBI. Analysis of the experiment in terms of the Minkowski diagrams is given by Kopeikin in the paper [@cqg] and in the proceedings of the 14th Midwest Relativity Meeting [@mw]. Minkowski diagram of the experiment shown in Fig. 1 of the present paper clearly demonstrates the origin of the observed retarded position of Jupiter as caused by the Lorentz-invariant nature of the gravitational field and its finite speed of propagation. A null direction connecting observer and Jupiter in the relativistic time delay Eq. (\[tde\]) is that corresponding to the null characteristic of the gravitational field. This is because of two facts: (1) it originates from the retarded Lienard-Wiechert solution of the gravity field equations, and (2) there is no radio or light wave emitted or reflected by and propagating from Jupiter towards observer during the time of the experiment. Thus, the measurement of the direction of the null characteristic of the gravitational field through the retarded coordinate of Jupiter, ${\bm x}_J(s)$, through the relativistic time delay is a direct confirmation of Einstein’s prediction that gravitational field has the same speed of propagation as the speed of light. The reader should not interpret the retarded position of Jupiter shown in figure 2 of Samuel’s paper [@s2] as caused by propagation of sunlight reflected from Jupiter as Samuel [@s1; @s2] erroneously used to believe. Unfortunately, this was either overlooked or ignored by referees of papers [@s1; @ass] despite of our persistent attempts to draw their attention to this, inconsistent with observations, fact. As explained in [@apj; @cqg] the radio emission of Jupiter or the sunlight reflected by Jupiter can not be, and was not, observed at any VLBI station due to the specific technical limitations of VLBI. Anyone who discusses the nature of the Jupiter-quasar experiment must learn how VLBI operates and detect radio signals before making any statements about the nature of the measured effect of the retardation of gravity. The retarded position of Jupiter, measured in the experiment through the best fit of the observed VLBI time delay to its theoretical value, is due to the finite speed of gravity and reflects the fundamental fact that the gravitational field is Lorentz-invariant and its null characteristics coincide with the null characteristics of the electromagnetic field withing 20% [@apj].
On the speed of gravity parameter $c_g$
---------------------------------------
Einstein’s theory of general relativity does not require introduction of any parameter for the speed of gravity $c_g$ because it is Lorentz invariant and the speed of gravity is equal to the speed of light. All general-relativistic effects associated with the speed of gravity can be easily and unambiguously identified through the retarded positions of the gravitating bodies in the post-Minkowskian solutions of the gravity field equations and equations of motion. Will [@book] has decided to introduce the parameter $c_g$ for the speed of gravity to facilitate discussion of the relativistic effects associated with the speed of gravity from those caused by the “speed of light” $c$. Introduction of $c_g$ to the Einstein equations is not trivial and requires to retain all differential relationships of general relativity for any value of $c_g$. This task was not fulfilled in [@book].
We have found [@cqg] the $c_g$-parametrization of Einstein’s equations which preserves all differential and algebraic properties of those equations for any value of the parameter $c_g$. This $c_g$-parametrization requires introduction of a global unit vector field, $V^\alpha$, which goal is to keep the Lorentz invariance of the gravity field equations. The difference between $c_g$ and $c$ leads to one kind of observable effects while existence of spatial components, ${\bm V}$, of the vector field $V^\alpha$ leads to apperance of the preferred frame effects (the PPN parameters $\alpha_1$, $\alpha_2$) which have been strongly limited by pulsar timing and lunar laser ranging observations [@book]. Hence, in the paper [@cqg] we did not analyzed the preferred frame effects caused by ${\bm V}$ and worked in the coordinate system, where $V^\alpha=(1,0,0,0)$, to concentrate on the discussion of the retardation of gravity effect [@apj]. Therefore, spatial vector field ${\bm V}$ does not appear explicitly in our derivation of the relativistic time delay equation. However, correct transformation of our time delay equation, derived in [@cqg] for the case of $c_g\not=c$, from one frame to another requires accounting for the preferred frame vector filed ${\bm V}$, which was not done by Samuel in deriving his equation (7.2) in [@s2]. Thus, Samuel’s criticism of my paper [@cqg] is completely unfounded and is based on his misunderstanding of the mathemtical formalism which has been worked out in [@cqg].
[00]{} Kopeikin S. M., Sch[" a]{}fer G., Gwinn C. R., and Eubanks T. M.,[*Phys. Rev. D*]{} [**59**]{} 84023 (1999) Kopeikin S.M. and Schäfer G., [*Phys. Rev. D*]{} [**60**]{} 124002 (1999) Kopeikin S.M. and Mashhoon B., [*Phys. Rev. D*]{} [**65**]{} 064025 (2002) Kopeikin S.M., [*Astrophys. J. Lett.*]{} [**556**]{} L1 (2001) Fomalont E.B. and Kopeikin S.M., in: Proc. 6th European VLBI Network Symp., eds. E. Ros, R.W. Porcas, A.P. Lobanov & J.A. Zensus, (Bonn: MPIfR) 53 (2002) Fomalont E.B. and Kopeikin S.M.,[*Astrophys. J.*]{} [**598**]{} 704 (2003) Samuel S., Intern. J. Mod. Phys., [**D13**]{} 1753 (2004) Samuel S., [*Phys. Rev. Lett.*]{} [**90**]{} 231101 (2003) Damour T., “The problem of motion in Newtonian and Einsteinian gravity”, in: [*300 Years of Gravitation*]{}, eds. S.W. Hawking and W. Israel (Cambridge: Cambridge Univ. Press 1987) pp. 128–198 Landau L.D. and Lifshitz E.M., “The Classical Theory of Fields” (Oxford: Pergamon 1971) Kopeikin S.M., [*Class. Quantum Grav.*]{} [**21**]{} 3251 (2004) Asada H., [*Astrophys. J. Lett.*]{} [**574**]{} L69 (2002) Klioner S.A., [*Astron. Astrophys.*]{} [**404**]{} 783 (2003) Bel L, Deruelle N., Damour T., Ibanez J. and Martin J., [*Gen. Rel. Grav.*]{} [**13**]{} 963 (1981) Will C.M., [*Astrophys. J.*]{} [**590**]{} 683 (2003) Fomalont E.B. and Kopeikin S.M., “Aberration and the Speed of Gravity in the Jovian Deflection Experiment”, astro-ph/0311063 (2003) Kopeikin S.M. and Fomalont E.B., “The Ultimate Speed of Gravity and $v/c$ Correction to the Shapiro Time Delay”, gr-qc/0310065 (2003) Carlip, S., [*Class. Quantum Grav.*]{} [**21**]{} 3803 (2004) S. Kopeikin “Jovian Light-Deflection Experiment and Its Results”, online publications, http://www.lsc-group.phys.uwm.edu/mwrm14/ Ellis, G.F.R. and Uzan, J.-P., “$c$ is the speed of light, isn’t it?”, gr-qc/0305099 (2003) Kopeikin S.M., [*Phys. Lett. A*]{} [**312**]{} 147 (2003) Will C.M., “Theory and experiment in gravitational physics” (Cambridge: Cambridge Univ. Press 1993 )
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[^1]: Greek indices run from 0 to 3. Roman indices run from 1 to 3. The Greek indices are rised and lowered with the Minkowski metric. Bold letters denote spatial vectors. Repeated indices mean the Einstein summation rule. Euclidean dot and cross products of two vectors are denoted as ${\bm a}\cdot{\bm b}$ and ${\bm a}\times{\bm b}$ respectively.
[^2]: Because Eq. (\[2\]) is the null cone of the gravitational field of moving Jupiter we used [@apj; @cqg] a symbol $c_g$ in there instead of $c$ to mark the parameter which we fit to observations in the data processing procedure.
[^3]: Notice that vector $k^\alpha$ does not coincide with the null characteristic of the gravitational field given by the interval $x^\alpha-x^\alpha_J(s)$ (see Fig. 1).
[^4]: These quantities appear in Eq. (1.3) in Samuel’s paper [@s2] along with the “distance of the closest approach” $\xi$ spontaneously without rigorous mathematical description so that their meaning is fuzzy.
[^5]: Similar mistake has been done also by Will in [@will] who used insufficiently elaborated $c_g$-parametrization of the Einstein gravity field equations.
[^6]: Critical discussion of the formally correct, but conceptually misleading point of view on the gravitational physics of the process of light scattering by the gravitational field of a moving body presented in [@car], requires more elaborated mathematical technique than that presented in the present paper, and will be given somewhere else. Some details are available in [@mw].
[^7]: See Fig. 1 for graphical illustration of this point.
[^8]: The same jargon is used by Asada [@ass] and Will [@will] that prevents them to see the true nature of the retardation of gravity effect measured in the Fomalont-Kopeikin experiment [@apj].
[^9]: The same assumption was also used by Will [@will]
[^10]: We emphasize once again that there is no light propagating from the retarded position of the light-ray deflecting body in the time delay Eq. (\[tde\]) to observer as it was erroneously deduced by Asada [@ass], it is gravity which propagates.
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abstract: 'This paper investigates the design of a robust output-feedback linear parameter-varying (LPV) gain-scheduled controller for the speed regulation of a surface permanent magnet synchronous motor (SPMSM). Motor dynamics is defined in the $\alpha - \beta$ stationary reference frame and a parameter-varying model formulation is provided to describe the SPMSM nonlinear dynamics. In this context, a robust gain-scheduled LPV output-feedback dynamic controller is designed to satisfy the asymptotic stability of the closed-loop system and meet desired performance requirements, as well as, guarantee robustness against system parameter perturbations and torque load disturbances. The real-time impact of temperature variation on the winding resistance and magnet flux during motor operations is considered in the LPV modelling and the subsequent control design to address demagnetization effects in the motor response. The controller synthesis conditions are formulated in a convex linear matrix inequality (LMI) optimization framework. Finally, the validity of the proposed control strategy is assessed in simulation studies, and the results are compared to the results of the conventional field-oriented control (FOC) method. The closed-loop simulation studies demonstrate that the proposed LPV controller provides improved transient response with respect to settling time, overshoot, and disturbance rejection in tracking the velocity profile under the influence of parameter and temperature variations and load disturbances.'
author:
- 'Shahin Tasoujian, Jaecheol Lee, Karolos Grigoriadis, and Matthew Franchek'
title: 'Robust linear parameter-varying output-feedback control of permanent magnet synchronous motors'
---
Introduction
============
Permanent magnet synchronous motors (PMSMs) are prevalent in industry and in various electromechanical applications, such as, electrical appliances, robotic systems and electric vehicles, due to their compact structure, high torque density, high power density and high-efficiency [@boldea1992vector; @zhong1997analysis; @yanliang2001development]. However, because of the inherent nonlinear dynamics, strong coupling effects and significant system parameter variability [@pillay1988modeling; @cai2017optimal], the precise speed and position control of a PMSM is a challenging task. Traditionally, the field-oriented control (FOC) method has been employed as a vector control of both magnitude and angle of the flux enabling independent control of torque and speed. Consequently, fast and high precision motor control can be achieved. For this reason, the motor drives implemented with the FOC method are typically comprised of two loops in a cascade manner in the $d-q$ rotating reference frame [@zhu2019performance]. The current control loop is the inner loop for the stator current to follow its reference value while the speed control loop is the outer loop taking into account speed error signals and providing reference signals to the inner loop [@giri2013ac].
In the FOC method, a proportional-integral (PI) controller is typically implemented for both current and speed control due to its simple design structure. The PI controller gains are typically determined through nominal motor parameters to satisfy motor performance specifications [@kim2016self]. However, PI control is not suitable for applications where high performance and high precision is required. When motor parameter variations and disturbances are present, robustness and stability issues inevitably arise. As an additional challenge, varying motor temperature has shown to have a significant impact on PMSM speed, current and torque resulting from the reversible demagnitization of the permanent magnet (NdFeB or SmCo) and the temperature-dependence of the stator winding resistance. Hence, traditional PI controllers typically fail to maintain the desired closed-loop motor response in high performance applications. Various robust and nonlinear control methods have been adopted to address the parameter variability, as well as, to cope with the nonlinearity in the PMSM model [@zhao2019robust]. The sliding mode control (SMC) method has been proposed to assure fast response and robustness in the presence of nonlinearity in the model. However, the SMC method inherently causes a chattering problem due to the signum function, which leads to deteriorating performance at steady-state [@baik1998robust; @kim2010high; @zhang2012nonlinear]. Disturbance observers (DOBs) have been proposed to estimate the disturbance for its compensation. [@zhao2015adaptive] studied the case with unknown load torque and model parameters and proposed an adaptive observer-based control method for the speed tracking in the PMSMs. Although DOBs can help improve the capability of a motor to reject disturbances, a disadvantage lies in the fact that the methodology is required to have full knowledge of the PMSM parameters to ensure the stability of the DOBs [@solsona2000nonlinear; @chang2010robust]. Additionally, the fuzzy logic control method has been proposed for the control of PMSMs and has shown an improved performance regarding robustness to disturbance rejection. However, shortcomings reside in the fact that membership functions rely solely on the designer’s experience and it demands heavy computations [@yu2007fuzzy; @chaoui2011adaptive].
Recently, the use of linear parameter-varying (LPV) gain-scheduling control techniques for the PMSM control problem has drawn the attention of researchers due to the controller’s scheduling nature providing the ability to handle system parameter variations and nonlinearities in a systematic framework. In this regard, a static fixed-gain state-feedback LPV controller with an estimator has been proposed for PMSM control [@lee2017lpv], where the estimator is utilized to provide the state-feedback controller with the full-state information needed to generate the control input. The authors in [@lee2017lpv] used the polytopic LPV description resulting in a relatively conservative control design, especially for the case of slow parameter variations. The gain-scheduling control technique is an extension of the linear control design to handle nonlinear and time variations, where a scheduling parameter vector captures the information about the nonlinearities or time-varying behavior of the system. The LPV gain-scheduling control methodology was first introduced in [@shamma1991guaranteed] to overcome the shortcoming of conventional gain-scheduling control techniques, namely, lack of closed-loop stability and performance guarantees. Unlike conventional gain-scheduling design methods which are based on interpolation between several independently designed LTI controllers for different fixed operating points, LPV gain-scheduling control design provides a direct, efficient, systematic and global control approach, which also guarantees closed-loop stability and performance. Stability analysis and control synthesis of LPV systems have been addressed extensively in the control literature in the past decade [@apkarian1998advanced; @wu2001lpv; @tasoujian2019delay; @tasoujian2019robust; @tasoujian2020robust].
In the present paper, first, the $\alpha-\beta$ stationary reference framework is considered for the surface permanent magnet synchronous motors (SPMSMs) modeling. The SPMSM model is assumed to be subject to varying parameters and torque load disturbances that impair the response of the closed-loop system to track a reference speed profile. Subsequently, we develop a LPV representation to describe the SPMSM dynamics. Resistance and magnetic fluxes in SPMSMs vary with temperature. To this end, temperature variation is taken into consideration in the LPV modeling as an LPV scheduling parameter. The presented formulation allows a systematic control design seeking to handle the temperature-dependent parameter variations and the model uncertainties in SPMSMs. To minimize the conservatism of the control design in meeting performance specifications, a parameter-dependent Lyapunov function approach is utilized to design an LPV gain-scheduled dynamic output-feedback controller to track the commanded reference speed profile and minimize the effect of disturbances and parameter variations over the entire operating envelope of the motor. The proposed dynamic LPV control design method guarantees asymptotic stability and robustness against disturbances and uncertainties in terms of the closed-loop system’s induced $\mathcal{L}_2$-norm performance index. A linear matrix inequality (LMI) framework is adopted to formulate the proposed $\mathcal{H}_{\infty}$ control synthesis problem in a convex, computationally tractable setting, which can be solved efficiently using numerical optimization algorithms. Finally, the performance of the proposed method is evaluated and validated in a computer simulation environment and compared to the conventional FOC method with a fixed-gain PI controller.
The notation to be used in the paper is standard and as follows: $\mathbb{R}$ denotes the set of real numbers, and $\mathbb{R}^n$ and $\mathbb{R}^{k \times m}$ are used to denote the set of real vectors of dimension $n$ and the set of real $k \times m$ matrices, respectively. $M \succ \mathbf{0}$ shows the positive definiteness of the matrix $M$ and the transpose of a real matrix $M$ is shown as $M^{\text{T}}$ . Also, $\mathbb{S}^{n}$ denotes the set of real symmetric $n \times n$ matrix. In a symmetric matrix, terms denoted by asterisk, $\star$, will be induced by symmetry as shown below: $$\left[\!\!\begin{array}{cc}
S\! +\! W +\! J +\! (\star) & \!\!\!\star \\
Q &\!\!\! R
\end{array}\!\!\right] :=\! \left[\!\!\begin{array}{cc}
S \!+ W \!+ W^\text{T} \!+ J \!+ J^\text{T} & \!\! Q^\text{T} \\
Q & \!\! R
\end{array}\!\!\right]$$ where $S$ is symmetric. $\mathbf{He} [\mathbf{M}]$ is Hermitian operator defined as $\mathbf{He} [\mathbf{M}] \triangleq \mathbf{M} + \mathbf{M}^{\text{T}}$ and $\mathcal{C} (J,\: K)$ stands for the set of continuous functions mapping a set $J$ to a set $K$.
The outline of the paper is as follows. Section presents the mathematical modeling for the SPMSMs and the proposed LPV model formulation. In Section , the output-feedback LPV gain-scheduling control technique is described considering scheduling parameters that capture the nonlinearity and temperature-dependent variability of the SPMSM model. Section outlines the closed-loop results and describes the performance evaluation of the proposed LPV controller in a computer simulation environment. Finally, Section concludes the paper.
SPMSM modeling {#sec:modeling}
==============
SPMSM dynamics
--------------
We consider a three-phase synchronous motor with permanent magnets where the magnetic coupling between the phases and the inductance variation due to magnetic saturation are assumed to be negligible. Additionally, the magnetic flux ganerated by the excitation is assumed to have an ideal sinusoidal density distribution. Consequently, the simplified dynamic model for SPMSMs can be expressed in the $\alpha - \beta$ stationary reference frame as follows [@Hwang2014H2CB]: $$\begin{aligned}
\dot{\theta} & = \omega, \nonumber\\[0.0cm]
\dot{\omega} & = \dfrac{1}{J_{m}}\left(-B\omega-K_{t}\textrm{sin}(p\theta)i_{\alpha}+K_{t}\textrm{cos}(p\theta)i_{\beta}-\tau_{L}\right), \nonumber\\[0.0cm]
\dot{i}_{\alpha}&=\dfrac{1}{L_{s}} \left( -R_{s} i_{\alpha} + p\lambda_{pm} \omega \textrm{sin}(p\theta) + v_{\alpha} \right), \nonumber\\[0.0cm]
\dot{i}_{\beta}&=\dfrac{1}{L_{s}} \left(-R_{s}i_{\beta}-p\lambda_{pm} \omega\textrm{cos}(p\theta)+v_{\beta}\right),
\label{PMSM_model} \end{aligned}$$ where $\theta$ stands for the mechanical rotor angular position \[rad\], $\omega$ is the mechanical rotor speed \[rad/sec\], $v_{\alpha}$, $v_{\beta}$ and $i_{\alpha}$, $i_{\beta}$ are the voltages \[V\] and currents \[A\] in the $\alpha - \beta$ stationary reference frame. In this model, $L_{s}$ denotes the stator inductance \[H\], $R_{s}$ is the stator resistance \[$\Omega$\], $\lambda_{pm}$ is the magnetic flux of the motor \[Wb\], $J_m$ is the moment of inertia $[kg.m^2]$, $p$ denotes the number of magnet pole pairs, $B$ is the viscous friction coefficient \[$\textrm{N}\cdot\textrm{m}\cdot\textrm{sec/rad}$\], $\tau_{L}$ is the load torque \[$\textrm{N}\cdot\textrm{m}$\], and $K_{t}=\dfrac{3}{2} p \lambda_{pm}$ is the torque constant $[V\cdot\textrm{rad/sec}]$. To assess the closed-loop SPMSM performance, the following tracking errors are defined for the quantities of interest: $$\begin{aligned}
e_{w}&=\omega^* - \omega, \nonumber\\
\displaystyle e_{z}&=\int_{0}^{t} e_{\omega} dx, \nonumber\\
e_{\alpha}&=i_{\alpha}^*-i_{\alpha}, \nonumber\\
e_{\beta}&=i_{\beta}^*-i_{\beta},
\label{tracking_error}\end{aligned}$$ where $\omega^*$ is the desired motor speed, and $i_{\alpha}^*$ and $i_{\beta}^*$ are the desired currents in the stationary reference $(
\alpha - \beta)$ frame, respectively. Additionally, $e_{z}$ represents the integral of speed error and $e_{\alpha}$ and $e_{\beta}$ are the current errors in the $\alpha - \beta$ stationary reference frame, respectively. The desired torque, $\tau^*$, the desired currents, $i_{\alpha}^*$ and $i_{\beta}^*$, and the voltage inputs to the motor, $v_{\alpha}$ and $v_{\beta}$ are defined as follows $$\begin{aligned}
\tau^*&=J_{m}\dot{\omega}^*+B\omega^*, \nonumber\\[0.0cm]
i_{\alpha}^*&=-\dfrac{\tau^*\textrm{sin}(p\theta)}{K_{t}}, \nonumber\\[0.0cm]
i_{\beta}^*&=\dfrac{\tau^*\textrm{cos}(p\theta)}{K_{t}}, \nonumber\\[0.0cm]
v_{\alpha}&=L_{s}\dot{i}_{\alpha}^*+R_{s}i_{\alpha}^*-p\lambda_{pm}\omega^*\textrm{sin}(p\theta)-u_{\alpha}, \nonumber\\[0.0cm]
v_{\beta}&=L_{s}\dot{i}_{\beta}^*+R_{s}i_{\beta}^*+p\lambda_{pm}\omega^*\textrm{cos}(p\theta)-u_{\beta},
\label{5}\end{aligned}$$
where $u_{\alpha}$ and $u_{\beta}$ are the control inputs. Hence, the error dynamics can be obtained by combining (\[PMSM\_model\]), (\[tracking\_error\]), and (\[5\]) as follows $$\begin{aligned}
\dot{e}_{z}&=e_{w}, \nonumber\\[0.0cm]
\dot{e}_{\omega}&=\dfrac{1}{J_{m}}\left(-Be_{\omega}-K_{t}\textrm{sin}(p\theta)e_{\alpha}+K_{t}\textrm{cos}(p\theta)e_{\beta}+\tau_{L} \right), \nonumber\\[0.0cm]
\dot{e}_{\alpha}&=\dfrac{1}{L_{s}} \left(-R_{s}e_{\alpha}+p\lambda_{pm}\textrm{sin}(p\theta)e_{\omega}+u_{\alpha} \right), \nonumber\\[0.0cm]
\dot{e}_{\beta}&=\dfrac{1}{L_{s}}\left(-R_{s}e_{\beta}-p\lambda_{pm}\textrm{cos}(p\theta)e_{\omega}+u_{\beta}\right).
\label{final_error_dynamics}\end{aligned}$$ Subsequently, an LPV representation for the introduced SPMSM error dynamics is developed to enable LPV control design:
LPV model formulation
---------------------
In order to be able to implement the proposed LPV control methodology to the SPMSM dynamics case study, we first rewrite the described system (\[final\_error\_dynamics\]) as a proper LPV model. LPV systems correspond to a class of linear systems, whose dynamics depend on time-varying parameters, known as the scheduling parameters. Therefore, considering (\[final\_error\_dynamics\]), the first two LPV scheduling parameters are defined as follows $$\begin{aligned}
\rho_1(\theta(t)) &= p\lambda_{pm}\textrm{sin}(p\theta(t)), \nonumber\\
\rho_2(\theta(t)) &= p\lambda_{pm}\textrm{cos}(p\theta(t)).
\label{eq:sch_par2}\end{aligned}$$ Since the scheduling parameters in (\[eq:sch\_par2\]) are trigonometric functions, they can be bounded as follows $$\begin{array}{cc}
-p\lambda_{pm}\leq\rho_{1}(\theta(t))\: \textrm{and} \: \rho_{2}(\theta(t)) \leq p\lambda_{pm}. \end{array}$$ It is known that temperature variation has a significant effect on SPMSM performance. Consequently, we define temperature as the third scheduling parameter: $$\begin{array}{cc}
\underline{T} \leq \rho_{3}(t) = T(t) \leq \overline{T},
\end{array}
\label{8}$$ where $\underline{T}$, and $\overline{T}$ are the minimum and maximum motor operating temperatures in $^\circ C$, respectively. Resistance and magnetic fluxes vary considerably throughout the motor operation as a function of temperature. Embedded insulate temperature sensors or estimation algorithms can be used to provide instantaneous measurements or estimates of stator winding temperature [@jun2018temperature]. The following relations can be used to obtain empirical expression for these motor parameter variations as functions of temperature $$\begin{aligned}
R_{s}(\rho_{3}(t))&=R_{s0} \left( \dfrac{235+\rho_{3}(t)}{310} \right), \nonumber\\[0.2cm]
\lambda_{pm}(\rho_{3}(t))&=\lambda_{pm0} \left( 1+\dfrac{\alpha(\rho_{3}(t)-30)}{100} \right),
\label{9}
\end{aligned}$$ where $R_{s0}$ is the resistance value of the winding at 75$^\circ C$, $\lambda_{pm0}$ is the flux of the magnet at 30$^\circ C$, and $\alpha$ is the temperature coefficient of the magnet in $\%/^\circ C$ [@sul2011control]. After defining the scheduling parameters, the scheduling parameter vector is represented as, $\boldsymbol{\rho}(t)=[\begin{array}{ccc}
\rho_1(t) & \rho_2(t) & \rho_3(t)\end{array} ]^{\text{T}}$. Subsequently, the LPV representation of the SPMSM dynamics takes the following matrix-vector form $$\begin{aligned}
\dot{\mathbf{e}}(t) &= \mathbf{A}(\boldsymbol{\rho}(t))\mathbf{e}(t) +\mathbf{B}_{1}\tau_{L}(t) + \mathbf{B}_{2}\mathbf{u}(t), \nonumber\\
\mathbf{y}(t) &= \mathbf{C}\: \mathbf{e}(t),
\label{eq:lpverrordynamics}
\end{aligned}$$ where the augmented state vector is defined as $\mathbf{e}(t) = \left[\,e_{z}(t)\quad e_{\omega}(t)\quad e_{\alpha}(t)\quad e_{\beta}(t) \right]^{\text{T}}$, the control input is $\mathbf{u}(t)=\left[\, u_{\alpha}(t)\quad u_{\beta}(t) \, \right]^{\text{T}}$, $\mathbf{y}(t)$ is the measured signal vector, and the state-space matrices of the LPV system (\[eq:lpverrordynamics\]) are as follows
$$\begin{aligned}
\mathbf{A}(\boldsymbol{\rho}(t)) = & \begin{bmatrix}
\quad 0 & 1 & 0 & 0 \\[0.1cm]
\quad 0 & -\dfrac{B}{J_{m}} & -\dfrac{3}{2}\dfrac{\rho_{1}(t)}{J_{m}} & \dfrac{3}{2}\dfrac{\rho_{2}(t)}{J_{m}} \\[0.3cm]
\quad 0& \dfrac{\rho_{1}(t)}{L_{s}}& -\dfrac{R_{s}(\rho_{3}(t))}{L_{s}}& 0 \\[0.3cm]
\quad 0& -\dfrac{\rho_{2}(t)}{L_{s}} & 0 & -\dfrac{R_{s}(\rho_{3}(t))}{L_{s}}
\end{bmatrix},\nonumber\\
\mathbf{B}_{1}\! =\! \begin{bmatrix}
\, 0 \, \\
\, 1 \, \\
\, 0 \,\\
\, 0 \,
\end{bmatrix}\!&,
\mathbf{B}_{2} = \begin{bmatrix}
\, 0& 0 \, \\
\, 0 & 0 \, \\
\, \dfrac{1}{L_{s}} & 0 \, \\
\, 0 & \dfrac{1}{L_{s}} \,
\end{bmatrix}\!,
\, \mathbf{C} = \begin{bmatrix}
\, 1&0&0&0 \, \\
\, 0&1&0&0 \,
\end{bmatrix}\!.
\label{eq:matrices1}\end{aligned}$$
Next, the proposed output-feedback LPV gain-scheduling control design method is described.
LPV control design {#sec:control}
==================
We aim to design an output-feedback LPV gain-scheduled controller for the SPMSM model (\[eq:lpverrordynamics\]) in the context of induced $\mathcal{L}_2$-norm performance specifications. To this end, we consider a generic LPV open-loop system with the following state-space realization $$\begin{aligned}
\dot{\mathbf{x}}(t) & = \mathbf{A}(\boldsymbol{\rho}(t)) \mathbf{x}(t)+ \mathbf{B}_1(\boldsymbol{\rho}(t))\mathbf{w}(t)+ \mathbf{B}_2(\boldsymbol{\rho}(t))\mathbf{u}(t), \nonumber\\[0.10cm]
\mathbf{z}(t) & = \mathbf{C}_1(\boldsymbol{\rho}(t)) \mathbf{x}(t) + \mathbf{D}_{11}(\boldsymbol{\rho}(t)) \mathbf{w}(t) + \mathbf{D}_{12}(\boldsymbol{\rho}(t)) \mathbf{u}(t),\nonumber\\[0.10cm]
\mathbf{y}(t)& = \mathbf{C}_2(\boldsymbol{\rho}(t)) \mathbf{x}(t)+ \mathbf{D}_{21}(\boldsymbol{\rho}(t)) \mathbf{w}(t),\nonumber\\[0.10cm]
\mathbf{x}(0) & = \mathbf{x}_0,
\label{LPVsystem} \end{aligned}$$ where $\mathbf{x} \in \mathbb{R}^n$ is the system state vector, $\mathbf{w} \in \mathbb{R}^{n_w}$ is the vector of exogenous disturbances with finite energy in the space $\mathcal{L}_2[0, \:\: \infty]$, $\mathbf{u} \in \mathbb{R}^{n_u}$ is the control input vector, $\mathbf{z}(t) \in \mathbb{R}^{n_z}$ is the vector of controlled output, $\mathbf{y}(t) \in \mathbb{R}^{n_y}$ is the vector of measured output, $\mathbf{x}_0 \in \mathbb{R}^n$ is the initial system condition. The state space matrices $\mathbf{A}(\cdot)$, $\mathbf{B}_1(\cdot)$, $\mathbf{B}_2(\cdot)$, $\mathbf{C}_1(\cdot)$, $\mathbf{C}_2(\cdot)$, $\mathbf{D}_{11}(\cdot)$, $\mathbf{D}_{12}(\cdot)$, and $\mathbf{D}_{21}(\cdot)$ have rational dependence on the time-varying scheduling parameter vector, $\boldsymbol{\rho}(\cdot) \in \mathscr{F}^\nu _\mathscr{P}$, which is also measurable in real-time. $\mathscr{F}^\nu _\mathscr{P}$ is the set of allowable parameter trajectories defined as $$\begin{aligned}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\mathscr{F}^\nu _\mathscr{P} \triangleq \{\boldsymbol{\rho}(t) \in \mathcal{C}(\mathbb{R}_{+},\mathbb{R}^{n_s}):\boldsymbol{\rho}(t) \in \mathscr{P},\qquad\quad \nonumber\\
|\dot{\rho}_i (t)| \leq \nu_i, i=1,2,\dots,n_s\},
\label{eq:parametertraj}\end{aligned}$$ wherein $n_s$ is the number of parameters and $\mathscr{P}$ is a compact subset of $\mathbb{R}^{s}$, $i.e.$ the parameter trajectories and parameter variation rates are assumed bounded as defined. The output-feedback LPV gain-scheduled control design procedure consists of finding a full-order dynamic LPV controller in the form of $$\begin{aligned}
\dot{\mathbf{x}}_K(t) & = \mathbf{A}_K (\boldsymbol{\rho}(t)) \mathbf{x}_K(t)+\mathbf{B}_K (\boldsymbol{\rho}(t))\mathbf{y}(t),\nonumber\\[0.1cm]
\mathbf{u}(t) & = \mathbf{C}_K (\boldsymbol{\rho}(t)) \mathbf{x}_K(t)+\mathbf{D}_K (\boldsymbol{\rho}(t))\mathbf{y}(t),
\label{controller} \end{aligned}$$ where $\mathbf{x}_K (t) \in \mathbb{R}^{n}$ is the controller state vector. By substituting the controller (\[controller\]) in the open-loop system (\[LPVsystem\]), and assuming $\mathbf{x}_{cl}(t) = [\begin{array}{cc}
\mathbf{x}(t) &\mathbf{x}_K (t)\end{array} ]^{\text{T}}$, the interconnected closed-loop system ($\mathbf{T}_{\mathbf{z}\mathbf{w}}$) is obtained as follows $$\begin{aligned}
\dot{\mathbf{x}}_{cl}(t)& = \mathbf{A}_{cl} (\boldsymbol{\rho}(t)) {\mathbf{x}}_{cl}(t) + \mathbf{B}_{cl} (\boldsymbol{\rho}(t)) \mathbf{w}(t),\nonumber\\[0.1cm]
\mathbf{z}(t) & = \mathbf{C}_{cl} (\boldsymbol{\rho}(t)) {\mathbf{x}}_{cl}(t) + \mathbf{D}_{cl} (\boldsymbol{\rho}(t)) \mathbf{w}(t),
\label{eq:closed-loop system} \end{aligned}$$ with $$\begin{aligned}
& \mathbf{A}_{cl}=\begin{bmatrix}
\mathbf{A} + \mathbf{B}_2 \mathbf{D}_K \mathbf{C}_2 & \mathbf{B}_2 \mathbf{C}_K\\
\mathbf{B}_K \mathbf{C}_2 & \mathbf{A}_K
\end{bmatrix},\\
& \mathbf{B}_{cl}=\begin{bmatrix}
\mathbf{B}_1 + \mathbf{B}_2 \mathbf{D}_K \mathbf{D}_{21} \\
\mathbf{B}_K \mathbf{D}_{21}
\end{bmatrix}, \\
& \mathbf{C}_{cl}=\begin{bmatrix}
\mathbf{C}_1 + \mathbf{D}_{12} \mathbf{D}_K \mathbf{C}_2 & \mathbf{D}_{12} \mathbf{C}_K
\end{bmatrix}, \\
& \mathbf{D}_{cl}= \mathbf{D}_{11} + \mathbf{D}_{12} \mathbf{D}_K \mathbf{D}_{21}, \end{aligned}$$ where the dependence on the scheduling parameter has been dropped for brevity. The final designed controller should be able to meet the following objectives for the closed-loop system:
- Input-to-state stability (ISS) of the closed-loop system (\[eq:closed-loop system\]) in the presence of parameter variations and disturbances, and
- Minimization of the worst-case amplification of the induced $\mathcal{L}_2$-norm of the mapping from the disturbances $\mathbf{w}(t)$ to the controlled output $\mathbf{z}(t)$, given by $$\Vert \mathbf{T}_{\mathbf{z}\mathbf{w}}\Vert_{i,2} = \underset{\boldsymbol{\rho}(t) \in \mathscr{F}^\nu _\mathscr{P}}{\sup} \:\:\: \underset{\Vert \mathbf{w}(t) \Vert_2 \neq 0}{\sup}\:\: \frac{\Vert \mathbf{z}(t) \Vert_2}{\Vert \mathbf{w}(t) \Vert_2}.
\label{eq:Performance Index}$$
Accordingly, in this paper, we utilize an extended form of the Bounded Real Lemma [@briat2014linear] and a quadratic parameter-dependent Lyapunov functions of the form $V(\mathbf{x}_{cl}(t), \boldsymbol{\rho}(t)) = \mathbf{x}^{\text{T}}_{cl}(t) \mathbf{P}(\boldsymbol{\rho}(t)) \mathbf{x}_{cl}(t)$ to obtain less conservative results that are valid for arbitrary bounded parameter variation rates [@apkarian1998advanced]. To this end, considering the closed-loop system (\[eq:closed-loop system\]), the following result provides sufficient conditions for the synthesis of a output-feedback LPV controller, which is formulated as convex optimization problems with LMI constraints. The designed LPV gain-scheduled controller guarantees closed-loop asymptotic parameter-dependent quadratic (PDQ) stability and a specified performance level as defined in (\[eq:Performance Index\]).
\[thm:thm1\][@briat2014linear] Considering the given open-loop LPV system (\[LPVsystem\]), there exists a gain-scheduled dynamic full-order output-feedback controller of the form (\[controller\]) that guarantees the closed-loop asymptotic stability and satisfies the induced $\mathcal{L}_2$-norm performance condition $\Vert \mathbf{z}(t) \Vert_2 \leq \gamma \Vert \mathbf{w}(t) \Vert_2$, if there exist continuously differentiable parameter-dependent symmetric matrices $\mathbf{X}, \mathbf{Y}:\mathbb{R}^{s}\rightarrow\mathbb{S}^{n}$, parameter-dependent matrices $\widehat{A} \in \mathbb{R}^{s}\rightarrow \mathbb{R}^{n \times n}$, $\widehat{B} \in \mathbb{R}^{s}\rightarrow \mathbb{R}^{n \times n_y}$, $\widehat{C} \in \mathbb{R}^{s}\rightarrow \mathbb{R}^{n_u \times n}$, $\widehat{D} \in \mathbb{R}^{s}\rightarrow \mathbb{R}^{n_u \times n_y}$, and a scalar $\gamma > 0$ such that the following LMI conditions hold for all $\boldsymbol{\rho} \in \mathscr{F}^\nu _\mathscr{P}$. $$\begin{array}{l}
\left[\begin{array}{cc}
\dot{\mathbf{X}} + \mathbf{X} \mathbf{A} + \widehat{B}\mathbf{C}_2 + (\star) & \star \\
\widehat{A}^{\text{T}} + \mathbf{A} + \mathbf{B}_2 \widehat{D} \mathbf{C}_2 & -\dot{\mathbf{Y}} + \mathbf{A}\mathbf{Y} + \mathbf{B}_2 \widehat{C} + (\star) \\
(\mathbf{X} \mathbf{B}_1 + \widehat{B} \mathbf{D}_{21})^{\text{T}} & (\mathbf{B}_1 + \mathbf{B}_2 \widehat{D} \mathbf{D}_{21})^{\text{T}} \\
\mathbf{C}_1 + \mathbf{D}_{12} \widehat{D} \mathbf{C}_2 & \mathbf{C}_1 \mathbf{Y} + \mathbf{D}_{12} \widehat{C}
\end{array}\right.\\[8mm]
\quad \qquad\qquad\quad\quad\quad\qquad\left.\begin{array}{cc}
\star & \star \\
\star & \star \\
-\gamma \mathbf{I} & \star \\
\mathbf{D}_{11} + \mathbf{D}_{12} \widehat{D} \mathbf{D}_{21} & -\gamma \mathbf{I}
\end{array}\right]\prec\mathbf{0},
\end{array}
\label{eq:LMI1}$$ $$\left[\begin{array}{cc}
\mathbf{X} & \mathbf{I} \\
\mathbf{I} & \mathbf{Y}
\end{array}\right] \succ \mathbf{0}.
\label{eq:LMI2}$$
Subsequently, the LPV control design is expanded to guarantee robustness against modeling mismatch and parameter uncertainties. To this end, $\mathbf{A}$ and $\mathbf{B}_2$ in (\[LPVsystem\]) are considered to be uncertain system matrices, $\mathbf{A}_{\Delta}(\boldsymbol{\rho}(t)) = \mathbf{A}(\boldsymbol{\rho}(t)) + \boldsymbol{\Delta} \mathbf{A}(t)$, $\mathbf{B}_{2, \Delta}(\boldsymbol{\rho}(t)) = \mathbf{B}_2(\boldsymbol{\rho}(t)) + \boldsymbol{\Delta} \mathbf{B}_2(t)$, where $\boldsymbol{\Delta} \mathbf{A}(t)$ and $\boldsymbol{\Delta} \mathbf{B}_2(t)$ are bounded matrices containing parametric uncertainties. The norm-bounded uncertainties are assumed to satisfy the following relation $$\left[\begin{array}{c}
\boldsymbol{\Delta} \mathbf{A}(t)\\
\boldsymbol{\Delta} \mathbf{B}_2(t)
\end{array}
\right] = \mathbf{H} \boldsymbol{\Delta} (t) \left[ \begin{array}{c}
\mathbf{E}_1\\
\mathbf{E}_2\end{array} \right],
\label{uncertainties}$$ where $\mathbf{H} \in \mathbb{R}^{n \times i} $, $\mathbf{E}_1 \in \mathbb{R}^{j \times n} $, $\mathbf{E}_2 \in \mathbb{R}^{j \times n_u} $ are known constant matrices and $\boldsymbol{\Delta} (t) \in \mathbb{R}^{i \times j}$ is an unknown time-varying uncertainty matrix function satisfying inequality $$\boldsymbol{\Delta}^T (t) \boldsymbol{\Delta} (t) \preceq \mathbf{I}.
\label{Delta1}$$ By substituting $\mathbf{A}_{\Delta}(\boldsymbol{\rho}(t))$ and $\mathbf{B}_{2, \Delta}(\boldsymbol{\rho}(t))$ for $\mathbf{A}$ and $\mathbf{B}_2$ in (\[eq:LMI1\]), the following result presents a condition for ensuring closed-loop stability and performance in the presence of norm-bounded uncertainties via an LPV control design of the form (\[controller\]).
\[thm:thm2\] There exists a full-order robust output-feedback LPV controller of the form (\[controller\]), over the sets $\mathscr{F}^\nu _\mathscr{P}$ with all admissible uncertainties $\boldsymbol{\Delta} \mathbf{A}(t)$ and $\boldsymbol{\Delta} \mathbf{B}_2(t)$ of the form (\[uncertainties\]) and all $\boldsymbol{\Delta}(t)$ satisfying (\[Delta1\]), that guarantees the closed-loop asymptotic stability and satisfies the induced $\mathcal{L}_2$-norm performance condition $\Vert \mathbf{z}(t) \Vert_2 \leq \gamma \Vert \mathbf{w}(t) \Vert_2$, if there exist continuously differentiable parameter dependent symmetric matrices $\mathbf{X}, \mathbf{Y}:\mathbb{R}^{s}\rightarrow\mathbb{S}^{n}$, parameter dependent real matrices $\widehat{A} \in \mathbb{R}^{s}\rightarrow \mathbb{R}^{n \times n}$, $\widehat{B} \in \mathbb{R}^{s}\rightarrow \mathbb{R}^{n \times n_y}$, $\widehat{C} \in \mathbb{R}^{s}\rightarrow \mathbb{R}^{n_u \times n}$, $\widehat{D} \in \mathbb{R}^{s}\rightarrow \mathbb{R}^{n_u \times n_y}$, and a positve scalars $\gamma$, and $\boldsymbol{\epsilon}$ such that the LMI (\[eq:robustLMIClosedloop\]) is feasible.
$$\left[\begin{array}{cccccccc}
\!\dot{\mathbf{X}} \!+\! \mathbf{X} \mathbf{A} + \widehat{B}\mathbf{C}_2 \!+\! (\star) & \star & \star & \star & \star & \star & \star & \star \\[.05cm]
\widehat{A}^{\text{T}} + \mathbf{A} + \mathbf{B}_2 \widehat{D} \mathbf{C}_2 & -\dot{\mathbf{Y}} + \mathbf{A}\mathbf{Y} + \mathbf{B}_2 \widehat{C} + (\star) & \star & \star & \star & \star & \star & \star \\[.05cm]
(\mathbf{X} \mathbf{B}_1 + \widehat{B} \mathbf{D}_{21})^{\text{T}} & (\mathbf{B}_1 + \mathbf{B}_2 \widehat{D} \mathbf{D}_{21})^{\text{T}} & -\gamma \mathbf{I} & \star & \star & \star & \star & \star \\[.05cm]
\mathbf{C}_1 + \mathbf{D}_{12} \widehat{D} \mathbf{C}_2 & \mathbf{C}_1 \mathbf{Y} + \mathbf{D}_{12} \widehat{C} & \mathbf{D}_{11} + \mathbf{D}_{12} \widehat{D} \mathbf{D}_{21} & -\gamma \mathbf{I} & \star & \star & \star & \star \\[.05cm]
\mathbf{H}^\text{T} \mathbf{X} & \mathbf{H}^\text{T} & \mathbf{0} & \mathbf{0} & -\boldsymbol{\epsilon} \mathbf{I} & \star & \star & \star \\[.05cm]
\boldsymbol{\epsilon} \mathbf{E}_1 & \boldsymbol{\epsilon} \mathbf{E}_1 \mathbf{Y} & \mathbf{0}& \mathbf{0} & \mathbf{0} & -\boldsymbol{\epsilon} \mathbf{I} & \star & \star \\[.05cm]
\mathbf{0} & \mathbf{H}^\text{T} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & -\boldsymbol{\epsilon} \mathbf{I} & \star \\[.05cm]
\boldsymbol{\epsilon} \mathbf{E}_2 \widehat{D} \mathbf{C}_2 & \boldsymbol{\epsilon} \mathbf{E}_2 \widehat{C} & \boldsymbol{\epsilon} \mathbf{E}_2 \widehat{D} \mathbf{D}_{21} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & -\boldsymbol{\epsilon} \mathbf{I}
\end{array}\right] \prec \mathbf{0}
\label{eq:robustLMIClosedloop}$$
By substituting the matrices with additive norm-bounded uncertainties in the LMI condition (\[eq:LMI1\]) given by Theorem \[thm:thm1\], *i.e.*, $\mathbf{A}_{\Delta}(\boldsymbol{\rho}(t))$ for $\mathbf{A}$ and $\mathbf{B}_{2, \Delta}(\boldsymbol{\rho}(t))$ for $\mathbf{B}_2$ in (\[eq:LMI1\]), the new LMI condition will be as follows $$\begin{array}{l}
(\ref{eq:LMI1}) +
\mathbf{He}\Bigg(
\left[\begin{array}{c}
\mathbf{X}\mathbf{H}\\
\mathbf{H}\\
\mathbf{0}\\
\mathbf{0}\end{array}\right]
\boldsymbol{\Delta}(t)
\left[\begin{array}{cccc}
\mathbf{E}_1 & \mathbf{E}_1\mathbf{Y} & \mathbf{0} & \mathbf{0}
\end{array}\right]\Bigg)\\[.8cm]
\!+\mathbf{He}\Bigg(\!
\left[\!\!\begin{array}{c}
\mathbf{0}\\
\mathbf{H}\\
\mathbf{0}\\
\mathbf{0}\end{array}\!\!\right]
\!\boldsymbol{\Delta}(t)\!
\left[\!\!\begin{array}{cccc}
\mathbf{E}_2 \widehat{D}\mathbf{C}_2 & \!\mathbf{E}_2\widehat{C} & \!\mathbf{E}_2 \widehat{D} \mathbf{D}_{21} & \mathbf{0}
\end{array}\!\!\right]\!\!\Bigg)
\!\!\prec\!\mathbf{0}.
\end{array}$$ Finally, using the following inequality [@xie1996output] $$\boldsymbol{\Theta} \boldsymbol{\Delta} (t) \boldsymbol{\Phi} + \boldsymbol{\Phi}^{\text{T}} \boldsymbol{\Delta}^{\text{T}} (t) \boldsymbol{\Theta}^{\text{T}} \leq \boldsymbol{\epsilon}^{-1} \boldsymbol{\Theta} \boldsymbol{\Theta}^{\text{T}} + \boldsymbol{\epsilon} \boldsymbol{\Phi} ^{\text{T}} \boldsymbol{\Phi},
\label{eq:ineq}$$ which holds for all scalars $\boldsymbol{\epsilon}>0$ and all constant matrices $\boldsymbol{\Theta}$ and $\boldsymbol{\Phi}$ of appropriate dimensions, and using the Schur complement [@boyd1994linear], the final LMI condition (\[eq:robustLMIClosedloop\]) is obtained.
Once the parameter-dependent LMI decision matrices, $\mathbf{X}$, $\mathbf{Y}$, $\widehat{A}$, $\widehat{B}$, $\widehat{C}$, and $\widehat{D}$ satisfying the LMI conditions (\[eq:LMI1\]) and (\[eq:LMI2\]) are obtained, the output-feedback LPV controller matrices can be readily computed following the steps below:
1\. Determine $\mathbf{M}$ and $\mathbf{N}$ from the factorization problem $$\mathbf{I} - \mathbf{X}\mathbf{Y} = \mathbf{N} \mathbf{M}^{\text{T}},$$ where the obtained $\mathbf{M}$ and $\mathbf{N}$ matrices are square and invertible in the case of a full-order controller. 2. Compute the controller matrices in the following order: $$\begin{aligned}
\mathbf{D}_K&=\widehat{D},\nonumber\\
\mathbf{C}_K&= (\widehat{C} - \mathbf{D}_K \mathbf{C}_{2} \mathbf{Y}) \mathbf{M} ^{-\text{T}},\nonumber\\
\mathbf{B}_K&= \mathbf{N}^{-1} (\widehat{B} - \mathbf{X} \mathbf{B}_2 \mathbf{D}_K),\nonumber\\
\mathbf{A}_{K}&= -\mathbf{N}^{-1} (\mathbf{X} \mathbf{A} \mathbf{Y} + \mathbf{X} \mathbf{B}_2 \mathbf{D}_K \mathbf{C}_{2} \mathbf{Y} + \mathbf{N} \mathbf{B}_K \mathbf{C}_{2} \mathbf{Y}\nonumber\\
&+ \mathbf{X} \mathbf{B}_2 \mathbf{C}_{K} \mathbf{M} ^{\text{T}} - \widehat{A}) \mathbf{M} ^{-\text{T}}.
\label{eq:controllermatrices} \end{aligned}$$
\[remark1\] Theorem \[thm:thm1\] results in an infinite-dimensional convex optimization problem with an infinite number of LMIs and decision variables since the scheduling parameter vector belongs to a continuous real vector space, $\boldsymbol{\rho} \in \mathscr{F}^\nu _\mathscr{P}$. To address this obstacle, the gridding method of the parameter space is utilized to convert the infinite-dimensional problem to a finite-dimensional convex optimization problem [@apkarian1998advanced]. In this regard, we choose the matrix parameter functional dependence as $\mathbf{M}(\boldsymbol{\rho}(t))=\mathbf{M}_0 + \sum\limits_{i=1}^{s}\rho_i(t) \mathbf{M}_{i_1}$, where $\mathbf{M}(\boldsymbol{\rho}(t))$ represents any of the parameter-dependent matrices appearing in the LMI conditions (\[eq:LMI1\]), and (\[eq:LMI2\]). Subsequently, by gridding the scheduling parameter space at appropriate intervals we obtain a finite set of LMIs to be solved for the unknown matrices and $\gamma$. The MATLAB^^ toolbox YALMIP can be used to solve the introduced optimization problem [@lofberg2004yalmip]. Also, it should be noted that due to the presence of derivatives of the parameter-dependent matrices in the LMI condition (\[eq:LMI1\]), *i.e.* $\dot{\mathbf{X}}$, and $\dot{\mathbf{Y}}$, the parameter variation rate $\dot{\rho}$, enters affinely in the LMIs, and it is sufficient to check the LMI only at the vertices of the $\dot{\rho}$ parameter range.
SPMSM LPV control design {#sec:results}
========================
We examine the application of the proposed LPV gain-scheduled control design method to the SPMSM speed regulation. The SPMSM error dynamics (\[eq:lpverrordynamics\]) is formulated in an LPV framework (shown in Section ) which is suitable for the proposed LPV control design synthesis. Considering the generic LPV system state-space realization (\[LPVsystem\]) for the SPMSM model described in (\[eq:lpverrordynamics\]), the LPV state-space matrices of the SPMSM are as shown in (\[eq:matrices1\]). Moreover, the vector of the target outputs to be controlled is defined as follows $$\begin{array}{l}
\!\!\mathbf{z}^{\!\text{T}}(t) = \left[\!\!\begin{array}{ccc}
\phi \cdot e_z (t)&
\sigma \cdot e_\omega (t) &
\xi \cdot e_\alpha (t)\end{array}\right.\\[.1cm]
\qquad\qquad\quad\quad\left.\begin{array}{ccc}
\psi \cdot e_\beta (t)&
\eta \cdot u_\alpha (t)&
\mu \cdot u_\beta (t)
\end{array}\!\!\right].
\end{array}$$ The velocity tracking error which is included in the second state $x_2(t) = e_w(t)$ is penalized by the weighting scalar $\sigma$ and the control efforts $u_\alpha(t)$, $u_\beta(t)$ are penalized by the weighting scalars $\eta$ and $\mu$, respectively. The choice of the weighting scalars $\phi$, $\sigma$, $\xi$, $\psi$, $\eta$, and $\mu$ determine the relative weighting in the optimization scheme and depends on the desired performance objectives that the designer seeks to achieve [@lee2015h]. Now, based on the definition of the desired controlled vector $\mathbf{z}(t)$, the output-feedback controller is designed for the SPMSM to minimize the induced $\mathcal{L}_2$ gain (or $\mathcal{H}_{\infty}$ norm) (\[eq:Performance Index\]) of the closed-loop LPV system (\[eq:closed-loop system\]). The design objective is to guarantee closed-loop stability and minimize the worst case disturbance amplification over the entire range of model parameter variations.
In order to demonstrate the improved performance of the proposed control with respect to the desired velocity profile tracking and load torque disturbance rejection, closed-loop simulations are performed in the MATLAB/Simulink environment. The model parameters of the SPMSM are listed in Table \[table: MotorParameters\]. For comparison purposes, we evaluate the closed-loop tracking performance of the proposed controller against the FOC method with fixed gains. The FOC tuned PI controller transfer functions are selected as follows [@kim2017electric]:
Parameter Value Unit
---------------------------------------- ------------------------- --------------------------
Number of pole pairs ($p$) 4 -
Stator resistance ($R_{s}$) 0.2 $\Omega$
Stator inductance ($L_{s}$) 0.4 mH
Magnetic flux linkage ($\lambda_{pm}$) 16.3 mWb
Moment of inertia ($J_{m}$) 3.24 $\times$ 10$^{-5}$ kg $\cdot$m$^2$
Coefficient of friction ($B$) 0.004 N$\cdot$ m $\cdot$ s/rad
: Parameters of the SPMSM.\[table: MotorParameters\]
$$\begin{split}
G_{cs} (s)& = 0.533 + \dfrac{61.4}{s}, \\
G_{cc} (s)& = 1.38 + \dfrac{691}{s},
\end{split}$$
where $G_{cs} (s)$ and $G_{cc} (s)$ indicate the speed controller in the $q$ axis and the current controllers in the $d$ and $q$ axis respectively. The gains of these controllers are obtained based on the nominal parameters of the motor and the desired bandwidth of the controllers.
In order to evaluate the closed-loop tracking performance of the proposed LPV method, we consider a desired velocity reference profile, as shown in Figure \[fig:Reference\_step\]. Figures \[fig:trackingstep1\] and \[fig:trackingstep2\] present the magnified plots of the tracking error result of the LPV and PI controllers in the absence of any disturbances, for the first step change, when the velocity reference accelerates from 0 r/min to 300 r/min, and for the second step change, when the velocity reference decelerates from 300 r/min to 100 r/min, respectively. As anticipated, the proposed LPV controller outperforms the PI controller with respect to the overshoot/undershoot, rise time, and speed of the response in both acceleration and deceleration intervals due to its scheduling structure. Figures \[fig:nodistcurrent\] and \[fig:nodistvoltage\] show the currents and control input voltages of the proposed LPV controller, both in the $\alpha - \beta$ axis.
![Desired velocity reference profile.[]{data-label="fig:Reference_step"}](Reference2.eps){width="\columnwidth" height="2.1in"}
![Closed-loop velocity tracking performance of the LPV controller and the fixed structure PI controller with no disturbance during acceleration period.[]{data-label="fig:trackingstep1"}](tracking_nodist_1.eps){width="\columnwidth" height="2.1in"}
![Closed-loop velocity tracking performance of the LPV controller and the fixed structure PI controller with no disturbance during deceleration period.[]{data-label="fig:trackingstep2"}](tracking_nodist_2.eps){width="\columnwidth" height="2.2in"}
![Currents of the LPV controller in the $\alpha - \beta$ axis.[]{data-label="fig:nodistcurrent"}](tracking_nodist_lpv_currents_alphabeta.eps){width="\columnwidth" height="2.2in"}
![Control input voltages of the LPV controller in the $\alpha - \beta$ axis.[]{data-label="fig:nodistvoltage"}](tracking_nodist_lpv_voltages_alphabeta.eps){width="\columnwidth" height="2.2in"}
Next, we assume that the SPMSM is experiencing temperature variation with a temperature profile shown in Figure \[fig:temp\] and an output disturbance. The temperature variation affects the model’s resistance and magnet flux as described in (\[9\]). The disturbance under consideration is a constant torque load disturbance as shown in Figure \[fig:dist\]. The closed-loop performance of the proposed LPV controller and the PI controller in tracking a given ramp-type velocity reference command with a step disturbance is shown in Figure \[fig:trackingdist\]. Additionally, Figures \[fig:distcurrent\] and \[fig:distvoltage\] demonstrate the currents and the input voltages of the LPV controller in the $\alpha - \beta$ phases, respectively. In order to evaluate the robustness of the proposed design, the closed-loop response of the proposed robust LPV gain-scheduling controller is investigated in the presence of model parameter variations. To this end, we select the stator inductance $L_s$ and the moment of inertia $J_m$ to be under-estimated by $50 \%$, and the stator resistance $R_s$ and viscous friction coefficient $B$ to be over-estimated by $50 \%$, which corresponds to a worst-case perturbation scenario. The closed-loop velocity tracking performance of the system with the proposed robust LPV control design (obtained through condition (\[eq:robustLMIClosedloop\]) and Theorem \[thm:thm2\]) is compared to the response of the LPV controller designed without considering uncertainty obtained using the results of Theorem \[thm:thm1\]. As per Figure \[fig:trackingdist\_robust\], the control without considering uncertainty in the design demonstrates significant oscillatory behavior, higher overshoots and settling time, which are undesirable. Hence, as the results demonstrate, the proposed robust LPV control design is capable of compensating for parameter uncertainties and modeling mismatches. Therefore, by investigating the presented results, we conclude that the proposed LPV control method demonstrates superior results in terms of velocity tracking, disturbance rejection and robustness under different simulated scenarios in the presence of parameter variations, disturbances and model uncertainty.
![Load torque disturbance.[]{data-label="fig:dist"}](dist.eps){width="\columnwidth" height="2.2in"}
![SPMSM operating temperature variation.[]{data-label="fig:temp"}](temperature.eps){width="\columnwidth" height="2.2in"}
Conclusion {#sec:conclusion}
==========
In the present paper, a linear parameter-varying (LPV) gain-scheduled output feedback controller has been proposed for the speed control of the surface permanent magnet synchronous motors (SPMSMs). The dynamic model of the motor has been developed in the $\alpha - \beta$ stationary reference frame, and an LPV model representation has been utilized to capture the nonlinear SPMSM dynamics. The effect of temperature on the variability of SPMSMs model parameters is taken into account in the model. The linear matrix inequality (LMI) framework has been used to formulate the controller synthesis conditions as numerically tractable convex optimization computational problem. Subsequently, the proposed controller was designed to guarantee the closed-loop stability and minimize the disturbance amplification in terms of the induced $\mathcal{L}_2$-norm performance specification of the closed-loop system. The effectiveness of the proposed controller was validated via comparisons with a conventional PI controller in the MATLAB/Simulink environment. The results demonstrated the effectiveness and superiority of the proposed approach in improving the transient performances in terms of settling time, overshoot, disturbance attenuation, and parametric uncertainty compensation. Future research would focus on designing a disturbance observer to empower the control design to cope better with unknown disturbances. Hence, the estimated disturbance can be compensated in the controller output to improve the stability of the motor and speed tracking performance.
![Closed-loop velocity tracking performance of the LPV controller and the fixed structure PI controller subject to load torque disturbance.[]{data-label="fig:trackingdist"}](tracking_dist.eps){width="\columnwidth" height="2.4in"}
![Currents of the LPV controller in the $\alpha - \beta$ axis.[]{data-label="fig:distcurrent"}](tracking_withdist_lpv_currents_alphabeta.eps){width="\columnwidth" height="2.3in"}
![Control input voltages of the LPV controller in the $\alpha - \beta$ axis.[]{data-label="fig:distvoltage"}](tracking_withdist_lpv_voltages_alphabeta.eps){width="\columnwidth" height="2.3in"}
![Closed-loop velocity tracking performance of the robust LPV controller in the presence of model parameter uncertainty subject to torque load disturbance.[]{data-label="fig:trackingdist_robust"}](tracking_dist_robust.eps){width="\columnwidth" height="2.4in"}
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abstract: 'Self-adaptive software systems (SASS) are equipped with feedback loops to adapt autonomously to changes of the software or environment. In established fields, such as embedded software, sophisticated approaches have been developed to systematically study feedback loops early during the development. In order to cover the particularities of feedback, techniques like one-way and in-the-loop simulation and testing have been included. However, a related approach to systematically test SASS is currently lacking. In this paper we therefore propose a systematic testing scheme for SASS that allows engineers to test the feedback loops early in the development by exploiting architectural runtime models. These models that are available early in the development are commonly used by the activities of a feedback loop at runtime and they provide a suitable high-level abstraction to describe test inputs as well as expected test results. We further outline our ideas with some initial evaluation results by means of a small case study.'
author:
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bibliography:
- 'references.bib'
title: 'A Testing Scheme for Self-Adaptive Software Systems with Architectural Runtime Models'
---
Introduction
============
Traditionally, software development follows an open-loop structure that requires human supervision when software systems are exposed to changing environments . To reduce human supervision, software systems are equipped with feedback loops to adapt autonomously to changing environments. Such closed-loop systems are designated as self-adaptive software systems (SASS) [@SEfSAS2-ROADMAP] and they are often split in two parts, an *adaptation engine* realizing the feedback loops and controlling the *adaptable software* . As pointed out by Calinescu [@Calinescu2013], such systems will become important for safety-critical applications, where they have to fulfill high-quality standards.
Testing is an established technique for ensuring quality in traditional systems, even safety-critical ones [@Nair2014689], and processes for testing such systems exist. For instance, embedded software with its feedback loops is often systematically tested in three stages [@broekman2003testing pp. 193–208]:
a simulation stage that tests the models (specification) of the software under development in a simulated or real-life environment,
a prototyping stage that tests the real software in a simulated environment, and finally,
pre-production stage that tests the real software in the real environment.
With each stage the software is more and more refined to the final product while testing continuously provides assurances for the software and particularly early in the development.
However, a similar systematic testing process providing continuous and early assurances does not exist for SASS. In contrast, models for substituting the environment or parts of a SASS usually cannot be obtained easily and therefore, a generic simulation environment for SASS does not exists. Consequently, testing SASS typically requires that the implementations of the feedback loops and adaptable software with its sensors and effectors are available. This impedes testing early in the development and makes it costly to remove faults in the feedback loops discovered late in the development.
Furthermore, approaches used for traditional systems are not as easily applicable to SASS as the interface between the adaptation engine and the adaptable software is often quite different from that of embedded software. SASS are usually not restricted to observing and adjusting parameters but additionally monitor and adapt the architecture of the software , thus requiring support of structural adaptations [@McKinley+2004].
Some approaches address the testing of SASS but only for later development stages [@Goldsby2008; @Zhao2006; @Eberhardinger2014; @Fredericks2015; @Fredericks2014a] when the systems have already been deployed. Others do promote testing in earlier stages but they still assume an executable and complete SASS to run the tests against [@Pueschel2014; @Wang2007; @Camara2014]. Testing of only parts of the feedback loop is not supported. In contrast, we consider testing parts of a SASS as a precondition to early validation since a system with the completely implemented adaptable software and feedback loop is only available in the latest development stages.
Therefore, we propose a systematic testing scheme for SASS that allows engineers to test the feedback loops (adaptation behavior) early in the development by exploiting runtime models. Such models represent the adaptable software and environment and they are typically used at runtime to drive the adaptation [@MC.2009.326]. Our approach leverages early testing of SASS by using *architectural* runtime models that are available early in the development and are commonly used by the activities of a feedback loop. Therefore, feedback loop activities such as monitor, analyze, plan, and execute (cf. [@Kephart2003]) can be individually tested while the whole feedback loop and the adaptable software do not have to be implemented yet. In contrast, the non-implemented parts are simulated based on the runtime models. Consequently, the feedback loop can be modularly tested while the different parts of the loop can be incrementally refined and implemented until they replace the simulated parts. Moreover, we expect reduced costs of testing since we do not require final or experimental implementations of certain feedback loops parts to test other parts.
The rest of the paper is structured as follows. We describe preliminaries in Section \[sec:preliminaries\] and the benefits of runtime models for testing in Section \[sec:runtime-models\]. Then, we discuss our approach by means of one-way (Section \[sec:one-way-testing\]), in-the-loop (Section \[sec:in-the-loop-testing\]), and online (Section \[sec:online-testing\]) testing. Finally, we sketch an initial evaluation in Section \[sec:evaluation\], contrast our approach with related work in Section \[sec:related-work\], and conclude the paper in Section \[sec:conclusion\].
Preliminaries {#sec:preliminaries}
=============
In this section we discuss preliminaries of the presented testing: MAPE-K feedback loops and architectural RTM.
MAPE-K Feedback Loops
---------------------
The development of SASS typically follows the *external approach* that separates adaptation from domain concerns by splitting up the software in two parts: an *Adaptation Engine* for the adaptation concerns and an *Adaptable Software* for the domain concerns while the former *senses* and *effects* and thus, controls the latter. This constitutes a *feedback loop* that realizes the self-adaptation (see Figure \[fig:mapeks\]). The engine senses as well the *Environment* with which the adaptable software *interacts*.
![MAPE-K Feedback Loop with a Runtime Model (RTM).[]{data-label="fig:mapeks"}](img/SAS-MapeKLoopII.pdf){width="0.65\columnwidth"}
The resulting feedback loop between the engine and the software can be refined according to the reference model [@Kephart2003]. This model considers the activities of **M**onitoring and **A**nalyzing the software and environment and, if needed, of **P**lanning and **E**xecuting adaptations to the software. All activities share a **K**nowledge base as illustrated by a runtime model (*RTM*) in Figure \[fig:mapeks\] and discussed in the following.
Architectural Runtime Models
----------------------------
The external approach as previously discussed requires that the adaptation engine has a representation of the adaptable software and environment to perform self-adaptation. This representation is often realized by a causally connected **R**un**t**ime **M**odel (*RTM*) [@MC.2009.326]. A causal connection means that changes of the software or environment are reflected in the model and changes of the model are reflected in the software (but, not in the environment being a non-controllable entity).
Considering Figure \[fig:mapeks\], an RTM can be used as a knowledge base on which the MAPE activities are operating. The monitor step observes the software and environment and updates the RTM accordingly. The analyze step then reasons on the RTM to identify any need for adaptation. Such a need is addressed by the plan step to prescribe an adaptation in the RTM, which is eventually enacted to the software by the execute step.
Using RTMs in self-adaptive software provides the benefits of creating appropriate abstractions of runtime phenomena that are manageable by the feedback loops and of applying automated model-driven engineering (MDE) techniques [@MC.2009.326].
The software architecture has been identified as such an appropriate abstraction level for representing the adaptable software and environment and for supporting structural adaptation . Hence, *architectural* RTMs of the adaptable software are used by a feedback loop to reflect on the state of the software and environment. Such state-aware models can be enriched by a feedback loop to cover, for instance, the history or time series of states and executed adaptations, which results in history-/time-aware models.
In our research on self-adaptive software such as [@VG-TAAS-EUREMA], we evaluate our work by using *mRUBiS*[^1], an internet marketplace on which users sell or auction products, as the adaptable software. A single shop on the marketplace consists of 18 components and we may scale up the number of shops. For a self-healing scenario, we created architectural runtime models of mRUBiS and defined different types of failures based on the models. These failures have to be handled by the adaptation engine. Examples of such failures are exceptions emitted by components, unwanted life-cycle changes of components, the complete removal of components because of crashes, and repeated occurrences of these failures. Based on that, we experiment with different adaptation mechanisms and can also exploit the models for testing as discussed in the following.
Exploiting Runtime Models for Testing {#sec:runtime-models}
=====================================
In the following, we assume a SASS that follows the cycle with runtime models (RTMs) as schematically depicted in Figure \[fig:mapeks\]. If the RTMs are just self-aware and reflect the current state of the adaptable software and environment, we can make the following two observations:
\(1) The behavior of the system can be described by a sequence of steps $(\to_{AS} or \to_{ENV})^* \to_M \to_A \to_P \to_E (\to_{AS} or \to_{ENV})^*; \dots$ where $\to_{AS}$ denotes a step of the adaptable software, $\to_{ENV}$ denotes a step of the environment, $\to_M$ denotes the complete monitoring step, $\to_A$ denotes the complete analysis step, $\to_P$ denotes the complete planning step, and $\to_E$ denotes the complete execute step. (2) The interface between those steps can be described by different states $S_{i}$ of the RTM if we do not consider the input of the monitoring and the output of the execute step: $(\to_{AS} or \to_{ENV})^* \to_M S_1^M \to_A S_1^A \to_P S_1^P \to_E (\to_{AS} or \to_{ENV})^*; \to_M S_2^M\dots$ where $S^M_i$ denotes the RTM state after the $i$-th monitoring, $S^A_i$ the RTM state after the $i$-th analysis, and $S^P_i$ the RTM state after the $i$-th planning.
![Example Trace for a Self-Healing Scenario.[]{data-label="fig:trace"}](img/RTM-Trace.pdf){width="1\columnwidth"}
Consider the self-healing example in Figure \[fig:trace\]. An intact architecture is monitored and results in RTM $S_{i-1}^M$. For now, analysis and planning are not required to take action since the architecture is not broken. Without an adaptation, the execute step will do nothing either. We can directly proceed with the next steps in the environment or adaptable software. Due to either an environmental influence or some failure in the adaptable software ($\to_{ENV}$ or $\to_{AS}$), a component of the architecture is removed. In the next step, this is monitored as RTM $S_{i}^M$. The result of the analysis step $\to_A$ is the annotated RTM $S_{i}^A$ that marks the missing component. The planning step $\to_P$ constructs a repaired RTM $S_{i}^P$ which will be applied to the adaptable software in the next step by $\to_E$.
These two observations indicate that the different states of the RTM are the key element to describe the input/output behavior of the MAPE activities concerning their communication with the adaptable software. Moreover, the RTMs also facilitate considering the required behavior of the adaptation engine at a much higher level of abstraction than the events observed by the monitoring step and the effects triggered by the execute step.[^2] Consequently, we suggest exploiting the RTMs to systematically test the adaptation engine and its parts in form of one-way testing of individual steps and fragments, in-the-loop testing of the analysis and planning steps, and online testing of the analysis and planning steps. We further study how we can validate the model which is required for the in-the-loop testing.
One-Way Testing {#sec:one-way-testing}
===============
We define *One-Way Testing* as the following: An input RTM and an expected oracle RTM are provided. One or more steps are tested in a single execution of a partial feedback loop. The tested parts receive the input RTM and are supposed to produce an output RTM. The output RTM is compared against the oracle. In this kind of testing the steps $\to_{AS}$, $\to_{ENV}$, $\to_{M}$, $\to_{A}$, $\to_{P}$ or $\to_{E}$ will happen at most once.
One-Way Testing single MAPE Activities
--------------------------------------
The most basic approach is to test each of the steps/activities that process the RTM on their own. Obviously, these tests need to be run before testing combinations of feedback-loop steps to better locate faults and tell single-step errors from errors that arise due to problems in the interaction of steps.
### One-Way Testing the Analysis
If we want to test the analysis step, we simply provide an input RTM $S_1^M$, run step $\to_A$, and compare the resulting RTM $S_1^A$ with an oracle RTM $S_o^A$. Applied to the example in Figure \[fig:trace\], we choose $S_{i}^M$ with the removed component as an input RTM. We then define an oracle RTM $S_o^A$ that contains an annotation where the missing component has been marked. Applying $\to_A$ on $S_{i}^M$ would give us $S_{i}^A$ which is compared to $S_o^A$. If both RTMs are the same, that is, both especially contain the same “missing component” annotation, the test would pass, otherwise fail.
### One-Way Testing the Planning
Similar to the analysis step, we provide an input RTM $S_1^A$, run step $\to_P$, and check whether the ouput of $\to_P$ is equal to an oracle RTM $S_o^P$ that was defined before. In the example of Figure \[fig:trace\], we start out with the annotated RTM $S_{i}^A$. The oracle $S_o^P$ would be defined as the intact architecture from the beginning ($S_{i-1}^M$) and we would expect $\to_P$ to return an RTM equal to $S_o^P$, that is, the plan step has re-created the removed component in the RTM.
One-Way Testing MAPE Fragments
------------------------------
We now discuss one-way testing of fragments by jointly testing the analyze and plan or the monitor and execute steps.
### One-Way Testing the Analysis and Planning
As a precondition to the separate test of the analysis and planning, it is necessary to have knowledge about the way the analysis works and what kind of models to expect. Obviously it would be hard to create a valid oracle model $S_o^A$ or input model $S_1^M$ if this knowledge is not available. In a simple scenario like the self-healing one presented before this should not pose a problem. But there are also more complex analysis algorithms, which will not result in models that can be tested as easily. Furthermore, some errors might only appear if the analysis and planning are tested together.
Consequently, we propose to test the analyze and plan steps as the next unit. Again we can benefit from the same pattern of testing, that is, by providing an input model $S_1^M$ and an oracle model in state $S_o^P$. In terms of the example trace (Figure \[fig:trace\]), this means to start with the broken monitored input model $S_i^M$, construct an expected model $S_o^P$ where the removed component is redeployed and check whether the resulting model of the application of $S_i^M \to_A \to_P$ $S_i^P$ is equal to $S_o^P$.
### One-Way Testing the Execute and Monitor
The separate testing of the monitor and execute steps via the runtime models is not feasible as the effect of the execute step cannot be directly observed. If we follow the same pattern as with the analysis and planning, we would end up with no result model for the execute step and no input model for the monitor step. The effect of the execute step cannot be directly observed since it is part of the concrete adaptable software. Likewise, the monitor step’s input is directly obtained from the software. Instead of the separate testing, we propose to test the monitor and execute steps together. In this setup we need a working adaptable software and the tested execute and monitor steps are effecting and sensing the software. The test input is provided by a model $S_1^P$ to the execute step $\to_E$ which will effect the adaptable software. The adaptable software is monitored $\to_M$ and a new runtime model is obtained $S_2^M$.
Equality and inequality of these two models can be interpreted in different ways: (1) equal models may indicate that the monitor and execute steps work correctly, (2) equal models may also mean that a failure in the execute step is masked by a failure in the monitor step (or the other way round), or (3) that the adaptable software or the environment mask a fault of the execute and/or monitor steps. If $S_1^P$ and $S_2^M$ are not equal, then either (4) the execute step, (5) the monitor step or (6) both do not work properly or (7) the environment introduced an error or the adaptable software showed erroneous behavior.
Cases (3) and (7) can be ruled out by applying the test several times. It is unlikely that the environment will introduce the same error for all test runs and if the adaptable software was tested before, it is equally unlikely that it will constantly show erroneous behavior. In the cases (4), (5) and (6) we can assume a broken monitor and/or execute step. Case (1) should be more likely than (2) since it is not impossible but hard to have two faults that mask each other. Case (2) should become less likely the more tests with different $S_1^P$ and $S_2^M$ are done. In the end, equal models are a good indicator of working execute and monitor steps and non-equal models show that at least one of them is broken.
With this test setup, only parts of the monitoring capabilities can be tested since its purpose is to detect not only correct but also incorrect states of the adaptable software. On the other hand, the execute step is not intended to have an effect on the software that causes an incorrect state. Therefore, we need to be able to impose an “incorrect” RTM on the adaptable software (such as $S_i^M$ in Figure \[fig:trace\]), so that we can test whether the monitor step is able to properly observe this incorrect state and create the according RTM. A special test adapter is needed, so that first a correct RTM can be imposed by the execute step and then the incorrect parts are added by the test adapter. The incorrect input RTM $S_{1 err}$ needs to be split into $S_{1 valid}$ which will be provided to $\to_E$ and $S_{1 invalid}$ which is given to the test adapter. The oracle $S_o^M$ for this test looks like $S_{1 err}$ and the monitor should observe an incorrect RTM.
In-the-Loop Testing {#sec:in-the-loop-testing}
===================
Considering the analysis and planning, one-way testing is effective to find errors that always show up, independent from their previous executions in the feedback loop. If we want to identify errors that arise from an accumulated state of the system, we need to test them with sequences of inputs. It would be a cumbersome task to construct these sequences by hand. Instead we propose to provide a simulation that captures the behavior of the adaptable software (AS), environment (ENV), monitor (M) and execute (E) steps. This simulation will provide sequences of RTMs to the analyze step and will read back the RTMs from the plan step. We define such a **r**un**t**ime **m**odel **s**imulation by an automaton ${RTMS} = (\mathcal{S}_{RTMS}, \to_{RTMS})$ that comprises the combined behavior of AS, ENV, M, and E. Note that ${RTMS}$ is a simulation for testing purposes. The provided input RTM and the way the simulation model reacts to the output of $\to_A$ and $\to_P$ are supposed to be realistic but not an exact replacement for the real AS, ENV, M and E. It also means that it may behave non-deterministically to reflect realistic AS and ENV and therefore involves some random component.
In order to decide whether a test is successful, we also need an oracle. In the simplest case the oracle is given by a state property $\phi$ for the model. In more complex cases $\phi$ may be even a sequence property or ensemble property. With respect to our example, the oracle may be the sequence property that some architectural constraints for our RTM are only violated for at most $n$ subsequent states.
Black-Box In-the-Loop Testing of Analysis and Planning
------------------------------------------------------
With ${RTMS}$ at hand we can test the feedback loop already in an early stage when neither the adaptable software or the monitor and execute steps are available or ready. The analyze and plan steps combined with ${RTMS}$ can be simulated together and produce observable sequences: $\to_{RTMS} S_1 \to_A S_1^A \to_P S_1^P \to_{RTMS} S_2 \dots$. From these we consider only the traces of states: $\pi = S_1; S_1^P; S_2 \dots$ and check whether $\pi \models \phi$ to ensure that $\to_A$ and $\to_P$ as a black box work as expected.
Grey-Box In-the-Loop Testing of Analysis and Planning
-----------------------------------------------------
We can also aim for a better fault location if we consider the result of $\to_A$ (i.e., the analyze and plan steps as a grey box). The sequence, we would like to look at, is the following: $\to_{RTMS} S_1 \to_A S_1^A \to_P S_1^P \to_{RTMS} S_2 \dots$. Here we will inspect the trace $\pi' = S_1; S_1^A; S_1^P; S_2 \dots$. In order to test these traces, we need a property $\phi'$ that covers $S_i^A$ as well. We now require $\pi' \models \phi'$ to ensure that $\to_A$ and $\to_P$ work as expected.
Online Testing and Validation {#sec:online-testing}
=============================
In a later development stage we can reuse the simulation model ${RTMS}$ and the properties $\phi$ and $\phi'$ alongside the running system for online testing and validation.
Online Testing
--------------
If $\to_A$ and $\to_P$ in the running system will expose $S_i^A$ and $S_i^P$ in the same way as in the development stage, we can check $\phi$ and $\phi'$ online or against a recorded trace. The simulation is simply replaced with the real system. Whether online or offline testing is to be preferred will depend on available resources on the system under test and the existence of logging facilities. Both approaches, black-box and grey-box testing, are applicable and can be carried out in the same way as with the simulation.
The sequences will be $(\to_{AS} or \to_{ENV})^* \to_M S_1^M \to_A S_1^A \to_P S_1^P \to_E (\to_{AS} or \to_{ENV})^*; \to_M S_2^M \dots$ and the traces will be the exchanged RTMs: $\pi = S_1^M; S_1^A; S_1^P; S_2^M \dots$
Validation
----------
The in-the-loop testing heavily depends on ${RTMS}$. If an error is detected during in-the-loop testing, it is likely that it is caused by an erroneous adaptation ($\to_A$, $\to_P$ or both). But the ${RTMS}$ itself might also be the source of an error or might mask an erroneous adaptation. The validation of ${RTMS}$ in this later stage can give an indication about the quality of ${RTMS}$ and therefore the suitability for testing. Additionally, if the real system produces sequences not covered by ${RTMS}$ which cause errors in the adaptation, we exactly know which sequence reveals the error and it can be added to ${RTMS}$ for regression tests.
The idea behind validating ${RTMS}$ is to observe $(\to_{AS} or \to_{ENV})^* \to_M S_1^M \to_A S_1^A \to_P S_1^P \to_E (\to_{AS} or \to_{ENV})^*; \to_M S_2^M \dots$ and look at the traces $\pi' = S_1^M; S_1^P; S_2^M \dots$. If our simulation model ${RTMS}$ is correct, it should cover the observed behavior: $\pi' \in \mathcal{L}({RTMS})$.
Initial Evaluation {#sec:evaluation}
==================
In this section, we report on our initial evaluation of the testing scheme for SASS we are proposing in this paper. This evaluation shows the benefits of using (architectural) runtime models with respect to implementing a test framework by means of reusing MDE techniques. Moreover, it gives us preliminary confidence about the effectivity of the scheme when developing feedback loops.
One-Way Testing {#one-way-testing}
---------------
To realize one-way testing, we developed a generic test adapter that loads the input model, triggers the adaptation steps such as analysis and planning to be tested, and finally, compares the resulting model with the oracle model. Developing such a test adapter has been simplified due to MDE principles as realized by the *Eclipse Modeling Framework* (EMF)[^3]. EMF provides mechanisms to generically load and process models and particularly of comparing models[^4]. Hence, we easily obtain matches and differences between two models such as the output model of adaptation steps and the oracle model to obtain the testing result. This result, that is, the output of the comparison, is also a model that can be further analyzed. For instance, the *Object Constraint Language* (OCL)[^5] can be used to check application-specific constraints such as mission-critical components like for authenticating users on the mRUBiS marketplace are not missing in the architecture.
In-the-Loop Testing {#in-the-loop-testing}
-------------------
For the internet marketplace mRUBiS we developed a simulator based on an architectural runtime model. It simulates the marketplace itself (i.e., the adaptable software) thereby injecting failures as well as the monitor and execute steps. The simulator maintains the runtime model against which the analyze and plan steps are developed.
Using this simulator, we can test the analyze and plan steps as follows:
the simulator injects failures into the runtime model (this simulates the behavior of the adaptable software and environment as well as the monitor step that reflects the failure in the model).
the analyze and plan steps to be tested are executed and they analyze and adjust the model according to the adaptation need.
the simulator performs the execute step that emulates the effects of the adaptation as performed by the analyze and plan steps in the runtime model. For instance, response times are updated in the model if the configuration of the architecture is adapted.
After one run of the feedback loop and before injecting the next failures, the simulator checks whether the analyze and plan steps performed a well-defined adaptation (e.g., by checking whether the life cycle of components has not been violated when adding or removing components) and it checks whether the state of the runtime model represents a valid architecture (e.g., components are not missing or there are no unsatisfied required interfaces, that is, no dangling edges). These checks are performed based on constraints and properties that the runtime models must fulfill and the results of these checks are given as feedback to the engineer.
This simulator has been used in research and in courses to let students develop and test different adaptation techniques (e.g., hard-coded event-condition-action, graph transformation, or event-driven rules) for the analyze and plan steps. Though the simulator helped in finding faults in the adaptation logic, the randomness included in the simulator and its basic logging facilities impeded the automated reproducibility of traces and therefore, the retesting of “interesting” edge cases.
Online Testing and Validation {#online-testing-and-validation}
-----------------------------
So far, we have not worked on testing the adaptation online. However, our experience with runtime models and employing MDE techniques at runtime for self-adaptation [@VG-TAAS-EUREMA; @Vogel-ICAC09] gives us promising confidence to achieve the online testing. For instance, our EUREMA interpreter [@VG-TAAS-EUREMA] that executes feedback loops already maintains the runtime models used within the loops and passes them along the loop’s adaptation activities. Thus, when passing models along the activities, the interpreter may defer the execution of the next activity. Before proceeding, the interpreter can either (1) hand over to an online testing activity that will compare the current RTM to one which is derived from a simulation model that runs in parallel or (2) log the RTM for later comparison in a simulation module. As discussed earlier, (1) has the advantage of immediate revelation of errors but needs computation resources on the system while (2) can benefit from more resources offline but needs persistance resources for the logs. In both cases, working only on changes of the RTM might reduce cost.
Related Work {#sec:related-work}
============
Testing of SASS has been addressed by others as well. This related work could usually be assigned to one of the following categories:
\[inpar:formal\] The adaptation is formally specified and verified with special constructs regarding the adaptation [@Sama2008; @Iftikhar2012],
\[inpar:online\] the SASS is tested/verified at runtime/online and the verification expressions are adapted to properties unique to adaptation [@Goldsby2008; @Zhao2006; @Eberhardinger2014],
\[inpar:evolved\] tests are evolved at runtime in an attempt to test for requirement fulfillment even when the environment or the adaptable software changes [@Fredericks2015; @Fredericks2014a], and
\[inpar:destime\] testing is carried out at design time addressing the special issues of adaptive systems [@Pueschel2014; @Wang2007; @Camara2014].
\[inpar:combined\] Combined assessment of quality assurance for self-adaptive systems from more than one direction has also been done [@Weyns2012].
The work presented in [@Weyns2012] already shows that a single quality assurance technique is not enough as an adequate approach to achieve high-quality SASS. Testing and formal verification have long been known as complementing techniques for most kinds of systems. We assume that quality assurance for SASS can benefit in the same way from the combination of approaches like the one presented by us and approaches of category \[inpar:formal\]. Likewise, we see early testing as a complementary technique to online and adaptive online testing (cf. categories \[inpar:online\] and \[inpar:evolved\]). To our understanding, this specifically holds for SASS where unknown circumstances may arise at runtime and need to be adequately taken care of. Nevertheless, testing still needs to be done before a system is to be deployed to ensure at least an initial and basic quality of the SASS.
Approaches of category \[inpar:destime\] also address testing of SASS at design time. We differ from these approaches by not being dependent on a complete system. Using RTM as the test interface allows us to test already when there are only fragments of the system available, which is in the earlier development stages. Also our approach allows to test in a bottom-up manner, starting from the smallest testable units of a SASS and proceeding to the entire system.
Conclusion {#sec:conclusion}
==========
In this paper we presented a systematic testing scheme for SASS. It encompasses a staged testing process inspired by the engineering of embedded software. Exploiting architectural runtime models with their various states allows us to address the different stages of one-way, in-the-loop, and online testing. Supporting early development stages with tests, we may find errors early. Furthermore, looking at the individual MAPE-K activities and their different integrations, we should be able to locate faults more easily. In this context, our initial evaluation gives us preliminary confidence about the scheme’s effectivity.
There are several directions to evolve the presented testing scheme in future. As of now we employ an ad hoc simulator for ${RTMS}$. We could instead make use of a formal model to automatically derive test cases by using coverage criteria, which includes the generation of test inputs and oracles (runtime models and properties) taking the uncertainty of SASS and its environment into account. Useful formalisms range from simple finite state machines to timed, hybrid or even probabilistic automata. Such a formal approach will further ease a thorough evaluation of the testing scheme. Another direction would be to address the neglected case that the adaptable software and environment change while the MAPE loop is running. We will study special test setups for this case.
[^1]: Modular Rice University Bidding System: <http://www.mdelab.de>
[^2]: \[fot:ignorepar\]We ignore here the case that the adaptable software and environment change while the feedback loop is running. While this case could not be excluded in general, we may neglect it due to the considered abstraction level as supported by architectural runtime models. That is, oftentimes the architecture does not change very frequently, for instance, due to failures.
[^3]: *EMF*: <https://eclipse.org/modeling/emf/>
[^4]: *EMF Compare*: <https://www.eclipse.org/emf/compare/>
[^5]: *Eclipse OCL*: <http://projects.eclipse.org/projects/modeling.mdt.ocl>
|
---
abstract: 'In [@Salce] L. Salce introduced the notion of a cotorsion pair $(\cal{F},\cal{C}) $ in the category of abelian groups. But his definitions and basic results carry over to more general abelian categories and have proven useful in a variety of settings. A significant result of cotorsion theory proven by Eklof & Trlifaj is that if a pair $(\cal{F},\cal{C})$ of classes of $R$-modules is cogenerated by a set, then it is complete [@Eklof]. Recently Herzog, Fu, Asensio and Torrecillas developed the ideal approximation theory [@Herzog], [@idealapp]. In this article we look at a result motivated by the Eklof & Trlifaj argument for an ideal ${\cal{I}\ }$ when it is generated by a set of homomorphisms.'
author:
- |
Furuzan Ozbek\
[Department of Mathematics, University of Kentucky, Lexington, Kentucky, USA]{}
title: '**Precovering and preenveloping Ideals [^1] [^2]**'
---
Introduction
============
The concepts of preenvelope and precover were introduced by Enochs [@pre] for classes of modules. Since then the definition has been applied to different classes of categories. One of the recent application is given by Herzog, Fu, Asensio and Torecillas [@idealapp]. Herzog first defined and looked at phantom morphisms [@phantom]. Some properties of phantom morphisms raise interest in the subfunctors of the bifunctor $\Hom$ which they call ideals [@Herzog]. Herzog then applied the definition of preenvelope(precover), special preenvelope(special precover) and cotorsion pair to the ideal case in [@Herzog].
We see that several nice results in category of modules such as Salce’s lemma [@Salce] carry over to the ideal case in Herzog’s paper [@Herzog]. One significant result in cotorsion theory proven by Eklof & Trlifaj is that if a pair $(\cal{F},\cal{C})$ of classes of $R$-modules is cogenerated by a set, then it is complete [@Eklof]. We look at how this partially carries over to the ideal case.
Throughout this paper, we will focus on the ideals which are generated by a set of homomorphisms. First we examine how the elements of such ideals look like (Remark \[setgenerated\]). This helps us to characterize the elements of ${\cal{I}^{\perp}\ }$ (Lemma \[iperp\]). We see that every homomorphism $g$ in ${\cal{I}^{\perp}\ }$ has a small enough factorization (Lemma \[factor\]) which is motivated by the Proposition(5.2.2) from Enochs and Jenda’s Relative Homological Algebra [@RHA]. With all the tools in hand we prove that if an ideal ${\cal{I}\ }$ is generated by a set then ${\cal{I}^{\perp}\ }$ is a preenveloping class (Theorem \[preenveloping\]).
In section 5, we give the definition for an ideal to be closed under sums. We first observe that being closed under sums is necessary for an ideal to be a precovering. Then we see that this property is sufficient under certain conditions (Theorem \[precovering\]).
Finally we revise the definition of being generated by a set of homomorphisms. We see that if we allow infinite direct sums in the factorization of elements of an ideal, the results we have luckily still hold. We conclude with some immediate questions that still need to be answered.
Preliminaries
=============
Let $R$ be an arbitrary ring and $R$-mod denote the category of left $R$ modules. Throughout the paper we will denote $\Ext^1_{R}$ as $\Ext$ and interpret $\Ext$ in terms of extensions of $R$ modules when necessary. An extension of $U$ by $V$ denoted by an element $\xi \in \Ext(U,V)$ is a short exact sequence, $$\xymatrix{0 \ar[r] & U \ar[r] & X \ar[r] & V \ar[r] & 0}$$ Two extensions are said to be equivalent if there is a homomorphism making the diagram, $$\xymatrix{0 \ar[r] & U \ar@{=}[d] \ar[r] & X \ar[d] \ar[r] & V \ar@{=}[d] \ar[r] & 0\\
0 \ar[r] & U \ar[r] & Y \ar[r] & V \ar[r] & 0}$$ commutative. Then one can easily see that there is an equivalence relation on $\Ext(U,V)$ for any given pair of modules and that $\Ext:R-mod \times R-mod \rightarrow Ab$ is a bifunctor covariant in the first component and contravariant in the second. Then $\Ext(f,g)$ will be calculated by using a pushout along $g$ followed by a pullback along $f$ (or equivalently a pullback along f followed by a pushout along g). Note that $\Ext(f,g)(\xi)=0$ means that the extension $\Ext(f,g)(\xi)$ is split exact.
Throughout the paper an additive subfunctor ${\cal{I}\ }$ of the $\Hom_{R}$ functor will be called an ideal. This is Herzog’s definition first given in [@Herzog]. As a consequence one can easily observe that a class ${\cal{I}\ }$ of $R$-homomorphisms will form an ideal if it satisfies the following conditions,
- If $f,g$ is in ${\cal{I}\ }$ with the same domain and codomain then $f+g$ is in ${\cal{I}\ }$.
- If $g$ is in ${\cal{I}\ }$ then for any $R$-homomorphisms $f,h$, $f \circ g \circ h$ is in ${\cal{I}\ }$ (assuming the domains and codomains are suitable for the composition).
The definitions of a precover(right-approximation) and preenvelope(left-approximation) are carried over to the ideal case as given below.
Let ${\cal{I}\ }\subseteq \Hom_{R}$ be an ideal and $M$ be a left R-module. An ${\cal{I}\ }$-precover of $M$ is a morphism $i:I \rightarrow M$ such that any $i': I' \rightarrow M$ from $I$ factors through $i$ as seen in the following diagram, $$\xymatrix{& I' \ar@{.>}[ld] \ar[d]^{i'} \\
I \ar[r]^{i} & M}$$ A $\cal{J}$-preenvelepe is defined similarly.
Given two ideals ${\cal{I}\ }, \cal{J} \subseteq \Hom_{R}$ of R-modules define,\
$ {\cal{I}^{\perp}\ }$$= \{ j | \Ext^{1}(i,j)=0 \text{ for all \,} i \in {\cal{I}\ }\}$\
$^{\perp} \cal{J}$$=\{i | \Ext^{1}(i,j)=0 \text{ for all \,} j \in \cal{J} \}$
An ideal cotorsion pair in R-Mod is a pair $({\cal{I}\ },\cal{J})$ of ideals such that ${\cal{I}\ }^{\perp}=\cal{J}$ and $^{\perp}\cal{J}={\cal{I}\ }$.
Properties of an ideal generated by a set
=========================================
In this section we observe how the elements of an ideal generated by a set can be factored through a certain kind of homomorphism. This observation helps us to identify the elements of ${\cal{I}^{\perp}\ }$. We finish the section by Lemma \[factor\] which will be the main tool while proving ${\cal{I}^{\perp}\ }$ to be preenveloping.
Let ${\cal{I}\ }$$=<f>$ where $f: M \rightarrow N$ then $\varphi:U \rightarrow V$ is in $I$ if and only if it has a factorization of the form, $$\xymatrix{U \ar[r] &M^{m} \ar[r]^{f_{ji}} & N^{n} \ar[r] & V }$$ for some $1 \leq m,n$ where $f_{ji}$ has entries either equal to $f$ or $0$.
Let $S=\{\varphi \, | \, \varphi \, \text{has the desired factorization property}\}$. Clearly $f \in {\cal{I}\ }$ and $S \subseteq {\cal{I}\ }$, so it is enough to prove that $S$ is an ideal. Let $g,g' \in S$ where, $$\xymatrix{g: U \ar[r]^{\alpha} &M^{m} \ar[r]^{g_{ji}} & N^{n} \ar[r]^{\beta} & V }$$ and $$\xymatrix{g': U \ar[r]^{\tilde{\alpha}} & M^{\tilde{m}} \ar[r]^{g'_{ji}} & N^{\tilde{n}} \ar[r]^{\tilde{\beta}} & V }$$ then $g+g'$ has the following factorization, $$\xymatrix{U \ar[r]^-{(\alpha,\tilde{\alpha})} & M^{m+\tilde{m}} \ar[r]^{h} & N^{n+\tilde{n}} \ar[r]^-{(\beta,\tilde{\beta})} & V }$$ where $$(\alpha,\tilde{\alpha})(u)=(\alpha(u),\tilde{\alpha}(u)),$$ $$(\beta,\tilde{\beta})(n,\tilde{n})=\beta(n)+\tilde{\beta}(n)$$ and $$h=\begin{bmatrix}
(g_{ji}) & 0 \\
0 & (g'_{ji}) \\
\end{bmatrix}$$ for each $1 \leq j \leq n+\tilde{n}$ and $1 \leq i \leq m+\tilde{m}$. Given $u\in U$ say $\alpha(u)=(x_{1},...,x_{m})$ and $\tilde{\alpha}(u)=(y_{1},...,y_{\tilde{m}})$ then, $$\xymatrix{u \ar@{|->}[r]^-{(\alpha,\tilde{\alpha})} & (\alpha(u),\tilde{\alpha(u)}) \ar[r]^-{h} & (g_{ji}(x_{j}),g'_{ji}(y_{j})) \ar[r]^-{(\beta,\tilde{\beta})} &\beta(g_{ji}(x_{j}))+ \tilde{\beta}(g'_{ji}(y_{j})) }$$ where notice that, $$g(u)+ g'(u)= \beta(g_{ji}(x_{j}))+ \tilde{\beta}(g'_{ji}(y_{j}))$$ Hence we conclude that we have a factorization of $g+g'$. $\Box$
Let ${\cal{I}\ }$$=<f^{k}>_{k \in K}$ be generated by a set of homomorphisms where $f^{k}:M_{k}\rightarrow N_{k}$ then $\varphi:U \rightarrow V$ is in ${\cal{I}\ }$ if and only if it has a factorization as following, $$\xymatrix{U \ar[r] & M_{k_{1}}^{m_{1}}\oplus ...\oplus M_{k_{t}}^{m_{t}} \ar[r]^{(h_{ji})} & N_{k_{1}}^{n_{1}}\times ...\times N_{k_{t}}^{n_{t}} \ar[r] &V }$$ where $k_{1},...,k_{t} \in K$ and
$$\begin{aligned}
h_{ji}&=f^{k_{1}} \text{\, or \,} 0 \text{\, for \,} 1 \leq j \leq n_{1}, 1\leq i \leq m_{1} ,\\
\hspace*{0.5in} . \\
\hspace*{0.5in} .\\
\hspace*{0.5in} . \\
h_{ji}&=f^{k_{t}} \text{\,or \,} 0 \text{\, for \,} (n_{1}+...+n_{t-1})\leq j \leq(n_{1}+...+n_{t}), \, (m_{1}+...+m_{t-1})\leq i \leq ( m_{1}+...+m_{t})\end{aligned}$$
and $h_{ji}$ can be viewed as a matrix with entries, $$\begin{bmatrix}
f_{ji}^{k_{1}} & 0 & \cdots & 0 \\
0 & f_{ji}^{k_{2}} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & f_{ji}^{k_{t}}
\end{bmatrix}$$ \[setgenerated\]
Very similar to the case where ${\cal{I}\ }$ is generated by a single homomorphism.
Let ${\cal{I}\ }$$=< f_{k} \, | \, k \in K >$ and $f: M_{k} \rightarrow N_{k}$, $g: U \rightarrow V$ be homomorphisms of R-modules then the following are equivalent,
1. $g$ is in ${\cal{I}^{\perp}\ }$.
2. Given any s.e.s., $$\xymatrix{ 0 \ar[r] & U \ar[r] & X \ar[r] & Y \ar[r] & 0}$$ the s.e.s. we get by using the pushout along $g$, $$\xymatrix{ 0 \ar[r] & U \ar[r] \ar[d]^{g} & X \ar[r] \ar[d] & Y \ar[r] \ar@{=}[d] & 0\\
0 \ar[r] & V \ar[r] & Q \ar[r] & Y \ar[r] & 0}$$ can be completed to a commutative diagram for any $N_{k} \rightarrow Y$ where $k \in K$ as shown below, $$\xymatrix{ & & & M_{k} \ar@{.>}^{\displaystyle{\circlearrowleft}}[ddl] \ar[d]^{f_{k}} & \\
& & & N_{k} \ar[d] & \\
0 \ar[r] & V \ar[r] & Q \ar[r] & Y \ar[r] & 0}$$
3. Let $0\rightarrow V\rightarrow Q\rightarrow Q/V\rightarrow 0$ be the s.e.s. obtained by using the pushout along $g$, then for any $k \in K$ the composition $ \Hom(N_{k}, Q/V) \rightarrow \Hom(M_{k} , Q/V) \rightarrow \Ext^{1}(M_{k},V)$ (obtained from $\xymatrix{ M_{k} \ar[r]^{f_{k}}& N_{k} \ar[r]& Q/V}$) is the zero map for any homomorphism $N_{k}\rightarrow Q/V$.
\[iperp\]
($1 \Rightarrow 2$) Assume that the first statement is correct. Since $g \in {\cal{I}^{\perp}\ }$, then $\Ext^{1}(\tilde{f},g)(\xi)=0$ for any $\tilde{f} \in {\cal{I}\ }$. That is the resulting s.e.s we get by computing the pushout along $g$ followed by the pullback along $\tilde{f}$ is split exact. But since $M_{k} \rightarrow N_{k} \rightarrow Y$ is in ${\cal{I}\ }$, $$\xymatrix{ 0 \ar[r] & V \ar[r] \ar@{=}[dd] & P \ar[dd] \ar[r]& M_{k} \ar@/_/[l] \ar@{.>}[ddl] \ar[d]^{f_{k}} \ar[r]& 0\\
& & & N_{k} \ar[d] & \\
0 \ar[r] & V \ar[r] & Q \ar[r] & Y \ar[r] & 0}$$ we get a split exact sequence on the upper row. Hence we can complete the triangle with the dotted map (which is obtained by the composition $M_{k} \rightarrow P \rightarrow Q$) above.
($2 \Rightarrow 1$) Assume now that the second property holds for $g: U \rightarrow V$. We need to prove that $\Ext^{1}(\tilde{f},g)(\xi)=0$ where $$\xymatrix{ \tilde{f}: M_{j} \oplus M_{i} \ar[r]^-f & N_{j} \oplus N_{i} \ar[r] & Y }$$ in ${\cal{I}\ }$ and for any s.e.s. $\xi: 0 \rightarrow U \rightarrow X \rightarrow Y \rightarrow 0$ (then the proof follows very similarly for an arbitrary $\tilde{f} \in {\cal{I}\ }$). By the previous proposition, $f=\left(
\begin{array}{cc}
f_{j} & 0 \\
0& f_{i}\\
\end{array}
\right)$ where $f_{j},f_{i}$ are from the set of generators of ${\cal{I}\ }$. Given any $\varphi: N_{j}\oplus N_{i} \rightarrow Y$, we define $\varphi_{j}: N_{j}\rightarrow Y $ and $\varphi_{i}: N_{i}\rightarrow Y $ such that $\varphi_{j}$ is the restriction of $\varphi$ to $N_{j} \oplus 0$ and similarly $\varphi_{i}$ is the restriction of $\varphi$ to $0 \oplus N_{i}$. Then by assumption there exists $\alpha_{j}$ and $\alpha_{i}$ making the following diagrams commutative, $$\xymatrix{ & & & M_{j} \ar@{.>}^{\displaystyle{\circlearrowleft}}[ddl]_{\alpha_{j}} \ar[d]^{f_{j}} & \\
& & & N_{j} \ar[d]^{\varphi_{j}} & \\
0 \ar[r] & V \ar[r] & Q \ar[r] & Y \ar[r] & 0}$$ and $$\xymatrix{ & & & M_{i} \ar@{.>}^{\displaystyle{\circlearrowleft}}[ddl]_{\alpha_{i}} \ar[d]^{f_{i}} & \\
& & & N_{i} \ar[d]^{\varphi_{i}} & \\
0 \ar[r] & V \ar[r] & Q \ar[r] & Y \ar[r] & 0}$$ Then it is easy to see that the map $\alpha: M_{j} \oplus M_{i} \rightarrow Q $ defined as $\alpha(x_{j},x_{i})=\alpha_{j}(x_{j})+\alpha_{i}(x_{i})$ makes the following diagram commutative, $$\xymatrix{ & & & M_{j}\oplus M_{i} \ar[ddl]_{\alpha} \ar[d]^{f} & \\
& & & N_{j}\oplus N_{i} \ar[d]^{\varphi} & \\
0 \ar[r] & V \ar[r] & Q \ar[r] & Y \ar[r] & 0}$$ If we compute the pullback along $\tilde{f}$ we get the following diagram, $$\xymatrix{ P \ar[r] \ar[d] & M_{j}\oplus M_{i} \ar[d]^{\tilde{f}} \\
Q \ar[r] & Y }$$ Now using the commutativity of the previous diagram, we get the following commutative diagram, $$\xymatrix{ M_{j}\oplus M_{i} \ar@/^/^{id}_{\circlearrowleft}[drr] \ar@/_/_{\alpha}^{\circlearrowright}[ddr] \ar@{.>}[dr]^{\psi} & &\\
&P \ar[r] \ar[d] & M_{j}\oplus M_{i} \ar[d] \\
&Q \ar[r] & Y }$$ So by the universal property of pushout diagrams we conclude that there exists a homomorphism $\psi$ such that $\xymatrix{ M_{j}\oplus M_{i} \ar^-\psi[r] & P \ar[r] & M_{j}\oplus M_{i} } $ is the identity morphism. Hence looking at the s.e.s. obtained by the pullback along $\tilde{f}$, $$\xymatrix{0 \ar[r] & U \ar[r] \ar@{=}[d] & P \ar[r] \ar[d] & M_{j}\oplus M_{i} \ar[r] \ar^{\tilde{f}}[d] & 0 \\
0 \ar[r] & U \ar[r] & Q \ar[r] & Y \ar[r] & 0}$$ we conclude that the upper row is split exact. That is $\Ext^{1}(\tilde{f},g)$ maps $\xi$ to a split exact sequence, i.e. $\Ext^{1}(\tilde{f},g)(\xi)=0$ for any s.e.s. $\xi$.
($2 \Leftrightarrow 3$) Assume that the second property holds. Given any s.e.s. $\xi: 0 \rightarrow U \rightarrow X \rightarrow Y \rightarrow 0$ by using the pushout along $g$ we get, $$\xymatrix{ 0 \ar[r] & U \ar[r] \ar[d]^{g} & X \ar[r] \ar[d] & Y \ar[r] \ar@{=}[d] & 0\\
0 \ar[r] & V \ar[r] & Q \ar[r] & Y\cong Q/V \ar[r] & 0}$$ by assumption the lower row can be completed to a commutative diagram for any $N_{k} \rightarrow Q/V$ as shown below, $$\xymatrix{ & & & M_{k} \ar@{.>}^{\displaystyle{\circlearrowleft}}[ddl] \ar[d]^{f_{k}} & \\
& & & N_{k} \ar[d] & \\
0 \ar[r] & V \ar[r] & Q \ar[r] & Q/V \ar[r] & 0}$$ So we get the diagram, $$\xymatrix{ &\Hom(N_{k}, Q/V) \ar[d] & \\
\Hom(M_{k},Q)\ar[r] &\Hom(M_{k},Q/V) \ar[r] & \Ext^{1}(M_{k},V)}$$ with an exact row. But now our assumption holds if and only if the following composition, $$\xymatrix{\Hom(N_{k},Q/V)\ar[r] &\Hom(M_{k},Q/V) \ar[r] & \Ext^{1}(M_{k},V) }$$ is the zero map. $\Box$
Let ${\cal{I}\ }$ be as in lemma \[iperp\] and $g: U \rightarrow V$ be in ${\cal{I}^{\perp}\ }$. If $V^{'} \subseteq V$ is a submodule such that $g(U)\subseteq V^{'} \subseteq V$ and the map $\Ext^{1}(M_{k},V^{'}) \rightarrow \Ext^{1}(M_{k},V)$ is an injection for any $k \in K$ then $g: U \rightarrow V^{'} $ is in ${\cal{I}^{\perp}\ }$ as well.\[extinjection\]
Notice that by lemma \[iperp\] we conclude that $g:U \rightarrow V' \subseteq V$ is in ${\cal{I}^{\perp}\ }$ if and only if the following composition is $0$ for any given $N_{k} \rightarrow Q^{'}/V^{'}$ and any given $k \in K$, $$\xymatrix{\Hom(N_{k},Q^{'}/V^{'})\ar[r] &\Hom(M_{k},Q^{'}/V^{'}) \ar[r] & \Ext^{1}(M_{k},V^{'}) }$$ which is induced from following s.e.s., $$0\rightarrow V^{'}\rightarrow Q^{'}\rightarrow Q^{'}/V^{'}\rightarrow 0$$ Then the following diagram, $$\xymatrix{ 0 \ar[r] & U \ar[r] \ar[d]^{g} & E(U)\ar[r] \ar[d] & E(U)/U \ar[r] \ar[d]^{\cong} & 0\\
0 \ar[r] & V^{'} \ar[r] \ar@{^{(}->}[d] & Q^{'} \ar[r] \ar[d] & Q^{'}/V^{'} \ar[r] \ar[d]^{\cong} & 0\\
0 \ar[r] & V \ar[r] & Q \ar[r] & Q/V \ar[r] & 0}$$ gives us, $$\xymatrix{\Hom(N_{k},Q'/V^{'})\ar[r] \ar[d]^{\cong} &\Hom(M_{k},Q'/V^{'}) \ar[r] \ar[d]^{\cong} & \Ext^{1}(M_{k},V^{'}) \ar[d] \\
\Hom(N_{k},Q/V)\ar[r] &\Hom(M_{k},Q/V) \ar[r] & \Ext^{1}(M_{k},V)}$$ We notice that if $\Ext^{1}(M_{k},V^{'}) \rightarrow \Ext^{1}(M_{k},V)$ is an injection for every $k \in K$ then the composition on the top row is $0$ for every $k \in K$. By lemma \[iperp\] we conclude that $g:U \rightarrow V'$ is in ${\cal{I}^{\perp}\ }$. $\Box$
Let ${\cal{I}\ }$ be as in lemma \[iperp\]. If $g_{i}: U \rightarrow V_{i}$, $i \in I$ are each in ${\cal{I}^{\perp}\ }$ then $g: U \rightarrow \prod\limits_{i \in I} V_{i}$ where $g(x)=(g_{i}(x))_{i \in I}$ is in ${\cal{I}^{\perp}\ }$.\[prod\]
Assume $g_{i}: U \rightarrow V_{i}$, $i \in I$ are each in ${\cal{I}^{\perp}\ }$. That is $\varphi_{i}=\Ext^{1}(\tilde{f},g_{i})$ is the zero map for any $\tilde{f}:Z \rightarrow Y $ in ${\cal{I}\ }$. Since $\Ext^{1}(Z,\prod\limits_{i \in I} V_{i})\cong \prod\limits_{i \in I}\Ext^{1}(Z,V_{i})$ we have the following diagram, $$\xymatrix{ & \Ext^{1}(Y,U) \ar^{\varphi_{j}}[ddr] \ar_{\varphi_{i}}[ddl] \ar@{.>}[d]& \\
& \prod\limits_{i \in I}\Ext^{1}(Z,V_{i}) \ar_{\pi_{j}}[dr] \ar^{\pi_{i}}[dl] & \\
\Ext^{1}(Z,V_{i}) & & \Ext^{1}(Z,V_{j})}$$ So there exists $\varphi: \Ext^{1}(Y,U) \rightarrow \prod\limits_{i \in I}\Ext^{1}(Z,V_{i}) $ which makes the above diagram commutative. Now given any $\eta \in \Ext^{1}(Y,U)$ and say $\varphi(\eta)=(\xi_{i})_{i \in I}$ then, $$0=\varphi_{j}(\eta)=\pi_{j}(\varphi(\eta))=\pi_{j}((\xi_{i})_{i \in I})=\xi_{j}$$ That is $\eta=0$, which gives us that $\varphi$ or $\Ext(\tilde{f},g)$ is the zero map. Hence $g \in {\cal{I}^{\perp}\ }$. $\Box$
Let ${\cal{I}\ }$ be as in lemma \[iperp\] and $g: U \rightarrow V$ in ${\cal{I}^{\perp}\ }$. Then $g$ can be factored through $V^{'}$ such that, $$\xymatrix{U \ar[r] \ar@/^1pc/^{g}[rr] &V^{'} \ar@{^{(}->}[r] & V}$$ where the cardinality of $V^{'}$ is bounded by a cardinal number $\kappa$ which depends only on $|U|$ and ${\cal{I}\ }$. \[factor\]
First we need to show that $g$ is in ${\cal{I}^{\perp}\ }$ if and only if $\Ext(\tilde{f},g)(\xi^{'})=0$ for the short exact sequence $\xi^{'}: 0 \rightarrow U \rightarrow E(U) \rightarrow E(U)/U \rightarrow 0 $ where $E(U)$ is the injective hull of $U$. One way is obvious. To show the other way let, $$\xi: 0 \rightarrow U \rightarrow X \rightarrow Y \rightarrow 0$$ be any short exact sequence. Since $E(U)$ is injective we get the following commutative diagram, $$\xymatrix{ 0 \ar[r] & U \ar[r] \ar@{=}[d] & X \ar[r] \ar[d] & Y \ar[r] \ar@{.>}^{\exists k}[d] & 0\\
0 \ar[r] & U \ar[r] & E(U) \ar[r] & E(U)/U \ar[r] & 0}$$ where $k: Y \rightarrow E(U)/U$ is induced from $X \rightarrow E(U)$. Now we look at $\Ext^{1}(k,id_{U})(\xi^{'})$, $$\xymatrix{ 0 \ar[rr] && U \ar[rr] \ar@{=}[dd] && X \ar[rr] \ar[dd] && Y \ar[r] \ar[dd]^{k} & 0 && \\
& 0 \ar[rr] && U \ar[rr] \ar[dl]^{id_{U}} && E(U) \ar[rr] \ar[dl] && E(U)/U \ar[r] \ar[dl] & 0 \\
0 \ar[rr] && U \ar[rr] && E(U) \ar[rr] && E(U)/U \ar[r] & 0 &&}$$ That is $\Ext^{1}(k,id_{U})(\xi^{'})=\xi$. Note that the upper row is a pullback along $k$ since $X \rightarrow Y$ and $E(U) \rightarrow E(U)/U$ are epimorphisms and $\xymatrix{U \ar[r]^{id_{U}} & U}$ is an isomorphism. Then, $$\Ext^{1}(\tilde{f},g)(\xi)= \Ext^{1}(\tilde{f},g) \circ \Ext^{1}(k,id_{U})(\xi^{'}) = \Ext^{1}(k \circ \tilde{f},g)(\xi^{'})=0$$ Hence $g \in {\cal{I}^{\perp}\ }$.\
Now we want to construct a “small” enough $V^{'}$ such that $g: U \rightarrow V^{'} \subset V$ is in ${\cal{I}^{\perp}\ }$. By corollary \[extinjection\] it is enough to show that $\Ext^{1}(M_{k},V^{'}) \rightarrow \Ext^{1}(M_{k},V)$ is an injection for any $k\in K$. But notice that this holds if the following map is an injection, $$\prod\limits_{k \in K} \Ext^{1}(M_{k},V^{'}) \rightarrow \prod\limits_{k \in K} \Ext^{1}(M_{k},V)$$ But we have the following commutative diagram, $$\xymatrix{ \Ext^{1}(\bigoplus\limits_{k\in K}M_{k},V^{'}) \ar[r]^{\cong} \ar[d] & \prod\limits_{k \in K} \Ext^{1}(M_{k},V^{'}) \ar[d] \\
\Ext^{1}(\bigoplus\limits_{k\in K}M_{k},V) \ar[r]^{\cong} & \prod\limits_{k \in K} \Ext^{1}(M_{k},V)}$$ So we conclude that to show $g:U \rightarrow V^{'}$ in ${\cal{I}^{\perp}\ }$ it is enough to show the left column is injective.
Now we will construct the desired $V^{'}$. Given $i\in K$. Let $0 \rightarrow K_{i} \rightarrow P_{i} \rightarrow M_{i} \rightarrow 0$ be a partial projective resolution of $M_{i}$. Then we obtain the following partial projective resolution for $\bigoplus\limits_{i \in K} M_{i} $, $$\xymatrix{0 \ar[r] & K=\bigoplus\limits_{i \in K} K_{i} \ar[r] & P=\bigoplus\limits_{i \in K} P_{i} \ar[r] & \bigoplus\limits_{i \in K} M_{i} \ar[r] &0 }$$ We will construct an ascending chain of modules to obtain such a “small” $V^{'}$. Let $g(U)=V_{0}$ and for every $K \rightarrow V_{0}$ morphism that has an extension $P \rightarrow V$, choose one such extension $\alpha: P \rightarrow V$. So we have the following diagram, $$\xymatrix{P \ar[rd] & \\
K \ar@{^{(}->}[u] \ar[r]& V_{0}\subset V }$$ Define $V_{1}= \sum \alpha(P)$ where the sum is over all such chosen extensions $\alpha: P \rightarrow V$ for each $K \rightarrow V_{0} $. Then $V_{0} \subset V_{1}$. Now we construct $V_{2}$ in a similar way. So we get an ascending chain of modules, $$V_{0} \subset V_{1} \subset ... \subset V_{\omega} \subset V_{\omega+1} \subset ...\subset V_{\beta}$$ where $\beta$ is the least cardinal number with $|K|< \beta$ . Define $V_{\lambda}=\bigcup\limits_{\alpha < \lambda} V_{\alpha}$ if $\lambda \leq \beta$ is a limit ordinal and let $V^{'}=V_{\beta}$.
Now we will show that $g: U \rightarrow V^{'}$ is in ${\cal{I}^{\perp}\ }$. By the previous observation all we need is, $$\Ext^{1}(\bigoplus\limits_{k\in K}M_{k},V^{'}) \rightarrow \Ext^{1}(\bigoplus\limits_{k\in K}M_{k},V)$$ to be injective. Given a morphism $\xymatrix{K \ar[r]^-{\varphi} & V^{'} \subset V}$ that has an extension $\xymatrix{P \ar[r]^{\Phi} & V} $ we want to show that then there is an extension $P\rightarrow V^{'}$. But now since $|K| < \beta$ we conclude that $Im(\varphi)\subseteq V_{\alpha}$ for some $\alpha$ such that $|\alpha| < \beta$ hence by the construction of the ascending chain we can extend $\varphi$ to $P \rightarrow V_{\alpha+1} \subset V^{'}$. This shows that $$\Ext^{1}(\bigoplus\limits_{k\in K}M_{k},V^{'}) \rightarrow \Ext^{1}(\bigoplus\limits_{k\in K}M_{k},V)$$ is an injection. Now by corollary \[extinjection\] we conclude that $g: U \rightarrow V^{'} \subset V$ is in ${\cal{I}^{\perp}\ }$. Moreover note that the cardinality of $V_{\alpha+1}= \sum \Phi(P)$ (where sum is over all chosen extensions $\Phi: P \rightarrow V$ for each $K \rightarrow V_{\alpha}$) is bounded by, $$|V_{\alpha+1}| \leq |P|^{|V_{\alpha}|^{|K|} }$$ Since, $$|\Hom(K,V_{\alpha}) | \leq |V_{\alpha}^{K}|=|V_{\alpha}|^{K}$$ and $$|\Phi(P)| \leq |P|$$ We find a bound on $|V^{'}| \leq \sum\limits_{\alpha < \beta} |V_{\alpha}|$. So we conclude that for any given $U$, if $g: U \rightarrow V$ is in ${\cal{I}^{\perp}\ }$ we can find a factorization $ U \rightarrow V^{'} \rightarrow V$ where $|V^{'}| \leq \kappa$ for some cardinal number $\kappa$ that depends on $|U|$, $|K|$ and $|P|$ and so only on $|U|$ and ${\cal{I}\ }$. $\Box$
Main result
===========
We now prove the main theorem. This result was motivated by the theorem of Eklof-Trlifaj’s (Thm.10, [@Eklof]).
If ${\cal{I}\ }$ is generated by a set then ${\cal{I}^{\perp}\ }$ is a preenveloping class.\[preenveloping\]
Given a R-module $U$ by lemma \[factor\] we find a cardinal number $\kappa$ with the desired properties given in the lemma . We will use a similar argument to Rada-Saorín’s from [@RadaSaorin]. Let $\{ g_{j}\}_{j \in J}$ be the set of all the homomorphisms $g_{j}: U \rightarrow V_{j}$ in ${\cal{I}^{\perp}\ }$ (up to isomorphism) where $|V_{j}| \leq \kappa$. Then by lemma \[prod\] the homomorphism, $$\xymatrix{U \ar[r]^{\prod g_{j}} & \prod\limits_{j \in J} V_{j} }$$ is in ${\cal{I}^{\perp}\ }$, moreover we claim that it is an ${\cal{I}^{\perp}\ }$-preenvelope of $U$.\
Given any $g: U \rightarrow V$ in ${\cal{I}^{\perp}\ }$ by lemma \[factor\] we get a factorization, $$\xymatrix{U \ar[r] \ar@/^1pc/[rr]^{g} & V' \ar@{^{(}->}[r]& V}$$ where $|V'| \leq \kappa$. Then $g: U \rightarrow V' \subset V$ is isomorphic to $g_{j}$ for some $j$. That is there exists a map making the following commutative, $$\xymatrix{ U \ar[r]^{g_{j}} \ar[d]_{g} & V_{j} \ar[dl]_{\displaystyle\circlearrowright} \\
V' & }$$ Now we get the following commutative diagram, $$\xymatrix{ U \ar[r]^{g_{j}} \ar[d] & \prod\limits_{j \in J} V_{j} \ar[dl] \\
V_{j} \ar[d]& \\
V' \ar[d]& \\
V & }$$ We conclude that the following diagram is commutative where we use the composition of maps $\prod\limits_{j \in J} V_{j} \rightarrow V_{j} \rightarrow V' \rightarrow V$ , $$\xymatrix{ U \ar[r]^{g_{j}} \ar[d]_{g} & \prod\limits_{j \in J} V_{j} \ar[dl]_{\displaystyle\circlearrowright} \\
V & }$$ That is ${\cal{I}^{\perp}\ }$ is a preenveloping class. $\Box$
A necessary condition for ${\cal{I}\ }$ to be precovering
=========================================================
The ideal ${\cal{I}\ }$ is said to be closed under sums if it satisfies one of the following equivalent conditions,
- If $(f_{j})_{j \in J}$, $f_{j}: M_{j} \rightarrow N_{j}$ is any family of elements of ${\cal{I}\ }$ then $\bigoplus\limits_{j \in J}f_{j}: \bigoplus\limits_{\j \in J} M_{j} \rightarrow \bigoplus\limits_{\j \in J} N_{j}$ is in ${\cal{I}\ }$.
- If $(g_{j})_{j \in J}$, $g_{j}: M_{j} \rightarrow N$ is any family of elements of ${\cal{I}\ }$ then $ g: \bigoplus\limits_{\j \in J} M_{j} \rightarrow N$ defined by $g((x_{j})_{j \in J})=\sum\limits_{j \in J}g_{j}(x_{j})$ is in ${\cal{I}\ }$.
If ${\cal{I}\ }$ is the closure under direct sums of the ideal generated by a single homomorphism $f:M \rightarrow N$ then ${\cal{I}\ }$ is a precovering ideal.\[single\]
Given an arbitrary $R$-module $V$, we consider the homomorphism $\alpha: M^{(\Hom(N,V))} \rightarrow V $ defined by, $$\alpha((x_{g})_{g\in \Hom(N,V)})=\displaystyle\sum\limits_{g\in \Hom(N,V)} g(f(x_{g}))$$ which is in ${\cal{I}\ }$ since it is closed under sums. Moreover we claim that it is a ${\cal{I}\ }$-precover of $V$. Given a homomorphism, $$\xymatrix{\bigoplus\limits_{j \in J} M \ar[r] & \bigoplus\limits_{j \in J}N \ar[r]^{h} & V }$$ in ${\cal{I}\ }$. Define $h_{j}: N \rightarrow V $ such that $h=\sum\limits_{j \in J} h_{j}$ and $\beta_{j}:M \rightarrow M^{(\Hom(N,V))} $ such that $\beta_{j}(x)$ is the element of $M^{(\Hom(N,V))}$ whose all entries are $0$, except the one that corresponds to $h_{j}$, which is $x$. With the maps defined above following diagram commutes, $$\xymatrix{ & M \ar[ddl]_{\beta_{j}} \ar[d]^{f} \\
& N \ar[d]^{h_{j}} \\
M^{(\Hom(N,V))} \ar[r]^-\alpha & V }$$
Now we define $\beta$, $$\xymatrix{ & \bigoplus\limits_{j \in J} M \ar@{.>}[ddl]_{\beta} \ar[d]^-{\bigoplus f} \\
& \bigoplus\limits_{j \in J}N \ar[d]^{h} \\
M^{(\Hom(N,V))} \ar[r]^-\alpha & V }$$ where $\beta((x_{j})_{j \in J})=\sum\limits_{j\in J} \beta_{j}((x_{j})$. Then notice that, $$\begin{aligned}
\alpha(\beta((x_{j})_{j \in J})) &= \alpha(\sum\limits_{j\in J} \beta_{j}(x_{j}))=\sum\limits_{j\in J} \alpha(\beta_{j}(x_{j})) \\
&=\sum\limits_{j\in J} h_{j}(f(x_{j}))= h((f(x_{j}))_{j\in J}) \\
&=(h ( \oplus f(x_{j})_{j \in J}))\end{aligned}$$ So the above diagram is commutative and we conclude that $V$ has an ${\cal{I}\ }$-precover. That is ${\cal{I}\ }$ is a precovering ideal. $\Box$
If ${\cal{I}\ }$ is the closure under direct sums of the ideal generated by a set then ${\cal{I}\ }$ is a precovering ideal.\[precovering\]
The proof follows very similarly to that of theorem \[single\]. $\Box$
Let ${\cal{I}\ }=$$< f_{s} >_{s\in S}$ generated by a set and ${\cal{I}\ }^{'}$ be the smallest ideal that contains ${\cal{I}\ }$ and closed under sums. Then ${\cal{I}^{\perp}\ }= ({\cal{I}\ }^{'})^{ \perp}$.\[extendedperp\]
One way of the inclusion is easy. By definition ${\cal{I}\ }\subseteq {\cal{I}\ }^{'}$ implies $({\cal{I}\ }^{'})^{ \perp} \subseteq {\cal{I}^{\perp}\ }$.
To prove the other way of inclusion let $g \in {\cal{I}^{\perp}\ }$ where $g: U \rightarrow V$. Then notice that $\Ext^{1}(f_{s},g)=0$ for all $f_{s}$. Now given any $T \subseteq S$ and a homomorphism, $$\xymatrix{\bigoplus M_{t} \ar[r]^{\oplus f_{t}} & \bigoplus N_{t} }$$ We want to prove that, $$\xymatrix{\Ext^{1}(\oplus f_{t}, g): \Ext^{1}(\oplus N_{t},U) \ar[r] & \Ext^{1}(\oplus M_{t},V)}$$ is the zero map. We observe the commutative diagram, $$\xymatrix{ \Ext^{1}(\bigoplus\limits_{t\in T}N_{t},U) \ar[r]^{\cong} \ar[d] & \prod\limits_{t\in T} \Ext^{1}(N_{t},U) \ar[d] \\
\Ext^{1}(\bigoplus\limits_{t\in T}M_{t},V) \ar[r]^{\cong} & \prod\limits_{t\in T} \Ext^{1}(M_{t},V) }$$ Notice that the right column is the zero map since $\Ext^{1}(f_{t},g)=0$ for all $t \in T$. Hence we conclude that $\Ext^{1}(\oplus f_{t}, g)=0$. Now given an arbitrary homomorphism in ${\cal{I}\ }^{'}$, $$\xymatrix{U \ar[r]^-{k} & \bigoplus M_{t} \ar[r]^-{\oplus f_{t}} & \bigoplus N_{t} \ar[r]^{h} & V }$$ where $t \in T \subseteq S$. But we have, $$\Ext^{1}(h\circ \oplus f_{t} \circ k,g)=\Ext^{1}(k,id)\circ \Ext^{1}(\oplus f_{t} ,g) \circ \Ext^{1}(h,id)=0$$ Hence we conclude that ${\cal{I}^{\perp}\ }\subseteq ({\cal{I}\ }^{'})^{ \perp}$. $\Box$
Ideals generated by a set in the extended sense
===============================================
We revise our definition of ${\cal{I}\ }$ being generated by a set.
Let $(f_{s})_{s \in S}$ be a set of homomorphisms where $f_{s}: M_{s} \rightarrow N_{s}$. ${\cal{I}\ }$ is said to be generated by $(f_{s})_{s \in S}$ in the extended sense if every $\tilde{f}:U \rightarrow V$ in ${\cal{I}\ }$ has a factorization, $$\xymatrix{U \ar[r] & \bigoplus\limits_{s\in S}M_{s}^{\kappa_{s}} \ar[r]^{\bigoplus f_{s}} & \bigoplus\limits_{s\in S}N_{s}^{\kappa_{s}} \ar[r] & V }$$ where $\kappa_{s}$ is a cardinal number for each $s \in S$.
If ${\cal{I}\ }$ is generated by a set in the extended sense, then it is closed under sums.\[closedsum\]
Let ${\cal{I}\ }$ be generated by a set of homomorphisms in the extended sense, then ${\cal{I}^{\perp}\ }$ is a preenveloping ideal.
Theorem \[preenveloping\] and proposition \[extendedperp\] gives us the result. $\Box$
Let ${\cal{I}\ }$ be generated by a set of homomorphisms in the extended sense, then ${\cal{I}\ }$ is a precovering ideal.
Theorem \[precovering\] and remark \[closedsum\] gives us the result. $\Box$
There are questions still need to be answered when ${\cal{I}\ }$ is an ideal generated by a set in the extended sense, such as whether $({\cal{I}\ },{\cal{I}^{\perp}\ })$ is a cotorsion ideal pair (i.e is ${\cal{I}\ }=$$^{\perp}$$({\cal{I}^{\perp}\ })$) and what the necessary conditions are for completeness if $({\cal{I}\ },{\cal{I}^{\perp}\ })$ is a cotorsion ideal pair.
**Acknowledgements.** The author is very thankful to her PhD. advisor Prof. Edgar Enochs for his contributions and suggestions during the course of preparing this manuscript.
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[^1]: 2010 [*Mathematics Subject Classification*]{}. 18G25, 18G15 .
[^2]: [*Keywords*]{}. Precover and preenvelope; ideal approximation; cotorsion theory; set theoretic homological algebra.
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abstract: 'Magnetization measurements of LaCoO$_{3}$ have been carried out up to 133 T generated with a destructive pulse magnet at a wide temperature range from 2 to 120 K. A novel magnetic transition was found at $B>100$ T and $T>T^{*}=32\pm 5$ K which is characterized by its transition field increasing with increasing temperature. At $T<T^{*}$, the previously reported transition at $B\sim65$ T was observed. Based on the obtained $B$-$T$ phase diagram and the Clausius-Clapeyron relation, the entropy of the high-field phase at 80 K is found to be smaller for about $1.5$ J K$^{-1}$ mol$^{-1}$ than that of the low-field phase. We suggest that the observed two high-field phases may originate in different spatial orders of the spin states and possibly other degrees of freedom such as orbitals. An inherent strong correlation of spin states among cobalt sites should have triggered the emergence of the ordered phases in LaCoO$_{3}$ at high magnetic fields.'
author:
- Akihiko Ikeda
- Toshihiro Nomura
- 'Yasuhiro H. Matsuda'
- Akira Matsuo
- Koichi Kindo
- Keisuke Sato
bibliography:
- 'lco.bib'
title: 'Spin state ordering of strongly correlating LaCoO$_{3}$ induced at ultrahigh magnetic fields'
---
Due to the strong correlations between the electrons, the transition metal oxide serves as a vast field hosting rich electronic phases represented by high-temperature superconductivity, colossal magnetoresistance and magnetic-field-induced ferroelectorics [@Imada; @Tokura]. Among them, cobalt oxides are unique for their spin state degrees of freedom which not only bring about a magnetic crossover but also a metal-insulator transition (MIT) [@Tachibana] in the thermal evolution. Perovskite cobalt oxide, LaCoO$_{3}$, has attracted significant attention for more than five decades for its unusual magnetic and transport properties, namely, the crossover from a diamagnet to a Curie paramagnet at 100 K and the transition from a paramagnetic insulator to a paramagnetic metal at 500 K with increasing temperature [@Goodenough1958]. Within the ionic picture, possible spin states of Co$^{3+}$ are the low spin state (LS: $t_{2g}^{6}e_{g}^{0}$, $S=0$) and the high spin state (HS: $t_{2g}^{4}e_{g}^{2}$, $S=2$) that energetically lie close to each other due to the delicate balance of Hund’s coupling and crystal field splitting. Besides those, the intermediate spin state (IS: $t_{2g}^{5}e_{g}^{1}$, $S=1$) is also argued to be stabilized due to the strong hybridization with the O $2p$ state [@Korotin]. Representative ideas describing the spin states of LaCoO$_{3}$ in the temperature range above 100 K are (i) the LS-HS mixture state [@Raccah; @Haverkort; @Knizek; @Ropka2003; @Kyomen2003; @Kyomen2005] and (ii) the IS state [@Korotin; @Yamaguchi1997; @Ishikawa]. However, they are still controversial. It is notable that recent theoretical studies on the two-orbital Hubbard model have qualitatively reproduced the thermally induced spin crossover and MIT with paramagnetic local moments [@Kunes2011; @Kanamori2011; @Krapek]. On the other hand, they are inclined to predict the ordering of different spin states which is not found experimentally except for a few studies [@Doi].
The validity of the models on spin states should be well judged by their field effects. One can uncover magnetic excited states using high magnetic fields at low temperatures, eliminating the thermal effect. Thermodynamical properties of the magnetic phase can also be revealed by observing its temperature and magnetic field dependence [@Tokunaga; @Nomura]. In the case of LaCoO$_{3}$, a spin gap of about 100 K [@Yamaguchi1996] necessitates a high magnetic field amounting to 100 T. In fact, a first-order field-induced spin state transition [@Sato2009; @Moaz] accompanied by magnetostriction [@Moaz; @Rotter] has been found at $B=65$ and 70 T with magnetization measurements up to 100 T at below 4.2 K. The results are either understood in terms of the local spin crossover model [@Sato2009] or the formation of the spin state crystalline (SSC) phase, where the different spin states at Co$^{3+}$ and possibly the orbitals are spatially ordered [@Moaz; @Rotter] and further, the following two magnetization jumps at $B>100$ T are predicted by the Ising type SSC model [@Moaz]. With the explosive magnetic flux compression technique, magnetization up to 3.5$\mu_{\mathrm{B}}$ was observed at 500 T, 4.2 K, although the smeared transitions up to 100 T may be due to the fast sweeping rate ($>10$ T/$\mu$s) [@Platonov]. The $B$-$T$ range explored so far, however, has been limited to low temperatures. To verify the physical origins of the thermally induced magnetic phase and the field-induced magnetic phase of LaCoO$_{3}$, it is plausible to explore the properties of LaCoO$_{3}$ in even wider $B$ and $T$ ranges and clarify how those phases evolve and interact with each other on the $B$-$T$ plane.
In this Rapid Communication, we report a high-field magnetization study of LaCoO$_{3}$ up to 133 T at a wide range of temperatures from 2 to 120 K, from whose data a phase diagram in the wide $B$-$T$ range is constructed. We found first-order magnetic transitions at $B>100$ T at $T>T^{*}=32\pm 5$ K where the transition fields increased with increasing $T$, suggesting the existence of the low entropy phase at $B>100$ T and at $T>T^{*}$. We also confirmed the reported first order magnetic transition at $\sim65$ T at $T<T^{*}$ where the transition field was almost temperature independent. We obtained a rich phase diagram that contradicts the prediction based on the spin crossover in the local ion picture. We discuss the result in light of the formation of the field induced ordered phase due to strongly correlating spin states and other degrees of freedom such as orbitals.
High field magnetization measurements were carried out in the following manner. For the generation of a high field with a maximum field $B_{\mathrm{Max}}$ of 133 T, a horizontal type single-turn coil, a semi-destructive pulse magnet [@Miura], was employed. Helium flow type cryostats made of nonmetallic parts were used to cool the sample [@Takeyama1988; @Amaya]. The temperature at the sample space ranged from 10 to 120 K and was measured with a chromel-constantan thermocouple. We also used a vertical type single turn coil with a helium bath type cryostat for the measurements at $T=2.5$ and 4.2 K and $B_{\mathrm{Max}}\sim105$ T, as described in Ref. [@Takeyama2012]. The magnetization ($M$) of LaCoO$_{3}$ was obtained by measuring the induction voltage (proportional to $dM/dt$) of a well compensated pair of pickup coils, one of which held the sample inside. Small grains of single crystalline LaCoO$_{3}$ [@Sato2009] were put into a sample space of $\phi=0.9$ and $l=3$ mm with their crystal axis unoriented. The magnetic field $B$ was measured with a calibrated pickup coil placed close to the sample space.
Representative results of the time derivative of $M$, $dM/dt$, at 17 and 70 K are shown in Fig. \[fig1\](a), along with the time evolution of $B$. At 17 K, sharp peaks are seen at 70 and 60 T, respectively, whereas, at 70 K, the peaks were observed at higher fields, indicating that the transition fields are temperature dependent. $M$ curves were obtained by numerically integrating the $dM/dt$ data. $dM/dB$ curves were obtained by dividing the $dM/dt$ data with the $dB/dt$ data. They are plotted against $B$, as shown in Figs. \[fig1\](b) and \[fig1\](c), respectively. In Fig. \[fig1\](b), absolute values are evaluated by scaling the data to the $M$ data obtained using a nondestructive pulse magnet at ISSP, Univerisity of Tokyo. The obtained $M$ curves in Fig. \[fig1\](b) are in good agreement with the low-field magnetization data up to 33 T, as reported in Ref. \[\]. Whereas the magnetic transitions at $\sim65$ T and at below 30 K have been reported previously [@Sato2009; @Moaz; @Rotter], we show the magnetic transitions at above 37 K and above 100 T. One can clearly notice that the field-induced magnetic transitions in Fig. \[fig1\](b) and \[fig1\](c) are temperature dependent. This trend may have a common root with the observed positive temperature dependence of magnetic transitions at $B\simeq$ 60 T and at $T$ > 40 K in Ref. [@Moaz; @Rotter].
![(a) The $B$-$T$ phase diagram of LaCoO$_{3}$ based on the observed transition fields $B_{\mathrm{C1}}$ and $B_{\mathrm{C2}}$ in the present study. The dashed curve represents the predicted phase boundary based on the spincross over in the local ion picture [@Biernacki2005]. (b) $\Delta M$ and (c) $\Delta S$ at the phase boundary at each temperature. The same symbols are used as in (a). \[fig2\]](fig2.pdf)
We first focus on the obtained $M$ at 4.2 K in Fig. \[fig1\](b). The amount of the magnetization jump $\Delta M$ at the transition is $\sim 0.5 \mu_{\mathrm{B}}/\mathrm{f.u.}$, which is in good agreement with the reported values [@Sato2009; @Moaz]. The values of the transition field observed in the ascending and descending fields are $\sim75$ T and 65-60 T, respectively. The existence of a large hysteresis of about 15 T indicates that the transition is a first-order transition. The relatively smeared transition in the descending fields in Fig. \[fig1\](b) should originate from the heating effects during the first order phase transition, as suggested in Ref. [@Moaz]. With increasing temperature up to 27 K, the transitions become smeared, possibly due to the thermal effect, as seen in Fig. \[fig1\](c). Our result is inconsistent with the reports in Ref. \[\], where the first increase of $M$ of $\sim 0.5 \mu_{\mathrm{B}}/\mathrm{f.u.}$ at 63 T was followed by a second increase of $M$ of $\sim 0.5 \mu_{\mathrm{B}}/\mathrm{f.u.}$ at $\sim70$ T. The cause of the discrepancy is not clear at this moment, although it may be due to the sample or the field sweeping rate dependence. We regard that the second transition is absent in the present study. The sweep rate up to 100 T is faster in our case ($\sim$50 T/$\mu$s) than in the case of the explosive compression technique ($\sim$10 T/$\mu$s) [@Platonov]. Therefore, the smearing of the sharp transition in Ref. [@Platonov] may not be due to the intrinsic effects, such as thermal effects.
Next, we observe the temperature dependence of $M$ and $dM/dB$ in Figs. \[fig1\](b) and \[fig1\](c), respectively. Guided by the sudden change in the transition field at $T^{*}=32\pm5$ K, we term the transition fields for the ascending field and descending field at $T<T^{*}$ as $B^{\mathrm{up}}_{\mathrm{C1}}$ and $B^{\mathrm{down}}_{\mathrm{C1}}$ denoted with solid and open circles, and at $T>T^{*}$, $B^{\mathrm{up}}_{\mathrm{C2}}$ and $B^{\mathrm{down}}_{\mathrm{C2}}$ denoted with solid and open triangles, respectively. At $T>T^{*}$, we found that the novel magnetic transition is present at $B > 100$ T in the ascending field ($B^{\mathrm{up}}_{\mathrm{C2}}$) as denoted by the solid triangles in Fig. \[fig1\](c). With increasing temperature, the peaks at $B^{\mathrm{up}}_{\mathrm{C2}}$ in Fig. \[fig1\](c) are gradually sharpened and shifted towards higher fields. $B^{\mathrm{down}}_{\mathrm{C2}}$ also shifted to higher fields with increasing temperature at $T>T^{*}$. This is highly in contrast with $B^{\mathrm{up}}_{\mathrm{C1}}$ and $B^{\mathrm{down}}_{\mathrm{C1}}$ at $T<T^{*}$ being independent of temperature [@Sato2009; @Moaz].
We plot the obtained transition fields on the $B$-$T$ plane as shown in Fig. \[fig2\](a). The transitions at $T>T^{*}$ and $B>100$ T are reported (colored in blue). The hysteresis region is indicated by the shaded area. For clarity, we term the low-temperature low-field region and the high-temperature low-field region to be phases (A1) and (A2), respectively. We also term the high-field phases (B1) and (B2). It is evident that the high -field phases (B1) and (B2) are separated from the low field phases (A1) and (A2) by a first-order magnetic transition with hysteresis. Phases (B1) and (B2) are distinguished based on $T^{*}$.
By integrating the peaks in the $dM/dB$ curves, we obtained $\Delta M$ at each transition field for various temperatures, as shown in Fig. \[fig2\](b). The saturation magnetizations $M_{\mathrm{S}}$ expected for IS or HS Co$^{3+}$ are 2.0$\mu_{\mathrm{B}}/\mathrm{f.u.}$ or 4.0$\mu_{\mathrm{B}}/\mathrm{f.u.}$, respectively, provided $g=2$. $M_{\mathrm{S}}$ is not reached even after the magnetic transition at 70 K ($M\sim 1.0 \mu_{\mathrm{B}}/\mathrm{f.u.}$). With the observed values of $\Delta M$ in Fig. \[fig2\](b) and $dB/dT$ obtained from the phase boundary in Fig. \[fig2\](a), we deduced the entropy change $\Delta S$ at the field-induced transition based on the Clausius-Clapeyron relation [@Landau] $dB/dT=-\Delta S/\Delta M$, as shown in Fig. \[fig2\](c). At $T<T^{*}$, the slope is vertical resulting in $\Delta S \sim 0$ J K$^{-1}$ mol$^{-1}$. At $T>T^{*}$, $\Delta S$ gradually decreases from 0 and converges to $\sim -1.5$ J K$^{-1}$ mol$^{-1}$ at $T > 80$ K. This compares to the entropy increase at the thermally induced spin crossover in LaCoO$_{3}$ of $\sim 2.0$ J K$^{-1}$ mol$^{-1}$ from 13 to 80 K [@stolen]. Both of them are much smaller than the value of $R \ln 3=$ 9.13 J K$^{-1}$ mol$^{-1}$ expected for the thermal spin crossover from $S=0$ to $S=1$.
The magnetic transitions discovered at $B>100$ T and $T>T^{*}$ in the present study need temperature assistance to ascertain their origin. Our data set lacks the low-temperature ($T < 4.2$ K) high field ($B > 100$ T) data, which may make the absence of a magnetic transition at $100$ T $<B<140$ T and at $T<T^{*}$ inconclusive. However, the data at 4.2 K up to 140 T are actually provided in Ref. [@Moaz] by making use of the single-turn coil, evidencing that such a magnetic transition is absent up to 140 T. On this basis, we regard the magnetic transitions discovered at $B>100$ T and $T>T^{*}$ to have different origins from the predicted spin state cascade based on the Ising type SSC model at 0 K in Ref. [@Moaz], where the predicted magnetic transition at $B>100$ T should be present even at $T\sim$ 4.2 K, and it is not predicted that another ordered phase such as (B2) appears with increasing temperature.
Here, we argue that the most striking feature of the obtained phase diagram shown in Fig. \[fig2\](a) is that the transition field increases with increasing temperature. It is completely contrary to the shared tendency of the previous reports on spin crossover compounds such as cobalt oxides \[Sr$_{1-x}$Y$_{x}$CoO$_{3}$ [@Kimura2008], (Pr$_{1-y}$Y$_{y}$)$_{0.7}$Ca$_{0.3}$CoO$_{3}$ [@Marysko; @Naito2014]\] and coordinate compounds (Fe\[(phen)$_{2}$(NCS)$_{2}$\] [@Qi], \[Mn$^{\mathrm{III}}$(taa)\] [@Kimura2005]), where the transition fields are observed to decrease with increasing temperature, as schematically shown by the dashed curve in Fig. \[fig2\](a). This tendency can be readily anticipated by considering the spin crossover in the local ion picture, where the ground state is less magnetic (i.e., LS) and that the excited state is more magnetic (e.g., HS or IS). In this situation, the magnetic state will be occupied with increasing either $T$ or $B$ due to the entropy or Zeeman energy contribution, respectively. In LaCoO$_{3}$, the ground state is the LS phase [@Asai1994], denoted as phase (A1) in Fig. \[fig2\](a), whose entropy is considered to be small. The thermally induced paramagnetic state [@Asai1994], denoted as phase (A2) in Fig. \[fig2\](a), is considered to possess a larger entropy due to the magnetic, orbital, and phonon degrees of freedom of HS or IS species and the mixing entropy of the LS-HS or LS-IS complexes [@Biernacki2005]. Based on the local model for spin crossover, it is expected that the transition field decreases with increasing temperature and that phase (B1), (B2) merges with phase (A2) at the high-temperature and high-field region, as shown by the dashed curve in Fig. \[fig2\](a). It is clear that phase (A2) and phase (B1), (B2) are the distinct phases in the present result as shown in Fig. \[fig2\](a). It is now decisive that the local model for the spin crossover compounds [@Kyomen2003; @Biernacki2005] is not applicable to the $B$-$T$ phase diagram of LaCoO$_{3}$, suggesting that phasesr (A2) and (B1), (B2) are distinct in origin, which is contrary to the previous notion [@Sato2009; @Rotter].
We now discuss the origin of the observed high-field phases (B1) and (B2). In the present observation, the reduction of $S$ is observed in the transition from phase (A2) to phase (B2), as shown in Fig. \[fig2\](c). This may suggest that some order is present in phase (B2). The candidates for the order of phase (B2) are (i) antiferromagnetic order (AFM), (ii) spin state crystalline (SSC), and (iii) orbital order (OO). In the SSC, the spin states of Co$^{3+}$ are spatially ordered. Among them, we believe the SSC is the most plausible idea for the following reasons. First, because AFM becomes unstable under larger magnetic fields, its Néel temperature is expected to decrease with increasing magnetic field. However, as shown in Fig. \[fig2\](a), the transition temperature of (B2) increases with increasing magnetic field. Therefore, AFM is excluded. Next, we consider the SSC. In phase (A2) the spin states are disordered. At the magnetic transition, the number of Co$^{3+}$ in the magnetic spin states is increased and the spatial order of the spin states is obtained, forming the SSC, the spatial order of spin states. This scenario is in good agreement with experimental observations, namely, the sudden increase of magnetization and the decrease of entropy. Thus, we regard the SSC is present in phase (B2). Lastly, we consider OO. The orbital degree of freedom is quite spin state dependent. Therefore, if the spin states are disordered, it should be very difficult for the OO to appear. Besides, OO itself does not change the magnetization. Therefore, OO alone cannot be the order parameter of phase (B2). On the other hand, the OO on the background of the SSC should appear plausible. Such spin state ordering is also suggested in recent theories [@Knizek; @Kunes2011; @Kanamori2011; @Krapek] and high-field experiments [@Moaz; @Rotter].
Another feature found in the obtained phase diagram in Fig. \[fig2\](a) is the sudden change in the transition fields at $T^{*}$, making the two high-field phases (B1) and (B2) distinct. The phase boundary between phase (B1) and (B2) seems horizontal ($dT/dB=0$) at $T^{*}$. This means that $\Delta M/\Delta S=0$ in the virtual transition from phase (B1) to (B2) based on the Clausius-Clapeyron relation. We deduce $\Delta M=0$, assuming that $\Delta S$ is not so large. As a possible origin of the two distinctive phases (B1) and (B2), we discuss that, besides SSC, another order may be present in phase (B1) which does not change $M$. This is because the SSC of phase (B2) is expected to be even more stable in phase (B1) due to the lower temperature.
Possible origins for the order of (B1) in addition to the SSC are (i) AFM, (ii) OO, (iii) excitonic condensate (EC), and (iv) the SSC with a spatial pattern that is different from (B2). We note here that it is difficult at present to further qualify those possibilities, except for the AFM. The AFM in phase (B1) is excluded because $M$ should be smaller than that of phase (B2). This is in contradiction to the experimental observation. EC may be plausible, although further experimental evidence is needed to confirm it. EC has been recently proposed as the origin of the insulating phase of LaCoO$_{3}$ and Pr$_{0.5}$Ca$_{0.5}$CoO$_{3}$ [@Kunes2014; @Kunes001; @Kunes002; @Kaneko2012; @Kaneko2014; @Kaneko2015]. In a very recent report, it is predicted by a dynamical mean field model calculation that a field-induced EC is possible [@Sotnikov]. Switching between two different SSCs may also be possible. The SSCs with various spatial patterns were considered with generalized gradient approximation (GGA+$U$) calculations in Ref. \[\]. Some two SSCs with the same $M$ may undergo a temperature-induced transition from one to the other with the assistance of the entropy difference of those phases due to a lattice or orbital contribution.
The OO in phase (B1) is also in good agreement with the experimental results. Co$^{3+}$ in the IS or HS both possess orbital degrees of freedom at the $e_{g}$ and $t_{2g}$ orbitals, respectively. The formation of the OO at phase (B1) will stabilize it energetically, which may well result in the reduction of the transition field to phase (B1) as compared to that to phase (B2), being in accord with the observed change of the transition field at $T^{*}$. In addition, the flat phase boundary between (B1) and (B2) is also in good agreement with the order-disorder phase transition of orbitals [@Murakami] or the switching between different OO [@Mcqueen] because they can occur with $\Delta M=0$. In those cases, orbitals are ordered in phase (B1) and in (B2) the orbitals are disordered or forming the OO with different spatial pattern. For these reasons, we regard that, in phase (B1), OO may be present in addition to the SSC. Orbital ordering taking place along with the spin state ordering has also been claimed in YBaCo$_{2}$O$_{5}$ [@Vogt], Sr$_{3}$YCo$_{4}$O$_{10.5}$ [@Nakao], the thin film of LaCoO$_{3}$ [@Fujioka], and a previous high-field study on LaCoO$_{3}$ [@Moaz]. We note, however, the origin of phases (B1) and (B2) is still an open question to be explored in future studies.
In conclusion, high-field magnetization measurements of LaCoO$_{3}$ up to 133 T were carried out in a wide temperature range from 2 to 120 K. At $T>T^{*}$, we observed the novel magnetic transition at $B>100$ T. In addition, we observed the previously reported magnetic transition at $\sim65$ T with $T<T^{*}$. Based on the obtained $B$-$T$ phase diagram and the Clausius-Clapeyron relation, it was found that the high-field phases possess lower entropy than the low-filed phases, and that the high-field phases are separated into two phases at $T=T^{*}$. We argue that the observed magnetic transitions take place from the LS-HS or LS-IS disordered phase to the ordered SSC of LS-HS or LS-IS complex. At $T<T^{*}$, spatially different SSC or orbital order may develop.
The authors acknowledge M. Tokunaga and S. Ishihara for fruitful discussions, T. T. Terashima, S. Takeyama and K. Yoshikawa for experimental and various supports. This work was supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B) Grant Number 16K17738.
|
---
abstract: 'Hybrid locomotion, which combines multiple modalities of locomotion within a single robot, can enable robots to carry out complex tasks in diverse environments. This paper presents a novel method of combining graph search and trajectory optimization for planning multi-modal locomotion trajectories. We also introduce methods that allow the method to work tractably in higher dimensional state spaces. Through the examples of a hybrid double-integrator, amphibious robot, and the flying-driving drone, we show that our planner tractably gives full-state trajectories that are probabilistically optimal and dynamically feasible.'
author:
- 'H.J. Terry Suh^1^, Xiaobin Xiong^2^, Andrew Singletary^2^, Aaron D. Ames^2^, Joel W. Burdick^2^ [^1][^2]'
bibliography:
- 'thesisbib2.bib'
title: '**Optimal Motion Planning for Multi-Modal Hybrid Locomotion**'
---
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Introduction
============
A hybrid locomotor combines multiple movement modalities into a single platform. Examples of hybrid locomotion (HL) include amphibious vehicles with the ability to swim and drive, or flying cars with the ability to drive and fly. Hybrid locomotion can allow robots to tackle more complex tasks in complicated environments, while achieving greater performance, such as improved energy efficiency. For instance, a flying-car can readily fly over obstacles or uneven terrain via aerial mobility, while driving when possible to improve energy-efficiency. Prior works on hybrid locomotion [@Ambot; @aquabot; @picobug; @picobug2; @kevin; @robotbee; @flyingstar] have often investigated the design and feasibility of hybrid locomotion strategies, and examples of such robots are given in Fig. \[fig:platformexamples\].
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However, realizing the full potential of these robots not only depends on clever design, but also on autonomous planning of their complex motion strategies. For instance, Fig. \[fig:cityexample\] illustrates different paths that a flying-driving robot (such as the Drivocopter of Fig. \[fig:platformexamples\]) could use to traverse between two building rooftops. The different paths have entirely different energy-costs, travel times, and robustness, which ultimately dictate robot performance in this task.
Unlike conventional motion planning problems, the path cost depends not only on the optimality of the trajectory segments in each modality, but also on the switching sequence, time, and state coordinates. The problem of achieving combinatorial optimization of the switching sequences, as well as optimization of trajectories within each modality, makes this problem particularly challenging, as illustrated by the different paths in Fig. \[fig:cityexample\].
Many existing motion planning strategies cannot directly address the difficulty of multi-modal planning. Graph-based motion planning approaches (PRM [@prm], RRT [@rrt], RRT\* [@rrtstar]) excel at discrete optimization in sampled coordinates, but suffer from the curse of dimensionality. They are unable to produce full-state trajectories for high-dimensional systems and do not readily incorporate costs that are not functions of sampled coordinates.
Motion planning algorithms based on trajectory optimization (e.g., via direct collocation [@collocation2; @gpops]) either assume smooth dynamics, or use multi-phase optimization with pre-specified domain sequences [@hybridcontrolsystem]. This is due to the fact that standard nonlinear programming frameworks cannot handle the combinatorial optimization of discrete locomotion mode switches. Mixed-Integer Programming methods [@mixedinteger; @tobia] transcribe these problems well, but do not scale well enough to handle switching sequences and coordinates of realistic high-dimensional problems.
Existing works in multi-modal planning often bypass this problem by only considering discrete graph-based planning ([@hybridswarm],[@amphibiousplanning],[@HyFDR]). By ignoring the continuous dynamics of the robot, these planners often ignore dynamic feasibility in a single modality, and how the dynamic constraints affect the cost. In addition, although energy expenditure is often the most important cost in hybrid locomotion, direct calculations of this cost from change in position is not possible without making vast simplifications, such as using Cost of Transport (COT) to linearly map distance to energy ([@hybridswarm],[@HyFDR]).
To address this problem, we present a novel motion-planning method for hybrid locomotion that considers the problem as a graph-search with local trajectory optimization on the continuous dynamics between connected nodes. In low-dimensions, we propose a graph construction strategy on the modality-partitioned state-space such that no edge crosses the guard between partitions. The cost of traveling between sample points is found using optimal trajectory generation for the corresponding modality, and a graph-search determines the nearly optimal state-space path. To make the method tractable in higher dimensional state spaces, we also propose constraining some subspace of the state-space to be a function of sampled states via virtual constraints, and learning the cost function from offline trajectory optimization batches.
To the best of our knowledge, we present the first motion planner for multi-modal locomotion that considers direct representation of the energy cost as well as dynamic constraints, while retaining probabilistic optimality. The proposed method is primarily implemented in simulation: the hybrid double-integrator with viscous friction is shown as a low-dimensional case (Sec.\[hybridd\]). Then, example trajectories for more-realistic systems are given by considering amphibious (Sec.\[amphibious\]) and flying-driving locomotion (Sec.\[drivocopter\]).
Problem Formulation {#framework}
===================
The Hybrid Locomotion System
----------------------------
We define a hybrid locomotion system as a type of hybrid control system [@hybridcontrolsystem] with additional constraints. We define the hybrid locomotion system $\mathscr{HL}$ as a tuple $$\mathscr{HL}=(FG, \mathcal{D},\mathcal{U},\mathcal{S},\Delta)
\nonumber$$
In the below description of each element, $i$ denotes the index of the locomotion mode (i.e. flying or driving):
- $FG=\{(f_i,g_i)\}$ describes the dynamics associated with each locomotion mode. The dynamics are assumed to take a control-affine form: $\dot{x}=f_i(x)+g_i(x)u$ .
- $\mathcal{D}=\{\mathcal{D}_i\}$ is the set of domains, or state-spaces, associated with the continuous dynamics of each mode.
- $\mathcal{U}=\{\mathcal{U}_i\}$ is the set of admissible control inputs associated with each mode.
- $\mathcal{S}=\{S_{i,j}\}$ is the set of guard surfaces that describes the boundaries between domains of mode $i$ and $j$.
- $\Delta=\{\Delta_{i\rightarrow j}\}$ is the set of reset maps that describe discrete transformations on the guard surface $S_{i,j}$
We additionally assume that each state $x\in \bigcup \mathcal{D}_i$ belongs to a single mode $i$. I.e., the domains disjointly partition the reachable state-space.
Optimal Trajectories in the Hybrid Locomotion System
----------------------------------------------------
To define an optimal hybrid trajectory, we formulate a cost for each mode’s control-affine system in Bolza form: $$J_i=\Phi_i(x(t_0),t_0,x(t_f),t_f)+\smallint^{t_f}_{t_0}\mathcal{L}_i(x(t),u(t),t)dt
\nonumber$$ There also exists a constant switching cost $J(\Delta_{i\rightarrow j})$ to transition from one modality $i$ to $j$. We formulate the problem of finding the optimal trajectory for a hybrid locomotion system as the following two-point boundary value problem.
|s| [u]{}[J\_i+J(\_[ij]{})]{}
In words, we want to find a trajectory that is dynamically feasible within each modality, while optimizing the cost functional throughout the entire trajectory, which would also require optimizing the order of discrete modes to visit.
Planning Methodology {#lowdimension}
====================
Section \[sec:basic\] reviews our basic planning concept that combines sampling-based planning with local trajectory optimization. Because this approach may not scale well to practical situations, Section \[sec:higher\] introduces virtual constraints and off-line learning to improve real-time performance.
Dynamic Programming with Continuous Optimization {#sec:basic}
------------------------------------------------
### Graph Structure
First, we discretize the problem by sampling coordinates in each domain $\mathcal{D}_i$. The vertices, $V$, of a digraph, $G(V,E)$, are constructed from these samples. The edges represent locally optimal paths between the vertices. Each edge is weighted with the optimal transport cost. To avoid the situation where the two vertices of an edge lie in different locomotion modalities, we additionally impose the following constraints on the graph:
1. $e=(x_i\rightarrow x_j)\in E$, $x_i,x_j \in \mathcal{D}_k$ for some mode $k$. I.e., edges only connect states in the same mode.
2. We explicitly sample the guard surface and only allow paths to cross a guard through a guard sample point.
Fig.\[fig:graphstructure\]. A and B illustrate these conditions. The shortest-path search is tackled by Djikstra’s algorithm [@shortestpath] once the locally optimal trajectory costs are known.
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If there exists a switching cost to go from one modality to another, we augment the sample on the guard surface $x$ with two connected nodes $x_1$ and $x_2$ that shares the same state-space coordinates, and assign switching cost to the edge cost between the two samples, as illustrated in Fig. \[fig:graphstructure\].C.
### Continuous Optimization of Trajectory Segments
As each edge connects states in a single mode, we estimate the edge weight by solving the optimization problem:
|s| [u]{}[J\_i]{}[w(x\_1x\_2)=]{} , 4mu x\_i, 4mu u\_i
where $x_1,x_2\in \mathcal{D}_i$. This standard trajectory optimization problem can be tackled using existing methods, such as direct collocation [@gpops].
### Final Path Smoothing
The path(s) returned from graph-search are smoothed via trajectory optimization, knowing the switching sequence and the guard surface points. Given a path of samples $P=(x_1,x_2,\cdots, x_k)$ resulting from graph search, we partition the state space samples using their modalities, such that $$\bigcup P_i = \begin{cases}
P_1 = \{x_i | 0 \leq i \leq k_1, \forall x_i \in \mathcal{D}_{j_1}\}\mkern9mu\cup\cdots\cup\\
P_n = \{x_i | k_{n-1} \leq i \leq k_n, \forall x_i \in \mathcal{D}_{j_n}\}
\end{cases}
\nonumber$$ where $j_i$ denotes the mode of each partition, and $x_{k_i}$ denotes the sample on the guard surface where the trajectory crosses during a mode switch. The optimal trajectories between boundary points is then found by solving the following optimization problem for each partition:
|s| [u]{}[J\_[j\_i]{}]{}
The total trajectory is reconstructed by concatenating the trajectories $x^*=(x_1^*,x_2^*,\cdots,x_n)^*$ induced by the optimal control inputs $u_i^*$ for each partition.
Extension to High Dimensions {#sec:higher}
----------------------------
Although dynamic programming with segment-wise trajectory optimization shows good promise for hybrid locomotion, it is computationally expensive, requiring $O(|V|^2)$ instances of trajectory optimization. In high dimensions, the number of samples increases exponentially if the resolution is maintained, and trajectory optimization methods scale poorly. The two methods introduced in this section aim to make this method tractable for high-dimensional systems.
### Reduced-Order Coordinates via Virtual Constraints
We can reduce the dimensionality of the sample space by introducing virtual constraints that fix some coordinates as a function of the sampled-coordinates. The state-space is divided into sampled coordinates ($x^s$) and auxiliary coordinates ($x^a$). The full state is recovered from samples in the reduced space, $x^s$, by appending auxiliary coordinates: $$x=(x^s,x^a)^T=\left(x^s,v\left(x^s\right)\right)^T$$ The state partioning into $x^s$ and $x^a$ are problem-dependent, but can be understood in the context of model-order reduction and bisimilarity: if the original system and the virtually-constrained system show bounded difference in their evolution, it indicates a good choice of coordinates and constraints. Rigid body coordinates of position and velocity, or differentially flat coordinates [@minimumsnap] can be good choices. Eliminating the sampling of the subspace $x^a$ can significantly reduce computation, making the method tractable.
### Learning the cost function from offline optimization
To find the weight between two sampled coordinates $x^s_1$ and $x^s_2$ in the graph, let us first define a function $J:\mathbb{R}^{\dim(x^s)}\times \mathbb{R}^{\dim(x^s)}\rightarrow \mathbb{R}$, which is described by the following optimization problem:
|s| [u]{}[J\_i]{}[J(x\_1\^s,x\_2\^s)=]{}\[opt:optim\]
where $(x_1^s, v(x_1^s))^T,(x_2^s, v(x_2^s))^T \in \mathcal{D}_i$. Since this optimization problem has to be solved $O(|V|^2)$ times, we choose to learn this function off-line for faster evaluation online.
Using $(x_1^s,x_2^s)$ as feature vectors, and $J(x_1^s,x_2^s)$ as label, we first produce a batch $\left(\left(x_1^s,x_2^s\right),J\left(x_1^s,x_2^s\right)\right)$ from multiple trajectory optimization runs. Then, function approximators from supervised learning algorithms such as Support Vector Regression (SVR) [@svr] or Neural Nets are used to approximate $J(x_1^s,x_2^s)$. Denoting the approximated function as $\tilde{J}(x_1^s,x_2^s)$, the weights on the graph are assigned by $w(x_1^s\rightarrow x_2^s)=\tilde{J}(x_1^s,x_2^s)$.
Since $\tilde{J}$ is learned from data, its evaluation does not require a full instance of nonlinear programming, greatly reducing on-line computation. Yet, as $\tilde{J}$ is learned from trajectory optimization, all costs in Bolza form can be utilized, and dynamic constraints can be incorporated.
Case Study: Hybrid Double Integrator {#hybridd}
====================================
This section first verifies our low-dimensional method for 1D problem of a thrust-vectored mass on a linear rail. While the rail is lubricated and frictionless at $p<0$, viscous drag appears $p\geq 0$. This can be formulated as a simple hybrid locomotion system with the following dynamics: $$\ddot{p}=u \text{ if } p < 0 \quad ,\quad \ddot{p}=u - \dot{p} \text{ if } p \geq 0
\nonumber$$ In addition, consider that we have the input constraint $|u|\leq 1$ for both domains. Converting this to a first-order system $x=\begin{pmatrix} p, v \end{pmatrix}^T$, the system can be described as: $$\mathscr{HL}=\begin{cases}
FG & =\{(f_-,g_-),(f_+,g_+)\},\\\mathcal{D}&=\left\{\left\{x|p<0\right\},\left\{x|p\geq 0\right\}\right\} \\ \mathcal{U}&=\left\{\left\{u||u|\leq 1\right\},\left\{u||u|\leq 1\right\}\right\} \\ \mathcal{S}&=\left\{S_{+,-}=\left\{x|p=0\right\}\right\},\\
\Delta& = \left\{\Delta_{+,-}= x^+\rightarrow x^-\right\}
\end{cases}
\nonumber$$ where the dynamics are described by $$f_- = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \; , \;
f_+ = \begin{pmatrix} 0 & 1 \\ 0 & -1 \end{pmatrix} \; , \; g_+ = g_-=\begin{pmatrix} 0 \\ 1 \end{pmatrix}
\nonumber$$ Then, let us find a trajectory from $x_i$ to $x_f$ while minimizing the input $$J_- = J_+ = \int^{t_f}_{t_0} u^2 dt
\nonumber$$
Using our framework, we first place a graph structure on the state-space using knowledge of the domains $\mathcal{D}_i$, then optimize each continuous trajectory using GPOPS-II [@gpops] with IPOPT [@ipopt] solver. The trajectory obtained using graph-search, and the final smoothened trajectory using the knowledge of the switching sequence and the boundary points on the guard surface is displayed in Fig.\[fig:hybridintegrator\].
Finally, since the switching sequence is trivial to guess for this example, we utilize multi-phase optimization in GPOPS-II with IPOPT, which puts an equality constraint from the end of the first phase in $\mathcal{D}_+$ and the beginning of the second phase in $\mathcal{D}_-$ and compare the results. The trajectory using multi-phase optimization is displayed in Fig.\[fig:hybridintegrator\].
In addition, to show probabilistic convergence, we run the algorithm $10$ times with different inter-sample distances (controlled by Poisson sampling [@poissonsampling] on state-space), and show convergence in Fig. \[fig:hybridintegratordata\]. Fig. \[fig:hybridintegratordata\] shows that our method probabilistically converges, with a lower cost compared to multi-phase optimization using GPOPS-II with IPOPT. IPOPT is only local optimal, while a PRM framework searches more globally over the domain. Our final cost shows that we can produce probabilistically optimal and dynamically feasible trajectories with inter-sample distance as large as $0.3$.
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2D Case Study: Amphibious Tank (AmBot) {#amphibious}
======================================
This section describes optimal trajectories for the amphibious vehicle introduced in [@Ambot], which uses tank treads for ground locomotion (skid-steer), and marine locomotion (paddles). After describing the vehicle dynamics in both modes, we obtain optimal trajectories for a model environment.
Dynamics
--------
### Ground and Marine Dynamics
We derive the Newtonian mechanics for planar operation, and incorporate first-order armature motor dynamics. The ground states, $x_g\in\mathbb{R}^{6}$, and marine states, $x_m\in\mathbb{R}^8$, are defined as $$\begin{cases}
x_g&=(p^w_b, v^b,\theta^w_b,\omega^b)^T\\
x_m&=(p^w_b,v^b, \theta^w_b, \omega^b,\phi_L,\phi_R)^T
\end{cases}
\nonumber$$ where $p^w_b\in \mathbb{R}^2$ is the body position with respect to (wrt) a world frame, $v^b\in\mathbb{R}^2$ is the velocity in the body frame, and $\theta^w_b,\omega^b\in \mathbb{R}$ denote the orientation and angular velocity wrt a world frame. Finally, $\phi_L,\phi_R\in\mathbb{R}$ denote the left and right motor speeds. In both locomotion modes, the control action $u_g=u_m=(u_L,u_R)^T\in [-1,1]^2$ correspond to commanded motor speeds via fraction of applied motor voltage.
We model a no slip constraint for ground operation. A $1^{st}$-order motor model relates motor torque (which generates tractive forces on the vehicle) to command inputs. A drag force proportional to the square of vehicle speed and a similar $1^{st}$-order motor model are used in the aquatic domain.
### Hybrid Dynamics
The governing dynamical systems for each mode are represented by the hybrid dynamics $$FG=\begin{cases}
\dot{x}_g = f_g(x_g)+g_g(x_g)u_g & x \in \mathcal{D}_g \\
\dot{x}_m = f_m(x_m)+g_m(x_m)u_m & x \in \mathcal{D}_m
\end{cases}
\nonumber$$ where $g,m$ denotes ground and marine modes, the domains and guard surfaces $\mathcal{D}_{g},\mathcal{D}_m,\mathcal{S}_{m,g}$ are obtained from terrain, and $\mathcal{U}_g=\mathcal{U}_m=[-1,1]^2$ for both inputs. We apply the identity map to $\Delta_{m\rightarrow g}$ and $\Delta_{g\rightarrow m}$.
### Cost Function
We minimize the robot’s total energy expenditure, modeled as: $$\label{eq:groundcost}
J_g=J_m= \int^{t_f}_{t_i} \bigg[\sum_{i=L,R}V_{cc}u_i\cdot \frac{k_t}{R}(V_{cc}u_i-k_t\phi_i)+P_d \bigg]dt$$ where $V_{cc}$ is the battery voltage, $k_t$ is the motor torque-constant, $R$ the internal resistance, and $P_d$ the constant power drain. The first term models motor power dissipation, and the latter term models constant power drainage. We assume no switching costs associated with the discrete reset map, $J(\Delta_{m\rightarrow g})=J(\Delta_{g\rightarrow m})=0$
Cost Learning
-------------
For ground operation, we divide the state-space $x_g\in\mathbb{R}^6$ into sampled and auxiliary coordinates $$\label{eq:groundstates}
x^s_g = (p^w_x,p^w_y,v^b_x,\theta)^T \quad,\quad x^a_g =(v^b_y,\omega)^T = (0,0)^T$$
This division of coordinates recognizes that side-slip is constrained for skid-steer vehicles, and angular velocity is small.
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For marine operation, $x_m\in\mathbb{R}^8$ is divided into $$\begin{cases}
x^s_m = (p^w_x, p^w_y, v^b_x, \theta)^T \\
x^a_m = (v^b_y, \omega, \phi_L, \phi_R)^T = (0, 0, \phi_n,\phi_n)^T
\end{cases}
\nonumber$$ where track forces equal water drag at equilibrium speed $\phi_n$.
The cost function $J(x_i^s,x_f^s)$ in Eq. (\[opt:optim\]) is learned from multiple offline optimizations. Using $11520$ samples, the function $J(x_i^s,x_f^s)$ is evaluated using GPOPS-II [@gpops], and SVR with Gaussian kernel trains the function $\tilde{J}$ with Sequential Minimal Optimization [@SVM]. The process is repeated for both ground and marine locomotion. Fig.\[fig:trajectorylearning\] shows the generated trajectories and the contour of the learned function.
Results
-------
We sample position using the method of Sec.\[lowdimension\].A, and grid the states $v_x,\theta$ to create $x^s$. The edge weights are estimated from the learned function $\tilde{J}$. Finally, the shortest path is found by Djikstra’s algorithm [@shortestpath]. Fig. \[fig:amphibiousframework\] illustrates this process. The final smoothed trajectory is shown in Fig.\[fig:amphibiousfinal\].
The final trajectories differ noticeably from those produced by a shortest-path planner due to the differences in Costs of Transport. Since the robot expends more energy in water, it drives further on the ground until it switches to swimming. This example shows that our method can autonomously decide switching sequences and switching points.
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----------- ---------- ---------- ---------- --------- --------- --------- ---------- ---------------
Dist. (m) F DF DFD FDF DFDF DFDFD Sequence Cost (Joules)
110 12129.72 11991.18 11780.26 6612.62 6620.16 6642.41 DFDFD 6655.44
90 10167.32 9902.18 9867.10 6345.64 6558.94 6578.74 DFDFD 6600.45
70 8208.72 7941.78 7726.73 6470.98 6492.85 6511.88 DFD 8156.85
50 6248.72 6112.78 5894.58 6477.60 6449.91 6452.32 DF 6456.37
30 4244.2 4148.18 3815.68 6343.64 6365.65 6387.47 DFD 4033.92
----------- ---------- ---------- ---------- --------- --------- --------- ---------- ---------------
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3D Case Study: Drivocopter {#drivocopter}
==========================
This section models the [*Drivocopter*]{} flying-driving drone of Fig. \[fig:platformexamples\]. It uses skid-steer driving and quadrotor flight.
Dynamics
--------
We use the ground model of Sec.\[amphibious\] with different parameters, while the flight dynamics are based on [@omnicopter] and [@minimumenergy].
### Flight Dynamics
Standard rigid-body dynamics [@quadrotordynamics] describe flight motions driven by four rotor forces, which use a speed-squared-dependent lift term and $1^{st}$-order armature motor dynamics. The state vector $x_f\in\mathbb{R}^{16}$ is $$x_f= (p^w_b, v^b, \Theta^w_b,\omega^b,\phi_i)^T
\nonumber$$ where $p^w_b\in\mathbb{R}^3$ is the vehicle position wrt a world frame, $v^b\in\mathbb{R}^3$ is the 3D velocity in the body frame, $\Theta^w_b\in\mathbb{R}^3$ denotes vehicle orientation wrt world frame parametrized by ZYX Euler angles, $\omega^b\in\mathbb{R}^3$ is the body angular velocity, and $\phi_i=(\phi_1,\phi_2,\phi_3,\phi_4)\in\mathbb{R}^4$ are the motor rotational speeds.
### Hybrid Dynamics
Again, the two modalities of ground and flight are represented by a hybrid dynamical system $$FG=\begin{cases}
\dot{x}_f = f_f(x_f)+g_f(x_f)u_f & x_f\in \mathcal{D}_f \\
\dot{x}_g = f_g(x_g)+g_g(x_g)u_g & x_g \in \mathcal{D}_g
\end{cases}
\nonumber$$ where $f,g$ denotes flight and ground modes, the domains and guard surfaces $\mathcal{D}_{f},\mathcal{D}_g,\mathcal{S}_{f,g}$ are obtained from knowledge of the ground surface. The motor inputs are $\mathcal{U}_f=(u_1,u_2,u_3,u_4)=[0,1]^4$ with $\mathcal{U}_g=(u_L,u_R)=[-1,1]^2$. Finally, $\Delta_{f\rightarrow g}$ (landing) and $\Delta_{g\rightarrow f}$ (takeoff) are discrete transitions: $$\begin{cases}
\Delta_{f\rightarrow g}=(p^w_x, p^w_y, p^w_z, 0^{9}, \phi_n)\rightarrow (p^w_x, p^w_y, 0^{4}) \\
\Delta_{g\rightarrow f}=(p^w_x, p^w_y, 0^{2}, \theta^w_b, 0)\rightarrow (p^w_x, p^w_y, p^w_z, 0^{9}, \phi_n)
\end{cases}
\nonumber$$ where $\phi_n$ is the motor speed needed to provide hovering lift. During takeoff, we set $p^w_z$ to be a meter higher than the ground surface of the ground sample.
### Cost Function
We use the same ground energy cost as Eq.(\[eq:groundcost\]), and formulate the same energy for flight with different motor parameters. The costs for reset maps $J(\Delta_{f\rightarrow g})$ and $J(\Delta_{g\rightarrow f})$ are constant takeoff and landing energy costs obtained via trajectory optimization.
Cost Learning
-------------
-0.15 true in -0.1 true in
The ground states are divided into sampled / auxiliary coordinates via Eq. (\[eq:groundstates\]). Flight states are divided by: $$\begin{cases}
x^s_f=(p^w_b, v^b)^T \\ x^a_f=(\Theta^w_b,\omega^b,\phi_i)=(0^{1\times 3},0^{1\times 3}, \phi_n \cdot 1^{1\times 4})^T
\end{cases}
\nonumber$$ where the $\phi_n$ is the rotor rate at which the lift provided by the propellers allows the drone to hover in stable equilibrium. The cost $J(x_1^s,x_2^s)$ is learned as in Sec.\[amphibious\].B from $17016$ paths. Fig. \[fig:resultflight\] shows the trajectories and energy map. The ground energy cost is found with Drivocopter parameters.
Results
-------
-0.1 true in -0.15 true in
-0.1 true in
A CAD environment model, consisting of two raised platforms separated by a flat-bottom chasm, is meshed into drivable and undrivable regions (Fig.\[fig:drivofinal\].A), and the ground and free-space meshes are Poisson sampled (Fig.\[fig:drivofinal\].B). The result of a shortest-path (Fig.\[fig:drivofinal\].C) is smoothened (Fig.\[fig:drivofinal\].D). This process is depicted in Fig. \[fig:drivofinal\]. We hypothesized that when the platforms are nearby, the drone should not drive in the chasm, since gravitational losses exceed energy gains from by driving. As the platforms separate further, the drone saves energy by driving in the chasm. We tested this idea on 5 different terrains parametrized by the distance between platforms (see Fig.\[fig:4traj\]). Our planning results show correct qualitative behavior.
To show quantitative competence, we also generate few heuristic trajectories per given fixed sequence (illustrated in Fig.\[fig:drivofinal\].E) and tabulate the final costs in Table.\[tab:1\]. Our method produces a switching sequence that mostly agrees with lowest-cost producing sequences among heuristic trajectories, and costs are quantitatively comparable to the heuristically optimal trajectories.
Conclusion
==========
We presented a novel scheme to plan near-optimal hybrid locomotion trajectories. A double-integrator example showed that our method can generate probabilistically optimal and dynamically feasible trajectories in low-dimensional state-spaces. The Ambot and Drivocopter examples showed that virtual constraints and cost function learning renders our method practical in high-dimensional problems.
Improvements are possible by upgrading components of our framework. Better computational speed could be realized by adaptive sampling [@hsu] and the use of RRT [@rrt] search to achieve faster single-query tractability. An (A\*) [@astar] graph search would be enabled by transport energy heuristics, while other function approximations, such as Neural Nets, might improve the cost function learning module. Differential Dynamic Programming [@ddp] is another promising scheme for planning trajectory segments.
While we have demonstrated probabilistic optimality in a low-dimensional example, we acknowledge the lack of provable optimality in high dimensions. A better understanding and justification of virtual constraints should result from studying the bisimilarity between the full-state and reduced-order systems. Finally, efforts are underway to demonstrate our results on the Drivocopter of Fig. \[fig:platformexamples\].
[^1]: ^1^Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, USA, [hjsuh@mit.edu]{}
[^2]: ^2^Deptartment of Mechanical and Civil Engineering, California Institute of Technology, Pasadena, CA 91125, USA, [{xxiong,asinglet,]{}
|
---
abstract: 'New physics near the TeV scale could modify neutrino-matter interactions or generate a relatively large neutrino magnetic (transition) moment. Both types of effects have been discussed since the 1970’s as alternatives to mass-induced neutrino flavor oscillations. Nowadays, the availability of high-statistics data makes it possible to turn the idea around and ask: How well do the simple mass-induced oscillations describe solar neutrinos? At what level are the above-mentioned nonstandard effects excluded? Can we use solar neutrinos to constrain physics beyond the Standard Model? These notes review the sensitivity of the present-day solar neutrino experiments to the nonstandard neutrino interactions and transition moment and outline progress that may be expected in the near future. Based on a talk given at the Neutrino 2006 conference [@Neutrino06].'
address: 'Theoretical Division, T-8, MS B285, Los Alamos National Laboratory, Los Alamos, NM 87545'
author:
- Alexander Friedland
bibliography:
- 'friedland\_neutrino06.bib'
title: 'MSW Oscillations - LMA and Subdominant Effects'
---
Preprint: LA-UR-06-6774
Standard LMA solution: basic features {#sect:stdLMA}
=====================================
The most basic experimental fact about the neutrinos from the Sun is that the electron neutrino survival probability, $P_{ee}^{std}\equiv
P(\nu_e\rightarrow\nu_e)$, is measured to vary as a function of the neutrino energy. At the high end of the spectrum ($E_\nu\gtrsim 6-7$ MeV) the SNO [@SNO2005] and Super-Kamiokande [@SK2003] experiments have established that $P_{ee}^{std}$ is about $\sim
34\pm3$%. The gallium experiments [@GNO2000], however, which are sensitive to both high- and low-energy neutrinos, see a higher survival probability: the measured rate is $74\pm7$ SNU, whereas the standard solar model prediction [@BP04] (before oscillations) is $131_{-10}^{+12}$. This simple fact has highly non-trivial implications. Indeed, this behavior is not “generic” for mass-induced oscillations, even when they combine with the MSW [@Wolfenstein:1977ue; @Mikheev:1986gs] matter effect. [*A priori*]{}, one might have expected solar neutrinos to be in one of these regimes:
- matter dominates at the production point, on the way out of the Sun neutrino flavor evolves adiabatically $\rightarrow$ constant suppression (regime 1);
- matter dominates at the production point, on the way out of the Sun neutrino flavor evolves non-adiabatically $\rightarrow$ vacuum oscillations
- vacuum oscillation length $\ll$ 1 a.u. (astronomical unit) $\rightarrow$ oscillations average out $\rightarrow$ constant observed suppression (regime 2);
- vacuum oscillation length $\gg$ 1 a.u. (astronomical unit) $\rightarrow$ no time to oscillate $\rightarrow$ no suppression (regime 3);
- vacuum oscillations dominate everywhere, matter effects negligible even in the center of the Sun $\rightarrow$ oscillations average out $\rightarrow$ constant observed suppression (regime 4).
The observed energy-dependent $P_{ee}^{std}$ then implies that solar neutrinos are in one of the several “special regimes”: the transition between regimes 1 and 4 (Large Mixing Angle – LMA – solution); the transition between regimes 2 and 3 (vacuum/quasi-vacuum [@Friedland:2000cp] oscillation solution); the transition between regimes 1 and 2 (Small Mixing Angle – SMA – solution); the regime where the density in the Earth is close to resonant, so that the flavor regeneration in the Earth is large (the LOW solution). These solutions were known for many years, in particular all four were allowed as recently as 2000, see, e.g., [@deGouvea:2000cq].
We now of course know that only the LMA solution survives. Let us consider the survival probability $P_{ee}^{std}$ under the (*a posteriori* justified) assumption that the oscillations take place between just two eigenstates. One easily obtains that during the day time $$\label{eq:Peestd}
P_{ee}^{std,\;2\nu}=\cos^2\theta_\odot\cos^2\theta+\sin^2\theta_\odot\sin^2\theta.$$ The probability of finding the neutrino in eigenstate 1(2) is $\cos^2\theta_\odot$($\sin^2\theta_\odot$), where $\theta_\odot$ is the mixing angle at the production point; in turn, the probability of detecting the neutrino already in eigenstate 1(2) as $\nu_e$ is $\cos^2\theta$($\sin^2\theta$). The key physical ideas here are that the evolution is adiabatic (no level jumping) and incoherent (interferences between 1 and 2 disappear upon integration over energies for $\Delta m^2 \gtrsim 10^{-9}-10^{-8}$ eV$^2$ [@Pakvasa:1990gf; @Friedland:2000cp] and over the production region).
The angle $\theta_\odot$ is determined from the oscillation Hamiltonian $H_{\rm tot}= H_{\rm vac} + H_{\rm mat}$, where $$\begin{aligned}
H_{\rm vac} = \left(\begin{array}{rr}
-\Delta \cos 2\theta & \Delta \sin 2\theta \\
\Delta \sin 2\theta & \Delta \cos 2\theta
\end{array} \right),\;\;\;H_{\rm mat} = \left(\begin{array}{cc}
\sqrt{2} G_F n_e & 0 \\
0 & 0
\end{array} \right).
\label{eq:VAC_convention}\end{aligned}$$ Here $\Delta\equiv \Delta m^2/(4 E_\nu)$ and $\Delta m^2$ is the mass splitting between the first and second neutrino mass states: $\Delta m^2\equiv m^2_2-m^2_1$. The two limiting values are $\theta_\odot=\theta$ ($H_{\rm tot}$ is dominated by $H_{\rm
vac}$) and $\theta_\odot=\pi/2$ ($H_{\rm tot}$ is dominated by $H_{\rm mat}$). The probability $P_{ee}^{std}$ then varies from $\cos^4\theta+\sin^4\theta$ ($=1-(1/2)\sin^22\theta$, averaged vacuum oscillations) to $\sin^2\theta$.
![The $\nu_e$ survival probability and day/night asymmetry for the LMA solution.[]{data-label="fig:PeeLMA"}](PeeLMA.eps){width="77.00000%"}
The transition from one regime to another occurs when $\sqrt{2}
G_F n_e \sim 2 \Delta$ at the production point. To accommodate the data on $P_{ee}$, this transition must occur *right in the middle of the solar neutrino spectrum*, implying $\Delta m^2\sim
\mbox{a few}\times 10^{-5}$ eV$^2$. Moreover, $\Delta m^2$ cannot be lower than $\sim3\times10^{-5}$ eV$^2$ to avoid being close to the resonance condition in the Earth and resulting large day/night variations of $P_{ee}$. The situation is illustrated in Fig. \[fig:PeeLMA\]. Evidently, Nature chose to “tune” the mass splitting involved in solar neutrino oscillations to the density in the solar core! Remarkably, a completely independent reactor antineutrino measurement by KamLAND [@KamLAND2002flux] showed that $\Delta m^2$ is indeed in this range.
The preceding discussion assumed that mass eigenstate 3 is not involved in the evolution of solar neutrinos. The correction due its presence is trivially computed if we notice that the splitting between this state and eigenstates 1 and 2 is significantly larger than the matter potential even in the center of the Sun ($\Delta m_{atm}^2/2E \gg \sqrt{2}G_F N_e(0)$), so that the $\nu_e$ content of that state is always given by $\sin^2\theta_{13}$. Repeating the arguments that led to Eq. (\[eq:Peestd\]), one gets $$\label{eq:Peestd3nu}
P_{ee}^{std,\;3\nu}=\sin^4\theta_{13}+\cos^4\theta_{13}P_{ee}^{std,\;2\nu}.$$ Given the bound $\sin^2\theta_{13}\lesssim0.02$ from CHOOZ [@Apollonio:2002gd], the first term is negligibly small. The effect of the third state then is to multiply the two-neutrino survival probability by $\cos^4\theta_{13}$. The resulting correction is at most 4%; this correction is basically the probability that the original electron neutrino “disappears” into state 3. See, [*e.g.*]{}, [@Goswami:2004cn; @Fogli:2005cq] for recent data analyses and further references.
Searching for nonstandard neutrino interactions
===============================================
The impact of nonstandard neutrino interactions with matter on solar neutrino oscillations was discussed already in the classical paper by L. Wolfenstein [@Wolfenstein:1977ue] and subsequently elaborated on by many authors ([@Valle:1987gv; @Roulet:1991sm; @Guzzo:1991hi] and many others). The idea is that novel interactions due to a heavy vector and scalar exchange could modify the neutrino forward scattering amplitude and hence the oscillation Hamiltonian in matter. Regardless of their origin, at low energies relevant to neutrino oscillations, nonstandard interactions (NSI) are described by the effective Lagrangian $$\begin{aligned}
L^{NSI} = - 2\sqrt{2}G_F (\bar{\nu}_\alpha\gamma_\rho\nu_\beta)
(\epsilon_{\alpha\beta}^{f L}\bar{f}_L \gamma^\rho f_L +
\epsilon_{\alpha\beta}^{f R}\bar{f}_R\gamma^\rho f_{R})+ h.c.
\label{eq:lagNSI}\end{aligned}$$ Here $\epsilon_{\alpha\beta}^{f L}$ ($\epsilon_{\alpha\beta}^{f
R}$) denotes the strength of the NSI between the neutrinos $\nu$ of flavors $\alpha$ and $\beta$ and the left-handed (right-handed) components of the fermions $f$.
Neutrino scattering tests, like those of NuTeV [@Zeller:2001hh] and CHARM [@Vilain:1994qy], constrain mainly the NSI couplings of the muon neutrino, e.g., $|\epsilon_{e\mu}|\lesssim 10^{-3}$, $|\epsilon_{\mu\mu}|\lesssim
10^{-3}-10^{-2}$. In contrast, direct limits on $\epsilon_{ee}$, $\epsilon_{e\tau}$, and $\epsilon_{\tau\tau}$ are remarkably loose, e. g., $|\epsilon_{\tau\tau}^{uu R}|<3$, $-0.4<\epsilon_{ee}^{uu
R}<0.7$, $|\epsilon_{\tau e}^{uu}|<0.5$, $|\epsilon_{\tau
e}^{dd}|<0.5 $ [@Davidson:2003ha]. Stronger constraints exist on the corresponding interactions involving the charged leptons. Those, however, are model-dependent and do not apply if the NSI come from the underlying operators containing the Higgs fields [@Berezhiani:2001rs]. Here we only consider direct experimental bounds.
Even with the addition of NSI the splitting $\Delta m_{atm}^2/2E$ remains much greater than the matter potential anywhere along the neutrino trajectory. This means the solar neutrino analysis can still be reduced to two neutrino states, following the arguments of Sect. \[sect:stdLMA\]. Neglecting small corrections of order $\sin \theta_{13}$ or higher, the corresponding matter contribution to the two-neutrino oscillation Hamiltonian can be written as $$\begin{aligned}
H_{\rm mat}^{NSI} = \frac{G_F n_e}{\sqrt{2}}
\left(\begin{array}{cc}
1+\epsilon_{11} & \epsilon^\ast_{12} \\
\epsilon_{12} & -1-\epsilon_{11}
\end{array} \right), \mbox{ where }
\begin{array}{l}
\epsilon_{11}=\epsilon_{ee} - \epsilon_{\tau\tau}
\sin^2\theta_{23}, \\
\epsilon_{12}=-2\epsilon_{e\tau} \sin
\theta_{23}.
\end{array}
\label{eq:ceciconv}\end{aligned}$$ The epsilons are the sums of the contributions from the matter constituents: $\epsilon_{\alpha\beta}\equiv
\sum_{f=u,d,e}\epsilon_{\alpha\beta}^{f}n_f/n_e$. In turn, $\epsilon_{\alpha\beta}^{f}\equiv\epsilon_{\alpha\beta}^{fL}+\epsilon_{\alpha\beta}^{fR}$. Observe that only the vector component of the NSI enters the propagation effect; in contrast, the NC detection process at SNO depends on the axial coupling. The propagation and detection effects of the NSI are thus sensitive to different parameters, and the corresponding searches could be complementary.
Eq. (\[eq:ceciconv\]) shows that the flavor changing NSI effect in solar neutrino oscillations comes from $\epsilon_{e\tau}$, while the flavor preserving NSI effect comes from both $\epsilon_{ee}$ and $\epsilon_{\tau\tau}$. A useful parameterization is $$\begin{aligned}
\label{eq:MAT_convention}
H_{\rm mat}^{NSI} =
\left(\begin{array}{cc}
A \cos 2\alpha & A e^{-2i\phi} \sin 2\alpha \\
A e^{2i\phi} \sin 2\alpha & -A \cos 2\alpha
\end{array} \right),
\mbox{ where }
\begin{array}{l}
\tan 2\alpha = |\epsilon_{12}|/(1+\epsilon_{11}) ,\\
2\phi=Arg(\epsilon_{12}), \\
%&&\cos 2\phi = \frac{\Re e({\epsilon_{12}})}{|\epsilon_{12}|}~,~~~~~
%\sin 2\phi =-\frac{\Im m({\epsilon_{12}})}{|\epsilon_{12}|}\nonumber \\
A= G_F n_e \sqrt{[(1+\epsilon_{11})^2+|\epsilon_{12}|^2]/2}~.
\end{array}\end{aligned}$$ The effect of $\alpha$ is to change the mixing angle in the medium of high density from $\pi/2$ to $\pi/2-\alpha$. The angle $\phi$ (related to the phase of $\epsilon_{e\tau}$) is a source of CP violation. Solar neutrino experiments, just like terrestrial beam experiments [@Gonzalez-Garcia:2001mp; @Campanelli:2002cc], are sensitive to its effects [@Friedland:2004pp], while atmospheric neutrinos are not [@Friedland:2004ah; @Friedland:2005vy].
![The electron neutrino survival probability (*left*) and day/night asymmetry (*right*) for $\Delta m^2=7\times 10^{-5}$ eV$^2$, $\tan^2\theta=0.4$ and several representative values of the NSI parameters: (1) $\epsilon_{11}^{u}=\epsilon_{11}^{d}=\epsilon_{12}^{u}=\epsilon_{12}^{d}=0$; (2) $\epsilon_{11}^{u}=\epsilon_{11}^{d}=-0.008$, $\epsilon_{12}^{u}=\epsilon_{12}^{d}=-0.06$; (3) $\epsilon_{11}^{u}=\epsilon_{11}^{d}=-0.044$, $\epsilon_{12}^{u}=\epsilon_{12}^{d}=0.14$; (4) $\epsilon_{11}^{u}=\epsilon_{11}^{d}=-0.044$, $\epsilon_{12}^{u}=\epsilon_{12}^{d}=-0.14$. Reproduced from [@Friedland:2004pp].[]{data-label="fig:PeeAdn"}](PeeDN.eps){width="97.00000%"}
The main effects of NSI on $P_{ee}$ are as follows [@Friedland:2004pp]: (i) the low-energy limit stays the same (vacuum oscillations); (ii) the high-energy limit changes, according to Eq. (\[eq:Peestd\]), $P_{ee}\rightarrow
\sin^2\alpha\cos^2\theta+\cos^2\alpha\sin^2\theta$; (iii) at intermediate energies, the transition from vacuum to matter dominated regime can shift in energy, with changing $A$, and can become more or less abrupt, with changing $\alpha$ and $\phi$. The nonadiabatic regime occurs when $\theta\rightarrow\alpha$, rather than $\theta\rightarrow0$. Also, very importantly, the day/night effect can change with *all three parameters*. In particular, it becomes small either as $A\rightarrow0$ [@Guzzo:2004ue; @Miranda:2004nb] or as $\alpha\rightarrow\theta$ [@Friedland:2004pp]. Thus, the LMA-0 region that is normally excluded by the non-observation of day/night asymmetry may become allowed [@Friedland:2004pp; @Guzzo:2004ue; @Miranda:2004nb].
Fig. \[fig:PeeAdn\] illustrates the impact of the NSI on $P_{ee}$ and the day/night asymmetry. Curve 3 gives an example of parameters that can already be excluded by the current data. Curve 4 illustrates the suppression of the Earth effect described above. For technical details, including approximate analytical expressions for $P_{ee}$ and day/night asymmetry, see [@Friedland:2004pp].
![*Left panel*: A 2-D section $(\epsilon_{ee}=-0.15)$ of the allowed region of the NSI parameters (shaded). We assumed $\Delta m^2_\odot=0$ and $\theta_{13}=0$, and marginalized over $\theta$ and $\Delta m^2 $. The dashed contours indicate our analytical predictions. See text for details. *Right panel*: The effect of the NSI on the allowed region and best-fit values of the oscillation parameters. From [@Friedland:2004ah]. []{data-label="fig:scan"}](parab_inv3.eps "fig:"){width="45.00000%"} ![*Left panel*: A 2-D section $(\epsilon_{ee}=-0.15)$ of the allowed region of the NSI parameters (shaded). We assumed $\Delta m^2_\odot=0$ and $\theta_{13}=0$, and marginalized over $\theta$ and $\Delta m^2 $. The dashed contours indicate our analytical predictions. See text for details. *Right panel*: The effect of the NSI on the allowed region and best-fit values of the oscillation parameters. From [@Friedland:2004ah]. []{data-label="fig:scan"}](fig.nse-I-oscill3.eps "fig:"){width="49.00000%"}
The solar neutrino analysis of NSI cannot be done in isolation: the same NSI can also be probed with atmospheric neutrinos. Indeed, on general grounds, one expects the atmospheric neutrinos – particularly the high energy ones for which nonstandard matter effects can dominate over the vacuum oscillation effects – to be a very sensitive probe of NSI. Early two-neutrino ($\nu_\mu,\nu_\tau$) numerical studies [@Maltoni2001] yielded $\epsilon_{\mu\tau}\lesssim 0.08-0.12$ and $\epsilon_{\tau\tau}\lesssim 0.2$ [^1]. Clearly, these are very strong bounds; if they were to extend to $\epsilon_{e\tau}$, the large NSI effects on solar neutrinos discussed above would be excluded. It turns out, however, that this is not the case: when the analysis is properly extended to three flavors, one finds that very large values of both $\epsilon_{e\tau}$ and $\epsilon_{\tau\tau}$ are still allowed by the data [@Friedland:2004ah]. This is illustrated in Fig. \[fig:scan\] (*left panel*), which shows that NSI with strengths comparable to the Standard Model interactions can be compatible with all atmospheric data. It must be noted that the compatibility is achieved as a result of adjusting the vacuum oscillation parameters: large NSI imply a smaller mixing angle and larger $\Delta m_{atm}^2$, as can be see in the *right panel* of Fig. \[fig:scan\].
The addition of the K2K data helps constrain the allowed NSI region somewhat [@Friedland:2005vy]. While the addition of the first data from MINOS brings no further improvement [@Friedland:2006pi], the future high-statistics MINOS dataset will be a very valuable probe of this parameter space [@Friedland:2006pi].
Searching for neutrino transition moments
=========================================
The idea that solar neutrinos could be affected by the neutrino spin precession (NSP) in the solar magnetic fields is even older [@Cisneros:1970nq] than the NSI idea. Remarkably, this idea – much improved with time [@Voloshin:1986ty; @Okun1986short; @Okun1986; @Akhmedov1988; @LimMarciano; @Raghavan] – remained viable for the next three decades. While, by the late 1990’s, the lack of time variations in the Super-Kamiokande data gave strong evidence against large NSP in the solar convective zone, NSP in the radiative zone continued to give a good fit to all solar data [@ourmagnfit].
{width="77.00000%"}
Even after the confirmation of the LMA oscillation solution by KamLAND [@KamLAND2002flux], the possibility of the NSP happening at a *subdominant* level remains of great interest, as a probe of the neutrino electromagnetic properties and, at the same time, of the magnetic fields in the solar interior. NSP coupled with flavor oscillations could lead to conversion $\nu_e\rightarrow\bar\nu_e$ in the Sun, on which recently, KamLAND [@Eguchi:2003gg] reported an upper bound. It is very important to understand what this bound implies for the neutrino magnetic (transition) moment and how it compares with other available bounds on the neutrino transition moment.
The laboratory bounds on the neutrino magnetic (transition) moment come from measuring the cross sections of $\nu e^-$ or $\bar\nu e^-$ scattering in nearly forward direction. The recent bound for the interaction involving the Majorana electron antineutrino is $2\mu_{e\beta} < 0.9 \times 10^{-10}\mu_B$ at the 90% confidence level [@munu2005], where $\mu_B \equiv e/(2 m_e)$ is the Bohr magneton ($m_e$ is the electron mass, $e$ is its charge). Stronger bounds, $\mu \lesssim 3 \times 10^{-12}\mu_B$, exist from astrophysical considerations, particularly from the study of red giant populations in globular clusters [@Raffeltbound]. Larger values of the transition moment would provide an additional cooling mechanism – via plasmon decay to $\nu\bar\nu$ – for the red giant core and change the core mass at helium flash beyond what is observationally allowed.
An important theoretical consideration is that the magnetic moment operators will radiatively generate the neutrino mass. Requiring that the corresponding contribution to the neutrino masses be not much greater than their observed values, one obtains a model-independent “naturalness" bound on $\mu_{ab}$. For Dirac neutrinos, the bound obtained in this way turns out to be very stringent, $\mu_{ab}\lesssim10^{-14}\mu_B$ [@Bell:2005kz]. However, very importantly, for Majorana neutrinos the bound is much weaker, only $\mu_{ab}\lesssim10^{-10}\mu_B$ [@Davidson:2005cs; @Bell:2006wi], owing to the different flavor symmetry properties of the mass and the transition moment operators [@Voloshin:1987qy]. In fact, explicit models exploiting these very symmetry properties were discussed many years ago ([*e.g.*]{}, [@BabuMohapatra; @BabuMohapatra1990] [^2]).
It turns out that *for the measured LMA oscillation parameters* NSP in the radiative zone cannot produce the $\bar\nu_e$ flux above the KamLAND bound. This is illustrated in Fig. \[fig:scan\_magn\]. This is a remarkable example that knowing neutrino oscillation parameters precisely can be very valuable: the answer would qualitatively change if the mixing angle were 20$^\circ$ instead of 30$^\circ$.
For NSP in the convective zone, the analysis is very different, though in the end the conclusion is similar: one should not have expected the flux of $\bar\nu_e$ in excess of the published KamLAND bound. Put another way, the bound on the neutrino transition moment from the KamLAND bound is comparable to the direct laboratory and “naturalness” bounds, but still weaker than that from analysis of the red giant cooling. An updated analysis of a larger KamLAND dataset is needed. The reader is referred to [@Friedland:2005xh] for details and further references.
It is a great pleasure to acknowledge my collaborators – C. Lunardini, M. Maltoni and C. Peña-Garay. I owe special thanks to A. Gruzinov for countless – always very clear and helpful – discussions of the solar magnetic fields and plasma physics in general. I also thank M. Rempel for a very helpful discussion and for pointing me to an excellent set of references. I benefited greatly from stimulating conversations with V. Cirigliano, E. Akhmedov and T. Rashba. Finally, I thank R. Mohapatra for bringing to my attention several important references.
[^1]: Notice the difference in normalization: our $\epsilon$’s are normalized per electron, while [@Maltoni2001] gives $\epsilon$’s per $d$ quark.
[^2]: I thank R. N. Mohapatra for bringing the last two references to my attention.
|
---
abstract: |
One of the most challenging and long-standing problems in computational biology is the prediction of three-dimensional protein structure from amino acid sequence. A promising approach to infer spatial proximity between residues is the study of evolutionary covariance from multiple sequence alignments, especially in light of recent algorithmic improvements and the fast growing size of sequence databases. In this paper, we present a simple, fast and accurate algorithm for the prediction of residue-residue contacts based on regularized least squares. The basic assumption is that spatially proximal residues in a protein coevolve to maintain the physicochemical complementarity of the amino acids involved in the contact. Our regularized inversion of the sample covariance matrix allows the computation of partial correlations between pairs of residues, thereby removing the effect of spurious transitive correlations. The method also accounts for low number of observations by means of a regularization parameter that depends on the effective number of sequences in the alignment. When tested on a set of protein families from Pfam, we found the RLS algorithm to have performance comparable to state-of-the-art methods for contact prediction, while at the same time being faster and conceptually simpler.
The source code and data sets are available at <http://cms.dm.uba.ar/Members/slaplagn/software>
author:
- Massimo Andreatta
- Santiago Laplagne
- Shuai Cheng Li
- Stephen Smale
bibliography:
- 'mybib\_plain.bib'
date: 'March 26, 2014'
title: 'Prediction of residue-residue contacts from protein families using similarity kernels and least squares regularization'
---
Introduction {#section:introduction}
============
A major problem in computational biology is the prediction of the 3D structure of a protein from its amino acid sequence. Anfinsen’s dogma suggests that, in principle, the amino acid sequence contains enough information to determine the full three-dimensional structure [@anfinsen1973principles]. However, a few decades on, the mechanisms of protein folding are still not satisfactorily explained [@dill2012protein]. In particular, the space of possible spatial configurations given a certain amino acid 1D sequence is immense (the “Levinthal paradox”), yet an unfolded polypeptide chain is driven to its native 3D structure in a finite time, typically milliseconds to seconds, upon shifting to folding conditions [@rose2006backbone].
Such enormous search space poses important challenges to the development of *ab initio* methods for structure prediction. Therefore, it is essential to exploit different kinds of information that can help reduce the degrees of freedom in the configurational search space. A powerful way of inferring distance constraints is the prediction of residue-residue contacts from multiple sequence alignments (MSA). The underlying assumption is that contacting residues coevolve to maintain the physicochemical complementarity of the amino acids involved in the contact. That is, if a mutation occurs in one of the contacting residues, the other one is also likely to mutate, lest the fold of the protein may be disrupted. Methods based on residue coevolution aim at inferring spatial proximity between residues (contacts) from such signals of correlated mutations (Figure \[fig:contacts\]).
![Illustration of a residue-residue contact. The contact imposes a constraint on the evolution of residues $i$ and $j$. Vice versa, coevolution of $i$ and $j$ can be used to infer their physical proximity.[]{data-label="fig:contacts"}](Contact_diagram-eps-converted-to.pdf){width="45.00000%"}
Thanks to the recent exponential growth in sequence data collected in databases such as Pfam [@PFAM], algorithms for the prediction of contacting residues from MSA have enjoyed increasing attention. Different kinds of approaches have been recently applied for contact prediction, from mutual information (MI) between pairs of positions [@buslje2009correction; @dunn2008mutual; @wang2013predicting], to Bayesian network models [@burger2010disentangling], direct-coupling analysis [@balakrishnan2011learning; @morcos2011direct; @marks2011protein] and sparse inverse covariance matrix estimation [@jones2012psicov]. See also [@marks2012protein] and [@de2013emerging] for recent reviews. In particular, the more sophisticated and successful methods attempt to disentangle direct and indirect correlations, that is the artifactual correlations emerging from transitive effects of covariance analysis [@lapedes1999correlated; @weigt2009identification]. Morcos et al. [@morcos2011direct] and Marks et al. [@marks2011protein] tackle this problem using a maximum-entropy approach, whereas Jones et al. [@jones2012psicov] estimate partial correlations by inverting the covariance matrix. A very recent pseudo-likelihood method based on 21-state Potts models [@ekeberg2013improved] was shown to outperform other approaches for direct-coupling analysis. Kamisetty et al. [@kamisetty2013assessing] systematically analyzed the conditions under which predicted contacts are likely to be useful for structure prediction, and found several hundred families that meet their criteria.
Here, we propose a new approach for computing direct correlations that employs regularized least squares (RLS) regression to invert a sample covariance matrix $S$. We compute the regularized inverse by the formula $$\label{eq:theta}
\Theta = (S^2 + \eta \operatorname{Id})^{-1} S,$$ with fixed $\eta > 0$. It proves to be a very simple, direct and fast approach, and requires no assumption on probabilities distributions or sparsity in the correlations.
The RLS algorithm described in this paper was applied to three different sets of protein families, and we compared its performance to state-of-the-art methods for contact prediction. The RLS method achieves precision rates superior to PSICOV [@jones2012psicov] and comparable to plmDCA [@ekeberg2013improved] but it is considerably faster than either.
Approach
========
The covariance matrix {#section:covariance}
---------------------
Let ${\mathscr{A}}$ be the set of $20$ amino acids plus the gap symbol $-$ and $\PP = \{p^m = (p_1^m, \dots, p_L^m)\}_{m=1,\dots,M}$ a given Pfam family of $M$ aligned protein sequences, possibly with gaps, where $L$ denotes the length of the protein domains. On this set of proteins, the covariance between any pair of columns $(i,j)$ for the amino acids pair $(a,b)$ is given by $$S^0_{ij}(a,b) = f_{ij}(a,b) - f_i(a) f_j(b)$$ where the corrected frequencies are calculated as $$\label{eq:pseudo}
f_i (a) = \frac{1}{\lambda + M_{\text{eff}}} \Big( \frac{\lambda}{21} + \sum_p w(p) \delta(a, p_i) \Big)$$ $$f_{ij} (a,b) = \frac{1}{\lambda + M_{\text{eff}}} \Big( \frac{\lambda}{21^2} + \sum_p w(p) \delta(a, p_i) \delta(b, p_j) \Big)$$ The delta kernel takes value $\delta(a,b)=1$ if $a = b$ and $\delta(a,b)=0$ otherwise. $w(p)$ is the weight of protein $p$ and ${M_{\text{eff}}}= \sum_p w(p)$ (see section \[section:measure\] for details on sequence weighting).
The parameter $\lambda$ is the so-called pseudocount, a regularization parameter that accounts for non-observed pairs. We note that the same, or similar, constructions for the corrected amino acid frequencies have been proposed previously by other authors [@ekeberg2013improved; @jones2012psicov; @morcos2011direct].
### Modified covariance matrix
We set $S^0_{ii}(a,b) = 0$ for $a \neq b$, and call $S$ this new matrix. This modification also appears in the code of PSICOV [@jones2012psicov] although it is not stated in their paper. By setting those values to $0$, the resulting matrix contains in general negative eigenvalues (see Figures and ) and hence is not anymore semi-definite positive, but it is still symmetric. We do not fully understand this step, but it is noteworthy that Equation \[eq:theta\] still makes sense for any $\eta > 0$.
In general, working with $S$ instead of $S^0$ gives better results in our experiments. See Table S1 for the effect of this step on predictive performance.
Regularized inverse – the key algorithm
---------------------------------------
As we mentioned in the Introduction, the covariance between our random variables does not distinguish between direct and indirect correlations. To overcome this problem, a technique used by statisticians is to compute the so-called partial correlations, which can be obtained from the inverse of the covariance matrix using its associated correlation matrix.
Since the covariance matrix is usually singular or ill conditioned, regularization techniques must be used to compute a regularized inverse $\Theta$. We achieve this by solving the following optimization problem $$\label{eq:optimization}
\Theta = \operatorname*{argmin}_{X \in {\mathbb{R}}^{20L \times 20L}} \| SX - \operatorname{Id}\|_2^2 + \eta \|X\|_2^2,$$ where $\| \cdot \|_2$ denotes the Frobenius norm, and $\eta$ is a regularization parameter to be determined. Observe that the first term is minimized by the inverse of $S$ when it exists.
The problem has a unique solution for any $\eta > 0$ as we see in the next proposition.
\[prop:regularization\] For a symmetric matrix $S \in {\mathbb{R}}^{n \times n}$ and a regularization parameter $\eta > 0$, the optimization problem has a unique solution, which is also symmetric and given by equation \[eq:theta\]. When $S$ is semidefinite positive, then the solution also is.
Since the norms involved are coordinate norms, the problem can be decoupled into independent problems for each column of $X$: $$\Theta^{(i)} = \operatorname*{argmin}_{x \in {\mathbb{R}}^{n \times 1}} \| S^tx - e^{(i)} \|_2^2 + \eta \|x\|_2^2,$$ where $\Theta^{(i)}$ is the $i$-th column of $\Theta$ and $e^{(i)}$ is the $i$-th column of the identity matrix.
This is a well studied problem known as regularized least squares (also called Tikhonov regularization or Ridge regression in different areas, see [@tikhonov1943stability] and [@hoerl1962application]). The unique solution is $\Theta^{(i)} = (S^tS + \eta \operatorname{Id})^{-1} S^t e^{(i)}$.
Hence, the solution to our matrix problem is $\Theta = (S^tS + \eta \operatorname{Id})^{-1} S^t$. Since we are assuming $S$ symmetric, we get $$\Theta = (S^2 + \eta \operatorname{Id})^{-1} S.$$
The matrix $S$ is diagonalizable with all of its eigenvalues real. The eigenvalues of $S$ are transformed by the same formula defining $\Theta$. If $\lambda_k$, $1 \le k \le 20L$, are the eigenvalues of $S$ then the eigenvalues of $\Theta$ will be $$\gamma_k = f(\lambda_k) = \frac{\lambda_k}{\lambda_k^2 + \eta}$$ This function is well defined for all $\lambda \in {\mathbb{R}}$ when $\eta$ is positive, which proves that the matrix $S^2 + \eta \operatorname{Id}$ is invertible. The resulting matrix $\Theta$ is symmetric by standard matrix theory. Finally, $f$ preserves the sign of the eigenvalue and hence $\Theta$ will be a semidefinite positive matrix whenever $S$ is.
Note that $\Theta$ can be computed by solving the linear system $(S^2 + \eta \operatorname{Id}) \Theta = S$, which is faster and more accurate than inverting the matrix $S^2 + \eta \operatorname{Id}$.
For a better understanding of our regularization formula, we study the function $f$ in more detail. The derivative of $f$ is $f'(\lambda) = \frac{-\lambda^2 + \eta}{(\lambda^2+\eta)^2}$. Hence $f$ is increasing for $|\lambda| < \sqrt{\eta}$ and decreasing for $|\lambda| > \sqrt{\eta}$, with maximum value at $\lambda = \sqrt{\eta}$ and minimum value at $\lambda = -\sqrt{\eta}$. We show in Figure the plot of this function for $\eta = \eta'/{M_{\text{eff}}}= 1000/3912$ (see Section \[subsec:regularization\] for the choice of $\eta$).
As mentioned in the proof of Proposition \[prop:regularization\], the function is smooth at $0$, so using this regularization formula we deal in a simple way with the conditioning problem of inverting the covariance matrix.
Aggregation
-----------
The matrix $\Theta$ obtained is a $20L \times 20L$ matrix. Its entries are estimates of the partial correlation between pairs of columns $(i,j)$ for [*all*]{} pairs of amino acids $(a,b)$. Since our goal is to detect relations between pairs of columns in the alignment, we compute a coupling score aggregating the values of $\Theta$ using the $l_1$-norm on the $20 \times 20$ sub-matrices, as in [@jones2012psicov]. That is, $$P(i, j) = \sum_{1 \le a, b \le 20} | \Theta_{ij}(a, b) |.$$
The $l_2$-norm for aggregation showed poorer performance than the $l_1$-norm above. Finally, following[@dunn2008mutual] and [@jones2012psicov] we define a corrected score $P_{\text{APC}}(i, j) = P(i, j) - \frac{P(\cdot, j)P(i, \cdot)}{P(\cdot, \cdot)}$, where $\cdot$ stands for the average over all positions.
The prediction of contacts between pairs of residues can now be obtained by ranking the $P_{\text{APC}}(i, j)$, where higher scores identify more likely residue-residue contacts.
Method details
==============
In this section we give more details on the actual implementation of the algorithm described above.
Sequence weighting {#section:measure}
------------------
Families from the Pfam database contain some degree of redundancy. A common strategy to overcome this problem is sequence weighting, which weighs down groups of similar sequences and assigns higher weights to isolated sequences.
We first define a similarity measure between proteins, following [@smale2013introduction]. We start from the BLOSUM90 frequency substitution matrix $B_{90}(a,b)$ defined in [@henikoff1992amino] and call $\hat{B}_{90}(a,b)$ for a pair of amino acids $(a,b)$ the normalized matrix $$\hat{B}_{90}(a,b) = \frac{B_{90}(a,b)}{\sqrt{B_{90}(a,a) B_{90}(b,b) }}$$
We then proceed to construct a similarity kernel between pairs of proteins $$K^3(p, q) = \sum_{k=1}^{10} \left(\sum_{i=1}^{L-k+1} K^2\left((p_i\dots p_{i+k-1}), (q_i\dots q_{i+k-1})\right)\right)$$ where $$K^2\left((p_i\dots p_{i+k-1}), (q_i\dots q_{i+k-1})\right) = \prod_{j=1}^k \hat{B}_{90}(p_{i+j-1}, q_{i+j-1}),$$ for $p, q \in \PP$, $1 \le k \le L$ and $1 \le i \le L-k+1$;
The normalized version of $K^3$ is obtained using $$\hat K^3(p, q) = \frac{K^3(p, q)}{\sqrt{K^3(p, p)K^3(q,q)}}.$$
Note that, since Pfam families consist of pre-aligned sequences, our $K^3$ kernel definition differs slightly from [@smale2013introduction] as it only compares aligned amino acid $k$-mers. Also, we limit the $k$-mers considered in the construction of $K^3$ to lengths smaller or equal to $10$. This implies a substantial improvement in computation time, with no significant loss in predictive power.
We fix a threshold $\theta$ (in this paper, $\theta = 0.7$) and for any protein $p \in \PP$ we define the equilibrium measure $$\pi(p) = \sum_{\substack{q \in \PP \\ \hat K^3(p, q) > (1-\theta)}} \hat K^3(p, q).$$
The weight of a protein $p$ is then defined as the reciprocal of the equilibrium measure $w(p) = (\pi(p))^{-1}$ , and the effective number of sequences in the alignment is $M_{\text{eff}} = \sum_p w(p)$. The kernel $ \hat K^3(p, q)$ is a measure of similarity between pairs of proteins, and for a given protein $p$ the quantity $\pi(p)$ effectively counts the number of sequences with similarity larger than a threshold $1 -\theta$, thereby weighing down sequences that are over-represented in the data set.
#### Hamming distance weighting
As a term of comparison, we also applied a more traditional sequence weighting scheme based on the hamming distance between pairs of sequences. For each sequence $p$, we count the number of other sequences in the alignment that share more than $\theta$% sequence identity with $p$ $$m^p = | \{ b \in \{ 1, \dots, M \} : \text{\%id}(p,b) > \theta \} |$$ and then assign a weight $w(p) = 1 / m^p$ to sequence $p$.
This approach was used previously by several authors such as [@morcos2011direct], [@jones2012psicov] and [@ekeberg2013improved], but we note that these authors use different values for the theshold $\theta$ (respectively 0.8, 0.62 and 0.9). In this paper we choose $\theta=0.62$. See See Tables S2-S3 for the optimization of the two weighting schemes with respect to the regularization parameter $\eta'$, and Table S4 for a comparison of their performance.
Regularization parameter {#subsec:regularization}
------------------------
The matrix inversion in Equation \[eq:theta\] contains a regularization parameter $\eta$. We observed that families containing few sequences, where the number of sequences $M$ is comparable in size to the number of random variables ($20L$) require a larger regularization parameter compared to bigger families ($M \gg 20L$). We use then a regularization parameter of the form $\eta = \eta' / {M_{\text{eff}}}$, where ${M_{\text{eff}}}$ is the effective number of sequences defined above.
We tried different values of $\eta'$ over the 15 families from [@marks2011protein], and observed that in general roughly the same $\eta'$ appears to be optimal across families with different ${M_{\text{eff}}}$. Thus the normalization $\eta = \eta' / {M_{\text{eff}}}$ appears appropriate.
In Figure we show how the actual eigenvalues of the modified covariance matrix corresponding to PFAM family PF00028 are transformed when computing the regularized inverse.
Pseudocounts {#subsec:pseudocounts}
------------
The pseudocounts parameter $\lambda$ in Equation \[eq:pseudo\] accounts for non-observed pairs of amino acids. Following [@morcos2011direct], we set $\lambda=44$. However, we observed that the performance gain is small compared to setting $\lambda=0$ (see Table S5). In fact, pseudocounts have a similar regularizing effect to the parameter $\eta'$ described in the previous section, and probably for this reason the contribution of $\lambda$ is minimal.
Results and Conclusion
======================
The method and estimation of parameters described above were first applied to the 15 families studied in Marks et al. [@marks2011protein]. Performance was estimated in terms of the fraction of correct predicted contacts among the $L/5$, $L/3$, $L/2$ and $L$ pairs with highest $P_{\text{APC}}$ score, where $L$ is the length of the alignment. We considered as a true contact a pair of amino acids with beta-carbons (C$\beta$) with distance $< 8$ [Å]{} and at least 5 residues apart along the length of the protein. We find that on these 15 families the optimal value for the regularization parameter is around $\eta'=1000$ (see Table S2).
Table \[table:results15\] compares the performance of the RLS algorithm with PSICOV version 1.11 [@jones2012psicov] and the plmDCA method [@ekeberg2013improved]. We observe that on this set our method outperforms both PSICOV and DCA on the majority of families. Additionally, in Table S6 in shown the positive predictive value of the methods with respect to short range ($5 \leq i - j \leq 11$), medium range ($12 \leq i - j \leq 23$) and long range ($>$ 23) interactions for the prediction of the top $L/5$ contacts.
Next, we applied the three methods using the same parameters on an additional set of families from [@ekeberg2013improved]. This set partially overlapped with the families from [@marks2011protein] studied in Table \[table:results15\], and after removing the duplicates we are left with a set of 22 families. Table \[table:results22\] shows the positive predictive value of the RLS algorithm compared to PSICOV and plmDCA. On this set plmDCA obtains the highest average performance on three out of four ranking categories. However, note that plmDCA was optimized on this set of families, so there may be a bias in favor of this method.
Finally, we constructed an independent set of 10 families, selected randomly from Pfam with the only condition of containing at least 1,000 unique sequences. This set had not been used in the optimization of the algorithms, therefore constitutes a fair ground for comparison. The results (see Table \[table:results10\]) show that RLS outperforms the other two methods in the $L/5$ and $L/3$ subsets, whereas DCA obtains highest average performance for the prediction of the top $L/2$ and $L$ contacts.
A comparison of the running times of the two best algorithms (RLS and DCA) shows that RLS is at least one order of magnitude faster than DCA (Table \[table:timings\]). Note that we used the latest fast version of plmDCA [@ekeberg2014fast], termed “asymmetric plmDCA”, which improves considerably on previous pseudolikelihood methods in terms of speed. Our fast regularized inversion of the covariance matrix allows contact prediction on hundreds of amino acids-long domains in a matter of seconds, practically removing the limitations on the length of proteins that can be analyzed. In fact, the slowest step in the predictions is sequence weighting (not accounted in Table \[table:timings\] for either method), in particular the $K^3$-based weighting can be slow for very large families, but we showed that a simpler and faster weighting strategy does not affect too dramatically the performance (Table S4).
In general, we observed that the performance depends on the effective number of sequences ${M_{\text{eff}}}$ in the alignment. For instance, families PF00390 or PF00793 are composed of several thousand sequences, but they contain much redundancy, which brings down ${M_{\text{eff}}}$ to a few hundred units. Roughly, it appears that at least 1000 non-redundant sequences (${M_{\text{eff}}}> 1000$) are necessary to achieve a reasonable precision for contact prediction. This is in agreement with previous estimates [@marks2011protein; @kamisetty2013assessing] which place this number at about $5L$, where $L$ is the length of the alignment.
In conclusion, we demonstrated how our simple regularization scheme for covariance matrix inversion allows the fast and accurate prediction of residue-residue contacts. Currently, a major restriction to this kind of approach is the fairly high number of non-redundant sequences required to infer coevolution from a multiple sequence alignment, limiting the application to a relatively small subset of Pfam. However, as the number of protein sequences deposited in public databases increases, we expect a larger number of protein families to become accessible to our analysis, as well as improved performance on those that are already accessible.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work received funding from City University of Hong Kong grants RGC \#9380050 and \#9041544. Work by S.L. was partially supported by Ministerio de Ciencia, Tecnología e Innovación Productiva, Argentina.
|
---
abstract: 'This article presents a machinery based on polyhedral products that produces faithful representations of graph products of finite groups and direct products of finite groups into automorphisms of free groups ${{\rm Aut}}(F_n)$ and outer automorphisms of free groups ${{\rm Out}}(F_n)$, respectively, as well as faithful representations of products of finite groups into the linear groups ${{{\rm SL}}}(n,{{\mathbb{Z}}})$ and ${{{\rm GL}}}(n,{{\mathbb{Z}}})$. These faithful representations are realized as monodromy representations.'
address: 'Mathematical Sciences, Indiana University - Purdue University Indianapolis, IN 46202'
author:
- Mentor Stafa
bibliography:
- 'C:/Users/mstafa/Dropbox/Works-Papers/Bibliography/topology.bib'
title: 'Polyhedral products, flag complexes and monodromy representations'
---
Introduction
============
Studying the topology of a fibration sequence frequently involves understanding the monodromy action, which is the action of the fundamental group of the base space on the fibre. For example monodromy is important in topology when using the Lyndon-Hochschild-Serre spectral sequence to compute the (co)homology with coefficients in that representation. The homology of the fibre inherits a module structure induced by the action of the fundamental group of the base on the fibre. When using the spectral sequence to obtain the homology of the fibre, then one has to consider the homology of the base with coefficients in the homology of the fibre regarded as a module, called the homology of the base with local coefficients. Here the monodromy representation for a fibration $p:E\to B$ with fibre $F$ will mean the representation $\rho: \pi_1(B) \to {{\rm Out}}(H_\ast(F)).$ The goal of this paper is to study polyhedral products in connection with monodromy representations for fibrations that arise naturally in the field of *toric topology*. The problem of explicitly describing such representations was started in [@stafa.monodromy]. The main idea is to give geometric descriptions in terms of polyhedral products for the spaces of the fibrations under consideration.
Numerous results on graph products of groups and polyhedral products demonstrate that the underlying simplicial complex $K$ plays an important role in their study, and is not only a convenient way to describe graph groups and polyhedral products. For example Droms [@droms1987isomorphisms] proved that two graph groups are isomorphic if and only if the graphs are isomorphic. Servatius *etal.* [@servatius1989surface] determine the simplicial complexes for which the commutator subgroup of a right-angled Artin groups is free. Moreover, if $K$ is chosen carefully, one obtains classifying spaces for various important families of discrete groups, including right-angled Artin and Coxeter groups from geometric group theory [@stafa.monodromy; @davis.okun]. In another application Grbić, Panov, Theriault and Wu [@wu.grbic.panov] give conditions on the 1-skeleton of $K$ that determine when the face ring of $K$ is a Golod ring, or equivalently the corresponding moment-angle complex has the homotopy type of a wedge of spheres. Recently Panov and Veryovkin [@panov2016polyhedral] studied polyhedral products that have the homotopy type of classifying spaces of right-angled Artin groups and right-angled Coxeter groups.
In the present article we study further properties of the monodromy representations associated to the fibration sequences $$\label{eqn: D-S-fibration INTRO1}
(\underline{EG},\underline{G})^K \to (\underline{BG},\underline{1})^K
\to \prod_{i=1}^n BG_i.$$ Each space in is a polyhedral product, depending on a simplicial complex $K$, together with a sequence of finite groups $\underline{G}:=\{G_1,\dots,G_n\}$, their classifying spaces $\underline{BG}:=\{BG_1,\dots,BG_n\}$, and corresponding universal covers $\underline{EG}:=\{EG_1,\dots,EG_n\}$, see Definition \[defn: polyhedral product\].
We give explicit descriptions of monodromy representations for simplicial complexes $K$ with more than two vertices, which were described geometrically in [@stafa.monodromy]. To do this we generalize and use some results of Panov and Veryovkin [@panov2016polyhedral]. We give applications, in particular to spaces of commuting elements in *commutative transitive (CT) finite groups*, where commutativity is a transitive relation, studied in a celebrated paper of M. Suzuki [@suzuki1957nonexistence], and to a problem related to the Feit-Thompson theorem, which states that all groups of odd order are solvable. Finally, we give give examples, which can be generalized easily, using the `Magma` [@magma] code included in the appendix.
Main results {#main-results .unnumbered}
------------
For given finite discrete groups $G_1,\dots, G_n$, we use polyhedral products to construct monodromy representations $ \Phi: G_1 \times \cdots \times G_n \to {{\rm Out}}(F_N) $ into outer automorphism groups of free groups. In particular, we obtain explicit faithful representations of graph products of finite groups into automorphism groups of free groups, and faithful representations of their direct products into linear groups ${{{\rm SL}}}(k,{{\mathbb{Z}}})$ or ${{{\rm GL}}}(k,{{\mathbb{Z}}})$. This article presents a machinery based on polyhedral products to achieve this. The first result is the following theorem.
Let $G_1,\dots, G_n$ be finite groups and $K$ a simplicial complex with $n$ vertices with 1-skeleton $K^1$ a chordal graph. Then there are faithful representations $$\Theta_K: \prod_{K^1} G_i \to {{\rm Aut}}(F_{\rho_K})$$ and faithful monodromy representations $$\Phi_K: G_1\times \cdots \times G_n \to {{\rm Out}}(F_{\rho_K}),$$ where $\rho_K$ is the rank of the fundamental group of the fibre in equation (\[eqn: D-S-fibration INTRO1\]).
The case when the groups $G_1,\dots,G_n$ are abelian the representations can be described explicitely and convenient models of polyhedral products can then be used to show that the corresponding monodromy representations obtained for non-abelian finite groups are also faithful.
Let $G_1,\dots, G_n$ be finite abelian groups. Then the faithful monodromy representation $\Phi_K$ induces a faithful representation $$\Phi_K: G_1\times\cdots \times G_n \to {{{\rm SL}}}({\rho_K},{{\mathbb{Z}}}).$$ If $G_1,\dots,G_n$ are non-abelian then $\Phi_K$ maps into ${{{\rm GL}}}({\rho_K},{{\mathbb{Z}}})$.
Let $E(2,G)\subseteq EG$ and $B(2,G) \subseteq BG$ be the spaces defined in $\S$\[sec: B(2,G) and E(2,G)\] that classify commuting elements in a group $G$. In particular, we use polyhedral products to study the class of finite transitively commutative (CT) groups, a class of groups where commtutativity is transitive. The following theorem is then an application of polyhedral products to group theory.
\[thm: top. equiv. form CT groups INTRO\] Finite CT groups with trivial center are solvable if and only if the induced map $H_1(E(2,G);{{\mathbb{Z}}}) \to H_1(B(2,G);{{\mathbb{Z}}})$ is not surjective.
This theorem is motivated from a result of Adem, Cohen and Torres-Giese [@fredb2g], which states an equivalent topological condition to the the Feit-Thompson theorem, namely that the theorem is true if and only if the map $H_1(E(2,G);{{\mathbb{Z}}}) \to H_1(B(2,G);{{\mathbb{Z}}})$ is not surjective.
Acknowledgments {#acknowledgments .unnumbered}
---------------
The author thanks Alina Vdovina (University of Newcastle, UK), who was visiting the Institute for Mathematical Research at ETH Zürich, during the spring semester 2016, for our numerous conversations on the topic, for suggesting me to use `Magma` for some computations, and for providing the first codes.
Polyhedral products and related fibrations {#sec: polyhedral products and fibrations}
==========================================
Polyhedral products are a generalized version of [*moment angle complexes*]{} that appear in the work of Buchstaber and Panov [@buchstaber2002torus] in the context of *toric topology*. Polyhedral products were also studied for instance by Bahri, Bendersky, Cohen and Gitler [@cohen.macs] and many others, and are the main objects of study in toric topology, see the recent monograph by Buchstaber and Panov [@bp2015].
\[defn: polyhedral product\] Let $(\underline{X},\underline{A} )$ denote a sequence of pointed $CW$-pairs $\{(X_i,A_i)\}_{i=1}^n$ and $[n]$ denote the sequence of integers $\{1,2,\dots,n\}$.
- A **simplex** $\sigma$ is given by an increasing sequence of integers $\sigma = \{1 \leq i_1 < \cdots < i_q \leq n\}
\subseteq [n]$. A [**simplicial complex**]{} $K$ is a collection of simplices such that if $\tau\subset \sigma$ and $\sigma \in K,$ then $\tau \in K.$ In particular $\emptyset \in K$.
- The **polyhedral product** $(\underline{X},\underline{A})^K$ is the subspace of the product $X_1 \times \cdots \times X_n$ given by the colimit $$(\underline{X},\underline{A})^K :=
\underset{{\sigma \in K}}{{{{\rm colim}}}}\, \mathcal D(\sigma)
= \bigcup_{\sigma \in K} \mathcal D(\sigma) \subseteq \prod_{i=1}^n X_i,$$ where $\mathcal D(\sigma)=\{(x_1,\dots,x_n)\in \prod_{i=1}^n X_i |
x_i\in A_i \text{ if } i \notin \sigma\}$, the maps are the inclusions, and the topology is the subspace topology of the product. Another standard notation for polyhedral products is $Z_K(\underline{X},\underline{A})$. Sometimes polyhedral products are called **$K$-powers**. Since $\emptyset$ is in any $K$, we have $\prod_{i=1}^n A_i \subset (\underline{X},\underline{A})^K
\subset \prod_{i=1}^n X_i.$
- If the pairs $(X,A)$ are $(D^2,S^1)$ or $(D^1,S^0)$, then the polyhedral products are called **moment-angle complexes** and **real moment-angle complexes**, respectively.
- If all the pairs in the sequence $(X_1,A_1),\dots,(X_n,A_n)$ are equal to $(X,A)$, then we omit the underline in the notation of the polyhedral product, and write simply $({X},{A})^K$.
\[defn: many definitions\] Next we give some relevant definitions and notation:
- For a simplicial complex $K$, the complex $K^i$ denotes the **$i$-skeleton** of $K$.
- A simplicial complex $K$ is called a **flag complex** if for any complete subgraph $\Gamma \subset K^1$, it also contains the simplex spanned by these vertices. The structure of moment angle complexes (or polyhedral products in general) is better understood when $K$ is a flag complex [@bp2015 §8.5].
- For a simplicial complex $K$, let ${{{\rm Flag}}}(K)$ denote the **clique complex** of $K^1$, i.e. the simplicial complex whose simplices are complete subgraphs of $K^1$. For example a flag complex is the clique complex of its 1-skeleton.
- A graph is called **chordal** if every cycle of length greater than three has an edge (called a *chord*) connecting two nonconsecutive vertices. Chordal graphs are also called **triangulated graphs**, or we say they have **perfect elimination ordering.**
- For any group $G$ denote its abelianization by $\mathscr A(G):=G/[G,G]$, and the abelianization map by ${\rm ab}_G: G\twoheadrightarrow \mathscr A(G)$.
- Let $G_1,\dots,G_n$ be a sequence of groups and $\Gamma$ a simplicial graph on $[n]$. The **graph product** of $G_1,\dots,G_n$ over $\Gamma$ is the quotient of their free product by the normal closure of the relations $R_{\Gamma}:=\{[g_i,g_i]: \{i,j\}\text{ is an edge in }\Gamma\}.$ The group obtained this way will be called a **graph group**, even though in the literature this name is sometimes used for right-angled Artin groups. We denote it by $$\prod_\Gamma G_i := (G_1\ast \cdots \ast G_n)/\langle R_{\Gamma} \rangle.$$ In this notation right-angled Artin groups are graph products of the group ${{\mathbb{Z}}}$.
\[example: polyhedral products\]
1. Let $X$ be the unit interval $[0,1]$ and $A\subset [0,1]$ be the subset $\{0,1/2,1\}$. Let $K$ be the simplicial complex consisting of only two vertices $\{\{v_1\},\{v_2\}\}$. Then $\mathcal D(\{v_1\})=X\times A \subset [0,1]^2$ and $\mathcal D(\{v_2\})=A\times X \subset [0,1]^2$. Therefore, $(X,A)^K= \mathcal D(\{v_1\}) \cup \mathcal D(\{v_2\})
= X\times A \cup A\times X$ is a graph inside the square $[0,1]^2$, homotopy equivalent to a wedge of 4 circles $\bigvee_4 S^1$. Similarly, we can choose $A$ to be any finite subset of the unit interval and we obtain similar graphs.
2. Let $(X,A)=(D^2,S^1)$. If $K$ is a the boundary of the $n$-simplex then the moment-angle complex $(D^2,S^1)^K = \bigcup_{\sigma_i} \mathcal D(\sigma_i)$ is homeomorphic to the sphere $S^{2n+1}=\partial D^{2(n+1)}$. It is also known [@wu.grbic.panov Theorem 4.6] that if $K$ is a flag complex, then $(D^2,S^1)^K$ has the homotopy of a wedge of spheres if and only if $K^1$ is chordal.
3. If $K$ is any simplicial complex on $n$ vertices, and $\underline{\ast}=\{\ast_1,\dots,\ast_n\}$ is the sequence of basepoints then $\underline{X}$ then $(\underline{X},\underline{\ast})^{K^0}
=\bigvee_{i=1}^n X_i.$ Therefore, in general $\bigvee_{i=1}^n X_i \subseteq (\underline{X},\underline{\ast})^{K}
\subseteq \prod_{i=1}^n X_i$.
Let $G$ be a topological group with basepoint its identity element $\ast=1$, $BG$ be the classifying space of $G$, and $EG$ be a contractible space with a free action of $G$ such that the quotient map $EG \to BG$ is a principal $G$-bundle. G. Denham and A. Suciu [@denham Lemma 2.3.2] gave a natural fibration relating the polyhedral product for the pair $(BG,1)$ to the polyhedral product for the pair $(EG,G)$. That is, for a simplicial complex $K$ with $n$ vertices, the polyhedral product $(BG,1)^K$ fibres over the product $(BG)^n$ as follows $$\label{eqn: D-S-fibration 1}
(EG,G)^K \to (EG)^n \times_{G^n} (EG,G)^K \to (BG)^n ,$$ where the total space is homotopy equivalent to $(BG,1)^K$. Note that the group $G$ acts coordinate–wise on the fibre $(EG,G)^K \subset (EG)^n$.
This fibration is a generalization of the Davis-Januszkiewicz space [@davis.januszckiewicz], with topological group the circle $S^1$, given by the Borel construction $$\mathcal{DJ}(K) = (ES^1)^n \times_{(S^1)^n} (ES^1,S^1)^K,$$ which describes a cellular realization of the Stanley-Reisner ring of $K$, in the sense that the cohomology ring of $\mathcal{DJ}(K)$ is precisely the Stanley-Reisner ring of $K$ defined as the quotient of the polynomial ring $R[K]=R[x_1,\dots,x_n]/I_K$ by the Stanley-Reisner ideal $I_K=\langle x_{i_1}\cdots x_{i_t}| \{i_1,\dots,i_t\} \neq K \rangle$. A later result of V. Buchstaber and T. Panov [@buch.panov] showed that $\mathcal{DJ}(K)\simeq (BS^1,1)^K$ and the homotopy fibre of the natural inclusion $\mathcal{DJ}(K) \to ({{\mathbb{C}}}P^\infty)^n$ is equivalent to the polyhedral product $(ES^1,S^1)^K$.
If $G_1,\dots,G_n$ is a sequence of topological groups, then for any simplicial complex $K$ with $n$ vertices, the fibration sequence (\[eqn: D-S-fibration 1\]) can be generalized to obtain $$\label{eqn: D-S-fibration}
(\underline{EG},\underline{G})^K \to (\underline{BG},\underline{1})^K \to \prod_{i=1}^n BG_i.$$ Similarly, the fundamental group of the base space acts naturally coordinate-wise on the homotopy fibre. The monodromy representation of this fibration is the main object of study in this article.
Note that the homotopy type of a polyhedral product depends only on the relative homotopy type of the pairs $(X,A)$, as observed in [@denham]. We are mainly interested in the cases when $G_1,\dots,G_n$ are finite discrete groups. If the pairs $(\underline{EG},\underline{G})$ are replaced by $(\underline{I},\underline{F})$, where $I$ is the unit interval and $F_i\subset I$ has the cardinality of $G_i$, then there is a homotopy equivalence $(\underline{EG},\underline{G})^K\simeq (\underline{I},\underline{F})^K$. Moreover, if $K=K^0$ is the 0-skeleton of $K$, then it follows from Example \[example: polyhedral products\] that $(\underline{BG},\underline{1})^{K^0}=BG_1 \vee \cdots \vee BG_n$. The homotopy fibre $(\underline{I},\underline{F})^{K^0}$ has the homotopy type of a finite wedge of circles $(\underline{I},\underline{F}) ^{K^0}\simeq \bigvee_{\rho_{K^0}} S^1, $ as shown in [@stafa.fund.gp], where $$\label{eqn: rank of kernel for chordal graph}
\rho_{K^0}=(n-1)\prod_{i=1}^n |G_i| - \sum_{i=1}^n (\prod_{j\neq i} |G_j|)+1.$$ This gives a topological proof of a classical theorem of J. Nielsen [@nielsen] concerning the rank of the free group in the following short exact sequence of groups $$1 \to F_{\rho_{K^0}} \to G_1\ast \cdots \ast G_n \to \prod_{1\leq i \leq n} G_i \to 1.$$ Therefore, the rank $\rho_{K^0}$ depends only on the order of the groups $G_1,\dots,G_n$, and not on their group structure. For simplicity we simply write $\rho_{K^0}$ for the rank, when the orders of $G_i$ are clear from the context.
More generally, it was shown in [@stafa.fund.gp] that if $G_1,\dots,G_n$ are countable discrete groups then the spaces in the fibration sequence (\[eqn: D-S-fibration\]) are Eilenberg-MacLane spaces of the type $K(\pi,1)$ if and only if $K$ is a flag complex. Moreover, for any $K$, the fundamental group of the polyhedral product $(\underline{BG},\underline{1})^K$ is determined by the 1-skeleton $K^1$ and is isomorphic to $\pi_1((\underline{BG},\underline{1})^K) \cong \prod_{K^1} G_i$, the graph product of the groups $G_1,\dots,G_n$. Hence there is a short exact sequence of groups
$$\label{eqn: s.e.s.}
1 \to \pi_1((\underline{EG},\underline{G})^K) \to
\prod_{K^1} G_i \to \prod_{1\leq i \leq n} G_i \to 1.$$
We want to study simplicial complexes $K$ for which the kernel of the short exact sequence above is a free group. The following theorem shows exactly which simplicial complexes $K$ have this property.
Let $G_1,\dots,G_n$ be (countable) discrete groups and $K$ be a flag complex on $n$ vertices. Then $(\underline{EG},\underline{G})^K$ has the homotopy type of a graph if and only if $K^1$ is a chordal graph.
It was shown in [@stafa.fund.gp Theorem 1.1] that the space $(\underline{EG},\underline{G})^K$ is a $K(\pi,1)$ if and only if $K$ is a flag complex. Therefore, we get the short exact sequence of groups in (\[eqn: s.e.s.\]). Panov and Veryovkin [@panov2016polyhedral Theorem 4.3] showed that $\pi_1((\underline{EG},\underline{G})^K)$ is free if and only if the graph $K^1$ is a chordal graph, which completes the proof. See also [@servatius1989surface Theorem 4.2] for a relevant result.
Commutator subgroups of graph groups {#sec: commutator subgp}
====================================
The *commutator subgroup* $[G,G]$ of a group $G$ is generated by commutators $[g,h]:=ghg^{-1}h^{-1}$ with $g,h\in G$. Let $G_1,\dots,G_n$ be finite groups and $K$ be a flag complex with $K^1$ a chordal graph. From the previous section we know that under these assumptions the kernel of the projection map $$p: \prod_{K^1} G_i \to \prod_{1\leq i \leq n} G_i$$ is a free group. This kernel is generated by iterated commutators of the form $$[g_{j_1},[g_{j_2},[\dots,[g_{j_k},g_{j_{k+1}}]\dots]]],$$ where $g_{j_i}$ belong to distinct $G_{j_i}$. The kernel ${\rm ker}(p)$ is not necessarily the commutator subgroup of $\prod_{K^1} G_i$, if at least one of the $G_i$ is not abelian. However, ${\rm ker}(p)$ [coincides]{} with the commutator subgroup of the graph group if the groups $G_1,\dots,G_n$ are all abelian.
In this section we describe a basis for the free group ${\rm ker}(p)=F_{\rho_K}$ in terms of iterated commutators. A basis was given in [@panov2016polyhedral Lemma 4.7], where the groups under consideration had order 2, that is the graph groups were right-angled Coxeter groups. Another version of this basis of commutators was studied by Grbić, Panov, Theriault, and Wu in the context of exterior algebras in [@wu.grbic.panov Theorem 4.3].
Before we proceed it is important to note that the commutator subgroup of a free group can also be described by a generating set not consisting of commutators. One can obtain new presentations not involving commutators using Tietze transformations [@magnus2004combinatorial; @vdovina1995constructing].
Recall that the fibre in (\[eqn: D-S-fibration\]) depends only on the order of the finite groups $G_1,\dots,G_n$. Therefore, it suffices to describe the basis elements (i.e. iterated commutators) of ${\rm ker}(p)=F_{\rho_K}$ only when $G_1,\dots,G_n$ are cyclic groups. The basis for the general case of any finite groups $G_1,\dots,G_n$ can be obtained by considering the basis when $G_i$ are all cyclic and then replacing the entries in the commutators with the nontrivial elements of $G_i$. This observation will be used in Section \[sec: graph products abelian gps\].
\[prop: basis of pi\_1 for K\^0\] Let $G_1,\dots,G_n$ be finite groups and $K=K^0$. Then the fundamental group of the fibre $(\underline{EG},\underline{G})^K$ in (\[eqn: D-S-fibration\]) is the free group with basis consisting of the following iterated commutators $$[g_j,g_i],\,\, [g_{k_1},[g_j,g_i]],\,\,\dots\,,[g_{k_1},\dots,[g_{k_l},[g_j,g_i]]\dots],$$ where $g_t\in G_t$, with $k_1<\cdots<k_{l}<j$ and $j>i\neq k_r$, for all $r$.
We need to show that (1) this set of elements generates the fundamental group, and that (2) the number of elements in the set equals the rank of the free group in equation (\[eqn: rank of kernel for chordal graph\]). Since the first part of the proof is essentially the proof of [@panov2016polyhedral Lemma 4.7], we only give an outline here. First recall the Hall identities for group elements $a,b,c$ $$\label{eqn: Hall-Witt}
[a,bc]=[a,c][a,b][[a,b],c], \text{ and }
[ab,c]=[a,c][[a,c],b][b,c],$$ and if $x$ is a commutator we can write $$\label{eqn: Hall-Witt derived}
[g_j,[g_i,x]]=[g_j,x] [x,[g_i,g_j]] [g_j,g_i] [x,g_i] [g_i,[g_j,x]] [x,g_j] [g_i,g_j] [g_i,x].$$ Therefore, given the equations (\[eqn: Hall-Witt\]) and (\[eqn: Hall-Witt derived\]) we proceed as follows:
- we can use the identities above to switch between the commutators $[g_j,[g_i,x]]$ and $[g_i,[g_j,x]]$ by using other commutators of lower degrees, we can change the order of $g_{k_1},\dots,g_{k_l},g_j$ in the commutator $[g_{k_1},\dots,[g_{k_l},[g_j,g_i]]\dots]$ to have them in increasing order, so we can thus obtain the inequalities in the proposition;
- we can use the identities to eliminate commutators with two entries from the same group, since we can reorder the terms to have these two entries next to each other, then their product is in the same group, hence having a lower degree commutator;
- we can thus assume that the commutators have the prescribed order, and that $g_k,g_l$ are from different groups if $k\neq l$;
- finally we obtain a generating set for the the free group $F_{\rho(n)}$ in terms of commutators $[g_{k_1},\dots,[g_{k_l},[g_j,g_i]]\dots]$, with $k_1<\cdots<k_{l}<j$ and $j>i\neq k_r$, for all $r$. Call this set $\mathcal S$.
Note that $\mathcal S$ does not generate the commutator subgroup, unless all $G_i$ are abelian. Now we need to show that this generating set is minimal, that is $|\mathcal S|=\rho_K$ defined in (\[eqn: rank of kernel for chordal graph\]). For this we use induction on the number $n$ of vertices of $K$. Let us denote $\rho_K=\rho(n)$ when $K=K^0$ consists of only $n$ vertices.
Assume $G_1,\dots,G_n$ have orders $m_1,\dots,m_n$, respectively. For $n=2$, clearly $|\mathcal S|=|\{(g_j,g_i)|g_j,g_i\neq 1\}|=(m_1-1)(m_2-1)=\rho_K=\rho(2)$. Suppose this is true for $n=k$. For $n=k+1$ we claim that $$\rho(k+1)=m_{k+1} \rho(k) +(m_{k+1}-1)(\prod_{1\leq i \leq k}m_i - 1).$$ When we introduce a new group $G_{k+1}$, since it has the highest index, according to our assumption, its elements come only second from the last in the iterated commutator. This yields $m_{k+1} \rho(k)$ generators, by taking a commutator and placing the elements of $G_{k+1}$ second from last in the iterated commutators, giving a higher degree commutator. The insertion of a new non-trivial element, gives more freedom to the last element in the iterated commutator. For each non-trivial element $1\neq g_i \in G_{k+1}$, the last entry can take $\prod_{1\leq i \leq k}m_i - 1$ values. Now counting for each element of the new group, gives the second term in our claim, hence proving the claim. Combining equation (\[eqn: rank of kernel for chordal graph\]) and the claim, and rearranging the terms, the minimality of $\mathcal S$ follows.
Let $G_1={{\mathbb{Z}}}_2=\{1,x\},\, G_2={{\mathbb{Z}}}_2=\{1,y\},\,G_3={{\mathbb{Z}}}_3=\{1,z,z^2\},$ and $K=\{\{1\},\{2\},\{3\}\}.$ Then the fibre of the fibration has fundamental group the free group $F_9$ with a minimal generating set $\mathcal S$ given by $$\mathcal S = \{[z,x] , [z^2,x] , [z,y] , [z^2,y] , [y,x], [x,[z,y]] , [x,[z^2,y]] , [y,[z,x]] , [y,[z^2,x]]\}.$$
For simplicial complexes $K$ strictly larger than their 0-skeleton the following proposition holds.
\[prop: basis of pi\_1 for K=flag\] Let $G_1,\dots,G_n$ be finite groups and $K$ be a flag complex with $n$ vertices such that $K^1$ is a chordal graph. Then the fundamental group of the fibre in (\[eqn: D-S-fibration\]) is a free group with a basis the iterated commutators $$[g_j,g_i],\,\, [g_{k_1},[g_j,g_i]],\,\,\dots\,,[g_{k_1},\dots,[g_{k_l},[g_j,g_i]]\dots],$$ where $g_t\in G_t$, with $k_1<\cdots<k_{l}<j$ and $j>i\neq k_r$, for all $r$, and $i$ is the smallest vertex in a component not containing $j$ in the subcomplex of $K$ restricted to $\{k_1,\dots,k_{l},j,i\}.$
When we start introducing edges in $K^0$, then we start introducing commutator relations $[g_i,g_j]$ whenever $\{i,j\}$ is an edge. In the iterated commutator $[g_{k_1},\dots,[g_{k_l},[g_j,g_i]]\dots]$ if $i,j$ are in the same connected component of $K$ restricted to $\{k_1,\dots,k_{l},j,i\},$ then there is a path from $i$ to $j$ with coordinates from $\{k_1,\dots,k_{l}\},$ hence we can consider the iterated commutator induced by these vertices. Using relations from the edges we can reduce this commutator to another commutator of shorter length etc. Thus we can choose $i,j$ to be in different path components. If we have two commutators where the last coordinate is in the same component, one can show that we can write one in terms of the other. Hence, we choose the smallest between them. We leave it to the reader to check that the detailed arguments in the proof of [@panov2016polyhedral Theorem 4.5] work also for any selection of finite groups.
Let us consider an example with the symmetric group on 3 letters. Let $G_1=\Sigma_3:=\langle s,t | s^2=t^2, (st)^3\rangle=\{1,x_1,x_2,x_3,x_4,x_5\}$, $G_2={{\mathbb{Z}}}_2=\{1,y\}$, $G_3={{\mathbb{Z}}}_3=\{1,z,z^2\},$ and $K''=\{\{1,2\},\{3\}\}$ in Figure \[fig: K and K’\]. Then the fibre of the fibration has fundamental group the free group $F_{22}$ with $\mathcal S$ given by $$\begin{aligned}
\mathcal S = \{ & [z,x_1] , [z,x_2], [z,x_3], [z,x_4],[z,x_5],
[z^2,x_1] , [z^2,x_2],[z^2,x_3],[z^2,x_4],[z^2,x_5], \\
& [z,y] , [z^2,y] , [x_1,[z,y]] ,[x_2,[z,y]], [x_3,[z,y]], [x_4,[z,y]], [x_5,[z,y]] \\
& [x_1,[z^2,y]] ,[x_2,[z^2,y]], [x_3,[z^2,y]], [x_4,[z^2,y]], [x_5,[z^2,y]]\}.\end{aligned}$$ Note that the structure of the symmetric group $\Sigma_3$ was not needed to write the generating set $\mathcal S$. Therefore, if we replace $\Sigma_3$ with the cyclic group of order six $C_6$, then the corresponding generating set $\mathcal S$ has the same number and types of generators where in the commutators in $S$ we replace the elements of the symmetric group with those of the cyclic group.
Examples of monodromy representations {#section: e.g.}
=====================================
Let $G_1,\dots,G_n$ be finite discrete groups and $K$ be a flag complex with $K^1$ a chordal graph. Consider the following commutative diagram $$\begin{tikzcd}
1 \arrow{r} & F_{\rho_K} \arrow{r} \arrow{d}{\Psi = \rm iso} &
\prod_{K^1} G_i \arrow{r} \arrow{d}{\Theta_K}
& \prod_{i} G_i\arrow{r} \arrow{d}{\Phi_{K}} & 1 \\
1 \arrow{r} & {{\rm Inn}}(F_{\rho_K}) \arrow{r} &{{\rm Aut}}(F_{\rho_K}) \arrow{r} &
{{\rm Out}}(F_{\rho_K}) \arrow{r} & 1,
\end{tikzcd}$$ where $\Theta(g)(h)=ghg^{-1}$ and $\Psi(g)(h)=ghg^{-1}$. We are interested in describing the maps $\Theta_{K}$ and $\Phi_{K}.$
For examples concerning only two finite groups, i.e. $n=2$, see [@stafa.monodromy], where explicit answers are given. Using `Magma` we can explicitly describe faithful representations $$\Phi_K: G_1\times \cdots \times G_n \to {{{\rm SL}}}({\rho_K},{{\mathbb{Z}}}),$$ where $G_1, \dots , G_n$ are finite abelian groups and $n\geq 3$. In general, if $G_i$ are any finite groups (not necessarily abelian), we obtain faithful representations of graph products of finite groups $$\Theta_K: \prod_{K^1} G_i \to {{\rm Aut}}(F_{\rho_K})$$ as well as faithful monodromy representations of direct products of finite groups $$\Phi_K: G_1\times \cdots \times G_n \to {{{\rm GL}}}({\rho_K},{{\mathbb{Z}}})$$ as will be shown below. These include many interesting classes of discrete groups, such as right-angled Coxeter groups. If one of the groups is infinite discrete, then additional examples include hyperbolic groups as described in [@holt2012generalising Theorem 5.1], braid groups, right-angled Artin groups and more. Thus such representations can be realized as monodromy representations.
The rank $\rho_K$ increases [very fast]{} (\[eqn: rank of kernel for chordal graph\]) if we increase the order and the number of the groups in consideration. We concentrate on a couple of examples including right-angled Coxeter groups. In addition we select the simplicial complexes $K$ and $K'$ in Figure \[fig: K and K’\], to keep the rank of the free group small. It is clear that by modifying the same codes in `Magma` included in the appendix, it is possible to obtain many more explicit examples, which we leave to the interested reader. However, note that the basis generated in `Magma` is different (yet equivalent) from the basis we describe in Section \[sec: commutator subgp\].
\[scale=1, vertices/.style=[draw, fill=black, circle, inner sep=0.5pt]{}\] (f) at (0,0) ; (e) at (0.5,0) ; (g) at (1,0) ; (a) at (3,0) ; (b) at (3.5,0) ; (c) at (4,0) ; (d) at (4.5,0) ; (k) at (6,0) ; (l) at (6.5,0) ; (m) at (7,0) ; /in [a/b,b/c,c/d]{} /in [k/l]{} ()–();
\[ex: 1\] Consider three groups of order 2 and the following short exact sequence obtained from the fibration sequence (\[eqn: D-S-fibration\]) $$1 \to F_5 \to {{\mathbb{Z}}}_2\ast {{\mathbb{Z}}}_2, \ast {{\mathbb{Z}}}_2 \to {{\mathbb{Z}}}_2 \times {{\mathbb{Z}}}_2, \times {{\mathbb{Z}}}_2 \to 1,$$ corresponding to the simplicial complex $K$ in Figure \[fig: K and K’\], where each of the cyclic groups is generated by $a$, $b$ and $c$, respectively. Recall that the rank of the fibre is given by equation (\[eqn: rank of kernel for chordal graph\]) and in this case is 5. Then $ F_5$ has a generating set (see `Magma` in the appendix), thus a presentation given by $$F_5:=\langle (ba)^2,(ca)^2,(cb)^2,acbcba,bcacba \rangle=\langle x_1,x_2,x_3,x_4,x_5\rangle.$$ The action of ${{\mathbb{Z}}}_2 \times {{\mathbb{Z}}}_2, \times {{\mathbb{Z}}}_2$ on $F_5$ is determined by the following: $$\begin{aligned}
a\cdot x_i = \begin{cases}
x_1^{-1} \\
x_2^{-1} \\
x_4 \\
x_3 \\
x_5^{-1}
\end{cases}
b\cdot x_i = \begin{cases}
x_1^{-1} \\
x_5 x_1^{-1} \\
x_3^{-1} \\
x_1 x_4^{-1} x_1^{-1} \\
x_2 x_1^{-1}
\end{cases}
c\cdot x_i = \begin{cases}
x_3 x_5 x_4^{-1} x_2^{-1} & \text{if } i=1\\
x_2^{-1} & \text{if } i=2\\
x_3^{-1} & \text{if } i=3\\
x_2 x_4^{-1} x_2^{-1} & \text{if } i=4\\
x_3 x_1 x_4^{-1} x_2^{-1} & \text{if } i=5,
\end{cases}\end{aligned}$$ where $a,b,c$ act by conjugation. This gives also a faithful representation of the right-angled Coxeter group to the automorphism group ${{\rm Aut}}(F_5)$. It is straightforward to check that the induced action on the abelianization ${{\mathbb{Z}}}^5$ gives a faithful representation of the right-angled Coxeter group to the special linear group over the integers: $$\begin{aligned}
\Phi_K: {{\mathbb{Z}}}_2 \times {{\mathbb{Z}}}_2, \times {{\mathbb{Z}}}_2 & \to {{{\rm SL}}}(5,{{\mathbb{Z}}})\\
a,b,c &\mapsto X,Y,Z,\end{aligned}$$ where $X,Y,Z$ are the following matrices, respectively (example 1 in the appendix): [[[[ $$\begin{aligned}
\left(\begin{array}{rrrrr}
-1 & 0 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 &-1 \\
\end{array}\right),
\left(\begin{array}{rrrrr}
-1 & 0 & 0 & 0 & 0 \\
-1 & 0 & 0 & 0 & 1 \\
0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 \\
-1 & 1 & 0 & 0 & 0 \\
\end{array}\right),
\left(\begin{array}{rrrrr}
0 & -1 & 1 & -1 & 1 \\
0 & -1 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 \\
1 & -1 & 1 & -1 & 0 \\
\end{array}\right).\end{aligned}$$ ]{}]{}]{}]{}
Note that the generators of the fundamental group of the polyhedral product $({E{{\mathbb{Z}}}_2},{{{\mathbb{Z}}}_2})^K$ can be described using the loops in Figure \[fig: the loops\] sitting in the space $({I},{F})^K$.
(0,0)–(1,1); (1,1)–(3,1); (3,1)–(2,0); (2,0)–(0,0); (0,0)–(0,2); (0,2)–(1,3); (1,3)–(3,3); (3,3)–(2,2); (2,2)–(0,2); (1,1)–(1,3); (2,2)–(2,0); (3,3)–(3,1);
(0,0)–(1,1); (1,1)–(3,1); (3,1)–(2,0); (1,1)–(1,3); (2,0)–(0,0); (0,0)–(0,2); (0,2)–(1,3); (1,3)–(3,3); (3,3)–(2,2); (2,2)–(0,2); (2,2)–(2,0); (3,3)–(3,1);
(0,0)–(1,1); (1,1)–(1,3); (1,1)–(3,1); (3,1)–(2,0); (2,0)–(0,0); (0,0)–(0,2); (0,2)–(1,3); (1,3)–(3,3); (3,3)–(2,2); (2,2)–(0,2); (2,2)–(2,0); (3,3)–(3,1);
(0,0)–(1,1); (1,1)–(3,1); (3,1)–(2,0); (2,0)–(0,0); (0,0)–(0,2); (0,2)–(1,3); (1,3)–(3,3); (3,3)–(2,2); (2,2)–(0,2); (1,1)–(1,3); (2,2)–(2,0); (3,3)–(3,1);
(0,0)–(1,1); (1,1)–(1,3); (1,1)–(3,1); (3,1)–(2,0); (2,0)–(0,0); (0,0)–(0,2); (0,2)–(1,3); (1,3)–(3,3); (3,3)–(2,2); (2,2)–(0,2); (2,2)–(2,0); (3,3)–(3,1);
\[fig: the loops\]
\[ex: 2\] Now we consider four cyclic groups. Construct the right-angled Coxeter group over the simplicial complex $K'$ given in Figure \[fig: K and K’\]. Then equation (\[eqn: D-S-fibration\]) gives the following short exact sequence of groups $$1 \to F_{5} \to \prod_{K'}{{\mathbb{Z}}}_2 \to {{\mathbb{Z}}}_2^4 \to 1,$$ where $F_{5} = \langle x_1,\dots,x_{5}\rangle:=\langle
(ca)^2,
(da)^2,
(db)^2,
a d b d b a,
c d a d c a,
\rangle.$ The conjugation action is then described as follows:
$$\begin{aligned}
a\cdot x_i = \begin{cases}
x_1^{-1} \\
x_2^{-1} \\
x_4 \\
x_3 \\
x_5^{-1}
\end{cases}
b\cdot x_i = \begin{cases}
x_1 \\
x_3^{-1}x_2 x_4 \\
x_3^{-1} \\
x_4^{-1} \\
x_3^{-1} x_5 x_4
\end{cases} \,
c\cdot x_i = \begin{cases}
x_1^{-1} \\
x_5 x_1^{-1} \\
x_3 \\
x_1 x_4 x_1^{-1} \\
x_2 x_1^{-1}
\end{cases}\!\!
d\cdot x_i = \begin{cases}
x_5 x_2^{-1} \\
x_2^{-1} \\
x_3^{-1} \\
x_2 x_4^{-1} x_2^{-1} \\
x_1 x_2^{-1}
\end{cases}\end{aligned}$$
for all values of $i=1,2,3,4,5,$ respectively. Then there is a representation of the right-angled Coxeter group $\prod_{K'}{{\mathbb{Z}}}_2$ into the automorphism group ${{\rm Aut}}(F_5)$. $$\prod_{K'}{{\mathbb{Z}}}_2 \hookrightarrow {{\rm Aut}}(F_5) .$$ The induced action on the abelianization of the free group gives a faithful representation $$\begin{aligned}
\Phi_{K'}: {{\mathbb{Z}}}_2^4 & \to {{{\rm SL}}}(5,{{\mathbb{Z}}})\\
a,b,c,d &\mapsto A,B,C,D\end{aligned}$$ where $A,B,C,D$ are the following matrices, respectively (example 2 in the appendix): [[[[[ $$\begin{aligned}
A=&\left(\begin{array}{rrrrr}
1 & 0 & 0 & 0 & 0 \\
0 & 1 & -1 & 1 & 0 \\
0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 &-1 & 0 \\
0 & 0 & -1 & 1 & 1 \\
\end{array}\right),
B=\left(\begin{array}{rrrrr}
-1 & 0 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 &0 & 0 \\
0 & 0 & 0 & 0 & -1 \\
\end{array}\right), \\
C=&\left(\begin{array}{rrrrr}
-1 & 0 & 0 & 0 & 0 \\
-1 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 &1 & 0 \\
-1 & 1 & 0 & 0 & 0 \\
\end{array}\right),
D=\left(\begin{array}{rrrrr}
0 & -1 & 0 & 0 & 1 \\
0 & -1 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 &-1 & 0 \\
1 & -1 & 0 & 0 & 0 \\
\end{array}\right).\end{aligned}$$ ]{}]{}]{}]{}]{}
One can start with any finite groups $G_1,\dots , G_n$ with given presentations and any simplicial complex $K$ with on $n$ vertices such that $K^1$ is a chordal graph. If either of the groups is not abelian, then the representations obtained in the abelianization may not have images in ${{{\rm SL}}}(\rho_K,{{\mathbb{Z}}})$, but rather in ${{{\rm GL}}}(\rho_K,{{\mathbb{Z}}})$ as shown in [@stafa.fund.gp Example 2].
It is known that ${{{\rm GL}}}(n,{{\mathbb{Z}}})$ can be generated by four matrices $x,y,z,w$, see for instance [@delaharpe2000topics pp.44-49]. For the case $n=3$ we can write $$\begin{aligned}
x=\left(\begin{array}{*{3}r}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0 \\
\end{array} \right)
y=\left(\begin{array}{*{3}r}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1 \\
\end{array} \right)
z=\left(\begin{array}{*{3}r}
1 & 1 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array} \right)
w=\left(\begin{array}{*{3}r}
-1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array} \right).\end{aligned}$$ If we define $v:=wyz$, then the groups $G=\langle x,y,w\rangle$ and $H=\langle v\rangle$ are finite of order 48 and 6, respectively, and $G$ is nonabelian. It can be shown that ${{{\rm GL}}}(3,{{\mathbb{Z}}})$ is a quotient of the free product $G\ast H$. Moreover, if $n$ is odd, then the $n\times n$ matrices $$\begin{aligned}
x=\left(\begin{array}{*{5}c}
0 & 0 & \cdots & 1 \\
1 & 0 & \cdots & 0 \\
\vdots & \ddots& \ddots& 0\\
0 & \cdots & 1 & 0 \\
\end{array} \right),\,
y=\left(\begin{array}{*{5}c}
1 & 1 & 0 & \cdots & 0 \\
0 & 1 & 0 & \cdots & 0 \\
0 & 0 & 1 & \cdots & 0\\
\vdots & \ddots& \ddots& &0\\
0 & \cdots & 0 & \cdots & 1 \\
\end{array} \right)\end{aligned}$$ generate ${{{\rm SL}}}(n,{{\mathbb{Z}}})$. A similar example can be constructed using this information.
Graph products of abelian groups {#sec: graph products abelian gps}
================================
Every finite abelian group can be written as a finite direct sum of finite cyclic subgroups with order a power of a prime. Here we will describe how to think of a graph product of finite abelian groups over $K$ as a graph product of cyclic groups over a new simplicial complex $K_{\underline{G}}$, at the expense of having more vertices in the simplicial complex.
In [@stafa.monodromy Theorem 2.2] it was shown that two cyclic subgroups yield a faithful monodromy representation $$C_n\times C_m \to {{\rm Out}}(F_{\rho_K})$$ and a faithful representation $$\Phi_K: C_n\times C_m \to {{{\rm SL}}}({\rho_K},{{\mathbb{Z}}}),$$ where $K=\{\{1\},\{2\}\}$.
Consider two finite abelian groups $G,H$. Then we can write them as direct sums if cyclic groups $$G\cong \bigoplus_{i\in I} C_{n_i},\,\,\,\, H \cong \bigoplus_{j\in J} C_{m_j}.$$ We can then replace the simplicial complex $K=\{\{1\},\{2\}\}$ by the union of two simplices $$K'=\Delta[|I|-1] \sqcup \Delta[|J|-1].$$ This does not change the monodromy representation $G\times H \to {{\rm Out}}(F_{\rho_K})$, because the short exact sequences of groups $$1 \to F_{\rho_K} \to G \ast H \to G \times H \to 1,$$ and $$1 \to F_{\rho_{K'}} \to \prod_{(K')^1} C_{n_i,m_j} \to \prod_{i,j} C_{n_i,m_j} \to 1,$$ are equivalent; note that $\prod_{i,j} C_{n_i,m_j} \cong G\times H$, and $$\prod_{(K')^1} C_{n_i,m_j}\cong \prod_{(\Delta[|I|-1])^1}C_{n_i}
\ast \prod_{(\Delta[|J|-1])^1}C_{m_j}\cong G\ast H.$$
More generally, if $K$ is a flag complex with $K^1$ a chordal graph, and $G_1,\dots,G_n$ are finite abelian, write the direct sum decomposition of each group $G_i$ $$G_i\cong \bigoplus_{j\in J_i} C_{n_j^i}.$$ Replace each vertex $i$ on the chordal graph $K^1$ by the simplex $\Delta[|J_i|-1]$. Give unique names to all vertices in all these different simplices. Note that, if $\{i,j\}$ is an edge in $K$, and $\Delta[|J_i|-1]=\{\{v_0^i,\cdots,v_{|J_i|-1}^i\}\}$, then we need to add $\{v_k^i,v_t^j\}$ to the new simplicial complex for all $k,t$ since the $G_i,G_j$ commute if and only if all their subgroups commute. We can now define the following simplicial complex.
Let $\underline{G}:=\{G_1,\dots,G_n\}$ be finite abelian groups. Let $K$ be a simplicial complex on $n$ vertices with 1-skeleton $K^1$ a chordal graph. Define the **simplicial complex $K_{\underline{G}}$** to be the flag complex obtained from $K$ by the following procedure: replace each vertex $i$ of $K$ with the full simplex $\Delta[|J_i|-1]=\{\{v_0^i,\cdots,v_{|J_i|-1}^i\}\}$, add an edge between the vertices in $\Delta[|J_k|-1]$ and the vertices in $\Delta[|J_l|-1]$ if $G_k$ and $G_l$ commute, and take the corresponding clique complex of the 1-skeleton of this new simplicial complex.
We then have the following lemma.
\[lem: new flag complex from given K\] With the same assumptions, the graph $(K_{\underline{G}})^1$ is chordal.
This follows from the definition: The 1-skeleton of $\Delta[|J_i|-1]$ and $K^1$ are chordal graphs. If $c$ is a cycle of length greater than 3, then its edges are either all in $(\Delta[|J_i|-1])^1$ for some $i$ or it moves between various 1-skeleta $(\Delta[|J_k|-1])^1$. Suppose $c$ has length 4. If vertices of $c$ are all in a single $(\Delta[|J_i|-1])^1$ we are done. If vertices of $c$ lie in two distinct $(\Delta[|J_i|-1])^1$’s, then there is one edge between $(\Delta[|J_k|-1])^1$ and $(\Delta[|J_l|-1])^1$, thus there is an edge between all the vertices between these two simplices, in particular between nonconsecutive vertices. If vertices of $c$ lie in three distinct $(\Delta[|J_i|-1])^1$’s, the same argument holds. If vertices of $c$ lie in four distinct $(\Delta[|J_i|-1])^1$’s, then $c$ is a replica of a cycle in $K$. The same arguments show the triangulation of longer cycles $c$ .
Let $G_1,\dots, G_n$ be finite abelian groups and $K^1$ a chordal graph. Then the faithful monodromy representation $G_1\times\cdots \times G_n \to {{\rm Out}}(F_{\rho_K})$ induces a faithful representation $$\Phi_K: G_1\times\cdots \times G_n \to {{{\rm SL}}}({\rho_K},{{\mathbb{Z}}}).$$
By Lemma \[lem: new flag complex from given K\], the graph $(K_{\underline{G}})^1$ is chordal and by definition $K_{\underline{G}}$ is a flag complex. Therefore the spaces in fibration (\[eqn: D-S-fibration\]) are Eilenberg-MacLane spaces. Furthermore, the monodromy representation $$\Phi_K: G_1\times\cdots \times G_n \to {{\rm Out}}(F_{\rho_K})$$ is equivalent to the monodromy representation $$\Phi_{K_{\underline{G}}}: G_1\times\cdots \times G_n \to {{\rm Out}}(F_{\rho_{K_{\underline{G}}}}),$$ where $F_{\rho_{K_{\underline{G}}}}=F_{\rho_{K}}$ and we rewrite $$G_i\cong \bigoplus_{j\in J_i} C_{n_j^i}.$$ Each element in $G_i$ lies in a cyclic group, which by [@stafa.monodromy Theorem 2.2] maps faithfully into ${{{\rm SL}}}(\rho_K,{{\mathbb{Z}}}).$ The theorem follows.
If $K^1$ is a chordal graph, then there is a faithful representation of the graph product $\prod_{K^1} G_i$ into the automorphism group ${{\rm Aut}}(F_{\rho_K})$ of the free groups of rank ${\rho_K}$. In particular, this is true for any right-angled Coxeter group.
Recall that there is a short exact sequence of groups $$1 \to {{\rm IA}}_{N} \to {{\rm Aut}}(F_{N}) \to {{{\rm GL}}}(N,{{\mathbb{Z}}}) \to 1$$ induced by the abelianization of the automorphisms of free groups, that is the induced map on the first homology $H_1(F_{N})$. The group ${{\rm IA}}_{N}$ is the analogue of the Torelli group in mapping class groups of surfaces. Then we have the following immediate corollary.
If $K^1$ is a chordal graph and $G_i$ are finite discrete groups, then the images of $\prod_{K^1} G_i$ under the faithful representations above are not in ${{\rm IA}}_{\rho_K}$.
[**The automorphism group ${{\rm Aut}}(K)$.**]{} Let ${{\rm Aut}}(K)$ denote the group of simplicial automorphisms of the finite simplicial complex $K$, which is naturally a subgroup of the symmetric group $\Sigma_n$ if $K$ has $n$ vertices. For example, if $K$ is the full simplex or the trivial simplicial complex, then ${{\rm Aut}}(K)=\Sigma_n$, or if $K$ is an $n$-gon ${{\rm Aut}}(K)$ is the dihedral group $D_{2n}$.
The group ${{\rm Aut}}(K)$ acts on the fibre $(\underline{EG},\underline{G})^K$. It is natural to consider the induced action of this automorphism group on the monodromy representation $$\Phi_K : G_1\times \cdots \times G_n \to {{\rm Out}}(F_{\rho_K}).$$
The action of the automorphism group ${{\rm Aut}}(K)$ on a polyhedral product $(\underline{X},\underline{A})^K$ was studied by A. Al-Raisi in his thesis [@raisithesis], in particular in case $K$ is an $n$-gon with more than 3 vertices (not a chordal graph) and for $G_i \cong {{\mathbb{Z}}}_2.$ The automorphism group ${{\rm Aut}}(K)$ acts on $(\underline{X},\underline{A})^K$ by permuting coordinates $$\sigma \cdot (x_1,\dots,x_n):=(x_{\sigma(1)},\dots, x_{\sigma(n)}).$$ By definition, if $K^1$ is the a chordal graph, then this action permutes only coordinates in a way that preserves the fundamental group of $(\underline{BG},\underline{1})^K$.
More precisely, let $\sigma \in {{\rm Aut}}(K) \leq \Sigma_n$ and $K$ be a flag complex. Note that $K={{{\rm Flag}}}(K^1)$, hence $K$ is determined by its 1-skeleton. Since $\sigma$ permutes only the coordinates which already commute with each other, then $\sigma$ permutes the groups which already commute in the graph group $\pi_1((\underline{BG},\underline{1})^K)=\prod_{K^1} G_i$. Hence we obtain the following statement.
Let $K$ be a flag complex with $n$ vertices. Then ${{\rm Aut}}(K)$ preserves the homotopy type of $(\underline{BG},\underline{1})^K$, equivalently its fundamental group and that of the fibre. In particular it preserves the monodromy representation.
Induced maps in homology
========================
In this section we prove the following proposition.
Let $K$ be a flag complex and $G_1,\dots,G_n$ be finite groups. Then the induced map on first homology groups $$H_1((\underline{EG},\underline{G})^K;{{\mathbb{Z}}}) \to H_1((\underline{BG},\underline{G})^K;{{\mathbb{Z}}})$$ is not a surjection.
The main ingredient in this proof is the fact that the abelianizations of both the fundamental group $\pi_1((\underline{BG},\underline{G})^K) \cong \prod_{K^1} G_i$ and the product $\prod_{1\leq i \leq n} G_i$ are the same. For a group $G$ denote $\mathscr A(G):=G/[G,G]$ and the abelianization map $G \to \mathscr A(G)$ by ${\rm ab}_G$. Note that the abelianization of $\prod_{K^1} G_i$ factors through the group $\prod_{1\leq i \leq n} G_i$: $$\prod_{K^1} G_i \to \prod_{1\leq i \leq n} G_i
\xrightarrow{ab_G} \mathscr A(\prod_{K^1} G_i).$$ Let $G:=\prod_{K^1} G_i,\, H:=\prod_{1\leq i \leq n} G_i$, and $N=\pi_1 ((\underline{EG},\underline{G})^K).$ Note that in general, for any abelian group $A$ and a surjection $h:G \twoheadrightarrow A$, there is a unique map $\phi: \mathscr A(G) \twoheadrightarrow A$ such that $\phi \circ {\rm ab}_G = h$.
Consider the following commutative diagram
1 & N & G & H & 1,\
& &(G) & (H) &
where $p\circ i =0,$ and ${\rm ab}_G \circ i = {\rm ab}_H \circ p \circ i = 0$.
Now, since $\mathscr{A}(G)$ is an abelian group, then there is a unique map $f: \mathscr A(N) \to \mathscr A(G)$ such that ${\rm ab}_G \circ i = f \circ {\rm ab}_N$. Since ${\rm ab}_G \circ i=0 = f \circ {\rm ab}_N,$ and ${\rm ab}_N$ is clearly not trivial, then $f$ cannot be onto. Therefore $H_1((\underline{EG},\underline{G})^K;{{\mathbb{Z}}}) \to H_1((\underline{BG},\underline{G})^K;{{\mathbb{Z}}})$ is not a surjection.
This proposition is in the spirit of the induced maps in homology introduced in the next section, concerning the spaces $B(2,G)$ and $E(2,G)$ defined below. We seek a similar result in that case, too.
CT groups and Feit-Thompson theorem {#sec: B(2,G) and E(2,G)}
===================================
In this section we study commutative transitive groups defined below, and use some methods from polyhedral products to understand the interplay between topology and group theory, and characterize some group properties using topology. For any group $\pi$ the descending central series is given by a sequence of normal subgroups $$\pi=\Gamma^1 \triangleright \Gamma^2 \triangleright
\cdots \triangleright \Gamma^{n+1} \triangleright \cdots$$ where inductively $\Gamma^{n+1}=[\pi,\Gamma^{n}]$ for $n\geq 2.$ If $\pi=F_n$ is the free group of rank $n$, then for any topological group $G$ there is a filtration $$Hom(F_n/\Gamma^2,G) \subset Hom(F_n/\Gamma^3,G) \subset \cdots \subset G^n.$$ The sequences of spaces given by $$B_k(q,G):=Hom(F_k/\Gamma^q,G)\subset G^k$$ and $$E_k(q,G):=G\times Hom(F_k/\Gamma^q,G) \subset G^{k+1}$$ have the structure of simplicial complexes ([@fredb2g]), respectively, with respective geometric realizations defined as follows $$\begin{aligned}
B(q,G):= |B_\ast(q,G)|, \text{ and } E(q,G):=|B_\ast(q,G)|. \end{aligned}$$ The projections $E_k(q,G) \twoheadrightarrow B_k(q,G)$ induce a fibration $$\label{eqn: fibration of E(q,G) to B(q,G)}
E(q,G) \to B(q,G) \to BG$$ and in particular, for $q=2$ we have $$\label{eqn: fibration of E(2,G) to B(2,G)}
E(2,G) \to B(2,G) \to BG.$$
The total space $B(2,G)$ and the homotopy fibre $E(2,G)$ were studied by A. Adem, F. Cohen and E. Torres Giese [@fredb2g]. They posed the question whether for finite $G$ the space $B(2,G)$ is always a $K(\pi,1)$, having showed that these spaces are occasionally $K(\pi,1)$ for the case of [*commutative transitive groups*]{}. C. Okay [@okay; @okay2015colimits] gave classes of groups for which $B(2,G)$ is not a $K(\pi,1)$, such as extraspecial 2-groups of order $2^{2n+1}$, for $n\geq2$, hence answering their question. A brief survey is given in [@stafa.comm.2 §9].
\[defn: CT group\] A group $G$ is **commutative transitive** or *CT* if commutativity is a transitive relation in $G$. That is, if $[a,b]=[b,c]=1$, then $[a,c]=1$ for all non-central elements $a,\,b,\,c \in G$.
The class of CT groups played an important role in the classification of finite simple groups and were studied by M. Suzuki [@suzuki1957nonexistence; @suzuki1986group], among many others, who showed that every non-abelian simple CT-group is of even order and isomorphic to ${{{\rm PSL}}}(2, 2^f)$ for some $f\geq2$. Finite CT groups have been classified, see for example [@schmidt1994subgroup p. 519, Theorem 9.3.12].
In particular, if $G$ is a finite CT group with trivial center then the following is true.
If there are maximal abelian subgroups $G_1,\dots, G_n$ of $G$ that cover $G$, then there is a homotopy equivalence $B(2,G) \simeq \bigvee_i BG_i$.
With the assumptions of this proposition we have the following corollary.
$B(2,G)$ has the homotopy type of the polyhedral product $(\underline{BG},\underline{1})^{K^0}.$
In what follows $G$ is assumed to be finite.
Using the five term short exact sequence from the Lyndon-Hochschild-Serre spectral sequence Adem, Cohen and Torres Giese [@fredb2g Proposition 7.2] showed that the non-surjectivity of the induced map on first homology of the fibration (\[eqn: fibration of E(2,G) to B(2,G)\]) $$H_1(E(2,G)) \to H_1(B(2,G))$$ is equivalent to the Feit-Thompson theorem that groups of odd order are solvable. Hence the study of the fibration encodes fundamental information about the group $G$. We would like to use polyhedral products, i.e. topology, to extract more information about this equivalent form of the Feit-Thompson theorem [@feit1963chapter], which is algebraic in nature.
Let $G$ be a finite CT group with trivial center and let $G_1,\dots, G_n$ be its cover by maximal abelian subgroups as above ($n$ is called the [[*covering number*]{}]{} of $G$). Since all spaces are $K(\pi,1)$’s, we will move frequently between fundamental groups and their classifying spaces. Note that there are two commutative diagrams of short exact sequences of groups:
$$\label{eqn: first diagram}
\begin{tikzcd}
\pi_1(E(2,G)) \arrow{r}\arrow{d} & {\mathcal H} \arrow{r} \arrow{d} & {[G,G]} \arrow{d}\\
\pi_1(E(2,G)) \arrow{r}\arrow{d} & \pi_1(B(2,G)) \arrow{r} \arrow{d}& \pi_1(BG) \arrow{d}\\
\ast \arrow{r} & H_1(BG) \arrow{r} & H_1(BG),
\end{tikzcd}$$
and $$\label{eqn: commutative diagram}
\begin{tikzcd}
N \arrow{r}\arrow{d} & \pi_1(\underline{EG},\underline{G})^{K_0} \arrow{r} \arrow{d}{i_2} & {[G,G]} \arrow{d}\\
\pi_1(E(2,G)) \arrow{r}{i_1} \arrow{d}{p_1} & \pi_1(B(2,G)) \arrow{r}{p_2} \arrow{d}{p_3}& \pi_1(BG) \arrow{d}\\
Q \arrow{r} & \pi_1(\prod_i BG_i) \arrow{r} & H_1(BG),
\end{tikzcd}$$ where $Q$ is a finite abelian group and $N$ is a free group (we omit the trivial groups on each side of the short exact sequences). The existence of the first diagram is clear, whereas for the second diagram, even though for CT groups the map $\pi_1(B(2,G)) \to G$ does not factor through the product $\prod_i G_i$, the composition $\pi_1(B(2,G)) \to G \to H_1(G),$ being an epimorphism onto an abelian group, factors uniquely through the abelianization of $\pi_1(B(2,G))$, which is the direct product $\prod_i G_i$.
The map $p_1$ factors uniquely through the abelianization $H_1(E(2,G))$, hence there is a map $q_1$ such that $p_1 = q_1 \circ {\rm ab}$. Hence there is a diagram $$\label{eqn: small commutative diagram}
\begin{tikzcd}
\pi_1(E(2,G)) \arrow{r}{i_1} \arrow{d}{p_1} & \pi_1(B(2,G)) \arrow{d}{p_3}\\
H_1(E(2,G)) \arrow{d}{q_1} \arrow{r}{(i_1)_\ast} & \prod_i G_i \arrow{d}\\
Q \arrow{r} & \prod_i G_i ,
\end{tikzcd}$$ where the dotted map is the one we are interested in (we are not claiming that the lower triangle commutes). By [@fredb2g Proposition 8.8] the group $\pi_1(E(2,G))$ is free, with rank $$\mathcal{N}_G=1-|G:Z(G)| + \sum_{1\leq i \leq n} \left( |G:Z(G)|-|G:G_i|\right).$$ Since $Z(G)=\{1_G\}$, by rearranging the terms of $ \mathcal{N}_G$ we obtain the following: $$\begin{aligned}
\mathcal{N}_G &= 1-|G:Z(G)| + \sum_{1\leq i \leq n} \left( |G:Z(G)|-|G:G_i|\right)\\
&= 1-|G| + \sum_{1\leq i \leq n} \left( |G|-|G|/|G_i|\right)\\
&= 1-|G| + n|G| - \sum_{1\leq i \leq n} |G|/|G_i|\\
&= 1+ (n-1)|G| - \sum_{1\leq i \leq n} |G|/|G_i|.\end{aligned}$$ Note that this is a more general version of the formula for $\rho_K$ in equation (\[eqn: rank of kernel for chordal graph\]), with the special case of $|G|=\prod_i|G_i|$ giving the rank $\rho_K$ when $K$ is only a set of $n$ points; let us use the notation $\rho_K= \rho(n)$ since $K$ becomes irrelevant. In general $l.c.m.(|G_1|,\dots,|G_n|) \leq |G|< \prod_i |G_i| $, since the groups $G_i$ cover $G$. Actually they divide each other from left to right. Since $|G|$ divides $\prod_i |G_i|,$ then $\prod_i |G_i|=C |G|$. Therefore, we have $$\begin{aligned}
\rho(n)-\mathcal N_G &= (n-1)|G|(C-1) - \sum_{1\leq j\leq n}(|G|(C-1))/|G_j|\\
& = |G|(C-1)\left( n-1 - \sum_{1\leq j\leq n}1/|G_j| \right).\end{aligned}$$ Since all $|G_i|\geq 2$, then we have $\rho(n)-\mathcal N_G \geq |G|(C-1)(n/2-1).$ If $n=2$ then $G$ is abelian, so assume that $n\geq 3$. Then we get $\rho(n)-\mathcal N_G >0$.
Let $G$ be a finite CT group with trivial center. Then $\rho(n)>\mathcal N_G.$
In addition to the above argument, this is also a direct consequence of the fact that the index of $N$ in each of the free groups is given by the following formula [@lyndon2015combinatorial p.16]: $$\begin{aligned}
& [ \pi_1(E(2,G)) : N] = \frac{{\rm rank} (N) - 1}{ {\rm rank} (\pi_1(E(2,G)))-1} = |Q| = \frac{\prod_i|G_i|}{|H_1(G)|},
\text{ and,}\\
& [\pi_1((\underline{EG},\underline{G})^{K_0}):N] = \frac{{\rm rank} (N) - 1}{ {\rm rank} (\pi_1((\underline{EG},\underline{G})^{K_0})-1} = |[G,G]| = \frac{|G|}{|H_1(G)|}.\end{aligned}$$ Since $|G| < {\prod_i|G_i|}$ the lemma follows.
Indeed the proof of this lemma tells us that $\rho(n)/\mathcal N_G \sim |Q|/|[G,G]|$.
Before we proceed, it is clear from the diagrams (\[eqn: first diagram\],\[eqn: commutative diagram\]) that if $G/[G,G]=1$, then the induced map on homology $H_1(E(2,G)) \to H_1(B(2,G))$ is onto (without using the 5-term sequence in homology).
[If $G$ is simple, then the following map is a surjection $$H_1(E(2,G);{{\mathbb{Z}}}) \to H_1(B(2,G);{{\mathbb{Z}}}).$$ ]{}
Instead, using only topology we want to prove the following equivalent statements: *if the map $H_1(E(2,G)) \to H_1(B(2,G))$ is onto, then $|G|$ is even,* or equivalently, *if $|G|$ is odd, then the map $H_1(E(2,G)) \to H_1(B(2,G))$ is not onto*.
Now, if $|G|$ is odd, then all $Q, \, \prod_i G_i$ and $[G,G]$ are odd. Also $N$ is a free group of odd index in both free groups $\pi_1(E(2,G))$ and $\pi_1(EG_i,G_i)^{K_0}$. The following results are immediate:
Either all $\rho(n),\, \mathcal N_G,\, {\rm rank}(N)$ are even, or, all $\rho(n),\,\mathcal N_G,\, {\rm rank}(N)$ are odd, such that the ratios $$\frac{{\rm rank}(N)-1}{\mathcal N_G-1},\, \frac{{\rm rank}(N)-1}{\rho(n)-1}, \, \frac{\rho(n)-1}{\mathcal N_G-1}$$ are odd.
Use the formulas in the proof of the previous Lemma.
Next note that the map $(i_1)_\ast$ in (\[eqn: small commutative diagram\]) can be a surjection only if $Q \lneq {\rm Im}((i_1)_\ast)$ (if not then their intersection is at most $Q$ and the image cannot be everything). Consider the following diagram
& [Ker]{}\_1 &\
[Ker]{}\_2 & H\_1(E(2,G)) & [Im]{}((i\_1)\_)\
& Q & [Im]{}((i\_1)\_).
The image ${\rm Im}((i_1)_\ast) $ has odd order. Since both kernels have full rank (the quotients are both finite groups) we have that ${\rm Ker}_2 \leq {\rm Ker}_1$. The kernels have bases as follows $${\rm Ker}_1=span\{\alpha_i e_i: i \in[N_G]\}, \text{ and }
{\rm Ker}_2=span\{\beta_i e_i: \beta_i|\alpha_i,\, \forall i \in[N_G]\}.$$ Here all $\alpha_i,\beta_i$ have to be odd numbers such that $\alpha_i|\beta_i $ for all $i$. Indeed this can be done for any (finite) sequence of subgroups $$Q < Q_1 < \cdots < {\rm Im}((i_1)_\ast) < \cdots <\prod_i G_i$$ as there are kernels $K_0,\, K_1,\dots,$ of full ranks corresponding to projections. The following theorem shows that ${\rm Im}((i_1)_\ast) \lneq \prod_i G_i$ for the case of CT groups with trivial center. We conclude this section with the following corollary.
\[cor: top. equiv. form CT groups\] Finite CT groups with trivial center are solvable if and only if the induced map $H_1(E(2,G);{{\mathbb{Z}}}) \to H_1(B(2,G);{{\mathbb{Z}}})$ is not a surjection.
Of course this is a special case of the condition in [@fredb2g Proposition 7.2], but for this corollary we use only the diagrams (\[eqn: commutative diagram\],\[eqn: small commutative diagram\]).
Consider the commutative diagram (\[eqn: commutative diagram\]) and (\[eqn: small commutative diagram\]). If the map $(i_1)_\ast$ is a surjection, then the composition $$H_1(E(2,G);{{\mathbb{Z}}}) \twoheadrightarrow H_1(B(2,G);{{\mathbb{Z}}}) \twoheadrightarrow G/[G,G]$$ is a surjection. On the other hand, from diagram (\[eqn: commutative diagram\]), the group $H_1(E(2,G);{{\mathbb{Z}}})$ maps trivially, hence $G/[G,G]$ is trivial. On the other hand, if $G/[G,G]$ is trivial, then $Q=\prod_i G_i$ and $(i_1)_\ast$ is a surjection.
The following question is still open for CT groups: [*Use Corollary \[cor: top. equiv. form CT groups\] to show that if $G$ is a simple finite TC group with trivial center, then $G$ has even order*]{}.
|
---
abstract: |
We give a combinatorial classification for the class of postcritically fixed Newton maps of polynomials and indicate potential for extensions.
As our main tool, we show that for a large class of Newton maps that includes all hyperbolic ones, every component of the basin of an attracting fixed point can be connected to $\infty$ through a finite chain of such components.
address: 'Jacobs University Bremen, School of Engineering & Science, Campus Ring 1, 28759 Bremen, Germany'
author:
- 'Yauhen Mikulich, Johannes Rückert'
title: A Combinatorial Classification of Postcritically Fixed Newton Maps
---
Introduction
============
One of the most important open problems in rational dynamics is understanding the structure of the space of rational functions of a fixed degree $d\geq 2$. This problem is today wide open.
Aside from being a useful tool for numerical root-finding, Newton maps of polynomials form an interesting subset of the space of rational maps that is more accessible for studying than the full space of rational maps. Hence, a partial goal in the classification of [*all*]{} rational maps can be to gain an understanding of the space of Newton maps.
In this paper, we present a theorem that structures the dynamical plane of postcritically finite Newton maps, and then use this result to construct a graph that classifies those Newton maps whose critical orbits all terminate at fixed points. Newton maps of degree $1$ and $2$ are trivial, and we exclude these cases from our investigation. Let us make precise what we mean by a Newton map.
\[Def\_NewtonMap\] A rational function $f:\Cc\to\Cc$ of degree $d\geq 3$ is called a [*Newton map*]{} if $\infty$ is a repelling fixed point of $f$ and for each fixed point $\xi\in\C$, there exists an integer $m\geq 1$ such that $f'(\xi)=(m-1)/m$.
This definition is motivated by the following observation, which is a special case of [@RS Proposition 2.8] (the case of superattracting fixed points, i.e. every $m=1$, goes back to [@Head Proposition 2.1.2]).
\[Prop\_Head\] A rational map $f$ of degree $d\geq 3$ is a Newton map if and only if there exists a polynomial $p:\C\to\C$ such that for $z\in\C$, $f(z)=z-p(z)/p'(z)$.
Observe that $f$ and $p$ have the same degree if and only if $p$ has $d$ distinct roots. Note also that some rational maps arise as Newton maps of entire functions, for example for $h=pe^q$ where $p,q$ are complex polynomials, the map $N_h$ is rational.
\[Def\_ImmediateBasin\] Let $f$ be a Newton map and $\xi\in\C$ a fixed point of $f$. Let $B_{\xi}=\{z\in\C\,:\, \lim_{n\to\infty}f^{\circ n}(z)=\xi\}$ be the [*basin (of attraction)*]{} of $\xi$. The component of $B_\xi$ containing $\xi$ is called the [*immediate basin*]{} of $\xi$ and denoted $U_\xi$.
Clearly, $B_\xi$ is open and by a theorem of Przytycki [@Przytycki], $U_{\xi}$ is simply connected and $\infty\in\partial U_\xi$ is an accessible boundary point (in fact, a result of Shishikura [@Shishikura] implies that every component of the Fatou set is simply connected).
Our first result is the following.
\[Thm\_PreimagesConnected\] Let $f:\Cc\to\Cc$ be a Newton map with attracting fixed points $\xi_1,\dots,\xi_d\in\C$, and let $U'_0$ be a component of some $B_{\xi_i}$. Then, $U'_0$ can be connected to $\infty$ by the closures of finitely many components $U'_1,\dots, U'_k$ of $\bigcup_{i=1}^d B_{\xi_i}$.
More precisely, there exists a curve $\gamma:[0,1]\to\Cc$ such that $\gamma(0)=\infty$, $\gamma(1)\in U'_0$ and for every $t\in[0,1]$, there exists $m\in\{0,1,\dots, k\}$ such that $\gamma(t)\in \ol{U}'_m$.
We will see that $\gamma$ can be chosen to consist of the closures of internal rays in the $U'_m$. Theorem \[Thm\_PreimagesConnected\] allows to describe how the components of the basins are connected to each other. Thus, it is a basis for a combinatorial classification of certain Newton maps: Theorems \[Thm\_NewtonGraph\] and \[Thm\_Realization\] show that the combinatorics of these connections suffice to describe postcritically fixed Newton maps uniquely.
We call a Newton map [*postcritically fixed*]{} if all its critical points are mapped onto fixed points after finitely many iterations. If $f$ is a postcritically fixed Newton map, Theorem \[Thm\_PreimagesConnected\] allows to structure the entire Fatou set, because each Fatou component is in the basin of some attracting fixed point. Then, we construct the channel diagram $\Delta$ of $f$, which is the union of the accesses from finite fixed points of $f$ to $\infty$ (see Section \[Sec\_Models\]). Denote by $\Delta_n$ the connected component of $f^{-n}(\Delta)$ containing $\infty$. Pulling back $\Delta$ several times we get a connected graph $\Delta_n$ that contains the forward orbits of all critical points, similar to the Hubbard tree of a postcritically finite polynomial.
We introduce the notion of an *abstract Newton graph*, which is a pair $(g,\Gamma)$ of a map $g$ acting on a graph $\Gamma$ that satisfies certain conditions, (see Definition \[Def\_NewtonGraph\]). In particular, the conditions on $(g,\Gamma)$ allow $g$ to be extended to the branched covering $\overline{g}$ of the whole sphere $\S^2$.
We show that for every [*abstract Newton graph*]{} there exists a unique up to affine conjugacy postcritically fixed Newton map realizing it.
The assignments of a Newton map to an abstract Newton graph and vice versa are injective and inverse to each other, so we give a combinatorial classification of postcritically fixed Newton maps by way of abstract Newton graphs. Thus, our main results are the following (see Sections \[Sec\_Graph\] and \[Sec\_Thurston\] for the precise definitions).
\[Thm\_NewtonGraph\] Every postcritically fixed Newton map $f$ gives rise to a unique up to equivalence relation abstract Newton graph. More precisely, there exists a unique $N\in\N$ such that $(\Delta_N,f)$ is an abstract Newton graph.
If $f_1$ and $f_2$ are Newton maps with channel diagrams $\Delta_1$ and $\Delta_2$ such that $(\Delta_{1,N},f_1)$ and $(\Delta_{2,N},f_2)$ are equivalent as abstract Newton graphs, then $f_1$ and $f_2$ are affinely conjugate.
\[Thm\_Realization\] Every abstract Newton graph is realized by a postcritically fixed Newton map which is unique up to affine conjugacy. More precisely, let $(\Gamma,g)$ be an abstract Newton graph. Then, there exists a postcritically fixed Newton map $f$ with channel diagram $\hat{\Delta}$ such that $(\ol{g},\Gamma')$ and $(f,\hat{\Delta}'_{N_{\Gamma}})$ are Thurston equivalent as marked branched coverings.
Moreover, if $f$ realizes two abstract Newton graphs $(\Gamma_1,g_1)$ and $(\Gamma_2,g_2)$, then the two abstract Newton graphs are equivalent.
Our construction of an abstract Newton graph can be done for all postcritically finite Newton maps, but will in general not contain the orbits of [*all*]{} critical points, and thus not describe the combinatorics of the entire Fatou set (note that there are rational maps with [*buried*]{} Fatou components that are not attached to any other Fatou component (it is not hard to find Newton maps with this property either). An extreme example of this behavior is provided by rational maps with Sierpinski Julia sets, see e.g. [@Milnor2 Appendix F]). It seems likely however that with Theorem \[Thm\_PreimagesConnected\] and additional combinatorial objects that describe any strictly periodic or preperiodic critical points, a classification of at least all hyperbolic Newton maps can be achieved. Thus, our results are a first step towards a combinatorial classification of Newton maps, and in particular of all hyperbolic components in the space of Newton maps. They may also be a basis for transporting the powerful concept of Yoccoz puzzles, which has been used to prove local connectivity of the Julia set for many classes of polynomials, to the setting of Newton maps beyond the cubic case (Roesch has successfully applied Yoccoz puzzles to cubic Newton maps [@Roesch]).
A number of people have studied Newton maps and used combinatorial models to structure the parameter spaces of some Newton maps. Janet Head [@Head] introduced the [*Newton tree*]{} to characterize postcritically finite cubic Newton maps. Tan Lei [@TanLei] built upon this work and gave a classification of postcritically finite cubic Newton maps in terms of matings and captures. Jiaqi Luo [@Luo] extended some of these results to “unicritical” Newton maps, i.e. Newton maps of arbitrary degree with only one [*free*]{} (non-fixed) critical value. The present work can be seen as an extension of these results beyond the setting of a single free critical value. The main differences to this setting are that the channel diagram is in general not a tree anymore and that in the presence of more than one non-fixed critical value, the iterated preimages of the channel diagram may be disconnected.
This article is structured as follows. In Section \[Sec\_Models\], we introduce the concept of a channel diagram for Newton maps and discuss some of its properties. We use the channel diagram and its preimages to prove Theorem \[Thm\_PreimagesConnected\] in Section \[Sec\_Proof\]. In Section \[Sec\_Graph\], we introduce abstract Newton graphs and prove Theorem \[Thm\_NewtonGraph\]. Theorem \[Thm\_Realization\] is proved in Section \[Sec\_Thurston\], following a review of some aspects of Thurston theory. We also give an introduction to the combinatorics of [*arc systems*]{} and state a result by Kevin Pilgrim and Tan Lei that restricts the possibilities of how arc systems and Thurston obstructions can intersect.
Notation
--------
Let $f$ be a Newton map. A point $z \in \C$ is called a [*pole*]{} if $f(z)=\infty$ and a [*prepole*]{} if $f^{\circ k}(z)=\infty$ for some minimal $k>1$. If $g:\Cc \to \Cc$ is a branched covering map, we call a point $z \in \Cc$ a [*critical point*]{} if $g$ is not injective in any neighborhood of $z$. For the Newton map $f$, this is equivalent to saying that $z\in\C$ and $f'(z)=0$, because $\infty$ is never a critical point of $f$. It follows from the Riemann-Hurwitz formula [@Milnor Theorem 7.2] that a degree-$d$ branched covering map of $\Cc$ has exactly $2d-2$ critical points, counting multiplicities.
\[Def\_PostCritFixed\] Let $g:\Cc \to \Cc$ be a branched covering map of degree $d\geq 2$ with (not necessarily distinct) critical points $c_1,\dots,c_{2d-2}$. We denote the set of [*critical values*]{} of $g$ by $$\crit(g) :=\{g(c_1),\dots,g(c_{2d-2})\}\;.$$ Then, $g$ is called [*postcritically finite*]{} if the set $$P_g := \bigcup_{n\geq 0} g^{\circ n}(\crit(g))$$ is finite. We say that $g$ is [*postcritically fixed*]{} if there exists $N\in\N$ such that for each $i\in\{1,\dots,2d-2\}$, $g^{\circ N}(c_i)$ is a fixed point of $g$.
\[Def\_Access\] Let $U_{\xi}$ be the immediate basin of the fixed point $\xi \in \C$. Consider a curve $\Gamma:[0,\infty)\to U_{\xi}$ with $\Gamma(0) =\xi$ and $\lim_{t\to\infty}\Gamma(t)=\infty$. Its homotopy class within $U_{\xi}$ defines an [*access to $\infty$*]{} for $U_{\xi}$, in other words a curve $\Gamma'$ with the same properties lies in the same access as $\Gamma$ if the two curves are homotopic in $U_{\xi}$ with the endpoints fixed.
\[Prop\_Access\] ([*c.f. [@HSS]*]{}) Let $f$ be a Newton map of degree $d\geq 3$ and $U_\xi$ an immediate basin for $f$. Then, there exists $k\in \{1,\dots,d-1\}$ such that $U_\xi$ contains $k$ critical points of $f$ (counting multiplicities), $f|_{U_\xi}$ is a covering map of degree $k+1$, and $U_\xi$ has exactly $k$ accesses to $\infty$.
The Channel Diagram {#Sec_Models}
===================
In the following, by a [*(finite) graph*]{} we mean a connected topological space $\Gamma$ homeomorphic to the quotient of a finite disjoint union of closed arcs by an equivalence relation on the set of their endpoints. The arcs are called [*edges*]{} of the graph, an equivalence class of endpoints a [*vertex*]{}.
Let $\Gamma_1,\Gamma_2\subset\S^2$ be two finite graphs and $g:\Gamma_1\to\Gamma_2$ continuous. We call $g$ a [*graph map*]{} if it is injective on each edge of $\Gamma_1$ and forward and inverse images of vertices are vertices. If the graph map $g$ is a homeomorphism, then we call it a [*graph homeomorphism*]{}.
In the following, the closure and boundary operators will be understood with respect to the topology of $\Cc$, unless otherwise stated. Also, we will say that a set $X\subset\Cc$ is [*bounded*]{} if $\infty\not\in\ol{X}$.
We say that a Newton map $f$ of degree $d\geq 3$ *satisfies* if it has the following property:
$$\tag{$\star$}
\label{Prop_PCF}
\left\{
\begin{aligned}
& \text{if } c \text{ is a critical point of } f \text{ with } c\in B_{\xi_j} \text{ for some } \\
& j\in\{1,\dots,d\}, \text{ then } c \text{ has finite orbit.}
\end{aligned}
\right.$$
We omit the easy proof of the following well-known fact.
\[Lem\_OnlyCritical\]
Let $f$ be a Newton map that satisfies and let $\xi\in\C$ be a fixed point of $f$ with immediate basin $U_{\xi}$. Then, $\xi$ is the only critical point in $U_{\xi}$.
For the following construction let us assume that $f$ satisfies $\eqref{Prop_PCF}$. It follows that each immediate basin $U_{\xi}$ has a global Böttcher map $\phi_\xi: (\disk,0)\to (U_\xi,\xi)$ with the property that $f(\phi_\xi(z)) = \phi_\xi(z^{k_\xi})$ for each $z\in
\disk$, where $k_\xi-1\geq 1$ is the multiplicity of $\xi$ as a critical point of $f$ [@Milnor Theorems 9.1 & 9.3]. The $k_\xi-1$ radial lines (or [*internal rays*]{}) in $\disk$ which are fixed under $z\mapsto z^{k_\xi}$ map under $\phi$ to $k_\xi-1$ pairwise disjoint, non-homotopic injective curves $\Gamma_{\xi}^1,\dots,\Gamma_{\xi}^{k_\xi-1}$ in $U_{\xi}$ that connect $\xi$ to $\infty$ and are each invariant under $f$. They represent all accesses to $\infty$ of $U_{\xi}$, see Proposition \[Prop\_Access\]. Hence if $\xi_1,\dots,\xi_d\in\C$ are the attracting fixed points of $f$, then the union $$\Delta := \bigcup_{i=1}^d\bigcup_{j=1}^{k_{\xi_i}-1} \ol{\Gamma_{\xi_i}^j}$$
of these invariant curves over all immediate basins forms a connected and $f$-invariant graph in $\Cc$ with vertices at the $\xi_i$ and at $\infty$. We call $\Delta$ the [*channel diagram*]{} of $f$. The channel diagram records the mutual locations of the immediate basins of $f$ and provides a first-level combinatorial structure to the dynamical plane. Figure \[Fig\_Degree6\] shows a Newton map and its channel diagram. The following definition is an axiomatization of the channel diagram.
(10,10)
The following useful observation is a special case of [@RS Theorem 4.8].
\[Lem\_Lefschetz\] Let $f$ be a Newton map and let $D\subset\Cc$ be a closed topological disk such that $\gamma:=f(\partial D)$ is a simple closed curve with the property that $\gamma\cap\mathring{D}=\emptyset$. Let $V$ be the unique component of $\Cc\setminus\gamma$ that contains $\mathring{D}$ and let $\{\gamma'_i\}_{i\in I}$ be the collection of boundary components of $f^{-1}(V)\cap D$. Then, the number of fixed points of $f$ in $D$ equals $$\sum_{i\in I} \left|\deg(f|_{\gamma'_i}:\gamma'_i\to\gamma)\right|\;.$$ In particular, if $f^{-1}(V)\cap D\neq\emptyset$, then $D$ contains a fixed point.
Since $f$ has no parabolic fixed points, we do not need to take multiplicities of fixed points into account.
Note also that the $\gamma'_i$ are exactly the components of $f^{-1}(\gamma)\cap D$, except possibly $\partial D$ itself. The boundary is excluded if points in $\mathring{D}$ near $\partial D$ are mapped out of $\ol{V}$. By the lemma, the only case in which $D$ does not contain a fixed point of $f$ is if all of $\mathring{D}$ is mapped outside of $\ol{V}$.
The following theorem shows a relation between poles and fixed points outside immediate basins. It considerably sharpens [@RS Corollary 5.2], which states that for an immediate basin $U_{\xi}$ of a Newton map, every component of $\C\setminus U_{\xi}$ contains at least one fixed point.
\[Thm\_FixedPoles\] Let $f$ be a Newton map and $U_\xi$ an immediate basin. If $W$ is a component of $\C\setminus U_\xi$, then the number of fixed points in $W$ equals the number of poles in $W$, counting multiplicities.
Let $d\geq 3$ be the degree of $f$. If $U_\xi$ does not separate the plane, i.e. if it has only one access to $\infty$, then the claim follows trivially: $W$ contains all $d-1$ finite poles and the $d-1$ other finite fixed points of $f$. So suppose in the following that there is a Riemann map $\phi:(\disk,0)\to (U_\xi,\xi)$ with $f(\phi(z))=\phi(g(z))$ for some Blaschke product $g:\disk\to\disk$ of degree $k:=\deg(f|_{U_{\xi}}:U_{\xi}\to U_{\xi})\geq 3$.
(9,5.2) (0,0) (3.9,3.1)[$\phi$]{} (1.6,2.3)[$0$]{} (0.8,3.4)[$\gamma$]{} (8.4,1.9)[$U_{\xi}$]{} (7.5,4)[$\Gamma$]{} (6.6,2.2)[$\xi$]{} (3.2,2.4)[$\zeta_1$]{} (0,1.6)[$\zeta_3$]{} (0.6,4.3)[$\zeta_2$]{} (5.6,4.5)[$W$]{} (2.5,4.1)[$I$]{} (4.6,2.1)[$p_1$]{} (5.8,4)[$p_2$]{} (7,4.4)[$Y$]{} (4.7,3.7)[$Y'$]{} (5.3,3.4)[$X$]{} (4.9,2.7)[$p_3$]{}
We may extend $g$ by reflection to a rational function $\hat{g}:\Cc\to\Cc$ of degree $k$ whose Julia set equals $\S^1$ and that has $k-1$ fixed points $\zeta_1,\dots,\zeta_{k-1}\in \S^1$. These fixed points correspond to the accesses to $\infty$ of $U_{\xi}$. Since $\hat{g}$ fixes $\disk$, the $\zeta_i$ have real positive multipliers and since $0$ and $\infty$ attract all of $\disk$ and of $\Cc\setminus\ol{\disk}$, respectively, none of the $\zeta_i$ can be attracting or parabolic. Hence they are pairwise distinct and repelling. For each $\zeta_i$, choose a linearizing neighborhood and choose $0<\rho<1$ large enough so that all critical values of $\hat{g}$ in $\disk$ have absolute value less than $\rho$ and so that the linearizing neighborhoods of all $\zeta_i$ intersect the circle at radius $\rho$. Let $\gamma\subset\disk$ be the unique curve with the following properties, see Figure \[Fig\_LefschetzCurve\]: there are adjacent fixed points $\zeta_j,\zeta_{j+1}\in\S^1$ and injective curves $\gamma_j$, $\gamma_{j+1}$, so that $\gamma_j$ connects $\zeta_j$ to the circle at radius $\rho$ and $\gamma_j$ is a straight line segment in linearizing coordinates of $\zeta_j$; the same for $\gamma_{j+1}$ and $\zeta_{j+1}$. Their closures separate the circle at radius $\rho$ into two arcs. Of those arcs, let $\gamma'$ be the one for which $\gamma:=\gamma_j\cup \ol{\gamma'}\cup \gamma_{j+1}$ has the property that $\phi(\gamma)\subset U_{\xi}$ separates $W$ from $\xi $. Let $\Gamma:=\phi(\gamma)\cup\{\infty\}$. Then, $\Gamma$ is a simple closed curve in $\ol{U}_{\xi}$ and contains no critical values, except possibly $\infty$.
Let us first suppose that $\infty$ is not a critical value. Then, every component $\Gamma'_i$ of $f^{-1}(\Gamma)$ is a simple closed curve and $\deg(f|_{\Gamma'_i}:\Gamma'_i\to\Gamma)$ equals the number of poles on $\Gamma'_i$ (here, we do not need to count multiplicities, because we have assumed that $f$ has no critical poles).
Let $\Gamma_1'$ be the component of $f^{-1}(\Gamma)$ containing $\infty$. We claim that $\Gamma'_1\cap U_{\xi}$ consists of two connected components, each of which is an injective curve that connects $\infty$ to a pole on $\partial U_{\xi}$; call these poles $p_1$ and $p_2$. Indeed, consider the situation in $\disk$-coordinates. Let $I\subset\S^1$ be the arc between $\zeta_j$ and $\zeta_{j+1}$ that is separated from $0$ by $\gamma$. Since $\mathring{I}$ contains no fixed points of $\hat{g}$, $\hat{g}(\ol{I})$ covers $\S^1\setminus \ol{I}$ exactly once and $\ol{I}$ itself exactly twice. Hence, it is easy to see that $\hat{g}^{-1}(\gamma)$ has exactly two connected components that intersect $\gamma$. This proves the claim.
Let $Y$ be the closure of the component of $\Cc\setminus \Gamma$ that contains $W$ and let $Y'$ be the closure of the component of $\Cc\setminus\Gamma'_1$ that intersects $W$ in an unbounded set. We distinguish two cases.
If $p_1=p_2$, then $\Gamma'_1\subset \ol{U}_{\xi}$ and $W\subset Y'$. Moreover, $Y'$ is a closed topological disk that contains exactly the same fixed points and poles of $f$ as $W$. Since $f(\partial Y')\cap \mathring{Y}'=\emptyset$, Lemma \[Lem\_Lefschetz\] gives that the number of fixed points in $Y'$ (including $\infty$) equals the number of poles (again including $\infty$), because on every component of $f^{-1}(\Gamma)$ in $Y'$, the degree of $f$ equals the number of poles it contains. Excluding $\infty$ again, the claim follows.
If $p_1\neq p_2$, then $\Gamma'_1\not\subset \ol{U}_{\xi}$ and $W\not\subset Y'$ (this is the situation pictured in Figure \[Fig\_LefschetzCurve\]). If the set $X:=W\setminus Y'$ contains neither poles nor fixed points of $f$, then we can proceed as before.
Indeed, $\Gamma'_1\cup U_{\xi}$ separates $X$ from $\infty$ and since every fixed point of $f$ is surrounded by its unbounded immediate basin, $X$ cannot contain a fixed point. Now suppose by way of contradiction that $X$ contains a pole $p_3$ of $f$. If $p_3\in \partial X$, then $p_3\in\partial U_{\xi}$. But this would imply the existence of an additional pre-fixed point of $\hat{g}$ on $I$, a contradiction. The other case is that $p_3\in\mathring{X}$. Then, there exists a component $\Gamma'_2$ of $f^{-1}(\Gamma)$ in $\mathring{X}$. Let $D$ be the bounded disk bounded by $\Gamma'_2$. We may assume without loss of generality that there is no component of $f^{-1}(\Gamma)$ separating $D$ from $\partial X$. Since points in $Y'$ near $\Gamma_1'$ are mapped into $Y$ under $f$, points in $X$ near $\Gamma_1'$ are mapped out of $Y$, and it follows that again $f(D)\cap Y\neq\emptyset$. Now, $\ol{D}$ contains a fixed point by Lemma \[Lem\_Lefschetz\]. This is a contradiction.
In the remaining case that $\infty$ is a critical value, we perturb $f$ slightly to avoid that situation. Since poles and fixed points of $f$ move continuously under perturbation, and $\Gamma$ does too, this does not change the count. Note that while $U_{\xi}$ and $W$ might move discontinuously, this does not pose a problem because we have actually counted poles and fixed points in $Y'$, after having established that we do not lose anything by this replacement.
\[Cor\_FixedComplement\] Let $f$ be a Newton map that satisfies and $\Delta$ be the channel diagram of $f$. For a component $V$ of $\Cc\setminus\Delta$ let $p$ be the number of poles of $f$ in $V$, counting multiplicities. Then $\partial V \cap \C$ contains $p+1$ fixed points.
If $V$ is the only component of $\Cc\setminus\Delta$, the claim follows trivially. If $\xi$ is the only fixed point on $\partial V$ whose immediate basin $U_\xi$ separates the plane, then the claim follows directly from Theorem \[Thm\_FixedPoles\]. Indeed, let in this case $R_1,R_2$ be the fixed internal rays of $U_\xi$ that are on $\partial V$ and let $V_1$ be the component of $\C\setminus (R_1\cup R_2\cup\{\xi\})$ such that $V\subset V_1$. Then, $V_1$ also contains $p$ poles and by Theorem \[Thm\_FixedPoles\], $V_1$ contains $p$ fixed points. Since $\xi\in\partial V$ as well, the claim follows.
(6,2.8) (0,0) (0.9,0.2)[$V$]{} (0.7,1)[$\xi_1$]{} (2.8,0.9)[$\xi_2$]{} (3.8,1.4)[$\xi_3$]{} (2,1.5)[$\xi_4$]{} (1.9,2.3)[$V_4^1$]{} (5.3,2)[$V_3^1$]{} (4.5,2.4)[$V_3^2$]{} (3.4,0.1)[$V_2^1$]{} (2.3,0.1)[$V_2^2$]{}
Now suppose that $\xi_1,\dots,\xi_k\in \partial V$ are the fixed points on $\partial V$ whose immediate basins separate the plane. Let $R_1, R_2$ be the fixed internal rays of $U_{\xi_1}$ on $\partial V$ and let $V_1$ be as above. Let $m$ be the number of poles in $V_1$. As before, it follows that $V_1$ contains $m$ fixed points. Let $m'=m-p$. For $j=2,\dots,k$, denote by $V_j^1,\dots,V_j^{i_j}$ all complementary components of the closures of the fixed internal rays of $U_{\xi_j}$ that do not contain $V$. By Theorem \[Thm\_FixedPoles\], each $V_j^i$ contains as many poles as fixed points, hence all $V_j^i$ combined contain $m'$ poles and $m'$ fixed points. Hence, $V_1\setminus(\bigcup_{j=2}^k\bigcup_{\ell=1}^{i_j} V_j^{\ell})$ contains $m-m'=p$ fixed points. The claim now follows, because including $\xi_1$, $\partial V\cap \C$ contains $p+1$ fixed points.
\[Cor\_SharedPoles\] Let $f$ be a Newton map and $\Delta$ be the channel diagram of $f$. If $V$ is a component of $\Cc\setminus\Delta$, then there is at least one pair of fixed points $\xi_1,\xi_2\in\partial V\cap\C$ such that $\partial U_{\xi_1}$ and $\partial U_{\xi_2}$ intersect in a pole.
Let $U_{\xi}$ be an immediate basin. Clearly, the components of $\partial U_{\xi}\cap\C$ are separated by the accesses to $\infty$. If $\partial U_{\xi}\cap\C$ has only one component, then in the conjugate dynamics $\hat{g}|_{\disk}$ we have $\hat{g}(\S^1)=\S^1$ therefore $S^1$ contains pre-fixed points of $\hat{g}$. In case $\partial U_{\xi} \cap \mathcal{C}$ consists of more than one component we have seen that in the conjugate dynamics $\hat{g}|_{\disk}$, for every arc $I\subset\S^1$ between two fixed points, $\hat{g}(I)=\S^1$. Therefore, $I$ contains pre-fixed points of $\hat{g}$. Since poles and $\infty$ are accessible boundary points of $U_{\xi}$, we conclude that every component of $\partial U_{\xi}\cap\C$ contains at least one pole. By Corollary \[Cor\_FixedComplement\], there has to be at least one pole in $V$ that is on the boundary of at least two immediate basins.
Figure \[Fig\_Degree6\] shows that a component of $\partial U_{\xi}\cap \C$ may contain more than one pole.
Note also that a simple pole is on the boundary of at most two immediate basins, because otherwise $f$ cannot preserve the cyclic order of the immediate basins near that pole. This was first observed by Janet Head [@Head].
Proof of Theorem \[Thm\_PreimagesConnected\] {#Sec_Proof}
============================================
Let $\xi_1,\dots,\xi_d \in \C$ be the finite fixed points of $f$ and let $U'_0$ be a component of some $B_{\xi_i}$. Let $\Delta$ be the channel diagram of $f$. Recall that it consists of invariant rays within the $U_{\xi_i}$ that connect the $\xi_i$ to $\infty$. Denote by $\Delta_n$ the connected component of $f^{-n}(\Delta)$ that contains $\Delta$ (with this convention, $\Delta=\Delta_0$). Every edge of $\Delta_n$ is then an internal ray of a component of some $B_{\xi_i}$, while every vertex is a preimage of a $\xi_i$, or a pole or prepole or $\infty$.
To prove Theorem \[Thm\_PreimagesConnected\], it suffices to show that there exists $n\in\N$ such that $\Delta_n$ contains all poles of $f$: then, every pole of $f$ can be connected to $\infty$ through a finite chain of internal rays in the $B_{\xi_i}$, and hence through a finite chain of components of the basins. By induction, each prepole is in $\Delta_m$ for sufficiently large $m$. Since $\infty$ is on the boundary of every immediate basin, $\partial U'_0$ contains a prepole. This finishes the argument.
It remains to show that there exists $n\in\N$ such that $\Delta_n$ contains all poles of $f$. If $\Delta_1$ contains all poles of $f$, then we are done. So assume in the following that there exists a component $C_1$ of $f^{-1}(\Delta)$ such that $C_1\cap\Delta_1=\emptyset$ (Figure \[Fig\_Degree6\] shows that this does occur). Equivalently, we may assume that there exists a component $V_0$ of $\Cc\setminus\Delta$ and a component $V_1$ of $f^{-1}(V_0)$ such that $V_1$ is multiply connected. Then, we choose $C_1$ so that it intersects $\partial V_1$.
Denote by $C_n$ the component of $f^{-n}(\Delta)$ containing $C_1$. We will assume that $C_n\cap\Delta_n=\emptyset$ for all $n\in\N$ (otherwise we would be done). We will lead this assumption to a contradiction.
\[Lem\_Inside\] With the above notation, $V_1\subset V_0$.
Since $\Delta\subset f^{-1}(\Delta)$, we either have $V_1\subset
V_0$ or $V_1\cap V_0=\emptyset$. In the latter case, let $\gamma\subset V_0$ be a simple closed curve near $\partial V_0$ that surrounds all critical values within $V_0$ (note that $V_0$ contains critical values, because it is the image of the multiply connected domain $V_1$). Then, $f^{-1}(\gamma)\cap V_1$ consists of several nested and non-contractible (in $V_1$) simple closed curves (at least two, since otherwise $V_0$ would be a homeomorphic image of $V_1$ and hence multiply connected). Let $\gamma'$ be the outermost of them, $D$ be the bounded component of $\Cc\setminus \gamma'$ and $V$ the complement component of $\Cc \setminus \gamma$ containing $D$. The choice of $\gamma'$ being the outermost preimage of $\gamma$ in $V_1$ implies that $f^{-1}(V) \cap D \neq \emptyset$. Hence by Lemma \[Lem\_Lefschetz\] we obtain that $\ol{D}$ contains a fixed point of $f$. This is a contradiction, because $\ol{D}$ is separated from $\Delta$ and all fixed points are contained in $\Delta$.
\[Lem\_V1\] Let $n\in\N$ and suppose that $W$ is an unbounded component of $\Cc\setminus\Delta_n$. If $W'$ is a component of $f^{-1}(W)$ with $W'\subset W$, then $W'$ is unbounded.
Let $\xi_1,\dots,\xi_k$ be the attracting fixed points of $f$ in $\partial W$. By Lemma \[Lem\_OnlyCritical\], $f$ has Böttcher coordinates near each $\xi_i$. For $i\in\{1,\dots, k\}$, choose a neighborhood $B_i$ of $\xi_i$ that has the following properties: $B_i$ contains no critical values except $\xi_i$; in Böttcher coordinates, $B_i$ is a round disk centered at $0$, small enough so that $\Delta_{n+1}\cap B_i$ consists of radial lines; $f$ is conjugate to $z\mapsto z^{k_i}$ on $B_i$. Since $\Delta_n$ is a graph, there exists a simple closed curve $\gamma\subset \partial
W\subset \Delta_n$ that surrounds $W$ and thus $W'$. By possibly modifying $\gamma$ within the $B_i$, we may assume that in Böttcher coordinates, every point $z\in \gamma\cap B_i$ is either in $\Delta_n$ or on a circle of constant radius centered at $0$. Let $D$ be the component of $\Cc\setminus \gamma$ that intersects $W$. Then, $\overline{D} \cap \gamma \subset \bigcup_{i=1}^k B_i$ .
Now suppose by way of contradiction that $W'\subset W$ is bounded and let $\gamma'$ be the outermost (in $\C$) simple closed curve in $f^{-1}(\gamma)$ that intersects $\partial W'$. Let $D'$ be the component of $\C\setminus\gamma'$ that is contained in $D$. Observe that $D'\cap\gamma=\emptyset$: $\gamma$ intersects $D'$ at most in the $B_i$. But there, $\gamma$ was chosen in such a way that $\gamma'$ is strictly further away from $\xi$ than $\gamma$, so in particular, $\gamma$ and $D'$ are disjoint. Now, either $f(D')\subset D$ or $\ol{D}'$ contains another component of $f^{-1}(\gamma)$. In both cases, Lemma \[Lem\_Lefschetz\] shows that $\ol{D}'$ contains a fixed point of $f$. This is a contradiction, because $\gamma$ was constructed in such a way that $\ol{D}'$ does not contain an attracting fixed point, while $\infty$ is not in $\ol{D}'$ by assumption.
\[Cor\_FreeNest\] Suppose that $\Delta_n\cap C_n=\emptyset$ for all $n\in\N$. Then for each $n\geq 1$, there exists an unbounded and multiply connected component $V_n$ of $f^{-n}(V_0)$ such that $V_n\subset V_{n-1}$ and $C_n\cap\partial V_n\neq\emptyset$, while the component of $\partial V_n$ that contains $\infty$ is in $\Delta_n$.
By definition, $V_1$ is multiply connected and $\partial V_1\cap C_1\neq\emptyset$. By Lemma \[Lem\_Inside\], $V_1\subset V_0$ and by Lemma \[Lem\_V1\], $V_1$ is unbounded.
Now suppose by induction that $V_{n}$ has the claimed properties. Clearly, every component of $f^{-1}(V_n)$ is multiply connected. Since $\infty \in \partial V_n$, there exists a component $V_{n+1}$ of $f^{-1}(V_n)$ such that $\partial V_{n+1}\cap
C_{n+1}\neq\emptyset$, since $C_1 \subset C_{n+1}$ and $C_1$ contains a pole by construction. Using the same argument as in Lemma \[Lem\_Inside\] we obtain that $V_{n+1} \subset V_n$.
Let $W$ be the component of $\Cc\setminus \Delta_n$ containing $V_n$ and let $W'$ be the component of $f^{-1}(W)$ containing $V_{n+1}$. If $V_{n+1}$ was bounded, then so would be $W'$, contradicting Lemma \[Lem\_V1\]. Since $\Delta_{n+1}$ is the only unbounded component of $f^{-(n+1)}(\Delta)$, the component of $\partial V_{n+1}$ that contains $\infty$ is in $\Delta_{n+1}$ and we are done.
We call the unbounded component of $\partial V_n$ the [*outer boundary*]{} and denote it with $B_n$.
Recall that to finish the proof of Theorem \[Thm\_PreimagesConnected\] in case of the finiteness condition , it suffices to show the following.
\[Thm\_PolesConnect\] There exists $n\in\N$ such that $\Delta_n$ contains all poles of $f$.
Suppose by way of contradiction that $C_n\cap \Delta_n=\emptyset$ for all $n\in\N$. Suppose first that for large enough $n$, $V_n$ surrounds only one pole, and this pole is in $C_n$.
For any $n\in\N$, we call a closed arc in $B_n$ a *bridge* if it connects two distinct finite fixed points $\xi_1$ and $\xi_2$ and contains no other fixed points, in particular not $\infty$. The *length* of a bridge is the number of edges of $\Delta_n$ it consists of. We say that two distinct finite fixed points $\xi_1, \xi_2\in B_n$ are *adjacent* if there exists a bridge in $B_n$ connecting them.
By Corollary \[Cor\_FreeNest\], all $V_n$ are unbounded and hence have at least one access to $\infty$. The number of fixed points in $\partial V_n$ is non-increasing: if there were a fixed point in $\partial V_n$, then it would also have been in its image $\partial
V_{n-1}$. Moreover, every unbounded edge of $B_n$ is contained in $\Delta$ and connects $\infty$ to a finite fixed point. It now follows from Definition \[Def\_ChannelDiagram\] , that each $B_n$ contains at least two finite fixed points. Hence, the number of fixed points and accesses in $\partial V_n$ cannot shrink infinitely often and there exists a minimal $n_0$ such that for $n\geq n_0$, $B_n$ and $B_{n_0}$ contain the same fixed points and $V_n$ and $V_{n_0}$ have the same accesses to $\infty$. Therefore, two distinct finite fixed points $\xi_1,\xi_2\in B_n$ are adjacent for $n\geq n_0$ if and only if they are adjacent in $B_{n_0}$: a bridge in $B_n$ between them that does not exist in $B_{n_0}$ would separate another fixed point on $B_{n_0}$ (or an access to $\infty$ of $V_{n_0}$) from $V_n$; the other direction is trivial.
\[Claim\_1\] If $\xi$ is a finite fixed point that is in $B_n$ for all $n\in\N$, then $\xi$ is adjacent to at most one finite fixed point.
Let $\phi:\disk\to U_{\xi}$ be a Böttcher map that conjugates $f$ to $g(z)=z^k$ for some $k\geq 2$ and let $R_1,\dots,
R_{k-1}\subset\disk$ be the fixed internal rays of $g$. For all $n\in\N$, $\phi^{-1}(V_n\cap U_{\xi})$ equals a sector $S_n$ of $\disk\setminus \bigcup_{i=1}^{k-1} \ol{g^{-n}(R_i)}$, such that $S_{n+1}\subset S_n$ and $g(S_{n+1})= S_n$. It is easy to see that this can only happen if one of the boundary arcs of all $S_n$ is a fixed ray. Indeed, the intersection $\bigcap_{n=1}^\infty \ol{S_n}$ must be a single ray $R$. Its image $g(R) \subset
\bigcap_{n=1}^\infty \ol{S_n}$ , hence $g(R)=R$. Since no fixed ray can be inside an $S_n$, it must be part of the boundary of all $S_n$.
It follows that $\xi$ and $\infty$ are connected in all $B_n$ by a fixed edge, and at most one other edge that ends at $\xi$ can be part of a bridge in $B_n$.
Observe also that $B_{n_0}$ contains a bridge: by minimality of $n_0$, there is a finite fixed point on $B_{n_0-1}$ or an access to $\infty$ of $V_{n_0-1}$ that is separated from $V_{n_0}$ by an arc in $B_{n_0}$. Possibly by extending this arc along $B_n$, we find a bridge $X_{n_0}\subset B_{n_0}$ between two finite fixed points (this is possible because the extension must hit a finite fixed point before $\infty$). If $n_0=1$, then $B_1$ contains a bridge $X_{n_0}$ between two finite fixed points by Corollary \[Cor\_SharedPoles\]. Let $\xi_1$ and $\xi_2$ be the endpoints of $X_{n_0}$.
By induction on $n\geq n_0$, let $X_{n+1}$ be the shortest arc in $f^{-1}(X_n)$ that ends at $\xi_1$ and connects $\xi_1$ to a point $\xi'_2\in f^{-1}(\{\xi_2\})$ within $B_{n+1}$. If $\xi'_2=\xi_2$, then $X_{n+1}$ is a bridge as well, otherwise we say that the bridge $X_{n_0}$ *disconnects* at time $n+1$.
Note that if $X_{n+1}$ is a bridge, then the length of $X_{n+1}$ can only be greater than the one of $X_{n}$ if $X_{n+1}$ contains a pole. Hence the length is constant for all sufficiently large $n$.
\[Claim\_2\] The bridge $X_{n_0}$ disconnects after finitely many pull-backs.
Suppose that $X_n$ is a bridge from $\xi_1$ to $\xi_2$ for all $n\geq n_0$. Near the endpoints $\xi_1$ and $\xi_2$, $X_{n}$ consists of internal rays $R^{n}_1\subset U_{\xi_1}$ and $R^{n}_2\subset U_{\xi_2}$, respectively. Hence, we can express $X_n$ as $R^n_1\cup R^n_2\cup X'_n$, where $X'_n$ consists of a bounded number of edges of $\Delta_n$ by the previous considerations. Considering the situation in Böttcher coordinates as in the previous claim, we see that $R^n_1$ converges to a fixed ray $R^0_1\subset U_{\xi_1}$, and $R^n_2$ converges to a fixed ray $R^0_2\subset U_{\xi_2}$. Let $r_n$ be the non-fixed endpoint of $R^n_1$. Then, $r_n\to\infty$ as $n\to\infty$.
Since $\infty$ is a repelling fixed point with multiplier $\lambda>1$, there exists a branched covering map $\psi:(\C,0)\to
(\Cc\setminus\{0\},\infty)$ such that $f(\psi(z))=\psi(\lambda z)$ for all $z\in\C$ (we may assume without loss of generality that $0\not\in \ol{V_0}$). Moreover, there exist a neighborhood $W$ of $\infty$ and a holomorphic branch $\phi$ of $\psi^{-1}$ on $W$ [@Milnor Corollary 8.10]. Let $N$ be sufficiently large so that for all $n\geq N$, $r_n\in W$ and $\ol{X}'_n$ contains no critical values of $f$. Then, $\ol{X}'_{n+1}$ is the lift of $\ol{X}'_n$ under the branch of $f^{-1}$ that maps $r_n$ to $r_{n+1}$. Observe that a point $z\in\C$ is a critical point of $\psi$ if and only if $f$ has a critical point at $f^{\circ j}(\psi(z/\lambda^m))$ for some $j\leq m$, where $m$ is chosen large enough so that $z/\lambda^m\in \phi(W)$. Therefore, none of the $X'_n$ for $n\geq
N$ contain a critical value of $\psi$ and we can pull back $X'_n$ under the branch of $\psi^{-1}$ that maps $r_n$ to $\phi(r_n)$. Since $f$ commutes with multiplication by $\lambda$, this pull-back operation commutes with division by $\lambda$ and it follows that $X'_n$ converges uniformly to $\infty$ as a set as $n\to\infty$. Therefore, $X_n\to \ol{R}^0_1\cup\ol{R}^0_2$ uniformly as a set as $n\to\infty$. (Note that since $X_n$ is connected by assumption, we could have as well argued starting with the endpoint of $R^n_2$.)
Let $U_n$ be an unbounded component of $\bar{C} \setminus
\overline{R^1_0 \cup R^2_0 \cup X_n}$, that contains $C_n$. Then $\bar{C} \setminus C_n$ has bounded components of some positive area, on other hand $area(U_n) \to 0$, because its boundary converges to an arc $R^1_0 \cup R^2_0$. This is a contradiction.
We can now finish the proof of Theorem \[Thm\_PolesConnect\].
If $X_{n_0}$ disconnects at time $n+1$, then consider its endpoint $\xi'_2\in f^{-1}(\{\xi_2\})$. If $\xi'_2\not\in \Delta_1$, then it must by assumption be in $C_1$. Thus, $X_{n+1}$ connects $C_{n}$ to $\Delta$. This means that $C_{n+1}\subset \Delta_{n+1}$ and we are done. If $\xi'_2\in \Delta_1$, we can extend $X_{n+1}$ to a bridge by connecting $\xi'_2$ to a pole $p \in V_n$ and to another fixed point $\xi_3$, different from $\xi_1$ (there always exists a bridge in $\partial{V_{n+1}}$ connecting two different fixed points) within $B_{n+1}$. Claim \[Claim\_1\] implies that $\xi_3\in\{\xi_1,\xi_2\}$, without loss of generality we assume that $\xi_3=\xi_2$.
But then we have found a new bridge between $\xi_1$ and $\xi_2$. According to Claim \[Claim\_2\] the new bridge also disconnects after taking some pullbacks. It follows from Claim \[Claim\_1\] that $V_{n+1} \cap U_{\xi_2} \subset V_n \cap U_{\xi_2}$, therefore each time the new constructed bridge connecting $\xi_1$ to $\xi_2$ in $\partial{V_n}$ disconnects we must encounter a new pole on the boundary of $V_n$. However, there are only finitely many poles of $f$ and we get a contradiction. Therefore at some level $n$, $\xi'_2
\in C_1$ and $X_n$ connects $C_n$ to $\Delta$.
If all $V_n$ surround several bounded components of $f^{-1}(\Delta)$, our arguments show that at least one of them is connected to $\Delta_k$ for some $k\in\N$. To finish the proof, it suffices to show that at time $k$, a new bridge is created that connects $\xi_1$ to some finite fixed point $\xi_3$. Then we can continue by induction. To see that this bridge exists, let $C_1$ be the component of $f^{-1}(\Delta)$ that was connected to $\Delta_k$ by $X_k$. Observe that $C_1$ also contains a preimage of $\xi_1$. We can extend the arc $X_k$ from $\xi_1$ to $\xi'_2\in C_1$ to this preimage of $\xi_1$ and then further to another preimage of $\xi_2$ etc., until we arrive at a finite fixed point, say $\xi_3$. As before, $\xi_3$ might equal $\xi_2$ if $X_k$ contains a multiple pole. The above arguments apply to all components of $\Cc\setminus\Delta$ that surround a pole.
This finishes the proof of Theorem \[Thm\_PreimagesConnected\] under the finiteness condition . In general, we can use a straightforward surgery construction to bring any Newton map into the desired form: all we require is finiteness of critical orbits in all basins of roots; there may well be infinite (or even periodic or preperiodic) critical orbits in the Julia set, or in attracting, parabolic, or Siegel components of the Fatou set.
Within any immediate basin $U_\xi$, we may replace the attracting dynamics (which may involve several critical points converging to the root $\xi$) by dynamics modeled after $z\mapsto z^k$ within the unit disk. This surgery procedure does not affect the correctness of Theorem \[Thm\_PreimagesConnected\]. A similar procedure can assure that all critical points in the entire basin $B_\xi$ of $\xi$ land on the fixed point $\xi$ after finitely many steps. Details are standard and thus omitted; compare for example Shishikura [@Shishikura]. This proves Theorem \[Thm\_PreimagesConnected\] in the general case.
The Newton Graph of a Newton Map {#Sec_Graph}
================================
In this section, we define abstract Newton graphs and use Theorem \[Thm\_PreimagesConnected\] to show that every postcritically fixed Newton map $f$ generates a unique abstract Newton graph in a natural way. We make some references to Thurston theory, which is discussed in more detail in Section \[Sec\_Thurston\].
Extending Maps on Finite Graphs
-------------------------------
The channel diagram motivates the definition of a Newton graph. For this, we first need to introduce some notation regarding maps on imbedded graphs and their extensions to $\S^2$, compare [@BFH Chapter 6]. We assume in the following without explicit mention that all graphs are imbedded into $\S^2$.
\[Def\_GraphMap\] Let $g:\Gamma_1\to\Gamma_2$ be a graph map. An orientation-preserving branched covering map $\ol{g}:\S^2\to\S^2$ is called a [*regular extension*]{} of $g$ if $\ol{g}|_{\Gamma_1}=g$ and $\ol{g}$ is injective on each component of $\S^2\setminus \Gamma_1$.
\[Lem\_IsotopGraphMaps\] [[@BFH Corollary 6.3]]{} Let $g,h:\Gamma_1\to\Gamma_2$ be two graph maps that coincide on the vertices of $\Gamma_1$ such that if $\gamma\subset\Gamma_1$ is an edge, then $g(\gamma)=h(\gamma)$ as a set. Suppose that $g$ and $h$ have regular extensions $\ol{g},\ol{h}:\S^2\to\S^2$. Then there exists a homeomorphism $\psi:\S^2\to\S^2$, isotopic to the identity relative the vertices of $\Gamma_1$, such that $\ol{g}=\ol{h}\circ \psi$.
Let $g:\Gamma_1\to\Gamma_2$ be a graph map. For the next proposition, we will assume without loss of generality that each vertex $v$ of $\Gamma_1$ has a neighborhood $U_v\subset\S^2$ such that all edges of $\Gamma_1$ that enter $U_v$ terminate at $v$; we may also assume that $U_v$ is a round disk of radius $r$ centered at $v$, where $r$ is chosen sufficiently small so that $U_{v_1}$ and $U_{v_2}$ are disjoint for different vertices $v_1 \neq v_2$, all edges entering $U_v$ are straight in $U_v$ and that $g|_{U_v}$ is length-preserving. We make analogous assumptions for $\Gamma_2$. Then, we can extend $g$ to each $U_v$ as in [@BFH]: for a vertex $v\in\Gamma_1$, let $\gamma_1$ and $\gamma_2$ be two adjacent edges ending there. In polar coordinates centered at $v$, these are straight lines with arguments, say, $\theta_1,\theta_2$ such that $0<\theta_2-\theta_1\leq 2\pi$ (if $v$ is an endpoint of $\Gamma_1$, then set $\theta_1=0$, $\theta_2=2\pi$). In the same way, choose arguments $\theta_1',\theta_2'$ for the image edges in $U_{g(v)}$ and extend $g$ to a map $\tilde{g}$ on $\Gamma_1\cup\bigcup_v U_v$ by setting $$(\rho,\theta)\mapsto \left(\rho, \frac{\theta_2'-\theta_1'}{\theta_2-\theta_1}\cdot\theta\right)\;,$$ where $(\rho,\theta)$ are polar coordinates in the sector bounded by the rays at $\theta_1$ and $\theta_2$. In other words, sectors are mapped onto sectors in an orientation-preserving way. Then, the following holds.
\[Prop\_RegExt\] [*[@BFH Proposition 6.4]*]{} The map $g:\Gamma_1\to\Gamma_2$ has a regular extension if and only if for every vertex $y\in\Gamma_2$ and every component $U$ of $\S^2\setminus\Gamma_1$, the extension $\tilde{g}$ is injective on $$\bigcup_{v\in g^{-1}(\{y\})} U_v \cap U\;.$$ In this case, the regular extension $\ol{g}$ may have critical points only at the vertices of $\Gamma_1$.
The Newton Graph
----------------
Let us first define an abstract channel diagram which plays the basement role for the main concept of an abstract Newton graph.
\[Def\_ChannelDiagram\] An [*abstract channel diagram*]{} of degree $d\geq 3$ is a graph $\Delta \subset \S^2$ with vertices $v_0,\dots,v_d$ and edges $e_1,\dots,e_l$ that satisfies the following properties:
1. $l\leq 2d-2$;
2. each edge joins $v_0$ to a $v_i$, $i>0$;
3. each $v_i$ is connected to $v_0$ by at least one edge;
4. \[necessaryCondition\] if $e_i$ and $e_j$ both join $v_0$ to $v_k$, then each connected component of $\S^2\setminus \ol{e_i\cup e_j}$ contains at least one vertex of $\Delta$.
We say that an abstract channel diagram $\Delta$ is [*realized*]{} if there exist a Newton map with channel diagram $\hat{\Delta}$ and a graph homeomorphism $h:\Delta\to\hat{\Delta}$ that preserves the cyclic order of edges at each vertex.
The channel diagram $\hat{\Delta}$ of a Newton map $f$, constructed in section \[Sec\_Models\] is an abstract channel diagram.
By construction, $\hat{\Delta}$ has at most $2d-2$ edges and it satisfies the properties (2) and (3) of the definition \[Def\_ChannelDiagram\]. Finally, $\hat{\Delta}$ satisfies (\[necessaryCondition\]), because for any immediate basin $U_{\xi}$ of $f$, every component of $\C\setminus U_\xi$ contains at least one fixed point of $f$ [@RS Corollary 5.2] (see also Theorem \[Thm\_FixedPoles\]).
Now, with these preparations, we are ready to introduce the concept of an abstract Newton graph. It turns out that it carries enough information to uniquely characterize postcritically fixed Newton maps.
\[Def\_NewtonGraph\] Let $\Gamma\subset\S^2$ be a connected graph, $\Gamma'$ the set of its vertices and $g:\Gamma\to\Gamma$ a graph map. The pair $(\Gamma,g)$ is called an [*abstract Newton graph*]{} if it satisfies the following conditions:
1. \[Cond\_Diagram\] There exists $d_{\Gamma}\geq 3$ and an abstract channel diagram $\Delta\subsetneq\Gamma$ of degree $d_\Gamma$ such that $g$ fixes each vertex and each edge of $\Delta$.
2. \[Cond\_Branch\] If $v_0,\dots,v_{d_\Gamma}$ are the vertices of $\Delta$, then $v_i\in \ol{\Gamma\setminus\Delta}$ if and only if $i\neq 0$. Moreover, there are exactly $\deg_{v_i}(g)-1\geq 1$ edges in $\Delta$ that connect $v_i$ to $v_0$ for $i\neq 0$, where $\deg_x(g)$ denotes the local degree of $g$ at $x\in\Gamma'$.
3. \[Cond\_Degree\] $\sum_{x\in\Gamma'} \left(\deg_x(g)-1\right) = 2d_{\Gamma}-2$.
4. \[Cond\_Enter\] There exists $N_{\Gamma}\in\N$ such that $g^{\circ {N_\Gamma}}(\Gamma)\subset\Delta$, where $N_\Gamma$ is minimal such that $g^{\circ (N_{\Gamma}-1)}(x)\in\Delta$ for all $x\in\Gamma'$ with $\deg_x(g)>1$.
5. \[Cond\_Connected\] The graph $\ol{\Gamma\setminus\Delta}$ is connected.
6. \[Cond\_Extension\] For every vertex $y\in\Gamma'$ and every component $U$ of $\S^2\setminus\Gamma$, the extension $\tilde{g}$ is injective on $$\bigcup_{v\in g^{-1}(\{y\})} U_v \cap U\;.$$
7. \[Cond\_Saturated\] $\Gamma$ equals the component of $\ol{g}^{-N_{\Gamma}}(\Delta)$ that contains $\Delta$.
If $(\Gamma,g)$ is an abstract Newton graph, $g$ can be extended to a branched covering map $\ol{g}:\S^2\to\S^2$ by (\[Cond\_Extension\]) and Proposition \[Prop\_RegExt\]. We use this implicitly in (\[Cond\_Saturated\]). Condition (\[Cond\_Degree\]) and the Riemann-Hurwitz formula ensure that $\ol{g}$ has degree $d_{\Gamma}$. An immediate consequence of Lemma \[Lem\_IsotopGraphMaps\] is that $\ol{g}$ is unique up to Thurston equivalence.
We say that two abstract Newton graphs $(\Gamma_1,g_1)$ and $(\Gamma_2,g_2)$ are [*equivalent*]{} if there exists a graph homeomorphism $h:\Gamma_1\to\Gamma_2$ that preserves the cyclic order of edges at each vertex of $\Gamma_1$ and conjugates $g_1$ to $g_2$.
Now we are ready to prove our first main result. Recall that for a Newton map $f$ with channel diagram $\Delta$, $\Delta_n$ denotes the component of $f^{-n}(\Delta)$ that contains $\Delta$.
Let $\Delta$ be the channel diagram of $f$. First observe that $\Delta$ connects every fixed point of $f$ to $\infty$. Since $f$ is postcritically fixed, each critical point of $f$ is connected to some prepole by an iterated preimage of $\Delta$. Theorem \[Thm\_PolesConnect\] shows that there exists $n\in\N$ such that $\Delta_n$ contains all poles of $f$. Since $f$ is postcritically fixed, it follows by induction that there exists a minimal $N\in\N$ such that $\Delta_{N-1}$ contains all critical points of $f$.
It is easy to see that $(\Delta_N,f)$ satisfies all conditions of Definition \[Def\_NewtonGraph\] except possibly (\[Cond\_Connected\]). Note that if all critical points are in $\Delta_{N-1}$, we need to pull back one more step to ensure that condition (\[Cond\_Degree\]) is satisfied.
To show (\[Cond\_Connected\]), suppose by way of contradiction that the bounded set $\ol{\Delta_{N-1}\setminus\Delta}$ is not connected. Then, there exists an unbounded component $V$ of $\Cc\setminus\Delta_{N-1}$ that separates the plane, i.e. $V$ has at least two accesses to $\infty$. Let $W$ be a neighborhood of $\infty$ that is a round disk in linearizing coordinates and satisfies $W\cap \Delta_{N-1}\subset \Delta$. Let $V_1,\dots,V_k$ be the components of $V\cap W$. Then, $f$ acts injectively on each $V_i$ and there exists a branch $g_i$ of $f^{-1}$ that maps $V_i$ into itself (recall that $\infty$ is a repelling fixed point of $f$, so it is attracting for the $g_i$). By assumption, $V$ is simply connected and contains no critical values of $f$, so the $g_i$ extend to all of $V$ by holomorphic continuation on lines. Since $\Delta_{N-1}\subset\Delta_{N}$, we get $g_i(V)\subset V$ for all $i$. If there are $i\neq j$ such that $g_i(V)\cap
g_j(V)\neq\emptyset$, then it follows that $g_i=g_j$ and we have found a holomorphic self-map of $V$ for which $\infty\in \partial V$ is attracting through two distinct accesses, contradicting the Denjoy-Wolff theorem [@Milnor Theorem 5.4].
Hence, the $g_i(V)$ are pairwise disjoint and if $w\in \partial
g_i(V)$ for some $i$, then $f(w)\in\partial V$, for otherwise the map $g_i$ would be defined in a neighborhood of $f(w)$. Hence $w\in
f^{-1}(\Delta_{N-1})$ and since all $g_i(V)$ are open disks, we even get that $w\in \Delta_N$. It follows that no component of $\Cc\setminus \Delta_N$ has more than one access to $\infty$, and $\ol{\Delta_N\setminus\Delta}$ is connected.
To prove the last claim, suppose that there exists a graph homeomorphism $h:\Delta_{1,N}\to \Delta_{2,N}$ such that $h(f_1(z)) = f_2(h(z))$ for all $z\in \Delta_{1,N}$. Since all complementary components of $\Delta_{1,N}$ are disks, we can extend $h$ to a homeomorphism $\ol{h}:\S^2\to\S^2$ that conjugates $f_1$ to $f_2$ up to isotopy relative $\Delta_{1,N}$ [@BFH Chapter 6]. By Theorem \[Thm\_Thurston\], $f_1$ and $f_2$ are conjugate by a Möbius transformation that fixes $\infty$.
Abstract Newton Graphs Are Realized {#Sec_Thurston}
===================================
In this section, we recall some fundamental notions of Thurston’s characterization of rational maps. Then, we use Thurston’s theorem to show that every abstract Newton graph is realized by a postcritically fixed Newton map, which is unique up to affine conjugation.
Thurston’s Criterion For Marked Branched Coverings
--------------------------------------------------
Thurston’s theorem provides a necessary and sufficient condition on the existence of a rational map with certain combinatorial behavior in terms of linear maps generated from a (potentially very large) collection of simple closed curves.
-1 The notations and results in this section are based on [@DH] and [@PT]. Before we can state Thurston’s criterion, we need several definitions. Recall that for a branched covering map $g:\S^2\to\S^2$, $P_g$ denotes the postcritical set.
A [*marked branched covering*]{} is a pair $(g,X)$, where $g:\S^2\to\S^2$ is a postcritically finite branched covering map and $X$ is a finite set containing $P_g$ such that $g(X)\subset X$.
Let $(g,X)$ and $(h,Y)$ be two marked branched coverings. We say that they are [*(Thurston) equivalent*]{} if there are two homeomorphisms $\phi_0,\phi_1:\S^2\to\S^2$ such that $$\phi_0\circ g = h\circ \phi_1$$ and there exists an isotopy $\Phi: [0,1]\times\S^2\to\S^2$ with $\Phi(0,.)=\phi_0$ and $\Phi(1,.)=\phi_1$ such that $\Phi(t,.)|_{X}$ is constant in $t\in [0,1]$ with $\Phi(t,X)=Y$.
If $(g,X)$ is a marked branched covering and $\gamma$ a simple closed curve in $\S^2\setminus X$, then the set $g^{-1}(\gamma)$ is a disjoint union of simple closed curves.
Let $(g,X)$ be a marked branched covering. We say that a simple closed curve $\gamma\subset\S^2$ is a [*simple closed curve in $(\S^2,X)$*]{} if $\gamma\subset\S^2\setminus X$. It is called [*peripheral*]{} if there exists a component of $\S^2\setminus\gamma$ that intersects $X$ in at most one point, and [*non-peripheral*]{} otherwise.
Two simple closed curves $\gamma_1,\gamma_2$ in $(\S^2,X)$ are called [*isotopic (relative $X$)*]{} (write $\gamma_1\simeq\gamma_2$) if there exists a continuous one-parameter family $\gamma_t$, $t\in [1,2]$, of such curves joining $\gamma_1$ to $\gamma_2$. We denote the isotopy class of $\gamma_1$ by $[\gamma_1]$.
A finite set $\Gamma=\{\gamma_1,\dots,\gamma_m\}$ of disjoint, non-peripheral and pairwise non-isotopic simple closed curves in $(\S^2,X)$ is called a [*multicurve*]{}.
Let $(g,X)$ be a marked branched covering and $\Gamma$ a multicurve. Denote by $\R^\Gamma$ the real vector space spanned by the isotopy classes of the curves in $\Gamma$. Then, we associate to $\Gamma$ its [*Thurston transformation*]{} $g_\Gamma:\R^\Gamma\to\R^\Gamma$ by specifying its action on representatives $\gamma\in\Gamma$ of basis elements: $$\label{Eqn_ThurstonTransform}
g_{\Gamma}(\gamma) := \sum_{\gamma'\subset g^{-1}(\gamma)} \frac{1}{\deg(g|_{\gamma'}:\gamma'\to\gamma)}[\gamma']\;.$$ The sum is taken to be zero if there are no preimage components isotopic to a curve in $\Gamma$.
The linear map given by equation (\[Eqn\_ThurstonTransform\]) is represented by a square matrix with non-negative entries and thus its largest eigenvalue $\lambda(\Gamma)$ is real and non-negative by the Perron-Frobenius theorem.
A square matrix $A_{i,j}\in\R^{n\times n}$ is called [*irreducible*]{} if for each $(i,j)$, there exists $k\geq 0$ such that $(A^k)_{i,j}>0$. We say that the multicurve $\Gamma$ is [*irreducible*]{} if the matrix representing $g_\Gamma$ is.
An irreducible multicurve $\Gamma$ is called an [*irreducible (Thurston) obstruction*]{} if $\lambda(\Gamma)\geq 1$.
\[Def\_HyberbolicOrbifold\] Let $(g,X)$ be a marked branched covering of degree $d>1$. The orbifold $O_g$ is a pair $(g,v_g)$, where $v_g:X \to \N \cup \{ \infty \}$ is the smallest function such that $v_g(x)$ is a multiple of $v_g(y)\deg_y g$ for each $y \in g^{-1}(\{x\})$.
The orbifold $O_g$ is called *hyperbolic* if the Euler charactersitic $$\chi(O_g) = 2-\sum_{x \in X} \left( 1-\frac{1}{v_g(x)} \right)$$ is negative.
In most cases $O_g$ is hyperbolic but few exceptions which can be easily studied [@DH]. If $g$ has at least three fixed branched points, then it will have hyperbolic orbifold. For Newton maps we consider in this paper it is always the case. In general, $\# P_g \geq 5$ suffices to make the orbifold of $g$ hyperbolic.
Now we are ready to state Thurston’s theorem for marked branched coverings as given in [@PT Theorem 3.1] and proved in [@DH].
\[Thm\_Thurston\] Let $(g,X)$ be a marked branched covering with hyperbolic orbifold. It is Thurston equivalent to a marked rational map $(f,Y)$ if and only if it has no irreducible Thurston obstruction, i.e. if $\lambda(\Gamma)<1$ for each irreducible multicurve $\Gamma$. In this case, the rational map $f$ is unique up to automorphism of $\Cc$.
Arcs Intersecting Obstructions
------------------------------
We present a theorem of Kevin Pilgrim and Tan Lei that is useful to show that certain marked branched coverings are equivalent to rational maps. Again, we first need to introduce some notation.
Let $(g,X)$ be a marked branched covering of degree $d\geq 3$.
An [*arc*]{} in $(\S^2,X)$ is a map $\alpha:[0,1]\to \S^2$ such that $\alpha(\{0,1\})\subset X$, $\alpha((0,1))\cap X=\emptyset$, $\alpha$ is a continuous mapping, injective on $(0,1)$. The notion of [*isotopy relative $X$*]{} extends to arcs and is also denoted by $\simeq$.
A set of pairwise non-isotopic arcs in $(\S^2,X)$ is called an [*arc system*]{}. Two arc systems $\Lambda,\Lambda'$ are [*isotopic*]{} if each curve in $\Lambda$ is isotopic relative $X$ to a unique element of $\Lambda'$ and vice versa.
Note that arcs connect marked points (the endpoints of an arc need not be distinct) while simple closed curves run around them. We will see that this leads to intersection properties that will give us some control over the location of possible Thurston obstructions. Since arcs and curves are only defined up to isotopy, we make precise what we mean by arcs and curves intersecting.
\[Def\_IntersectionNumber\] Let $\alpha$ and $\beta$ each be an arc or a simple closed curve in $(\S^2,X)$. Their [*intersection number*]{} is $$\alpha\cdot\beta := \min_{\alpha'\simeq\alpha,
\,\beta'\simeq\beta} \#\{(\alpha'\cap\beta')\setminus X\}\;.$$
The intersection number extends bilinearly to arc systems and multicurves.
If $\lambda$ is an arc in $(\S^2,X)$, then the closure of a component of $g^{-1}(\lambda\setminus X)$ is called a [*lift*]{} of $\lambda$. Each arc clearly has $d$ distinct lifts. If $\Lambda$ is an arc system, an arc system $\tilde{\Lambda}$ is called a [*lift*]{} of $\Lambda$ if each $\tilde{\lambda}\in\tilde{\Lambda}$ is a lift of some $\lambda\in\Lambda$.
If $\Lambda$ is an arc system, we introduce a linear map $g_{\Lambda}$ on the real vector space $\R^{\Lambda}$ similar as for multicurves: for $\lambda\in\Lambda$, set $$g_{\Lambda}(\lambda):=\sum_{\lambda'\subset g^{-1}(\lambda)} [\lambda']\;,$$ where $[\lambda']$ denotes the isotopy class of $\lambda'$ relative $X$. Again, the sum is taken to be zero if $\lambda$ has no preimages in the isotopy class of $\lambda'$. We say that $\Lambda$ is [*irreducible*]{} if the matrix representing $g_{\Lambda}$ is.
Denote by $\tilde{\Lambda}(g^{\circ n})$ the union of those components of $g^{-n}(\Lambda)$ that are isotopic to elements of $\Lambda$ relative $X$, and define $\tilde{\Gamma}(g^{\circ n})$ in an analogous way. Note that if $\Lambda$ is irreducible, each element of $\Lambda$ is isotopic to an element of $\tilde{\Lambda}(g^{\circ n})$.
The following theorem is Theorem 3.2 of [@PT]. It shows that up to isotopy, irreducible Thurston obstructions cannot intersect the preimages of irreducible arc systems (except possibly the arc systems themselves). We will use this theorem to show that the extended map of an abstract Newton graph is Thurston equivalent to a rational map.
\[Thm\_ArcsInterOb\] Let $(g,X)$ be a marked branched covering, $\Gamma$ an irreducible Thurston obstruction and $\Lambda$ an irreducible arc system. Suppose furthermore that $\#(\Gamma\cap\Lambda)=\Gamma\cdot\Lambda$. Then, exactly one of the following is true:
1. $\Gamma\cdot\Lambda=0$ and $\Gamma\cdot g^{-n}(\Lambda)=0$ for all $n\geq 1$.
2. $\Gamma\cdot\Lambda\neq 0$ and for $n\geq 1$, each component of $\Gamma$ is isotopic to a unique component of $\tilde{\Gamma}(g^{\circ n})$. The mapping $g^{\circ
n}:\tilde{\Gamma}(g^{\circ n})\to \Gamma$ is a homeomorphism and $\tilde{\Gamma}(g^{\circ n})\cap (g^{-n}(\Lambda)\setminus
\tilde{\Lambda}(g^{\circ n}))=\emptyset$. The same is true when interchanging the roles of $\Gamma$ and $\Lambda$.
The Realization of Abstract Newton Graphs
-----------------------------------------
We conclude with a proof of Theorem \[Thm\_Realization\]. If $\hat{\Delta}$ is the channel diagram of the postcritically fixed Newton map $f$, recall that $\hat{\Delta}_n$ denotes the component of $f^{-n}(\hat{\Delta})$ that contains $\hat{\Delta}$. By $\hat{\Delta}'_n$ we denote the set of vertices of $\hat{\Delta}_n$.
Let $\Delta\subsetneq\Gamma$ be the abstract channel diagram of $\Gamma$. First observe that by condition (\[Cond\_Branch\]) of Definition \[Def\_NewtonGraph\], the vertices $v_1,\dots,v_{d_\Gamma}$ of $\Delta$ are branch points of $\ol{g}$. Since $d_{\Gamma}\geq 3$, $\ol{g}$ has hyperbolic orbifold and it suffices to show that $(\ol{g},X)$ has no irreducible Thurston obstruction: it then follows from Theorem \[Thm\_Thurston\] that $\ol{g}$ is Thurston equivalent to a rational map $f$ of degree $d_\Gamma$, which is unique up to Möbius transformation. Then, $f$ has $d_\Gamma+1$ fixed points, $d_\Gamma$ of which are superattracting because $\ol{g}$ has marked fixed branch points $v_1,\dots,v_{d_\Gamma}$. The last fixed point is repelling [@Milnor Corollary 12.7 & 14.5] and after possibly conjugating $f$ with a Möbius transformation, we may assume that it is at $\infty$. Now it follows from Definition \[Def\_NewtonMap\] that $f$ is a Newton map. It is unique up to a Möbius transformation fixing $\infty$, hence up to affine conjugacy.
So suppose by way of contradiction that $\Pi$ is an irreducible Thurston obstruction for $(\ol{g},X)$ and let $\gamma\in\Pi$. Then, $\gamma$ is a non-peripheral simple closed curve in $\S^2\setminus
X$. It is easy to see that each edge $\lambda$ of $\Delta$ forms an irreducible arc system and $\Pi \cdot \lambda = \# \left( \Pi \cap \lambda \right)$, hence Theorem \[Thm\_ArcsInterOb\] implies that $\gamma\cdot (\ol{g}^{-n}(\lambda)\setminus\lambda)=0$ for all $n\geq 1$. Since this is true for all edges of $\Delta$ and all vertices of $\Gamma$ are marked points, we get that $\gamma\cdot
(\ol{\Gamma\setminus\Delta})=0$. But since $\ol{\Gamma\setminus\Delta}$ is connected and contains $X\setminus\{v_0\}$, this means that $\gamma$ is peripheral, a contradiction.
In order to prove the last claim, note that $(\ol{g}_1,\Gamma'_1)$ and $(\ol{g}_2,\Gamma'_2)$ are Thurston equivalent as marked branched coverings. Let $h:(\S^2, \Gamma'_1)\to (\S^2,\Gamma'_2)$ be a homeomorphism that conjugates $\ol{g}_1$ to $\ol{g}_2$ on $\Gamma'_1$. If $e$ is an edge of $\Gamma_1$ with endpoints $x_1,x_2\in \Gamma'_1$, then $h(e)$ connects $h(x_1)$ with $h(x_2)$. Moreover, $h$ preserves the cyclic order at each vertex of $\Gamma_1$, because it is a homeomorphism of $\S^2$. So if $h':h(\Gamma_1)\to\Gamma_2$ is a homeomorphism that maps each $h(e)$ to the edge of $\Gamma_2$ between $h(x_1)$ and $h(x_2)$, then $h'\circ h$ realizes an equivalence between the two abstract Newton graphs.
Acknowledgements
================
We thank Dierk Schleicher for his everlasting support and for many fruitful discussions about the combinatorics of Newton maps. We also thank Tan Lei for her comments that helped to improve this paper.
[BFH]{} B. Bielefeld, Y. Fisher J. Hubbard: [*The classification of critically preperiodic polynomials as dynamical systems*]{}. J. Amer. Math. Soc. [**5**]{} (4) (1992), 721–762. A. Douady J. Hubbard: [*A proof of Thurston’s topological characterization of rational functions.*]{} Acta Math. [**171**]{} (1993), 263–297. J. Head: [*The combinatorics of Newton’s method for cubic polynomials*]{}, Thesis Cornell University (1987). J. Hubbard, D. Schleicher S. Sutherland: [*How to find all roots of complex polynomials by Newton’s method.*]{} Invent. Math. [**146**]{} (2001), 1–33. J. Luo: [*Newton’s method for polynomials with one inflection value*]{}, preprint, Cornell University 1993. J. Milnor, [*Geometry and dynamics of quadratic rational maps*]{}, with an appendix by the author and Tan Lei. Exp. Math. 2 (1993), no. 1, 37–83. J. Milnor: [*Dynamics in One Complex Variable*]{}, Vieweg (2000). K. Pilgrim T. Lei: [*Combining rational maps and controlling obstructions.*]{} Ergodic Theory Dynam. Systems [**18**]{} (1998) 221–245. F. Przytycki: [*Remarks on the simple connectedness of basins of sinks for iterations of rational maps*]{}, Collection: Dynamical systems and ergodic theory, Warsaw, 1986. Banach Center Publications [**23**]{} (1989), 229–235. P. Roesch, [*Topologie locale des métodes de Newton cubiques: plan dynamique.*]{} C. R. Acad. Sci. Paris Série I 326 (1998), 1221–1226. J. Rückert D. Schleicher, [*On Newton’s method for entire functions*]{}. Journal of the London Mathematical Society [**75**]{} 3 (2007), 659–676. ArXiv math.DS/0505652. M. Shishikura: [*The connectivity of the Julia set and fixed points*]{}, in “Complex dynamics: families and friends” (Ed. by D. Schleicher), A. K. Peters (2009). Tan Lei: [*Branched coverings and cubic Newton maps*]{}. Fund. Math. [**154**]{} (1997), 207–260.
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abstract: 'We consider the mechanical coupling between a two-dimensional Bose-Einstein condensate with a graphene sheet via the vacuum fluctuations of the electromagnetic field which are at the origin of the so-called Casimir-Polder potential. By deriving a self-consistent set of equations governing the dynamics of the condensate and the flexural (out-of-plane) modes of the graphene, we can show the formation of a new type of purely acoustic quasi-particle excitation, a phonon-polariton resulting from the coherent superposition of quanta of flexural and Bogoliubov modes.'
author:
- 'H. Terças'
- 'S. Ribeiro'
- 'J. T. Mendonça'
title: 'Phonon-polaritons in Bose-Einstein condensates induced by Casimir-Polder interaction with graphene'
---
Introduction
============
The Casimir effect is a consequence of the field-theoretical description of the quantum vacuum and results directly from the quantization of the electromagnetic field. Traditionally derived for two infinite uncharged metallic plates placed only a few nanometers apart [@casimir; @genet], this quantum-mechanical effect has also been investigated in the context of atoms interacting with surfaces [@babb; @milton]. Casimir-Polder (CP) forces have been a subject of research of its own, both in view of nano-technological applications [@chan] and motivated by the possibility of probing fundamental forces at the submicron scale [@dimopoulos]. Indeed, the effects of a dispersive potential due to a macroscopic surface on an atom, both at zero and finite temperature, are already quite well established [@StefanB_Book].
Moreover, experiments with Bose-Einstein condensates (BEC) near a surface have attracted a special attention since the early days of microtraps [@lin; @leanhardt; @harber]: if, in one hand, the understanding of the influence of vacuum forces is crucial to the operations in atomic microtraps, on the other, the measurements based on quantum optics of ultracold gases are very accurate, making them a very desirable candidate to probe vacuum forces [@antezza]. In fact, the influence of the solid-state substrate on the atomic dynamics of a cold gases has been the target of several experiments. As it has been shown, an atomic cloud near a rough surface (typical distances of 1 $\mu$m) may undergo a matter-wave Anderson localization in a random potential [@PRL105_210401_2010]. Also, by rotating a corrugated plate separated of a few microns from a BEC, the nucleation of quantized vortices is theoretically predicted [@EPL92_40010_2010]. More recently, Bender et al. have used BEC to harvest information about the shape of the potential landscape of a solid grating, in an excellent agreement with the theoretical predictions [@bender].
An important and recent activity with ultracold atoms includes mechanical coupling via vacuum forces. Experiments have put in evidence the resonant coupling of mechanical BEC modes to a micromechanical oscillator [@PRL104_143002_2010], as well as the backaction of the atomic motion onto the membrane [@PRL107_223001_2011]. There are also theoretical propositions of how creating a force in a graphene sheet due to highly excited (Rydberg) atomic states [@PRA88_052521_2013]. Actually, the interest in the family of fullerenes has growing since the recent propositions of sympathetic cooling of carbon nanotubes, via CP interaction, by a laser cooled atomic gas [@weiss]. However, to the best of our knowledge, a completed treatment of the coupling of the flexural modes of a graphene membrane and a BEC via vacuum forces has not yet been discussed. In this paper, we will provide a microscopic description of the problem.
![(color online) Schematic representation of the system. A two-dimensional BEC is placed at a distance $d$ from a monolayer graphene sheet. The a ripple at the position $\mathbf{x}'$ provoques a small deformation in the Casimir-Polder potential at a position $\mathbf{x}$ in the BEC and vice-versa.[]{data-label="fig1"}](fig1.pdf)
Benefiting from its remarkable mechanical and transport properties [@neto], monolayer graphene is a natural candidate to perform atom-surface interfaces. Due to thermal fluctuations, the membrane may undergo mechanical out-of-plane vibrations (flexural phonons), which can be well described within the Kirchoff-plate theory of elasticity [@amorim]. A recent experimental study makes use of a cavity optomechanical protocol to cool down the zero-point flexural mode of a suspended graphene sheet [@song]. By previously cooling the graphene sheet (with the help of in a dilution refrigerator, for example), the flexural modes can be quantized. Thus, in this paper, we investigate the dynamics of a two-dimensional BEC interacting with a monolayer graphene sheet via the Casimir-Polder potential (see Fig. (\[fig1\]) for a schematic illustration). We proceed to a full quantization of the system in order to harvest the coherent coupling between the two phonon modes, leading to a new type of polariton excitation: a purely acoustic phonon-polariton. We show that for sufficiently large separation distances $d$, heating of the condensate via vacuum fluctuations is negligible, showing that the polaritons may exist in the strong-coupling regime. In Sec. II, we present the governing equations in terms of the mean-field equations. The details of the CP potential for a $^{87}$Rb condensate are provided in Sec. III. In Sec. IV, the dispersion relation of the phonon-polariton modes and the BEC heating rate are derived. Finally, in Sec. V some concluding remarks are stated.
Governing equations: mean-field description
===========================================
In a general way, the potential between a single neutral atom and a surface is given by $$U_S=\frac{C_\nu}{d^\nu},$$vacuum where $d$ is the separation distance between the atom and the surface and $C_\nu$ is the strength of the interaction, which depends on the both the atomic polarizability and on the electromagnetic properties of the surface. Let $\Psi(\mathbf{r})=\psi(z)\psi(\mathbf{x})$ be the condensate order parameter, normalized to the number of atoms such as $N=\int \vert \Psi\vert^2 d\mathbf{r}$, where $\mathbf{x}=(x,y)$ is the plane coordinate. The dynamics of the two-dimensional BEC can then be described in terms of the Gross-Pitaevskii equation $$i\hbar \frac{\partial }{\partial t}\psi(\mathbf{x})=-\frac{\hbar ^2\nabla^2}{2m}\psi(\mathbf{x})+g\vert \psi(\mathbf{x}) \vert^2 \psi(\mathbf{x})+U\psi(\mathbf{x}),
\label{GP1}$$ where $g=g_\mathrm{3D}/(2\sqrt{2\pi}\ell_z)$, with $g_\mathrm{3D}=4\pi \hbar^2 a/m$, is the effective 2D coupling constant, $a$ is the atomic scattering length and $\ell_z=(\hbar/m\omega_z)^{1/2}$ is the transverse harmonic length. The atoms in the BEC feel a vacuum-field potential of the form $$U=\int \frac{d\mathbf{x}'}{\sqrt{\mathcal{V}}} \frac{C_\nu }{\left[\left(d-{\bm \eta}(\mathbf{x}')\right)^2+\vert \mathbf{x}-\mathbf{x'}\vert^2\right]^{\nu/2}},
\label{potential1}$$ where ${\bm \eta}(\mathbf{x})$ is a small deformation on the graphene surface (ripple) located at the position $\mathbf{x}$ and $\mathcal{V}$ is the quantization surface. On the other hand, the mechanical properties of the graphene sheet can be easily derived from the Kirchoff-Love plate theory [@amorim]. The Lagrangian density of the sheet can be written as $\mathcal{L}=\rho \dot {\bm \eta}^2/2-\mathcal{W}$, with $\rho$ standing for the graphene mass density, and the potential energy per unit surface can be expressed as $$\mathcal{W}=\frac{1}{2}D\left(\nabla^2{\bm \eta}\right)^2+ \gamma_x \left(\frac{\partial {\bm \eta}}{\partial x}\right)^2+\gamma_y \left(\frac{\partial {\bm \eta}}{\partial y}\right)^2+\frac{1}{2}\kappa \bm \eta^2,$$ where $D$ is the bending stiffness, $\gamma_x$ ($\gamma_y$) is the tension along the $x$ ($y$) direction to the clamping with the substrate and $\nabla^2{\bm \eta}$ is the local curvature. The term $\kappa=d^2U/d{\bm \eta}^2\vert_{{\bm \eta}=0}$ is the restoring force in the harmonic approximation, which acts as a charge on the surface of the sheet through the CP potential, $$\kappa=\nu C_\nu\int d\mathbf{x}' \vert \psi (\mathbf{x}')\vert^2 \frac{(1+\nu)d^2-\vert \mathbf{x}-\mathbf{x}'\vert^2}{\left(d^2+\vert \mathbf{x} - \mathbf{x}' \vert^2 \right)^{2+\nu/2}}.$$ From the Euler-Lagrange equations, we can obtain the Kirchoff-Love equation for the flexural modes $$\rho \frac{\partial ^2 {\bm \eta}}{\partial t^2}+D \nabla^4{\bm \eta} +\kappa{\bm \eta}=0,
\label{plate1}$$ where we assume, for simplicity, the self-suspended case $\gamma_x=\gamma_y=0$. Eqs. (\[GP1\]) and (\[plate1\]) form a self-consistent set of equations for the condensate field $\psi(\mathbf{x})$ and for the deformation field ${\bm \eta}(\mathbf{x})$, being the governing equations to be quantized in Sec. IV.
![(color online) Casimir-Polder potential of a $^{87}$Rb atom near the surface of a graphene sheet. For short distances, the potential scales as $\sim d^3$ (oblique line). The dependence on the distance as $\sim 1/d^4$ (horizontal line) is obtained for separations larger than $d=0.5~\mu$m.[]{data-label="fig2"}](fig2.pdf)
In the absence of coupling (or, equivalently, in the limit of very large distances $d\rightarrow \infty$), Eqs. (\[GP1\]) and (\[plate1\]) can be easily solved by taking the Bogoliubov prescription $\psi=e^{-i\mu t/\hbar}\left[\psi_0+\sum_k \left(u_k e^{i\mathbf{k}\cdot \mathbf{x}-i\omega t}+v_k^* e^{-i\mathbf{k}\cdot \mathbf{x}+i\omega t}\right) \right]$ and the expansion ${\bm \eta}= \sum_k {\bm \eta}_k e^{i\mathbf{k}\cdot \mathbf{x}-i\omega t} $, which respectively yield to two stable modes, $$\omega_B=\left(c_s^2k^2+\frac{1}{4}c_s^2\xi^2k^4\right)^{1/2}, \quad \omega_C=\left(\beta^2 k ^4+\omega_0^2\right)^{1/2}.$$ Here, $c_s=(g\vert \psi_0\vert^2/m)^{1/2}$ and $\xi=\hbar(g\vert \psi_0\vert^2m)^{-1/2}$ are the condensate sound speed and healing length, respectively, $\beta=(D/\rho)^{1/2}$ is the specific stiffness and $\omega_0=\sqrt{\kappa/\rho}$. In agreement with an experimentally feasible situation, we choose a $^{87}$Rb condensate with an areal density of $n_0\equiv \vert \psi_0 \vert^2\sim 10^8$ cm$^{-2}$, trapped along the $z$-direction by a harmonic trap frequency $\omega_z\sim 2\pi\times 1500$ Hz, $a=5.5$ nm, which yields $\xi\sim 0.1$ $\mu$m and $c_s\sim 0.1$ mm/s. On the other hand, for the graphene sheet we have $\rho=0.761$ mg/m$^2$ and $D\sim1.5 $ eV [@lambin; @fasolino1; @fasolino2], leading to $\beta=6.1\times 10^{-7}$m$^2$/s. This simply means that the BEC phonons and the ripples in the graphene sheet propagate at very different frequencies. In fact, for wavevectors of the order $k\sim 10~\mu$m$^{-1}$ (typical of the BEC phonon-like excitations), $\omega_B/\omega_C\sim c_s/\beta k\simeq 10^{-3}$. Such a picture is very similar to what happens in semiconductor microcavities, where strong coupling - allowing for the formation of exciton-polaritons - occurs between two modes of very different energy scales: photons and excitons. Because of their very small effective mass, photons exhibit a parabolic dispersion relation, while the excitonic dispersion is almost flat for the relevant wavevectors [@guillaumebook; @carusotto]. The coupling is then provided via the dipolar interaction between the photons and the excitons. In the next section, we show that the same thing happens here, where the Bogoliubov excitations in the condensate, possessing a very flat dispersion, are the analogue of the semiconductor excitons, while the mechanical vibrations of the graphene sheet work as massive photons. In what follows, we quantize Eqs. (\[GP1\]) and (\[plate1\]) to show that strong coupling between the condensate phonons (bogolons) and the flexural modes in the graphene (flexural phonons) is possible, leading to the formation of a sort of phonon-polaritons.
Casimir-Polder interaction with a graphene sheet
================================================
The theoretical approaches to determine the Casimir-Polder energy shift are usually based on second-order perturbation theory [@StefanB_Book]. Here, we will discuss how to evaluate the Casimir-Polder potential between a single graphene sheet and a $^{87}$Rb atom within the formalism of macroscopic QED [@acta2008]. To do so, we shall neglecting the possible effects that may arise from the finite size of graphene and assume it to be infinitely extended. For planar structures, the Casimir-Polder potential of an atom in a ground state $| 0 \rangle$ at a distance $d$ away from the macroscopic body can be written as [@StefanB_Book] $$\begin{array}{c}
U_S {\ensuremath{\left(d\right)}} = \frac{\hbar \mu_{0}}{8 \pi^{2}} \int_{0}^{\infty}
d \xi \xi^{2} \alpha {\ensuremath{\left(i \xi\right)}} \nonumber \\
\times \int\limits_{0}^{\infty} d k_{\parallel} \frac{e^{-2 k_{\parallel}
\gamma_{0z} d} }{\gamma_{0z}} \left[
\mathrm{R}_{\mathrm{TE}} + \mathrm{R}_{\mathrm{TM}} \left( 1- \frac{2
k_{\parallel}^{2} \gamma_{0z}^{2} c^{2} }{\xi^{2}} \right) \right] ,
\end{array}
\label{eq:Ucp_1}$$ where integration is done along the imaginary frequency axis $\omega= i \xi$ and $k_{\parallel}$ the wave vector in the plane of the interfaces. We have defined $\gamma_{iz} = \sqrt{1+ \varepsilon_{i} (i \xi)\xi^{2}/(c^{2} k_{\parallel}^2)}$, which is the $z$ component of the wave number in a medium with permittivity $\varepsilon_i$ (the index 0 refers to the medium in which the atom is placed). Here, $\alpha(\omega)$ is the atomic polarizability defined for an isotropic atom as $$\mathbf{\alpha} (\omega) = \lim_{\varepsilon \rightarrow 0} \frac{2}{3 \hbar}
\sum_{k \neq 0} \frac{\omega_{k0} \, |\mathbf{d}_{0 k}|^2}{\omega_{k0}^2-\omega^2-i\omega\varepsilon}\,.
\label{eq:atomicpol}$$ The latter is valid in the zero temperature limit and for atoms in the ground state, which will be the case of the present manuscript. All the relevant electromagnetic features of the graphene sheet are cast in the reflection coefficients $\mathrm{R}_{\mathrm{TE}}$ and $\mathrm{R}_{\mathrm{TM}}$ in Eq. \[eq:Ucp\_1\]. In Ref. [@PRB84_035446_2011], the reflection coefficients have been calculate assuming that the dynamics of the quasiparticles in graphene can be described within the $(2+1)-$dimensional Dirac model. Imposing appropriated boundary conditions to the EM field, it is possible to explicitly determine the reflection coefficients. Taking the contribution of the electrons near the Dirac cone, one obtains, for a self-suspend graphene sheet in vacuum, $$\begin{aligned}
\mathrm{R}_{\mathrm{TM}} &=& \frac{4 \pi \alpha \sqrt{k_{0}^{2} +
k_{\parallel}^{2}} }{4 \pi \alpha \sqrt{k_{0}^{2} + k_{\parallel}^{2}} + 8
\sqrt{k_{0}^{2} + \tilde{v}^{2} k_{\parallel}^{2}}}\,, \\
\mathrm{R}_{\mathrm{TE}} &=& - \frac{4 \pi \alpha\sqrt{k_{0}^{2} +
\tilde{v}^{2} k_{\parallel}^{2}} }{4 \pi \alpha \sqrt{k_{0}^{2} + \tilde{v}^{2}
k_{\parallel}^{2}} + 8 \sqrt{k_{0}^{2} + k_{\parallel}^{2}}} \,,
\label{eq:reflcoefG}\end{aligned}$$ where we have defined $k_{0}^{2} = \xi^{2}/c^{2}$ and $\tilde{v}=v_F/c=(300)^{-1}$, with $v_F$ being the Fermi velocity and $c$ the speed of light; $\alpha=1/137$ is the fine structure constant. More elaborated models could be performed, however, for the present conditions, Eq. (\[eq:reflcoefG\]) provides a very good approximation [@PRA88_052521_2013].
There are two regimes according to the atom-surface distances: the near-field, nonretarded limit, and the far-field (retarded) limit. In the nonretarded limit, $|\sqrt{\varepsilon (\omega)}| \omega d / c \ll 1$, and for the usual Fresnel reflection coefficients, successive approximations yield $U_S \simeq C_3 / d^3$, and we obtain a fitting to Eq. (\[eq:Ucp\_1\]) with $C_3 = -215.65 $ Hz$\mu$m$^3$. The CP potential for a ground-state atom is attractive for a non-magnetic medium [@acta2008]. We notice that the sign of the CP potential of a ground-state atom with a conducting plate can be easily understood with an image-dipole model. However, if a resonant coupling between the atomic dipole and a surface excitation occurs, it is possible to obtain a repulsive force [@PRL83_5467_1999].
The retarded limit corresponds to the situation where the atom-surface distance $d$ is large when compared to the effective transition wavelength. In this situation, one finds the approximation $U_S = C_4 / d^4$ also to be valid, with $C_4 = -14.26 $ Hz$\mu$m$^4$. The dependence of the CP potential on the separation distance $d$ as resulting from the numerical integration of Eq. (\[eq:reflcoefG\]) is depicted in Fig. (\[fig2\]). In the following, we will operate in the distance range $d\gtrsim 1~\mu$m, deep in the limit $U_C\sim 1/d^4$. Such a choice is a consequence of three major constraints: i) transverse trap size, as the two-dimensional BEC approximation is only valid if the transverse size of the BEC is much smaller than the distance $d$, ii) the de-confinement effect associated to the CP potential, and iii) the heating of the condensate for small distances. In the following, we assume condition i) to be satisfied (which is true for $\omega_z\sim 2\pi \times 1000$ Hz) and discuss the limitations imposed by ii) and iii) in Sec. V.
Phonon-polariton quantization and avoided crossing
==================================================
In order to proceed to a microscopic description of the coupling between the bogolons and the flexural phonons, we quantize the theory. The total Hamiltonian of the system can be defined as $\hat H=\hat H_B +\hat H_C+\hat H_{\rm int}$, where $$\hat H_B=\int d\mathbf{x} ~\hat \psi^\dagger (\mathbf{x})\left[ -\frac{\hbar^2\nabla^2}{2m}-\mu+g\hat \psi^\dagger (\mathbf{x}) \hat \psi (\mathbf{x})\right]\hat \psi (\mathbf{x}),
\label{HB}$$ is the condensate Hamiltonian, which can be solved in the Fourier basis with $\hat \psi(\mathbf{r})=\sum _\mathbf{k}\varphi_\mathbf{k}(\mathbf{r})\hat a_k$ and $\varphi_\mathbf{k}=\mathcal{V}^{-1/2}e^{i\mathbf{k}\cdot \mathbf{r}}$. Here, $\hat a_k$ represent the bosonic annihilation operator satisfying the canonical commutation relation $\left[\hat a_\mathbf{k},\hat a^\dagger_{\mathbf{k}'}\right]=\delta_{\mathbf{k}\mathbf{k}'}$. Using the Bogoliubov approximation $\hat a_{\bf k}\simeq \sqrt{N_0}+\hat a'_{\bf k}$, with $\hat a'_{\bf k} $ standing for the $k\neq 0$ fluctuations, the condensate Hamiltonian can be simply given by (expressing the summation in terms of the absolute value $k$ only) $$\begin{array}{ccl}
\hat H_B&=&E_0+\displaystyle{\frac{1}{2}\sum_{k\neq 0} \left[(\epsilon_k+\mu) \left(\hat a_k^\dagger \hat a_k+\hat a_{-k}^\dagger \hat a_{-k}\right)
\right.}\\
&+& \displaystyle{\mu \left.\left( \hat a_k^\dagger \hat a_{-k}^\dagger+\hat a_{k} \hat a_{-k} \right)\right] }.
\end{array}
\label{HB2}$$ Here, $\epsilon_k=\hbar^2k^2/2m$, $\mu=gn_0$ is the chemical potential and $E_0=N\mu/2$ is the condensate zero-point energy. Similarly, the Hamiltonian for the flexural modes easily follows from the canonical quantization of Eq. (\[plate1\]) $$\hat H_C=\frac{1}{2}\int d{\bf r} \left[ \left(\rho \partial _t \hat {\bm \eta}(\mathbf{r})\right)^2+D\left(\nabla^2 \hat {\bm \eta}(\mathbf{r})\right)^2+\kappa {\bm \eta}^\dagger(\mathbf{r}) \hat {\bm \eta}(\mathbf{r})\right].$$ Expressing the phonon operator in the form $$\hat {\bm \eta}(\mathbf{r})=\frac{1}{\sqrt{2}} \sum_{\bf k,\sigma} \varphi_{\bf k}(\mathbf{r})h_{\bf k}(\mathbf{r})\mathbf{e}_\sigma \left(\hat c_{\bf k,\sigma}+\hat c_{\bf k,\sigma}^\dagger \right),$$ with two polarizations $\sigma=(x,y)$ and satisfying the normalisation condition $\langle h_{\bf k}, h_{\bf k'}\rangle=\hbar/(M\omega_C)\delta _{\bf k k'}$, with $M=\rho A$ being the membrane mass, such that the phonon operators obey the bosonic commutation relation $\left[\hat c_{\mathbf{k},\sigma},\hat c^\dagger_{\mathbf{k}',\sigma'}\right]=\delta_{\mathbf{k}\mathbf{k}'}\delta_{\sigma, \sigma'}$, the graphene Hamiltonian reads $$\hat H_C=\sum_{k ,\sigma}\hbar \omega_C\hat c_{k,\sigma} ^\dagger \hat c_{k,\sigma} .$$ The quantization of the flexural modes is relevant for temperature scales given of the cryogenically cooled atom chip, on which the graphene sheet is supposed to be suspended. Typical experimental conditions involving a dilution refrigerator allow to cool down a carbon nanotube down to 4 K [@weiss]. Additionally, according to the recent experiments of Ref. [@song], optomechanical cooling can be used to bring the graphene down to 50 mK, which corresponds to roughly 40 flexural phonon quanta.
The interaction Hamiltonian is given by a convolution $$\hat H_{\rm int}=\int \frac{d\mathbf{x}}{\mathcal{V}}\int d\mathbf{x}'~\hat n (\mathbf{x}')U(\mathbf{x}-\hat {\bm \eta} (\mathbf{x}')).$$ Assuming small deformations, we can Taylor expand the potential to first order in $\hat {\bm \eta}$, $U(\mathbf{x}-\hat {\bm \eta} (\mathbf{x}'))\simeq U(\mathbf{x}-\mathbf{x'})+\bm\nabla U(\mathbf{x}-\mathbf{x'})\cdot \hat {\bm \eta} (\mathbf{x}')$, which yields $$\hat H_{\rm int}=\sum_k U_k\hat \rho_k+i \sum_k U_k \hat \rho_k{\bm k}\cdot \hat {\bm \eta}_k, \label{Hint}$$ with $U_k=\pi C_4 k n_0K_1(kd)/d$ being the Fourier transform of the potential and $\hat \rho_k=\sum_q \hat a_{k+q}^\dagger \hat a_q$ the density operator. The first term can be separated into the condensate and the fluctuation contributions, leading to a change in free-particle energy $$\sum_k U_k\hat \rho_k=\sum_q U_0\hat a_q^\dagger \hat a_q+\sum_{k\neq 0, q}\hat a_{k+q}^\dagger \hat a_qU_k.
\label{separation}$$ The second term of Eq. (\[separation\]) is of the order of the condensate depletion $\mathcal {O}(N')$, and therefore we should neglect it. By making the substitutions $\epsilon_k\rightarrow \tilde \epsilon_k\equiv ( \epsilon_k+U_kn_0)$ and $E_0\rightarrow E_0+U_0n_0$ in Eq. (\[HB2\]), and proceeding to a Bogoliubov-Valatin transformation to the excitation operators in the usual way, $\hat a_k'=u_k \hat b_k-v_k \hat b_k^\dagger$, we obtain the modified condensate Hamiltonian as $\hat H_B=\sum_{k}\hbar \omega_B\hat b_k \hat b_k^\dagger$, where $\hbar\omega_B=\sqrt{\tilde \epsilon_k(\tilde\epsilon_k+\mu)}$ is the renormalized Bogolubov spectrum and the transformation coefficients read $$u_k=\left(\frac{\tilde \epsilon_k+\mu}{\hbar\tilde\omega_B(k)}+\frac{1}{2}\right)^{1/2}, \quad v_k=\left(\frac{\tilde \epsilon_k+\mu}{\hbar\tilde\omega_B(k)}-\frac{1}{2}\right)^{1/2}.$$
The second term of Eq. (\[Hint\]) can be evaluated by noticing that in the Bogoliubov approximation $ \hat{\rho_k}\simeq \sqrt{N_0}(\hat a_k^\dagger +\hat a_{-k})$, which then yields $$\hat H_{\rm int}=\sum_{k,\sigma} M_{\kappa,\sigma}\left(\hat b_k^\dagger \hat c_{k,\sigma}+\hat b_k \hat c_{k,\sigma}^\dagger +\hat b_k^\dagger \hat c_{k,\sigma}^\dagger +\hat b_k \hat c_{k,\sigma}\right),
\label{Hint1}$$ where $M_{k,\sigma}$ is the $k-$dependent coupling $$M_{k,\sigma}=i\sqrt{\frac{N_0}{2}}U_k(u_k-v_k)k h_k \mathbf{e}_k\cdot \mathbf{e}_\sigma.$$ The latest two terms in Eq. (\[Hint1\]) do not conserve the total number of excitations, and therefore will be neglected in the present discussion. This approximation remains valid here as long as the interaction is small compared to the single-particle energy, $\vert M_{k,\sigma}\vert\ll \hbar\omega_B+\hbar \omega_C$, which we will confirm a posteriori. Finally, the total Hamiltonian of the system can be given as $$\begin{array}{ccl}
\hat H &\simeq& \displaystyle{\sum_{k}\left(\Delta \hat b_k^\dagger \hat b_k+\sum_{\sigma}\hbar\omega_C\hat c_{k,\sigma}^\dagger \hat c_{k,\sigma} \right. }\\
&+&\left. \sum_\sigma M_{k,\sigma}\hat b_k^\dagger \hat c_{k,\sigma}+{\rm h.c.}\right),
\end{array}$$ where $\Delta=\pi \vert C_4\vert n_0/d^2$ is the interaction-induced energy shift. The later Hamiltonian can be diagonalized using the so-called Hopfield transformations $$\hat a_k^{(L)}=\chi_{B,k}\hat b_k -\chi_{C,k} \hat c_k, \quad \hat a_k^{(U)}=\chi_{C,k}\hat b_k +\chi_{C,k} \hat c_k,
\label{pol}$$ where $\hat c_k=\sum_\sigma \hat c_{k,\sigma}$ and $\hat a_k{(L)}$ and $\hat a_k^{(U)}$ respectively denote the destruction operators for the lower (L) and upper (U) polaritons. In order to be bosonic, the operator in Eq. (\[pol\]) must satisfy the normalization condition $\vert \chi_{C,k} \vert ^2 +\vert \chi_{B,k} \vert ^2=1$, which allow us to write the Hamiltonian in the decoupled form $$\hat H=\sum_{P=L,U}\sum _k \hbar \omega_{P}(k)\hat a_k^{(P)\dagger}\hat a_k^{(P)},$$ where the eigenenergies are simply given by
$$\hbar \omega_{U,L}(k)=\frac{1}{2}\left[ \Delta +\hbar\omega_C\pm\sqrt{\left( \Delta -\hbar\omega_C\right)^2+4\vert M_k\vert^2}\right],$$
with $\vert M_k\vert^2 =\sum_\sigma M_{k,\sigma}^* M_{k,\sigma}$. The two-mode polariton dispersion is depicted in Fig. (\[fig\_dispersion\]). The graphene (condensate) fraction of the LP (UP) mode almost 1 near the bottom of the dispersion $k\sim 0$; on the contrary, for $k\rightarrow \infty$ the LP (UP) mode is essentially Bogoliubov- (graphene-) like. Moreover, the branches repeal each other at the wave vector $k_*=\sqrt{\Delta/\hbar \beta}$. The effective Rabi frequency $\Omega=2\vert M_{k_*}\vert$ measures the strength of the coupling and is equal to $\Omega\simeq 17$ Hz for $d=1.5~\mu$m, being much smaller that the free-energy scale of the system.
![(color online) Phonon-polariton dispersion for a system composed of a two-dimensional (2D) condensate of $^{87}$Rb atoms and a graphene sheet interacting via the Casimir-Polder potential $V_C\sim C_4/d^4$. Panels a): Upper (red solid line) and lower (blue solid line) polariton. Panel b): Amplification of the rectangle depicted in panel a). The bare dispersions for the flexural modes (dot-dashed red line) and for the condensate (dot-dashed blue line) are also shown. Both panels are obtained for $d=1.5~\mu$m.[]{data-label="fig_dispersion"}](fig3.pdf)
In order to demonstrate the conditions for which strong coupling effectively occurs, we must quantify the heating induced in the condensate. As illustrated in Fig. (\[fig\_coupling\] a)), the Casimir-Polder interaction modifies the trapping potential. If $d$ is small enough such that the first excited state of the trap lies above the cut-off energy $U_c=U(z_c)$, where $U(z)=m \omega_z^2z^2/2+C_4/(d-z)^4$ and $z_c\neq 0$ is the solution of the equation $U'(z_c)=0$, the heating may cause the particles to scape the trap [@PRL104_143002_2010]. In order to avoid such a situation, we should require the restriction $d>d_c$, where the critical distance is $d_c\simeq z_c+\{2\vert C_4\vert/(3\hbar \omega_z)\}^{1/4}$. For $\omega_z=2\pi \times1500$ Hz, we obtain $d\simeq 1.45~\mu$m. In this situation, the heating rate rules out trap losses and lis related to the creation of in-plane excitations only (condensate depletion), which we calculate with the help of Fermi’s Golden Rule $$\Gamma=\frac{2\pi}{\hbar}\sum_{i,f} \vert \langle i \vert \hat H_{\rm int}\vert f\rangle\vert ^2 \delta\left(\hbar \omega_B-\hbar \omega_C\right),$$ where the delta function accounts for the resonant terms only. Here, the initial state $\vert i\rangle=\vert 0_{k_i},1_{\sigma_i,k_i}\rangle$ describes the condensate in the ground state and the graphene phonon with energy $\epsilon_i\simeq \hbar \beta k_i^2$ and polarization $\sigma_i$; the final state $\vert f\rangle=\vert 1_{k_f},1_{\sigma_f,k_f}\rangle$ contains an extra excited state above with flat dispersion $\epsilon_i\simeq \Delta$. By proceeding that way, we capture only the inelastic processes, which after some simple algebra yields the following heating rate $$\Gamma=\frac{4\pi^2d^2\Delta^3}{M\hbar^2\beta^3}S(k_*)^2K_1^2(k_*d),$$ where $S(k)=(u_k-v_k)^{1/2}$ is the BEC static structure factor. The strong coupling regime is achieved for $\Omega\gg \Gamma$, which corresponds to separation distances of the order of 1.5 $\mu$m, deep in the regime where the CP potential considered here is valid (see Fig. (\[fig\_coupling\]b)). The weak coupling situation is also possible for shorter distances, where the heating of the BEC becomes quite appreciable. Due to fast variation of the CP potential, the transition between the strong and weak coupling regimes is quite sensitive. This requires a condensate to be confined at the sub-micro size corresponding to large values of $\omega_z$, therefore safely lying in the regime where trap deconfining becomes less critical.
![(color online) Panel a): Effective trapping potential illustrating the lose trapping effect. If the first transverse excited state lies above the cut-off energy $U_c$, the condensate particles can escape as a consequence of the mechanical heating. Panel b) Effective Rabi frequency (black solid line) and heating rate (red solid line). The shadowed region represents the strong-coupling regime. The dashed vertical line depicts the critical distance $d_c$ (see text). []{data-label="fig_coupling"}](fig4.pdf)
Conclusion
==========
We have demonstrated that the Casimir-Polder due to the interaction of a Bose-Einstein condensate and a graphene sheet can be perturbed by the mechanical out-of-plane vibrations and by the Bogoliubov excitations. At cryogenic temperatures, the Kirchoff-Love flexural modes can be quantized, therefore coupling to the Bogoliubov excitations of the condensate. As a result, a phonon-polariton is formed for sufficiently large separation distances, for which the effective Rabi frequency (the coupling strength) dominates over the heating rate. Our results may motivate a scheme for the sympathetic cooling of monolayer graphene via vacuum-fluctuation forces. In a feasible experimental situation, such a cooling scheme may be implemented in combination with an optomecanical cooling protocol, in order to pre-cool the membrane down to a temperature of few tens of mK [@song]. We may think that the sympathetic cooling via the Casimir-Polder interaction may then be used to further cool the zero-point mode, since the condensate temperature is of a few tens of nK. In fact, the major interest in cooling the mechanical modes of graphene lies in the perspective of controlling its transport properties, as the electronic response at low temperatures is intimately related by the scattering with the flexural phonons [@eduardo].
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abstract: 'Magnetic reconnection is a rapid energy release process that is believed to be responsible for flares on the Sun and stars. Nevertheless, such flare-related reconnection is mostly detected to occur in the corona, while there have been few studies concerning the reconnection in the chromosphere or photosphere. Here we present both spectroscopic and imaging observations of magnetic reconnection in the chromosphere leading to a microflare. During the flare peak time, chromospheric line profiles show significant blueshifted/redshifted components on the two sides of the flaring site, corresponding to upflows and downflows with velocities of $\pm$(70–80) km s$^{-1}$, comparable with the local Alfvén speed as expected by the reconnection in the chromosphere. The three-dimensional nonlinear force-free field configuration further discloses twisted field lines (a flux rope) at a low altitude, cospatial with the dark threads in 10830 images. The instability of the flux rope may initiate the flare-related reconnection. These observations provide clear evidence of magnetic reconnection in the chromosphere and show the similar mechanisms of a microflare to those of major flares.'
author:
- 'Jie Hong, M. D. Ding, Ying Li, Kai Yang, Xin Cheng, Feng Chen, Cheng Fang, and Wenda Cao'
title: Bidirectional outflows as evidence of magnetic reconnection leading to a solar microflare
---
Introduction
============
Magnetic reconnection is a fundamental physical process on the Sun, and is considered to be the cause of many kinds of solar activities, from drastic eruptions like flares and coronal mass ejections [@1964carmichael; @1966sturrock; @1974hirayama; @1976kopp; @1999shibata], to small-scale events like Ellerman bombs [@2009archontis; @2006fanga] and microflares [@1988tandberg]. Although magnetic reconnection is supposed to be the most important ingredient in theoretical models for various solar activities, in particular solar flares, its observational evidence is still elusive. Most major solar flares are believed to originate in the tenuous and fully ionized corona, a favorable place for magnetic reconnection to proceed. There have been some observations supporting the reconnection picture like double hard X-ray sources above the flare loop top [@1992sakao; @1994masuda; @1994bentley], cusp-like structure in soft X-ray loops [@1992acton; @1992tsuneta; @1995doschek], reconnection inflows [@2001yokoyama; @2015sun], and sudden changes in magnetic topology [@1994masuda; @2015sun; @2011cheng]. Recent observations reveal hot outflows with large velocities during flares, which can be seen in both images and spectra [@2007wang; @2013liu; @2015reeves]. Note that, the outflows in flare-related reconnection have been detected mostly in one direction at a time. In particular, simultaneous bidirectional outflows have been reported in transition region explosive events from an optically thin line, e.g., the 1393 line [@1997innes].
It is known that, theoretically, magnetic reconnection can also occur in the lower atmosphere [@1999sturrock]. The most recent observations of Ellerman bombs and hot explosions show bidirectional flows as revealed in chromospheric and coronal lines. These flows indicate magnetic reconnection in the photosphere or the temperature minimum region, which is locally heated to a high temperature comparable with that of the upper chromosphere or transition region [@2015vissers; @2014peter]. Nevertheless, we still lack enough observational evidence of magnetic reconnection in the lower atmosphere leading to flares.
In this Letter, we report observations of a reconnection scenario in the chromosphere that produces a microflare by investigating the H$\alpha$ and 8542 spectra. In Section 2, we present the observations. Data analysis and results are shown in Section 3, which is followed by a summary and discussions in Section 4.
Observations
============
A microflare of GOES B-class occurred in Active Region NOAA 12146 at 21:04 UT on 2014 August 24. This active region was very complex and had produced three C-class flares during the past three days. We observed this microflare with the Fast Imaging Solar Spectrograph (FISS, @2013chae) and the Near InfraRed Imaging Spectropolarimeter (NIRIS, @2012cao) installed at the 1.6 meter New Solar Telescope (NST, @2010cao [@2012goode]) of Big Bear Solar Observatory. FISS adopts an Echelle disperser, and can acquire both H$\alpha$ and 8542 line spectra along a slit simultaneously with the aid of dual cameras. The scan over the target region yields two-dimensional spectra and monochromatic images at different wavelengths. The spectral resolution is 0.19 m and 0.26 m for the two lines respectively, and the spatial sampling is $0\arcsec.16$. NIRIS can provide 10830 images with a high spatial resolution. It covers a field of view (FOV) of about $85\arcsec\times85\arcsec$ with a pixel size of about $0\arcsec.083$. The time cadence is 35 s for FISS and 10 s for NIRIS.
The Atmospheric Imaging Assembly (AIA, @2012lemen) on the Solar Dynamics Observatory (SDO, @2012pesnell) can provide extreme ultraviolet (EUV) images with a pixel size of $0\arcsec.6$ and a cadence of 12 s, and the Helioseismic and Magnetic Imager (HMI, @2012schou) provides both line-of-sight magnetograms and vector field data in the photosphere with the same pixel size but a cadence of 45 s and 12 min respectively.
Before data analysis, we need to co-align the images from different instruments. Note that for the NST data, a de-rotation of images should be performed first. The basic method used for co-alignment is to calculate the correlation coefficients of two images with different offsets, and to find the optimal offset value that corresponds to the maximum coefficient. A sub-region of images containing the most distinguishable feature like a sunspot is used for co-alignment. In practice, we choose a base (reference) image and co-align all the other images with that. As the morphology of the sunspot does not change much during our observations, this code can yield good results. For co-alignment of data from NST and SDO , we use the AIA 1700 image as the reference one. The accuracy of all the co-alignment is within $1\arcsec.2$.
Fig. \[fig1\](a) shows the lightcurves of the microflare in two channels of AIA, as well as the lightcurve of H$\alpha$ (wavelength-integrated intensity). It is seen that the microflare lasts for about ten minutes. At the beginning of the flare, a set of flare loops appear in both the 304 Å and 94 Å channels. In particular, one can notice two bright loops that are located closely in space (Fig. \[fig1\](b)-(c)). The left (southeastern) loop reaches its peak emission about two minutes earlier than the right (northwestern) one (Fig. \[fig1\](a)). After the flare peak, the loops gradually cool down. Here, we focus on the brightest part of the left loop, which produces the first peak of EUV emission and is likely the site of flare energy release (magnetic reconnection).
On the other hand, the magnetic field around the flare region (Fig. \[fig1\](d)) is very complicated, showing adjacent opposite polarities and parasitic polarities. It is known that such a complicated magnetic structure is a favorable place for flare occurrence.
The flare seen in H$\alpha$ and 8542 brightens nearly simultaneously and cospatially with that in 304 Å and 94 Å during the flare peak time (Fig. \[fig1\](e)-(f)). However, the H$\alpha$ and 8542 emission is relatively restricted to particular sites where the chromosphere (formation layer of the two lines) gets significantly heated. It is interesting that in the 10830 images, there appears a dark thread-like structure in addition to some brightenings seen below the threads (Fig. \[fig1\](g)). The dark threads seem to have a twist and experience an untwisting with the flare development.
Data Analysis and Results
=========================
Spectral Analysis of the Flare Region
-------------------------------------
In the flaring region, the chromospheric H$\alpha$ and 8542 lines show obvious asymmetries (see Figs. \[fig3\] and \[fig4\]). In particular, the H$\alpha$ line profile exhibits broadened wings that are higher than the continuum level, and the 8542 line profile shows a hump in either the red or the blue wing. For these asymmetric line profiles showing a net emission, their contrast profiles, the profiles on the flare subtracted by the pre-flare ones, can be used to study the change in profiles, which is caused by the flare [@1987canfield]. Applying the radiative transfer equation with a constant source function $S$ for the flare yields: $$I_{f}=I_{0}\exp(-\tau_{f})+S[1-\exp(-\tau_{f})],$$ where $I_{f}$ is the emergent intensity from the flare, $I_{0}$ is the background intensity from the photosphere, and $\tau_{f}$ is the optical depth of the flare-perturbed atmosphere. At the far wings, the chromosphere is normally transparent, so that the background intensity can roughly be replaced by the pre-flare intensity $I_{n}$, and $\tau_{f}$ is sufficiently less than unity [@1984ichimoto]. Then the equation above can be simplified as $$I_{f}-I_{n}\approx(S-I_{n})\tau_{f}.\label{eq2}$$ Therefore, such contrast profiles, in particular at the far wings, can be regarded as the net emission from the flaring plasma at a height where the magnetic energy is released, namely the reconnection site.
We first apply the bisector method to derive the mass flow velocities from the Doppler shifts of the H$\alpha$ line. The bisector method, also referred to as lambdameter, is a simple but often used method to derive Doppler velocities from optically thick line profiles [@2013chaeb]. This method makes a horizontal cut at a certain emission intensity on the profile $C(\lambda)$ so that $$C\left(\lambda_{m}-\frac{\delta\lambda_{b}}{2}\right)=C\left(\lambda_{m}+\frac{\delta\lambda_{b}}{2}\right),$$ where $\delta\lambda_{b}$ is the full width of the line cut, and $\lambda_{m}$ is considered as the observed line center. Then the Doppler velocities are derived from the shifts of the line center relative to the theoretical (reference) one. Since the bisector is usually not a straight line, different values of velocities can be obtained for horizontal cuts at different intensity levels [@2014hong], i.e. from line center to wings, which are supposed to represent different heights of the solar atmosphere. Choosing the level for the horizontal cut should balance two requirements as below. On one hand, one needs to choose as far as possible the line wings so that Equation (\[eq2\]) can still hold. On the other hand, the noise becomes large at the far wings that can yield a large uncertainty in the derived velocities. For our event, we choose the horizontal cut at line wings with a net emission intensity of 0.2 times the maximum intensity to avoid the invalidity of the method towards the line center as well as the noise at far wings [@1995ding]. Such an intensity level corresponds to the chromospheric level where microflares likely occur [@2006fang]. Note that our first purpose is to provide a velocity map through a preliminary analysis of the line profiles. Changing the level of the horizontal cut a little can only change slightly the absolute value but not the direction of the derived velocity.
The velocity map at the flare peak time (Fig. \[fig1\](e)) reveals clearly both the downflows and upflows in the flare region, which are located very close to the bright regions seen in H$\alpha$ and 8542 , also the brightest part of the left flare loop seen in 304 Å and 94 Å (Fig. \[fig1\](b)-(c)). Note that we actually get a time series of velocity maps during the flare, which show that such flows appear impulsively in the flare rise phase and become relatively steady for about two minutes. We select a region of interest (ROI) containing one brightest part with striking opposite velocity signs nearby for further study (the yellow square in Fig. \[fig1\](e)). The size of the ROI is $4\arcsec\times5\arcsec$.
We divide the ROI into $10\times20$ grids and show the H$\alpha$ contrast profiles at each grid point in a time sequence (Fig. \[fig2\]). Line asymmetries are clearly seen during the flare peak time. For example, in a number of subpanels (columns 6–8, row 8), blueshifted emission is seen in the top region while redshifted emission appears in the bottom region. Though the redshifted emission seems stronger than the blueshifted emission, the speed derived from them are similar. The red and blue asymmetries appear nearly simultaneously and at locations only separated by about $3\arcsec$, implying that they are physically related with each other.
For a more quantitative study, we focus on some typical H$\alpha$ line spectra with significant asymmetries along both the scan and slit directions (Fig. \[fig3\](a)), and plot the contrast profiles at two specific positions during the flare peak time (Fig. \[fig3\](c) and (f)). It is interesting that these asymmetric profiles consist of two components, a main (static) component and a second (redshifted or blueshifted) one. Based on the shapes of the line profiles observed in our event, we use a multi-component function, which comprises two Gaussian-shaped components plus a constant background, to fit the contrast profile: $$C(\lambda)=A_{0}+A_{1}\textrm{exp}\left[-\left(\frac{\lambda-\lambda_{1}}{\sigma_{1}}\right)^{2}\right]
+A_{2}\textrm{exp}\left[-\left(\frac{\lambda-\lambda_{2}}{\sigma_{2}}\right)^{2}\right],$$ where $A_{0}$ is the background, $A_{1}$ and $A_{2}$ are the strengths, and $\sigma_{1}$ and $\sigma_{2}$ are the widths of the two components, respectively. In particular, $\lambda_{1}$ and $\lambda_{2}$ are the fitted line centers of the two components.
The procedure ‘mpfitfun.pro’ in the Solar SoftWare package, which can fit a user-supplied model to data using the Levenberg-Marquadt algorithm to solve the least-squares problem, is used for the double Gaussian fitting. The results show Doppler velocities of $\pm$(70–80) km s$^{-1}$ from the second components. The related downflows and upflows should be located in the chromosphere where the H$\alpha$ line is formed [@2012leenaarts]. Such a speed is comparable with the local Alfvén speed in the chromosphere [@2008nishizuka]. Note that the 8542 profiles show similar asymmetries but with somewhat smaller velocities ($\pm$(40–50) km s$^{-1}$) from the second components (Fig. \[fig4\]). These significant bidirectional flows in the flare region are most likely the outflows of magnetic reconnection.
The Cause of Line Asymmetries
-----------------------------
Usually, it is difficult to interpret the asymmetries of optically thick lines, in particular the H$\alpha$ line. For the blue asymmetry (larger blue wing emission), it can either be caused by an upward motion of a heated plasma (in emission) or a downward motion of a relatively cool plasma (in absorption). Likewise, the red asymmetry (larger red wing emission) can be produced by either a downward moving plasma in emission or an upward moving plasma in absorption. Such a scenario has been clearly verified by non-LTE calculations [@1994heinzel] and more recently by radiative hydrodynamic simulations of the flare atmosphere [@2015kuridze]. However, we tend to verify from multi-aspects that in our event, the blue and red asymmetries are related to upflows and downflows of the heated plasma in emission, respectively. First, as can be seen from the asymmetric profiles, there appears a big hump in the blue wing (Fig. \[fig3\](c) and Fig. \[fig4\](c)) or in the red wing (Fig. \[fig3\](f) and Fig. \[fig4\](f)). This is a strong signature of existence of a Doppler-shifted emission component superposed on a static profile at the blue or red wings, while it can hardly be produced by an absorption at the opposite wings. Second, the time evolutions of the line wing intensities hint an emission cause of the line asymmetries. For the red asymmetric profiles, the intensity of the red wing rises more sharply than the blue wing when the flare begins (Fig. \[fig3\](d) and Fig. \[fig4\](d)). This is contrary to the line asymmetries of an absorption cause, in which one might expect a temporary decrease of the intensities at the opposite wings if the increase due to flare heating and the drop due to absorption are not exactly coincident. Third, previous model calculations showed that the line asymmetries may change their sign with the development of the flare owing to the different heating status of the moving plasma [@1994heinzel; @2015kuridze]. In our event, however, the line asymmetries do not change their sign during the whole flare. Fourth, the same asymmetries appear not only in the H$\alpha$ line but also in the 8542 line. The latter is formed somewhat lower than the former. To our knowledge, there are no model calculations up to now showing that the blue/red asymmetries of the 8542 line can be due to downward/upward moving cool plasma in absorption, though theoretically also possible. Finally, in order to further check the possibility of an absorption cause of the line asymmetries, we also do a test to fit the line profiles with a static component plus a negative shifted component (due to absorption) at the opposite wings. This fitting fails to yield reasonable results with the fitting errors much larger than the case of a static component plus a positive shifted component as done in Section 3.1. Considering the above facts, we think that the blue/red line asymmetries in this flare are caused by upward/downward motions of heated mass, which are likely related to reconnection outflows, as will be discussed below.
As for the cause of the mass flows in a flare, one would naturally think of the possibility of chromospheric evaporation and condensation. However, we can exclude this possibility for the following reasons. First, the line-of-sight velocity along a slit, derived from the sequence of H$\alpha$ and 8542 contrast profiles, shows a gradual (nearly linear) variation in magnitude (Fig. \[fig3\](a) and Fig. \[fig4\](a)). Such a spatial distribution can hardly be explained by the spatially different evaporations, but is more likely a result of plasma acceleration. Second, previous observations show that chromospheric condensation, thought to be responsible for the red asymmetries in H$\alpha$ profiles, rapidly decreases before the H$\alpha$ intensity reaches its maximum [@1984ichimoto]. But in our event, the H$\alpha$ red asymmetries seem to exist longer. Third, the most recent radiative hydrodynamic simulations of solar flares show that the H$\alpha$ and 8542 lines only exhibit red asymmetries in response to chromospheric condensation in the early heating phase, but that they hardly exhibit blue asymmetries responding to chromospheric evaporation containing plasma of very high temperatures [@2015rubiodacosta]. This is apparently different from what are revealed in our observations, where the red and blue asymmetries appear almost simultaneously in both chromospheric lines.
Magnetic Field Extrapolation
----------------------------
The three-dimensional magnetic field structure of this flare region is constructed using nonlinear force-free field extrapolations with the optimization method [@2000wheatland; @2004wiegelmann]. The vector magnetic field on the bottom boundary comes from the Space-weather HMI Active Region Patches (SHARP, @2014bobra), in which the $180^{\circ}$ ambiguity of transverse components has already been removed using the minimum energy method [@2009leka] and the projection effect has been corrected [@1990gary]. The data have been remapped to a heliographic cylindrical equal-area coordinate system. An additional preprocessing is applied for the vector field to meet the force-free and torque-free conditions [@2006wiegelmann]. The extrapolation box consists of $230\times200\times200$ uniform grid points in a box, corresponding to a domain of about $175\times152\times152$ Mm$^{3}$.
The extrapolation result shows that, before the flare onset, there exist twisted magnetic field lines above the parasitic polarities, which lie along the polarity inversion line (PIL) and overlap well with the dark thread-like structure seen in the 10830 images (Fig. \[fig5\]). This clearly indicates the existence of a flux rope in the lower solar atmosphere [@2015wang]. The height of the flux rope is about 800–1000 km above the solar surface, as judged from both the magnetic structure and the 10830 dark threads. Such a low-lying flux rope can serve as a cause of the onset of the reconnection in the chromosphere by its instabilities [@2005torok; @2010aulanier; @2006kliem]. Note that the flare is a confined one since the flux rope failed to erupt finally. And we did not see magnetic flux emergence in the flare region under present resolution.
Summary and Discussions
=======================
We study a microflare based on both high-resolution spectroscopic and imaging observations, focusing on the observational evidence of magnetic reconnection in the chromosphere. The H$\alpha$ contrast profiles show obvious excess emission in the red or blue wings, with a Doppler velocity of $\pm$(70–80) km s$^{-1}$. The 8542 contrast profiles show similar asymmetric patterns with, however, a slightly lower velocity. Such a difference could originate from the different formation heights of the two lines [@2012leenaarts; @2008cauzzi]. Nevertheless, the qualitatively similar behaviours shown in the two chromospheric lines do suggest that the mass flows lie in the chromosphere. The co-existence of both upflows and downflows, located closely in space and with speeds comparable to the local Alfvén speed, indicates that they are a pair of outflows resulting from the reconnection in the chromosphere.
Previous observations have yielded spectroscopic measurements of reconnection outflows. @2007wang showed fast flows with speeds of $\sim$900–3500 km s$^{-1}$ in a flare. Redshifted outflows of about 125 km s$^{-1}$ in a flare were also reported [@2014tian]. In addition, @1997innes showed bidirectional outflows ($\sim$100 km s$^{-1}$) in transition region explosive events. In our event, we find bidirectional outflows with a Doppler velocity of $\pm$(70–80) km s$^{-1}$ in the H$\alpha$ line. From the magnetic field extrapolation, the flux rope is found to lie at a height of $\sim$900 km, where the ambient magnetic field is $\sim$600 G and the mass density is $2.34\times10^{-10}$ g cm$^{-3}$ if we take the F1 model of a weak flare [@1980machado] as a reference model. Then, the local Alfvén speed at the reconnection site, if just below the flux rope, is calculated to be $\sim$110 km s$^{-1}$. The speeds of the bidirectional outflows are then roughly comparable with the local Alfvén speed, indicating that the outflows are most likely driven by the flare-related magnetic reconnection.
Therefore, our results show clear evidence of magnetic reconnection in the chromosphere leading to the occurrence of a microflare. The chain of evidence includes the trigger of reconnection, a twisted flux rope above the flaring site, and the effect of reconnection, bidirectional outflows. In fact, magnetic reconnection in the lower atmosphere has been suggested to be the cause of some small-scale activities like Ellerman bombs [@2009archontis; @2015vissers] and hot explosions [@2014peter]. Our observations suggest that flare-related magnetic reconnection can also proceed in the chromosphere. Moreover, we confirm that the basic mechanisms for a microflare are similar to those for a major flare occurring mostly in the corona, regardless of the quite different energies between them.
We are very grateful to the referee for valuable comments that helped improve the paper. The observation program was supported by the Strategic Priority Research Program — The Emergence of Cosmological Structures of the Chinese Academy of Sciences, Grant No. XDB09000000. The authors thank the BBSO staff for their help during the observations. SDO is a mission of NASA’s Living With a Star Program. This work was also supported by NSFC under grants 11303016, 11373023, 11403011 and 11533005, and NKBRSF under grants 2011CB811402 and 2014CB744203. W. C. acknowledges the support of the US NSF (AGS-0847126) and NASA (NNX13AG14G). J. H. would also like to thank Donguk Song for his help in FISS data analysis.
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|
---
abstract: 'We provide an analytic expression for the quantity described in the title. Namely, we perform a preferential attachment growth process to generate a scale-free network. At each stage we add a new node with $m$ new links. Let $k$ denote the degree of a node, and $N$ the number of nodes in the network. The degree distribution is assumed to converge to a power-law (for $k\geq m$) of the form $k^{-\gamma}$ and we obtain an exact implicit relationship for $\gamma$, $m$ and $N$. We verify this with numerical calculations over several orders of magnitude. Although this expression is exact, it provides only an implicit expression for $\gamma(m)$. Nonetheless, we provide a reasonable guess as to the form of this curve and perform curve fitting to estimate the parameters of that curve — demonstrating excellent agreement between numerical fit, theory, and simulation.'
author:
- Michael Small
title: Expected degree of finite preferential attachment networks
---
Expected degree {#expected-degree .unnumbered}
===============
Preferential attachment [@aB99] is the archetypal growth mechanism for scale-free networks. Asymptotically, under certain circumstances, such network produce a degree distribution which converges asymptotically to a power law with exponent $3$. But this is not true in general, and it is not true for arbitrary finite networks generated along the way. In this note we derive straightforward analytic results for the expected exponent $\gamma$ of a scale free network with power law degree distribution $p(k)\propto k^{-\gamma}$.
We assume that the network is grown with a Barabási-Albert attachment process as described in [@aB99]. With each new node we add $m$ links and the growth process is terminated when the network has $N$ nodes. We make the approximation that the degree distribution of this finite networks follows a truncated power-law with some exponent $\gamma$.
Hence, a preferential attachment (PA) network with minimum degree $m$ will add exactly $m$ new links for each new node. The expected degree $$\begin{aligned}
\label{spower2} E(k)&=&2m\end{aligned}$$ (since each link has two ends and contributed to the degree of two nodes). Conversely, the probability that a node has degree $k$ is given by $$\begin{aligned}
P(k|\gamma,d) &=&\left\{\begin{array}{cc}
0 & k< m\\
\frac{k^{-\gamma}}{K(\gamma)} & k\geq m
\end{array}
\right.\end{aligned}$$ where the normalization factor $K(\gamma)$ is inconvenient. However $$\begin{aligned}
\zeta({\gamma} )&=& \left(\sum_{k=1}^{m-1} +\sum_{k=m}^\infty\right)k^{-\gamma} \\
&=&\sum_{k=1}^{m-1}k^{-\gamma}+K(\gamma)\end{aligned}$$ and hence it is easily computable.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Left panel: Expected values of $\gamma$ as a function of $m$ (Eqn. (\[egam\])) (heavy line) and estimated values of $\gamma$ from $30$ independent realisations of PA networks of size $N$ (mean $\pm$ standard deviation). We take $d\in[1,10]$ and $N=10^3$ (red), $10^4$ (green), $10^5$ (blue). Right panel: $\gamma$ as a function of $m$ computed via the solution of (\[mgamm2\]) (stars) and estimated from a function fit of the form $\hat\gamma(m)= 3-(m+\alpha)^{-\beta}$. The best fit (obtained from a fit on $m\in[1,10]$) is then extrapolated over the domain. Parameter values are $\alpha= 0.9205$ and $\beta=0.9932$. []{data-label="fig_gd"}](fig_gd "fig:"){width="45.00000%"} ![Left panel: Expected values of $\gamma$ as a function of $m$ (Eqn. (\[egam\])) (heavy line) and estimated values of $\gamma$ from $30$ independent realisations of PA networks of size $N$ (mean $\pm$ standard deviation). We take $d\in[1,10]$ and $N=10^3$ (red), $10^4$ (green), $10^5$ (blue). Right panel: $\gamma$ as a function of $m$ computed via the solution of (\[mgamm2\]) (stars) and estimated from a function fit of the form $\hat\gamma(m)= 3-(m+\alpha)^{-\beta}$. The best fit (obtained from a fit on $m\in[1,10]$) is then extrapolated over the domain. Parameter values are $\alpha= 0.9205$ and $\beta=0.9932$. []{data-label="fig_gd"}](fig_gd2 "fig:"){width="45.00000%"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The expected degree is $$\begin{aligned}
\nonumber E(k) &=&\sum_{k=1}^\infty kP(k|\gamma)\\
\label{egam}
&=& \frac{\sum_{k=m}^\infty k^{1-\gamma}}{\zeta(\gamma)-\sum_{k=1}^{m-1}k^{-\gamma}}
\end{aligned}$$ Equating (\[spower2\]) and (\[egam\]), we have that the asymptotic value of $\gamma$ satisfies $$\begin{aligned}
\label{mgamm}
\zeta(\gamma) &=& \sum_{k=1}^{m-1}k^{-\gamma}+\frac{1}{2m}\sum_{k=m}^\infty k^{1-\gamma}.
\end{aligned}$$ Replacing the RHS of (\[mgamm\]) with the corresponding infinite sum and cancelling identical terms we obtain $$\begin{aligned}
\label{mgamm2}
\sum_{k=m}^\infty (2m-k)k^{-\gamma}&=&0\end{aligned}$$ Solving (\[mgamm2\]) allows us to determine the expected value of $\gamma $ for the PA algorithm with a particular choice of minimum degree $d$. In particular, for $m=1$ we recover $2\zeta({\gamma})=E(k).$
In Fig. \[fig\_gd\] we illustrate the agreement between sample preferential attachment networks of various sizes and the prediction of (\[mgamm2\]). The curve appears to be asymptotic to $\gamma=3$ and so we fit a function of the form $\hat\gamma(m)= 3-(m+\alpha)^{-\beta}$ to the solution of the series (\[mgamm2\]). We obtain that $$\gamma(m)\approx 3-\frac{1}{(m+0.925)^{0.9932}}.$$ These results are required to explain expected degree distributions observed in a related work [@stars], and in that case also show excellent agreement.
Acknowledgements {#acknowledgements .unnumbered}
================
MS is supported by an Australian Research Council Future Fellowship (FT110100896)
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12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [****, ()]{} @noop [**** ()]{}
|
---
abstract: 'We discuss the local index formula of Connes–Moscovici for the isospectral noncommutative geometry that we have recently constructed on quantum $SU(2)$. We work out the cosphere bundle and the dimension spectrum as well as the local cyclic cocycles yielding the index formula.'
author:
- |
Walter van Suijlekom,$^1$ Ludwik Dabrowski,$^1$ Giovanni Landi,$^2$\
Andrzej Sitarz,$^3$ Joseph C. Várilly$^4$\
$^1\,$Scuola Internazionale Superiore di Studi Avanzati,\
Via Beirut 2–4, 34014 Trieste, Italy\
$^2\,$Dipartimento di Matematica e Informatica, Università di Trieste,\
Via Valerio 12/1, 34127 Trieste\
and INFN, Sezione di Napoli, Napoli, Italy\
$^3\,$Institute of Physics, Jagiellonian University,\
Reymonta 4, 30–059 Kraków, Poland\
$^4\,$Departamento de Matemática, Universidad de Costa Rica,\
2060 San José, Costa Rica
title: 'The local index formula for $SU_q(2)$'
---
*Key words and phrases*: Noncommutative geometry, spectral triple, quantum $SU(2)$.
*Mathematics Subject Classification:* Primary 58B34; Secondary 17B37.
Introduction
============
Recent investigations show that the “quantum space” underlying the quantum group $SU_q(2)$ is an important arena for testing and implementing ideas coming from noncommutative differential geometry. In [@Naiad] it has been endowed with an isospectral tridimensional geometry via a bi-equivariant $3^+$-summable spectral triple $({\mathcal{A}}(SU_q(2)),{\mathcal{H}},D)$. Earlier, a “singular” (in the sense of not admitting a commutative limit) spectral triple was constructed in [@ChakrabortyPEqvt]. The latter geometry was put in the general theory of Connes–Moscovici [@ConnesMIndex] by a systematic discussion of the local index formula [@ConnesSUq]. In this paper, we present a similar analysis for the former geometry. It turns out that most of the results coincide with those of [@ConnesSUq].
The main idea of that paper is to construct a (quantum) cosphere bundle ${\mathbb{S}}_q^*$ on $SU_q(2)$, that considerably simplifies the computations concerning the local index formula. Essentially, with the operator derivation $\delta$ defined by $\delta(T) := |D|T - T|D|$, one considers an operator $x$ in the algebra ${\mathcal{B}}= \bigcup_{n=0}^\infty \delta^n({\mathcal{A}})$ up to smoothing operators; these give no contribution to the residues appearing in the local cyclic cocycle giving the local index formula. The removal of the irrelevant smoothing operators is accomplished by introducing a symbol map from $SU_q(2)$ to the cosphere bundle ${\mathbb{S}}_q^*$. The latter is defined by its algebra $C^\infty({\mathbb{S}}_q^*)$ of “smooth functions” which is, by definition, the image of a map $$\rho : {\mathcal{B}}\to C^\infty(D^2_{q+} {\times}D^2_{q-} {\times}{\mathbb{S}}^1)$$ where $D^2_{q\pm}$ are two quantum disks. One finds that an element $x$ in the algebra ${\mathcal{B}}$ can be determined up to smoothing operators by $\rho(x)$.
In our present case, the cosphere bundle coincides with the one obtained in [@ConnesSUq]; the same being true for the dimension spectrum. Indeed, using this much simpler form of operators up to smoothing ones, it is not difficult to compute the dimension spectrum and obtain simple expressions for the residues appearing in the local index formula. We find that the dimension spectrum is simple and given by the set $\{1,2,3\}$.
The cyclic cohomology of the algebra ${\mathcal{A}}(SU_q(2)$ has been computed explicitly in [@MasudaNW] where it was found to be given in terms of a single generator. We express this element in terms of a single local cocycle similarly to the computations in [@ConnesSUq]. But contrary to the latter, we get an extra term involving $P |D|^{-3}$ which drops in [@ConnesSUq], being traceclass for the case considered there. Here $P = {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}(1 + F)$ with $F = \operatorname{Sign}D$, the sign of the operator $D$.
Finally as a simple example, we compute the Fredholm index of $D$ coupled with the unitary representative of the generator of $K_1({\mathcal{A}}(SU_q(2)))$.
The isospectral geometry of $SU_q(2)$ {#sec:iso-ge}
=====================================
We recall the construction of the spectral triple $({\mathcal{A}}(SU_q(2)),{\mathcal{H}},D)$ of [@Naiad]. Let ${\mathcal{A}}= {\mathcal{A}}(SU_q(2))$ be the $*$-algebra generated by $a$ and $b$, subject to the following commutation rules: $$\begin{gathered}
ba = q ab, \qquad b^*a = qab^*, \qquad bb^* = b^*b,
{\nonumber}\\
a^*a + q^2 b^*b = 1, \qquad aa^* + bb^* = 1.
\label{eq:suq2-relns}\end{gathered}$$ In the following we shall take $0 < q < 1$. Note that we have exchanged $a {\leftrightarrow}a^*$, $b {\leftrightarrow}-b$ with respect to the notation of [@ChakrabortyPEqvt] and [@ConnesSUq].
The Hilbert space of spinors ${\mathcal{H}}$ has an orthonormal basis labelled as follows. For each $j = 0,{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}},1,\dots$, we abbreviate $j^+ = j + {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}$ and $j^- = j - {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}$. The orthonormal basis consists of vectors ${|j\mu n{{\mathord{\uparrow}}}\rangle}$ for $j = 0,{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}},1,\dots$, $\mu = -j,\dots,j$ and $n = -j^+,\dots,j^+$; together with ${|j\mu n{{\mathord{\downarrow}}}\rangle}$ for $j={{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}},1,\dots$, $\mu = -j,\ldots,j$ and $n = -j^-,\dots,j^-$. We adopt a vector notation by juxtaposing the pair of spinors $${|j\mu n\rangle\!\rangle} := \begin{pmatrix} {|j\mu n{{\mathord{\uparrow}}}\rangle} \\[2\jot]
{|j\mu n{{\mathord{\downarrow}}}\rangle} \end{pmatrix},
\label{eq:kett-defn}$$ and with the convention that the lower component is zero when $n = \pm(j + {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}})$ or $j = 0$. In this way, we get a decomposition ${\mathcal{H}}= {\mathcal{H}}^{{\mathord{\uparrow}}}\oplus {\mathcal{H}}^{{\mathord{\downarrow}}}$ into subspaces spanned by the “up” and “down” kets, respectively.
The spinor representation is the $*$-representation $\pi$ of ${\mathcal{A}}$ on ${\mathcal{H}}$ –denoted by $\pi'$ in [@Naiad]– defined as follows. We set $\pi(a) := a_+ + a_-$ and $\pi(b) := b_+ + b_-$, where $a_\pm$ and $b_\pm$ are the following operators in ${\mathcal{H}}$: $$\begin{aligned}
a_+ \,{|j\mu n\rangle\!\rangle}
&:= q^{(\mu+n-{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}})/2} [j + \mu + 1]^{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}\begin{pmatrix}
q^{-j-{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}} \, \frac{[j+n+{{\mathchoice{{{\tfrac{3}{2}}}}{{{\tfrac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}}}]^{1/2}}{[2j+2]} & 0 \\[2\jot]
q^{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}\,\frac{[j-n+{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}]^{1/2}}{[2j+1]\,[2j+2]} &
q^{-j} \, \frac{[j+n+{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}]^{1/2}}{[2j+1]}
\end{pmatrix} {|j^+ \mu^+ n^+\rangle\!\rangle},
{\nonumber}\\
a_- \,{|j\mu n\rangle\!\rangle}
&:= q^{(\mu+n-{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}})/2} [j - \mu]^{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}\begin{pmatrix}
q^{j+1} \, \frac{[j-n+{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}]^{1/2}}{[2j+1]} &
- q^{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}\,\frac{[j+n+{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}]^{1/2}}{[2j]\,[2j+1]} \\[2\jot]
0 & q^{j+{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}} \, \frac{[j-n-{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}]^{1/2}}{[2j]}
\end{pmatrix} {|j^- \mu^+ n^+\rangle\!\rangle},
{\nonumber}\\
b_+ \,{|j\mu n\rangle\!\rangle}
&:= q^{(\mu+n-{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}})/2} [j + \mu + 1]^{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}\begin{pmatrix}
\frac{[j-n+{{\mathchoice{{{\tfrac{3}{2}}}}{{{\tfrac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}}}]^{1/2}}{[2j+2]} & 0 \\[2\jot]
- q^{-j-1} \,\frac{[j+n+{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}]^{1/2}}{[2j+1]\,[2j+2]} &
q^{-{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}} \, \frac{[j-n+{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}]^{1/2}}{[2j+1]}
\end{pmatrix} {|j^+ \mu^+ n^-\rangle\!\rangle},
{\nonumber}\\
b_- \,{|j\mu n\rangle\!\rangle}
&:= q^{(\mu+n-{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}})/2} [j - \mu]^{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}\begin{pmatrix}
- q^{-{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}} \, \frac{[j+n+{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}]^{1/2}}{[2j+1]} &
- q^j \,\frac{[j-n+{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}]^{1/2}}{[2j]\,[2j+1]} \\[2\jot]
0 & - \frac{[j+n-{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}]^{1/2}}{[2j]}
\end{pmatrix} {|j^- \mu^+ n^-\rangle\!\rangle}.
\label{eq:spin-rep}\end{aligned}$$ Here $[N] := (q^{-N} - q^N)/(q^{-1} - q)$ is a “$q$-integer”.
The Dirac operator $D$ that was exhibited in [@Naiad] is diagonal in the given orthonormal basis of ${\mathcal{H}}$, and is one of a family of selfadjoint operators of the form $$D {|j\mu n\rangle\!\rangle} = \begin{pmatrix} d^{{\mathord{\uparrow}}}j + c^{{\mathord{\uparrow}}}& 0 \\
0 & d^{{\mathord{\downarrow}}}j + c^{{\mathord{\downarrow}}}\end{pmatrix} {|j\mu n\rangle\!\rangle},
\label{eq:linear-evs}$$ where $d^{{\mathord{\uparrow}}}, d^{{\mathord{\downarrow}}}, c^{{\mathord{\uparrow}}}, c^{{\mathord{\downarrow}}}$ are real numbers not depending on $j,\mu,n$. In order that the sign of $D$ be nontrivial we need to assume $d^{{\mathord{\downarrow}}}d^{{\mathord{\uparrow}}}< 0$, so we may as well take $d^{{\mathord{\uparrow}}}> 0$ and $d^{{\mathord{\downarrow}}}< 0$.
Apart from the issue of their signs, the particular constants that appear in are fairly immaterial: $c^{{\mathord{\uparrow}}}$ and $c^{{\mathord{\downarrow}}}$ do not affect the index calculations later on while $d^{{\mathord{\uparrow}}}$ and $|d^{{\mathord{\downarrow}}}|$ yield scaling factors on some noncommutative integrals. Thus little generality is lost by making the following choice, $$D {|j\mu n\rangle\!\rangle} = \begin{pmatrix} 2j + {{\mathchoice{{{\tfrac{3}{2}}}}{{{\tfrac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}}}& 0 \\
0 & -2j - {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}\end{pmatrix} {|j\mu n\rangle\!\rangle}.
\label{eq:classical-evs}$$ whose spectrum (with multiplicity!) coincides with that of the classical Dirac operator of the sphere ${\mathbb{S}}^3$ equipped with the round metric (indeed, the spin geometry of the 3-sphere can now be recovered by taking $q = 1$).
We let $D = F\,|D|$ be the polar decomposition of $D$ where $|D| := (D^2)^{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}$ and $F = \operatorname{Sign}D$. Explicitly, we see that $$F{|j\mu n\rangle\!\rangle}
= \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} {|j\mu n\rangle\!\rangle},
\qquad
|D|\,{|j\mu n\rangle\!\rangle} = \begin{pmatrix} 2j + {{\mathchoice{{{\tfrac{3}{2}}}}{{{\tfrac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}}}& 0 \\
0 & 2j + {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}\end{pmatrix} {|j\mu n\rangle\!\rangle}.$$ Clearly, $P^{{\mathord{\uparrow}}}:= {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}(1 + F)$ and $P^{{\mathord{\downarrow}}}:= {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}(1 - F) = 1 - P^{{\mathord{\uparrow}}}$ are the orthogonal projectors whose range spaces are ${\mathcal{H}}^{{\mathord{\uparrow}}}$ and ${\mathcal{H}}^{{\mathord{\downarrow}}}$, respectively.
\[pr:regular\] The triple $({\mathcal{A}}(SU_q(2)),{\mathcal{H}},D)$ is a regular $3^+$-summable spectral triple.
It was already shown in [@Naiad] that this spectral triple is $3^+$-summable: indeed, this follows easily from the growth of the eigenvalues in . The remaining issue is its regularity. Recall [@CareyPRS; @ConnesMIndex; @Polaris] that this means that the algebra generated by ${\mathcal{A}}$ and $[D,{\mathcal{A}}]$ should lie within the smooth domain $\bigcap_{n=0}^\infty \operatorname{Dom}\delta^n$ of the operator derivation $\delta(T) := |D|T - T|D|$.
Since $2j + {{\mathchoice{{{\tfrac{3}{2}}}}{{{\tfrac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}}}= 2j^+ + {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}$ and $2j + {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}= 2j^- + {{\mathchoice{{{\tfrac{3}{2}}}}{{{\tfrac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}}}$ and due to the triangular forms of the matrices in , the off-diagonal terms vanish in the $2 {\times}2$-matrix expressions for $\delta(a_+)$ and $\delta(a_-)$. Indeed one finds, $$\begin{aligned}
\delta(a_+){|j\mu n\rangle\!\rangle}
&= \begin{pmatrix} 2j + \tfrac{5}{2} & 0 \\
0 & 2j + {{\mathchoice{{{\tfrac{3}{2}}}}{{{\tfrac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}}}\end{pmatrix} a_+ {|j\mu n\rangle\!\rangle}
- a_+ \begin{pmatrix} 2j + {{\mathchoice{{{\tfrac{3}{2}}}}{{{\tfrac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}}}& 0 \\
0 & 2j + {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}\end{pmatrix} {|j\mu n\rangle\!\rangle},
\\
\delta(a_-){|j\mu n\rangle\!\rangle}
&= \begin{pmatrix} 2j + {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}& 0 \\
0 & 2j - {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}\end{pmatrix} a_- {|j\mu n\rangle\!\rangle}
- a_- \begin{pmatrix} 2j + {{\mathchoice{{{\tfrac{3}{2}}}}{{{\tfrac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}}}& 0 \\
0 & 2j + {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}\end{pmatrix} {|j\mu n\rangle\!\rangle}.\end{aligned}$$ In both cases we obtain $$\delta(a_+) = P^{{\mathord{\uparrow}}}a_+ P^{{\mathord{\uparrow}}}+ P^{{\mathord{\downarrow}}}a_+ P^{{\mathord{\downarrow}}}, \qquad
\delta(a_-) = - P^{{\mathord{\uparrow}}}a_- P^{{\mathord{\uparrow}}}- P^{{\mathord{\downarrow}}}a_- P^{{\mathord{\downarrow}}}.
\label{eq:delta-apm}$$ Replacing $a$ by $b$, the same triangular matrix structure leads to $$\delta(b_+) = P^{{\mathord{\uparrow}}}b_+ P^{{\mathord{\uparrow}}}+ P^{{\mathord{\downarrow}}}b_+ P^{{\mathord{\downarrow}}}, \qquad
\delta(b_-) = - P^{{\mathord{\uparrow}}}b_- P^{{\mathord{\uparrow}}}- P^{{\mathord{\downarrow}}}b_- P^{{\mathord{\downarrow}}}.
\label{eq:delta-bpm}$$ Thus $\delta(\pi(a)) = \delta(a_+) + \delta(a_-)$ is bounded, with $\|\delta(\pi(a))\| \leq \|\pi(a)\|$; and likewise for $\pi(b)$. Next, $\delta([D,a_+]) = [D,\delta(a_+)]$, so that $$\delta([D,a_+]) {|j\mu n\rangle\!\rangle}
= \begin{pmatrix} 2j + \tfrac{5}{2} & 0 \\
0 & -2j - {{\mathchoice{{{\tfrac{3}{2}}}}{{{\tfrac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}}}\end{pmatrix} \delta(a_+) {|j\mu n\rangle\!\rangle}
- \delta(a_+) \begin{pmatrix} 2j + {{\mathchoice{{{\tfrac{3}{2}}}}{{{\tfrac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}}}& 0 \\
0 & -2j - {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}\end{pmatrix} {|j\mu n\rangle\!\rangle},$$ since all matrices appearing are diagonal. This, together with the analogous calculation for $\delta([D,a_-])$, shows that $$\delta([D,a_+]) = P^{{\mathord{\uparrow}}}a_+ P^{{\mathord{\uparrow}}}- P^{{\mathord{\downarrow}}}a_+ P^{{\mathord{\downarrow}}}, \qquad
\delta([D,a_-]) = P^{{\mathord{\uparrow}}}a_- P^{{\mathord{\uparrow}}}- P^{{\mathord{\downarrow}}}a_- P^{{\mathord{\downarrow}}}.
\label{eq:delta-Dapm}$$ A similar argument for $b$ gives $$\delta([D,b_+]) = P^{{\mathord{\uparrow}}}b_+ P^{{\mathord{\uparrow}}}- P^{{\mathord{\downarrow}}}b_+ P^{{\mathord{\downarrow}}}, \qquad
\delta([D,b_-]) = P^{{\mathord{\uparrow}}}b_- P^{{\mathord{\uparrow}}}- P^{{\mathord{\downarrow}}}b_- P^{{\mathord{\downarrow}}}.
\label{eq:delta-Dbpm}$$ Combining , , and the analogous relations with $a$ replaced by $b$, we see that both ${\mathcal{A}}$ and $[D,{\mathcal{A}}]$ lie within $\operatorname{Dom}\delta$. An easy induction shows that they also lie within $\operatorname{Dom}\delta^k$ for $k = 2,3,\dots$.
This proposition continues to hold if we replace ${\mathcal{A}}(SU_q(2))$ by a suitably completed algebra, which is stable under the holomorphic function calculus.
Let $\Psi^0({\mathcal{A}})$ be the algebra generated by $\delta^k({\mathcal{A}})$ and $\delta^k([D,{\mathcal{A}}])$ for all $k \geq 0$ (the notation suggests that, in the spirit of [@ConnesMIndex] one thinks of it as an “algebra of pseudodifferential operators of order $0$”). Since, for instance, $$\begin{aligned}
P^{{\mathord{\uparrow}}}\pi(a) P^{{\mathord{\uparrow}}}&= {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}\delta^2(\pi(a)) + {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}\delta([D,\pi(a)]),
\\
P^{{\mathord{\uparrow}}}a_+ P^{{\mathord{\uparrow}}}&= {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}P^{{\mathord{\uparrow}}}\pi(a) P^{{\mathord{\uparrow}}}+ {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}P^{{\mathord{\uparrow}}}\delta(\pi(a)) P^{{\mathord{\uparrow}}},\end{aligned}$$ we see that $\Psi^0({\mathcal{A}})$ is in fact generated by the diagonal-corner operators $P^{{\mathord{\uparrow}}}a_\pm P^{{\mathord{\uparrow}}}$, $P^{{\mathord{\downarrow}}}a_\pm P^{{\mathord{\downarrow}}}$, $P^{{\mathord{\uparrow}}}b_\pm P^{{\mathord{\uparrow}}}$, $P^{{\mathord{\downarrow}}}b_\pm P^{{\mathord{\downarrow}}}$ together with the other-corner operators $P^{{\mathord{\downarrow}}}a_+ P^{{\mathord{\uparrow}}}$, $P^{{\mathord{\uparrow}}}a_- P^{{\mathord{\downarrow}}}$, $P^{{\mathord{\downarrow}}}b_+ P^{{\mathord{\uparrow}}}$, and $P^{{\mathord{\uparrow}}}b_- P^{{\mathord{\downarrow}}}$. Following [@ConnesSUq], let ${\mathcal{B}}$ be the algebra generated by all $\delta^n({\mathcal{A}})$ for $n \geq 0$. It is a subalgebra of $\Psi^0({\mathcal{A}})$ and it is generated by the diagonal operators $$\tilde a_\pm := \pm \delta(a_\pm)
= P^{{\mathord{\uparrow}}}a_\pm P^{{\mathord{\uparrow}}}+ P^{{\mathord{\downarrow}}}a_\pm P^{{\mathord{\downarrow}}}, \qquad
\tilde b_\pm := \pm \delta(b_\pm)
= P^{{\mathord{\uparrow}}}b_\pm P^{{\mathord{\uparrow}}}+ P^{{\mathord{\downarrow}}}b_\pm P^{{\mathord{\downarrow}}},
\label{eq:extended-pi}$$ and by the off-diagonal operators $P^{{\mathord{\downarrow}}}a_+ P^{{\mathord{\uparrow}}}+ P^{{\mathord{\uparrow}}}a_- P^{{\mathord{\downarrow}}}$ and $P^{{\mathord{\downarrow}}}b_+ P^{{\mathord{\uparrow}}}+ P^{{\mathord{\uparrow}}}b_- P^{{\mathord{\downarrow}}}$.\
For later convenience we shall introduce an approximate representation ${\underline{\pi}}$ found in [@Naiad], which coincides with $\pi$ up to compact operators. Note first, that the off-diagonal coefficients in give rise to smoothing operators in ${\mathrm{OP}}^{-\infty}$ (see Appendix \[sec:pdc\]), due to the terms appearing in their denominators; we can furthermore simplify the diagonal terms.
We set ${\underline{\pi}}(a) := {\underline{a}}_+ + {\underline{a}}_-$ and ${\underline{\pi}}(b) := {\underline{b}}_+ + {\underline{b}}_-$ with the following definitions: $$\begin{aligned}
{\underline{a}}_+ \,{|j\mu n\rangle\!\rangle}
&:= \sqrt{1-q^{2j+2\mu+2}} \begin{pmatrix}
\sqrt{1- q^{2j+2n+3}} & 0 \\ 0 & \sqrt{1- q^{2j+2n+1}}
\end{pmatrix} {|j^+ \mu^+ n^+\rangle\!\rangle},
{\nonumber}\\
{\underline{a}}_- \,{|j\mu n\rangle\!\rangle}
&:= q^{2j+\mu+n+{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}} \begin{pmatrix}
q & 0 \\ 0 & 1 \end{pmatrix} {|j^- \mu^+ n^+\rangle\!\rangle},
{\nonumber}\\
{\underline{b}}_+ \,{|j\mu n\rangle\!\rangle}
&:= q^{j+n-{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}} \sqrt{1-q^{2j+2\mu+2}} \begin{pmatrix}
q & 0 \\ 0 & 1 \end{pmatrix} {|j^+ \mu^+ n^-\rangle\!\rangle},
{\nonumber}\\
{\underline{b}}_- \,{|j\mu n\rangle\!\rangle}
&:= -q^{j+\mu} \begin{pmatrix}
\sqrt{1-q^{2j+2n+1}} & 0 \\ 0 & \sqrt{1-q^{2j+2n-1}}
\end{pmatrix} {|j^- \mu^+ n^-\rangle\!\rangle}.
\label{eq:approx-repn}\end{aligned}$$ These formulas can be obtained from by truncation, using the pair of estimates $$\begin{aligned}
\bigl((q^{-1} - q)[n] \bigr)^{-1} - q^n &= q^{3n} + O(q^{5n}),
\\
1 - \sqrt{1 - q^{\alpha}} & \leq q^{\alpha}, \qquad{\rm for ~any} \ {\alpha}\geq 0.\end{aligned}$$ The operators ${\underline{\pi}}(x) - \pi(x)$ are given by sequences of rapid decay, and hence are elements in ${\mathrm{OP}}^{-\infty}$ (as defined in Appendix \[sec:pdc\]). Therefore, we can replace $\pi$ by ${\underline{\pi}}$ when dealing with the local cocycle in the local index theorem in the next section.
\[rk:approx-repn\] These operators differ slightly from the approximate representation given in [@Naiad]. Using the inequality $1 - \sqrt{1-q^\alpha} \leq q^\alpha$, they can be seen to differ from the operators therein by a compact operator in the principal ideal ${\mathcal{K}}_q$ generated by the operator $L_q {\colon}{|j\mu n\rangle\!\rangle} \mapsto q^j {|j \mu n\rangle\!\rangle}$. Note that ${\mathcal{K}}_q \subset {\mathrm{OP}}^{-\infty}$.
Now, observe that $$\begin{aligned}
[|D|, {\underline{\pi}}(a)] &= {\underline{a}}_+ - {\underline{a}}_-, &
[D, {\underline{\pi}}(a)] &= F({\underline{a}}_+ - {\underline{a}}_-),
{\nonumber}\\
[|D|, {\underline{\pi}}(b)] &= {\underline{b}}_+ - {\underline{b}}_-, &
[D, {\underline{\pi}}(b)] &= F({\underline{b}}_+ - {\underline{b}}_-),
\label{eq:comm-DA}\end{aligned}$$ and also that $F$ commutes with ${\underline{a}}_\pm$ and ${\underline{b}}_\pm$. The operators ${\underline{a}}_\pm$ and ${\underline{b}}_\pm$ have a simpler expression if we use the following relabelling of the orthonormal basis of ${\mathcal{H}}$, $$\begin{aligned}
v_{xy{{\mathord{\uparrow}}}}^j &:= {|j,x-j,y-j-{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}},{{\mathord{\uparrow}}}\rangle} {\quad\mbox{for}\quad}
x = 0, \dots, 2j; \ y = 0, \dots, 2j + 1,
{\nonumber}\\
v_{xy{{\mathord{\downarrow}}}}^j &:= {|j,x-j,y-j+{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}},{{\mathord{\downarrow}}}\rangle} {\quad\mbox{for}\quad}
x = 0, \dots, 2j; \ y = 0, \dots, 2j - 1.
\label{eq:new-basis}\end{aligned}$$ We again employ the pairs of vectors $$v_{xy}^j := \begin{pmatrix} v_{xy{{\mathord{\uparrow}}}}^j \\ v_{xy{{\mathord{\downarrow}}}}^j \end{pmatrix},$$ where the lower component is understood to be zero if $y = 2j$ or $2j+1$, or if $j = 0$. The simplification is that on these vector pairs, all the $2 {\times}2$ matrices in become scalar matrices, $$\begin{aligned}
{\underline{a}}_+ v_{xy}^j
&= \sqrt{1 - q^{2x+2}} \sqrt{1 - q^{2y+2}} \, v_{x+1,y+1}^{j^+},
{\nonumber}\\
{\underline{a}}_- v_{xy}^j &= q^{x+y+1} \, v_{xy}^{j^-},
{\nonumber}\\
{\underline{b}}_+ v_{xy}^j &= q^y \sqrt{1 - q^{2x+2}} \, v_{x+1,y}^{j^+},
{\nonumber}\\
{\underline{b}}_- v_{xy}^j &= - q^x \sqrt{1 - q^{2y}} \, v_{x,y-1}^{j^-}.
\label{eq:approx-repn-bis}\end{aligned}$$ These formulas coincide with those found in [@ConnesSUq Sec. 6] up to a doubling of the Hilbert space and the change of conventions $a {\leftrightarrow}a^*$, $b {\leftrightarrow}-b$. Indeed, since the spin representation is isomorphic to a direct sum of two copies of the regular representation, the formulas in exhibit the same phenomenon for the approximate representations.
The cosphere bundle {#sec:cosphere}
===================
In [@ConnesSUq] Connes constructs a “cosphere bundle” using the regular representation of ${\mathcal{A}}(SU_q(2))$. In view of , the same cosphere bundle may be obtained directly from the spin representation by adapting that construction, as we now proceed to do. In what follows, we use the algebra ${\mathcal{A}}= {\mathcal{A}}(SU_q(2))$, but we could as well replace it with its completion $C^\infty(SU_q(2))$, which is closed under holomorphic functional calculus (see Appendix \[sec:pdc\]).
We recall two well-known infinite dimensional representations $\pi_\pm$ of ${\mathcal{A}}(SU_q(2))$ by bounded operators on the Hilbert space $\ell^2({\mathbb{N}})$. On the standard orthonormal basis ${\{\,{\varepsilon}_x : x \in {\mathbb{N}}\,\}}$, they are given by $$\pi_\pm(a) \,{\varepsilon}_x := \sqrt{1-q^{2x+2}} \,{\varepsilon}_{x+1}, \qquad
\pi_\pm(b) \,{\varepsilon}_x := \pm q^x \,{\varepsilon}_x.
\label{eq:pi-pm}$$ We may identify the Hilbert space ${\mathcal{H}}$ spanned by all $v_{xy{{\mathord{\uparrow}}}}^j$ and $v_{xy{{\mathord{\downarrow}}}}^j$ with the subspace ${\mathcal{H}}'$ of $\ell^2({\mathbb{N}})_x {\otimes}\ell^2({\mathbb{N}})_y {\otimes}\ell^2({\mathbb{Z}})_{2j} {\otimes}{\mathbb{C}}^2$ determined by the parameter restrictions in . Thereby, we get the correspondence $$\begin{aligned}
{\underline{a}}_+ &{\leftrightarrow}\pi_+(a) {\otimes}\pi_-(a) {\otimes}V {\otimes}1_2,
{\nonumber}\\
{\underline{a}}_- &{\leftrightarrow}- q\,\pi_+(b) {\otimes}\pi_-(b^*) {\otimes}V^* {\otimes}1_2,
{\nonumber}\\
{\underline{b}}_+ &{\leftrightarrow}- \pi_+(a) {\otimes}\pi_-(b) {\otimes}V {\otimes}1_2,
{\nonumber}\\
{\underline{b}}_- &{\leftrightarrow}- \pi_+(b) {\otimes}\pi_-(a^*) {\otimes}V^* {\otimes}1_2,
\label{eq:corr-smoothing}\end{aligned}$$ where $V$ is the unilateral shift operator ${\varepsilon}_{2j} \mapsto {\varepsilon}_{2j+1}$ in $\ell^2({\mathbb{Z}})$. This again, apart from the $2 {\times}2$ identity matrix $1_2$, coincides with the formula (204) in [@ConnesSUq], up to the aforementioned exchange of the generators.
The shift $V$ in the action of the operators ${\underline{a}}_\pm$ and ${\underline{b}}_\pm$ on ${\mathcal{H}}$ can be encoded using the ${\mathbb{Z}}$-grading coming from the one-parameter group of automorphisms ${\gamma}(t)$ generated by $|D|$, $${\gamma}(t) = \begin{pmatrix} {\gamma}_{{{\mathord{\uparrow}}}{{\mathord{\uparrow}}}}(t) & {\gamma}_{{{\mathord{\uparrow}}}{{\mathord{\downarrow}}}}(t) \\
{\gamma}_{{{\mathord{\downarrow}}}{{\mathord{\uparrow}}}}(t) & {\gamma}_{{{\mathord{\downarrow}}}{{\mathord{\downarrow}}}}(t) \end{pmatrix}, {\quad\mbox{where}\quad}
\left\{
\begin{aligned}
{\gamma}_{{{\mathord{\uparrow}}}{{\mathord{\uparrow}}}}(t)
&: P^{{\mathord{\uparrow}}}T P^{{\mathord{\uparrow}}}\mapsto P^{{\mathord{\uparrow}}}e^{it|D|} T e^{-it|D|} P^{{\mathord{\uparrow}}},
\\
{\gamma}_{{{\mathord{\uparrow}}}{{\mathord{\downarrow}}}}(t)
&: P^{{\mathord{\uparrow}}}T P^{{\mathord{\downarrow}}}\mapsto P^{{\mathord{\uparrow}}}e^{it|D|} T e^{-it|D|} P^{{\mathord{\downarrow}}},
\\
{\gamma}_{{{\mathord{\downarrow}}}{{\mathord{\uparrow}}}}(t)
&: P^{{\mathord{\downarrow}}}T P^{{\mathord{\uparrow}}}\mapsto P^{{\mathord{\downarrow}}}e^{it|D|} T e^{-it|D|} P^{{\mathord{\uparrow}}},
\\
{\gamma}_{{{\mathord{\downarrow}}}{{\mathord{\downarrow}}}}(t)
&: P^{{\mathord{\downarrow}}}T P^{{\mathord{\downarrow}}}\mapsto P^{{\mathord{\downarrow}}}e^{it|D|} T e^{-it|D|} P^{{\mathord{\downarrow}}},
\end{aligned} \right.
\label{eq:spectral-flow}$$ for any operator $T$ on ${\mathcal{H}}$. On the subalgebra of “diagonal” operators $T = P^{{\mathord{\uparrow}}}T P^{{\mathord{\uparrow}}}+ P^{{\mathord{\downarrow}}}T P^{{\mathord{\downarrow}}}$, the compression ${\gamma}_{{{\mathord{\uparrow}}}{{\mathord{\uparrow}}}} \oplus {\gamma}_{{{\mathord{\downarrow}}}{{\mathord{\downarrow}}}}$ detects the shift of $j$ of the restrictions of $T$ to ${\mathcal{H}}^{{\mathord{\uparrow}}}$ and ${\mathcal{H}}^{{\mathord{\downarrow}}}$ respectively. For example, ${\gamma}_{{{\mathord{\uparrow}}}{{\mathord{\uparrow}}}}(t) \oplus {\gamma}_{{{\mathord{\downarrow}}}{{\mathord{\downarrow}}}}(t)
: a_\pm \mapsto e^{\pm it} a_\pm$, so that the ${\mathbb{Z}}$-grading encodes the correct shifts $j \to j\pm {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}$ in the formulas for $a_\pm$; and likewise for $b_\pm$.
From equation it follows that $b - b^* \in \ker \pi_\pm$, and so the representations $\pi_\pm$ are not faithful on ${\mathcal{A}}(SU_q(2))$. We define two algebras ${\mathcal{A}}(D^2_{q\pm})$ to be the corresponding quotients, $$0 \to \ker \pi_\pm \to {\mathcal{A}}(SU_q(2))
\xrightarrow{r_\pm} {\mathcal{A}}(D^2_{q\pm}) \to 0.
\label{eq:q-disks}$$ We elaborate a little on the structure of the algebras ${\mathcal{A}}(D^2_{q\pm})$. For convenience, we shall omit the quotient maps $r_\pm$ in this discussion. Then $b = b^*$ in ${\mathcal{A}}(D_{q\pm}^2)$, and from the defining relations of ${\mathcal{A}}(SU_q(2))$, we obtain $$\begin{gathered}
b a = q\, a b, \qquad a^*b = q\, b a^*,
{\nonumber}\\
a^*a + q^2 b^2 = 1, \qquad aa^* + b^2 = 1.
\label{eq:Sq2-relns}\end{gathered}$$ These algebraic relations define two isomorphic quantum $2$-spheres ${\mathbb{S}}^2_{q+} \simeq {\mathbb{S}}^2_{q-} =: {\mathbb{S}}^2_q$ which have a classical subspace ${\mathbb{S}}^1$ given by the characters $b \mapsto 0$, $a \mapsto \lambda$ with $|\lambda| = 1$. A substitution $q \mapsto q^2$, followed by $b \mapsto q^{-2} b$ shows that ${\mathbb{S}}^2_q$ is none other than the equatorial Podleś sphere [@Podles]. Thus, the above quotients of ${\mathcal{A}}(SU_q(2))$ with respect to $\ker \pi_\pm$ either coincide with ${\mathcal{A}}({\mathbb{S}}^2_q)$ or are quotients of it. Now, from one sees that the spectrum of $\pi_\pm(b)$ is either real positive or real negative, depending on the $\pm$ sign. Hence, the algebras ${\mathcal{A}}(D^2_{q+})$ and ${\mathcal{A}}(D^2_{q-})$ describe the two hemispheres of ${\mathbb{S}}^2_q$ and may be thought of as quantum disks, thus justifying the notation $D_{q\pm}$.
There is a symbol map ${\sigma}{\colon}{\mathcal{A}}(D^2_{q\pm}) \to {\mathcal{A}}({\mathbb{S}}^1)$ that maps these “noncommutative disks” to their common boundary ${\mathbb{S}}^1$, which is the equator of the equatorial Podleś sphere ${\mathbb{S}}^2_q$. Explicitly, the symbol map is given as a $*$-homomorphism on the generators of ${\mathcal{A}}(D_{q,\pm}^2)$ by $$\begin{aligned}
{\sigma}(r_\pm(a)) := u; \qquad {\sigma}(r_\pm(b)) := 0,
\label{eq:symbol-map}\end{aligned}$$ where $u$ is the unitary generator of ${\mathcal{A}}({\mathbb{S}}^1)$.
Recall the algebra ${\mathcal{B}}$ defined around with generators $\tilde a_\pm$, $\tilde b_\pm$ and $P^{{\mathord{\downarrow}}}a_+ P^{{\mathord{\uparrow}}}+ P^{{\mathord{\uparrow}}}a_- P^{{\mathord{\downarrow}}}$, $P^{{\mathord{\downarrow}}}b_+ P^{{\mathord{\uparrow}}}+ P^{{\mathord{\uparrow}}}b_- P^{{\mathord{\downarrow}}}$. The following result emulates Proposition 4 of [@ConnesSUq] and establishes the correspondence . The results of [@Naiad] on the approximate representation are crucial to its proof.
\[pr:symbol-map\] There is a $*$-homomorphism $$\rho: {\mathcal{B}}\to {\mathcal{A}}(D^2_{q+}) {\otimes}{\mathcal{A}}(D^2_{q-}) {\otimes}{\mathcal{A}}({\mathbb{S}}^1)
\label{eq:symbol-map-bis}$$ defined on generators by $$\begin{aligned}
\rho(\tilde a_+) &:= r_+(a) {\otimes}r_-(a) {\otimes}u, &
\rho(\tilde a_-) &:= -q\,r_+(b) {\otimes}r_-(b^*) {\otimes}u^*,
\\
\rho(\tilde b_+) &:= - r_+(a) {\otimes}r_-(b) {\otimes}u, &
\rho(\tilde b_-) &:= - r_+(b) {\otimes}r_-(a^*) {\otimes}u^*.\end{aligned}$$ while the off-diagonal operators $P^{{\mathord{\downarrow}}}a_+ P^{{\mathord{\uparrow}}}+ P^{{\mathord{\uparrow}}}a_- P^{{\mathord{\downarrow}}}$ and $P^{{\mathord{\downarrow}}}b_+ P^{{\mathord{\uparrow}}}+ P^{{\mathord{\uparrow}}}b_- P^{{\mathord{\downarrow}}}$ are declared to lie in the kernel of $\rho$.
First note that the $j$-dependence of the operators in ${\mathcal{B}}$ is taken care of by the factor $u$. Thus, it is enough to show that the following prescription, $$\begin{aligned}
\rho_1(\tilde a_+) &:= \pi_+(a) {\otimes}\pi_-(a), &
\rho_1(\tilde a_-) &:= -q\,\pi_+(b) {\otimes}\pi_-(b^*),
\\
\rho_1(\tilde b_+) &:= - \pi_+(a) {\otimes}\pi_-(b), &
\rho_1(\tilde b_-) &:= - \pi_+(b) {\otimes}\pi_-(a^*),\end{aligned}$$ together with $\rho_1(P^{{\mathord{\downarrow}}}a_+ P^{{\mathord{\uparrow}}}+ P^{{\mathord{\uparrow}}}a_- P^{{\mathord{\downarrow}}}) =
\rho_1(P^{{\mathord{\downarrow}}}b_+ P^{{\mathord{\uparrow}}}+ P^{{\mathord{\uparrow}}}b_- P^{{\mathord{\downarrow}}}) := 0$, defines a $*$-homomorphism $\rho_1: {\mathcal{B}}\to {\mathcal{A}}(D^2_{q+}) {\otimes}{\mathcal{A}}(D^2_{q-})$. In the notation, we have replaced the representations $\pi_\pm$ of ${\mathcal{A}}(SU_q(2))$ by corresponding faithful representations of ${\mathcal{A}}(D^2_{q\pm})$ (omitting the maps $r_\pm$).
We define a map $\Pi: {\mathcal{H}}\to (\ell^2({\mathbb{N}}) \otimes \ell^2({\mathbb{N}})) \otimes {\mathbb{C}}^2$, which simply forgets the $j$-index on the basis vectors $v_{xy}^j$: $$\Pi: v_{xy}^j= \begin{pmatrix} v_{xy{{\mathord{\uparrow}}}}^j \\ v_{xy{{\mathord{\downarrow}}}}^j \end{pmatrix} \mapsto {\varepsilon}_{xy}:=\begin{pmatrix} {\varepsilon}_{xy{{\mathord{\uparrow}}}} \\ {\varepsilon}_{xy{{\mathord{\downarrow}}}} \end{pmatrix},$$ where ${\varepsilon}_{xy{{\mathord{\uparrow}}}}:={\varepsilon}_x \otimes {\varepsilon}_y$ and ${\varepsilon}_{xy{{\mathord{\downarrow}}}}:={\varepsilon}_x \otimes {\varepsilon}_y$ in the two respective copies of $\ell^2({\mathbb{N}}) \otimes \ell^2({\mathbb{N}})$ in its tensor product with ${\mathbb{C}}^2$.
For any operator $T$ in ${\mathcal{B}}$, we define the map $\rho_1$ by $$\rho_1(T) {\varepsilon}_{xy} = \lim_{j \to \infty} \Pi(T v^j_{xy}).$$ This map is well-defined, since $T$ is a polynomial in the generators of ${\mathcal{B}}$. Each such generator shifts the indices $x,y,j$ by $\pm {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}$, with a coefficient matrix that can be bounded uniformly in $x,y$ and $j$ (cf. [@Naiad]) so that the limit $j \to \infty$ exists.
First of all, it can be directly verified, using estimates given in [@Naiad Sec. 7], that the off-diagonal operators $P^{{\mathord{\downarrow}}}a_+ P^{{\mathord{\uparrow}}}+ P^{{\mathord{\uparrow}}}a_- P^{{\mathord{\downarrow}}}$ and $P^{{\mathord{\downarrow}}}b_+ P^{{\mathord{\uparrow}}}+ P^{{\mathord{\uparrow}}}b_- P^
{{\mathord{\downarrow}}}$ are in the kernel of $\rho_1$. Next, the differences between the generators and the approximate generators ${\underline{a}}_\pm - \tilde a_\pm$ (and similarly $\tilde b_\pm - {\underline{b}}_\pm$) lie in the kernel of $\rho_1$, as well. Hence we can replace $\tilde a_\pm$ and $\tilde b_\pm$ by ${\underline{a}}_\pm$ and ${\underline{b}}_\pm$, respectively.
Since the coefficients in the definition of ${\underline{a}}_\pm$ and ${\underline{b}}_\pm$ (equation ) are $j$-independent, we conclude that $\rho_1$ is of the desired form. For example, we compute: $$\begin{aligned}
\rho_1(\tilde a_+) {\varepsilon}_{xy}=\rho_1({\underline{a}}_+) {\varepsilon}_{xy} &= \lim_{j \to \infty} \sqrt{1-q^{2x+2}} \sqrt{1-q^{2y+2}} \Pi(v_{x+1, y+1}^{j^+})\\
&= \sqrt{1-q^{2x+2}} \sqrt{1-q^{2y+2}} {\varepsilon}_{x+1, y+1}= (\pi_+ (a) \otimes \pi_-(a) \otimes 1_2) {\varepsilon}_{xy}.\end{aligned}$$ Since a product of the operators ${\underline{a}}_\pm$ and ${\underline{b}}_\pm$ still does not contain $j$-dependent coefficients, $\rho_1$ respects the multiplication in ${\mathcal{B}}$. By linearity of the limit, $\rho_1$ is an algebra map.
The *cosphere bundle on $SU_q(2)$* is defined as the range of the map $\rho$ in ${\mathcal{A}}(D^2_{q+}) {\otimes}A(D^2_{q-}) {\otimes}{\mathcal{A}}({\mathbb{S}}^1)$ and is denoted by ${\mathcal{A}}({\mathbb{S}}_q^*)$.
Note that ${\mathbb{S}}_q^*$ coincides with the cosphere bundle defined in [@ConnesSUq; @ConnesCIME], where it is regarded as a noncommutative space over which $D^2_{q+} {\times}D^2_{q-} {\times}{\mathbb{S}}^1$ is fibred.
The symbol map $\rho$ rectifies the correspondence . Denote by $Q$ the orthogonal projector on $\ell^2({\mathbb{N}}){\otimes}\ell^2({\mathbb{N}}) {\otimes}\ell^2({\mathbb{Z}}) {\otimes}{\mathbb{C}}^2$ with range ${\mathcal{H}}'$, which is the Hilbert subspace previously identified with ${\mathcal{H}}$ just before . Using in combination with Proposition \[pr:symbol-map\], we conclude that $$T - Q (\rho(T) {\otimes}1_2) Q \in {\mathrm{OP}}^{-\infty}
{\quad\mbox{for all}\quad} T \in {\mathcal{B}}.
\label{eq:symbol-smoothing}$$ Here, the action of $\rho(T)$ on $\ell^2({\mathbb{N}}) {\otimes}\ell^2({\mathbb{N}}) {\otimes}\ell^2({\mathbb{Z}})$ is determined by regarding $\ell^2({\mathbb{Z}})$ as the Hilbert space of square-summable Fourier series on ${\mathbb{S}}^1$.
The dimension spectrum
======================
We again follow [@ConnesSUq] for the computation of the dimension spectrum. We define three linear functionals $\tau_0^{{\mathord{\uparrow}}}$, $\tau_0^{{\mathord{\downarrow}}}$ and $\tau_1$ on the algebras ${\mathcal{A}}(D_{q\pm}^2)$. Since their definitions for both disks $D_{q+}^2$ and $D_{q-}^2$ are identical, we shall omit the $\pm$ for notational convenience.
For $x \in {\mathcal{A}}(D^2_{q})$ we define, $$\begin{aligned}
\tau_1(x) &:= \frac{1}{2\pi} \int_{S^1} {\sigma}(x),
\\
\tau_0^{{\mathord{\uparrow}}}(x) &:= \lim_{N\to\infty} \operatorname{Tr}_N \pi(x) - (N+{{\mathchoice{{{\tfrac{3}{2}}}}{{{\tfrac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}}}) \tau_1(x),
\\
\tau_0^{{\mathord{\downarrow}}}(x) &:= \lim_{N\to\infty} \operatorname{Tr}_N \pi(x) - (N+{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}) \tau_1(x),\end{aligned}$$ where ${\sigma}$ is the symbol map , and $\operatorname{Tr}_N$ is the truncated trace $$\operatorname{Tr}_N(T) := \sum_{k=0}^N {\langle{\varepsilon}_k\mathbin|T{\varepsilon}_k\rangle}.$$ The definition of the two different maps $\tau_0^{{\mathord{\uparrow}}}$ and $\tau_0^{{\mathord{\downarrow}}}$ is suggested by the constants ${{\mathchoice{{{\tfrac{3}{2}}}}{{{\tfrac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}}}$ and ${{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}$ appearing in our choice of the Dirac operator; it will simplify some residue formulas later on. We find that $$\begin{aligned}
\operatorname{Tr}_N(\pi(a))
&= (N + {{\mathchoice{{{\tfrac{3}{2}}}}{{{\tfrac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}}})\tau_1(a) + \tau_0^{{\mathord{\uparrow}}}(a) + O(N^{-k})
\\
&= (N + {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}})\tau_1(a) + \tau_0^{{\mathord{\downarrow}}}(a) + O(N^{-k})
{\quad\mbox{for all}\quad} k > 0.\end{aligned}$$
Let us denote by $r$ the restriction homomorphism from ${\mathcal{A}}(D^2_{q+}) {\otimes}A(D^2_{q-}) {\otimes}{\mathcal{A}}({\mathbb{S}}^1)$ onto the first two legs of the tensor product. In particular, we will use it as a map $$r: {\mathcal{A}}({\mathbb{S}}_q^*) \to {\mathcal{A}}(D^2_{q+}) {\otimes}A(D^2_{q-}).$$ In the following, we adopt the notation [@ConnesMIndex]: $${\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}T := \operatorname{Res}_{z=0} \operatorname{Tr}T |D|^{-z}.$$
The dimension spectrum of the spectral triple $({\mathcal{A}}(SU_q(2)),{\mathcal{H}},D)$ is simple and given by $\{1,2,3\}$; the corresponding residues are $$\begin{aligned}
{\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}T |D|^{-3} &= 2(\tau_1 {\otimes}\tau_1) \bigl(r\rho(T)^0\bigr),
\\
{\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}T |D|^{-2} &= \bigl(\tau_1 {\otimes}(\tau_0^{{\mathord{\uparrow}}}+ \tau_0^{{\mathord{\downarrow}}})
+ (\tau_0^{{\mathord{\uparrow}}}+ \tau_0^{{\mathord{\downarrow}}}) {\otimes}\tau_1\bigr) \bigl(r\rho(T)^0\bigr),
\\
{\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}T |D|^{-1}
&= (\tau_0^{{\mathord{\uparrow}}}{\otimes}\tau_0^{{\mathord{\downarrow}}}+ \tau_0^{{\mathord{\downarrow}}}{\otimes}\tau_0^{{\mathord{\uparrow}}})
\bigl(r\rho(T)^0\bigr),\end{aligned}$$ with $T \in \Psi^0({\mathcal{A}})$.
If we identify ${\mathcal{H}}' \subset \ell^2({\mathbb{N}}){\otimes}\ell^2({\mathbb{N}}) {\otimes}\ell^2({\mathbb{Z}}) {\otimes}{\mathbb{C}}^2$ with ${\mathcal{H}}$ as above, the one-parameter group of automorphisms $\gamma(t)$ induces a ${\mathbb{Z}}$-grading on ${\mathcal{A}}({\mathbb{S}}^*_q)$, in its representation on ${\mathcal{H}}'$. We denote by $\rho(T)^0$ the degree-zero part of the diagonal operator $\rho(T)$, for $T \in {\mathcal{B}}$. For the calculation of the dimension spectrum we need to find the poles of the zeta function $\zeta_T(z) := \operatorname{Tr}(T |D|^{-z})$ for all $T \in \Psi^0({\mathcal{A}})$. From our discussion of the generators of $\Psi^0({\mathcal{A}})$, we see that we only need to adjoin $P^{{\mathord{\uparrow}}}{\mathcal{B}}$ to ${\mathcal{B}}$.
In the zeta function $\zeta_T(z)$ for $T \in {\mathcal{B}}$, we can replace $T$ by $Q(\rho(T) {\otimes}1_2)Q$ since their difference is a smoothing operator by . The operator $Q(\rho(T) {\otimes}1_2)Q$ commutes with the projector $P^{{\mathord{\uparrow}}}$ so we can first calculate $$\begin{aligned}
\operatorname{Tr}(P^{{\mathord{\uparrow}}}Q(\rho(T) {\otimes}1_2)Q\,|D|^{-z})
&= \sum_{2j=0}^\infty (2j + {{\mathchoice{{{\tfrac{3}{2}}}}{{{\tfrac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}}})^{-z}
(\operatorname{Tr}_{2j} {\otimes}\operatorname{Tr}_{2j+1}) (r \rho(T)^0)
{\nonumber}\\
&= (\tau_1 {\otimes}\tau_1) (r \rho(T)^0)\,\zeta(z - 2)
{\nonumber}\\
&\qquad + (\tau_1 {\otimes}\tau_0^{{\mathord{\downarrow}}}+ \tau_0^{{\mathord{\uparrow}}}{\otimes}\tau_1)
(r \rho(T)^0 )\,\zeta(z - 1)
{\nonumber}\\
&\qquad + (\tau_0^{{\mathord{\uparrow}}}{\otimes}\tau_0^{{\mathord{\downarrow}}})
(r \rho(T)^0)\,\zeta(z) + f_{{\mathord{\uparrow}}}(z),
\label{eq:res-compute-up}\end{aligned}$$ where $f_{{\mathord{\uparrow}}}(z)$ is holomorphic in $z \in {\mathbb{C}}$. Similarly, $$\begin{aligned}
\operatorname{Tr}(P^{{\mathord{\downarrow}}}Q(\rho(T) {\otimes}1_2)Q\,|D|^{-z})
&= \sum_{2j=0}^\infty (2j + {{\mathchoice{{{\tfrac{3}{2}}}}{{{\tfrac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}{{{\scriptstyle\frac{3}{2}}}}}})^{-z}
(\operatorname{Tr}_{2j+1} {\otimes}\operatorname{Tr}_{2j}) (r\rho(T)^0)
{\nonumber}\\
&= (\tau_1 {\otimes}\tau_1) (r\rho(T)^0)\,\zeta(z - 2)
{\nonumber}\\
&\qquad + (\tau_1 {\otimes}\tau_0^{{\mathord{\uparrow}}}+ \tau_0^{{\mathord{\downarrow}}}{\otimes}\tau_1)
(r\rho(T)^0)\,\zeta(z - 1)
{\nonumber}\\
&\qquad + (\tau_0^{{\mathord{\downarrow}}}{\otimes}\tau_0^{{\mathord{\uparrow}}})
(r\rho(T)^0)\,\zeta(z) + f_{{\mathord{\downarrow}}}(z),
\label{eq:res-compute-dn}\end{aligned}$$ where $f_{{\mathord{\downarrow}}}(z)$ is holomorphic in $z$. Since $\zeta(z)$ has a simple pole at $z = 1$, we see that the zeta function $\zeta_T$ has simple poles at $1$, $2$ and $3$.
From the above proof, we derive the following formulas which will be used later on: $$\begin{aligned}
{\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}P^{{\mathord{\uparrow}}}T |D|^{-3} &= (\tau_1 {\otimes}\tau_1) \bigl(r\rho(T)^0\bigr),
{\nonumber}\\
{\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}P^{{\mathord{\uparrow}}}T |D|^{-2}
&= \bigl(\tau_1 {\otimes}\tau_0^{{\mathord{\downarrow}}}+ \tau_0^{{\mathord{\uparrow}}}{\otimes}\tau_1 \bigr)
\bigl(r\rho(T)^0\bigr),
{\nonumber}\\
{\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}P^{{\mathord{\uparrow}}}T |D|^{-1}
&= (\tau_0^{{\mathord{\uparrow}}}{\otimes}\tau_0^{{\mathord{\downarrow}}}) \bigl(r\rho(T)^0\bigr),
\label{eq:expr-res-up}\end{aligned}$$ with $T$ any element in $\Psi^0({\mathcal{A}})$.
Local index formula ($d = 3$)
=============================
We begin by discussing the local cyclic cocycles giving the local index formula, in the general case when the spectral triple $({\mathcal{A}},{\mathcal{H}},D)$ has simple discrete dimension spectrum not containing $0$ and bounded above by $3$.
Let us recall that with a general (odd) spectral triple $({\mathcal{A}},{\mathcal{H}},D)$ there comes a Fredholm index of the operator $D$ as an additive map $\varphi : K_1({\mathcal{A}}) \to {\mathbb{Z}}$ defined as follows. If $F = \operatorname{Sign}D$ and $P$ is the projector $P = {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}(1 + F)$ then $$\varphi([u])= \operatorname{Index}(PuP),
\label{eq:fr-ind}$$ with $u \in \operatorname{Mat}_r({\mathcal{A}})$ a unitary representative of the $K_1$ class (the operator $PuP$ is automatically Fredholm). The above map is computed by pairing $K_1({\mathcal{A}})$ with “nonlocal” cyclic cocycles $\chi_n$ given in terms of the operator $F$ and of the form $$\chi_n(a_0, \dots, {\alpha}_n) = \lambda_n \operatorname{Tr}(a_0\,[F,a_1] \dots [F,a_n]),
{\quad\mbox{for all}\quad} a_j \in {\mathcal{A}},
\label{eq:nlcc}$$ where $\lambda_n$ is a suitable normalization constant. The choice of the integer $n$ is determined by the degree of summability of the Fredholm module $({\mathcal{H}},F)$ over ${\mathcal{A}}$; any such module is declared to be $p$-summable if the commutator $[F,a]$ is an element in the $p$-th Schatten ideal ${\mathcal{L}}^p({\mathcal{H}})$, for any $a \in A$. The minimal $n$ in needs to be taken such that $n \geq p$.
On the other hand, the Connes–Moscovici local index theorem [@ConnesMIndex] expresses the index map in terms of a local cocycle $\phi_{{\mathrm{odd}}}$ in the $(b,B)$ bicomplex of ${\mathcal{A}}$ which is a local representative of the cyclic cohomology class of $\chi_n$ (the cyclic cohomology Chern character). The cocycle $\phi_{{\mathrm{odd}}}$ is given in terms of the operator $D$ and is made of a finite number of terms $\phi_{{\mathrm{odd}}}= (\phi_1, \phi_3, \dots )$; the pairing of the cyclic cohomology class $[\phi_{{\mathrm{odd}}}] \in HC^{{\mathrm{odd}}}({\mathcal{A}})$ with $K_1({\mathcal{A}})$ gives the Fredholm index of $D$ with coefficients in $K_1({\mathcal{A}})$. The components of the cyclic cocycle $\phi_{{\mathrm{odd}}}$ are explicitly given in [@ConnesMIndex]; we shall presently give them for our case.
We know from Proposition \[pr:regular\] that our spectral triple $({\mathcal{A}},{\mathcal{H}},D)$ with ${\mathcal{A}}= {\mathcal{A}}(SU_q(2))$ has metric dimension equal to $3$. As for the corresponding Fredholm module $({\mathcal{H}},F)$ over ${\mathcal{A}}= {\mathcal{A}}(SU_q(2))$, it is $1$-summable since all commutators $[F,\pi(x)]$, with $x\in{\mathcal{A}}$, are off-diagonal operators given by sequences of rapid decay. Hence each $[F,\pi(x)]$ is trace-class and we need only the first Chern character $\chi_1(a_0,a_1) = \operatorname{Tr}(a_0\,[F,a_1])$, with $a_1,a_2 \in {\mathcal{A}}$ (we shall omit discussing the normalization constant for the time being and come back to it in the next section). An explicit expression for this cyclic cocycle on the PBW-basis of $SU_q(2)$ was obtained in [@MasudaNW].
The local cocycle has two components, $\phi_{{\mathrm{odd}}}= (\phi_1, \phi_3)$, the cocycle condition $(b + B)\phi_{{\mathrm{odd}}}= 0$ reading $B \phi_1=0,~ b \phi_1 + B \phi_3 =0,~b \phi_3= 0$ (see Appendix \[sec:cy-co\]); it is explicitly given by $$\begin{aligned}
\phi_1(a_0,a_1) &:= {\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}a_0\, [D,a_1] \,|D|^{-1}
- \frac{1}{4} {\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}a_0\, \nabla([D,a_1]) \,|D|^{-3}
+ \frac{1}{8} {\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}a_0\, \nabla^2([D,a_1]) \,|D|^{-5},
\\
\phi_3(a_0,a_1,a_2,a_3)
&:= \frac{1}{12} {\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}a_0\,[D,a_1]\,[D,a_2]\,[D,a_3]\,|D|^{-3},\end{aligned}$$ where $\nabla(T) := [D^2,T]$ for any operator $T$ on ${\mathcal{H}}$. Under the assumption that $[F,a]$ is traceclass for each $a \in {\mathcal{A}}$, these expressions can be rewritten as follows: $$\begin{aligned}
\phi_1(a_0,a_1) &= {\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}a_0 \,\delta(a_1) F|D|^{-1}
- \frac{1}{2} {\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}a_0 \,\delta^2(a_1) F|D|^{-2}
+ \frac{1}{4} {\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}a_0 \,\delta^3(a_1) F|D|^{-3},
{\nonumber}\\
\phi_3(a_0,a_1,a_2,a_3) &= \frac{1}{12} {\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}a_0 \,\delta(a_1)
\,\delta(a_2) \,\delta(a_3) F|D|^{-3}.
\label{eq:odd-cycle}\end{aligned}$$
We now quote Proposition 2 of [@ConnesSUq], referring to that paper for its proof.
\[pr:LIF-one\] Let $({\mathcal{A}},{\mathcal{H}},D)$ be a spectral triple with discrete simple dimension spectrum not containing $0$ and bounded above by $3$. If $[F,a]$ is trace-class for all $a \in {\mathcal{A}}$, then the Chern character $\chi_1$ is equal to $\phi_{{\mathrm{odd}}}- (b + B) \phi_{{\mathrm{ev}}}$ where the cochain $\phi_{{\mathrm{ev}}}= (\phi_0, \phi_2)$ is given by $$\begin{aligned}
\phi_0(a) &:= \operatorname{Tr}(Fa\,|D|^{-z}) \bigr|_{z=0},
\\
\phi_2(a_0,a_1,a_2)
&:= \frac{1}{24} {\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}a_0 \,\delta(a_1) \,\delta^2(a_2) F|D|^{-3}.\end{aligned}$$
The absence of $0$ in the dimension spectrum is needed for the definition of $\phi_0$. The cochain $\phi_{{\mathrm{ev}}}= (\phi_0, \phi_2)$ was named $\eta$-cochain in [@ConnesSUq]. In components, the equivalence of the characters means that $$\phi_1 = \chi_1 + b \phi_0 + B \phi_2 , \qquad
\phi_3 = b \phi_2 .$$
The following general result, in combination with the above proposition, shows that $\chi_1$ can be given (up to coboundaries) in terms of one single $(b,B)$-cocycle $\psi_1$.
\[pr:LIF-two\] Let $({\mathcal{A}},{\mathcal{H}},D)$ be a spectral triple with discrete simple dimension spectrum not containing $0$ and bounded above by $3$. Assume that $[F,a]$ is trace class for all $a \in {\mathcal{A}}$, and set $P := {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}(1 + F)$. Then, the local Chern character $\phi_{{\mathrm{odd}}}$ is equal to $\psi_1 - (b + B)\phi'_{{\mathrm{ev}}}$, where $$\psi_1(a_0,a_1) := 2 {\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}a_0 \,\delta(a_1) P|D|^{-1}
- {\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}a_0 \,\delta^2(a_1) P|D|^{-2}
+ \frac{2}{3} {\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}a_0 \,\delta^3(a_1) P|D|^{-3},$$ and $\phi'_{{\mathrm{ev}}}= (\phi'_0, \phi'_2)$ is given by $$\begin{aligned}
\phi'_0(a) &:= \operatorname{Tr}(a\,|D|^{-z}) \bigr|_{z=0},
\\
\phi'_2(a_0, a_1, a_2)
&:= - \frac{1}{24} {\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}a_0 \,\delta(a_1) \,\delta^2(a_2) F|D|^{-3}.\end{aligned}$$
One needs to verify the following equalities between cochains in the $(b,B)$ bicomplex: $$\begin{aligned}
\phi_1 + b \phi'_0 + B \phi'_2 &= \psi_1,
\\
\phi_3 + b \phi'_2 &= 0.\end{aligned}$$ The second equality follows from a direct computation of $b\phi'_2$ and comparing with equation . Note that this identity proves that $\psi_1$ is indeed a cyclic cocycle. One also shows that $$B \phi'_2(a_0,a_1) = \frac{1}{12} {\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}a_0\,\delta^3(a_1) F|D|^{-3}.$$ Then, using the asymptotic expansion [@ConnesMIndex]: $$|D|^{-z} a \sim \sum_{k\geq 0} \binom{-z}{k} \delta^k(a)\,|D|^{-z-k}$$ modulo very low powers of $|D|$, one computes $$b \phi'_0(a_0,a_1) = {\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}a_0\,\delta(a_1) |D|^{-1}
- \frac{1}{2} {\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}a_0 \,\delta^2(a_1) |D|^{-2}
+ \frac{1}{3} {\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}a_0 \,\delta^3(a_1) |D|^{-3},$$ and it is now immediate that $\phi_1 + b \phi'_0 + B \phi'_2$ gives the cyclic cocycle $\psi_1$.
The term involving $P |D|^{-3}$ would vanish if the latter were traceclass, which is the case in [@ConnesSUq] (this is the statement that the metric dimension of the projector $P$ is $2$).
Combining these two propositions, it follows that the cyclic $1$-cocycles $\chi_1$ and $\psi_1$ are related as: $$\chi_1 = \psi_1 - b \beta,
\label{eq:tau-psi}$$ where $\beta (a) = 2\operatorname{Tr}(Pa\,|D|^{-z})\bigr|_{z=0}$.
The pairing between $HC^1$ and $K_1$
====================================
In this section, we shall calculate the value of the index map when $U$ is the unitary operator representing the generator of $K_1({\mathcal{A}}(SU_q(2)))$, $$\varphi([U]) = \operatorname{Index}(PUP) := \dim\ker P UP - \dim\ker P U^* P,$$ with $$U = \begin{pmatrix} a & b \\ -q b^* & a^* \end{pmatrix},
\label{eq:uni-def}$$ acting on the doubled Hilbert space ${\mathcal{H}}{\otimes}{\mathbb{C}}^2$ via the representation $\pi {\otimes}1_2$. The projector $P$ was denoted $P^{{\mathord{\uparrow}}}$ in Section \[sec:iso-ge\]. One expects this index to be nonzero, since the $K$-homology class of $({\mathcal{A}},{\mathcal{H}},D)$ is non-trivial. This has been remarked also in [@ChakrabortyPDirac], where our spectral triple is decomposed in terms of the spectral triple constructed in [@ChakrabortyPEqvt].
We first compute the above index directly, which is possible due to the simple nature of this particular example. A short computation shows that the kernel of the operator $P U^* P$ is trivial, whereas the kernel of $P U P$ contains only elements proportional to the vector $$\begin{pmatrix} \quad {|0,0,-{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}, {{\mathord{\uparrow}}}\rangle} \\
-q^{-1} {|0,0,{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}},{{\mathord{\uparrow}}}\rangle}
\end{pmatrix},$$ leading to $\varphi([U]) = \operatorname{Index}(PUP) = 1$.
Recall that for ${\mathcal{A}}= {\mathcal{A}}(SU_q(2))$, our Fredholm module $({\mathcal{H}},F)$ over ${\mathcal{A}}(SU_q(2))$ is $1$-summable. From the previous section we know that $\operatorname{Index}(PUP)$ can be computed using the local cyclic cocycle $\psi_1$, see eqn. . To prepare for this index computation via $\psi_1$, we recall the following lemma [@Book IV.1.${\gamma}$], which fixes the normalization constant in front of $\chi_1$. For completeness we recall the proof.
Let $({\mathcal{H}},F)$ be a $1$-summable Fredholm module over ${\mathcal{A}}$ with $P = {{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}(1 + F)$; let $u \in \operatorname{Mat}_r({\mathcal{A}})$ be unitary with a suitable $r$. Then $P u P$ is a Fredholm operator on $P {\mathcal{H}}$ and $$\operatorname{Index}(P u P) = -{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}\operatorname{Tr}(u^* [F,u]) = -{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}\chi_1(u^*,u).$$
We claim that $P u^* P$ is a parametrix for $P u P$, that is, an inverse modulo compact operators on $P{\mathcal{H}}$. Indeed, since $P - u^* P u = -{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}u^*\,[F,u]$ is traceclass by 1-summability, by composing it from both sides with $P$ it follows that $P - P u^* P u P$ is traceclass. Therefore, $$\operatorname{Index}(P u P) = \operatorname{Tr}(P - P u^* P u P) - \operatorname{Tr}(P - P u P u^* P),
\label{eq:index-formula}$$ and the identities $P - P u^* P u P = -{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}P u^*\,[F,u] P$ and $[F,u]\,u^* + u\,[F,u^*] = 0$, together with $[F,[F,u]]_+=0$, imply the statement.
Thus, the index of $P U P$, for the $U$ of is given, up to an overall $-{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}$ factor, by $$\psi_1(U^{-1}, U)
= 2 {\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}U_{kl}^* \,\delta(U_{lk}) P |D|^{-1}
- {\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}U_{kl}^* \,\delta^2(U_{lk}) P |D|^{-2}
+ \frac{2}{3} {\mathop{\mathchoice{\copy\ncintdbox} {\copy\ncinttbox}{\copy\ncinttbox} {\copy\ncinttbox}}\nolimits}U_{kl}^* \,\delta^3(U_{lk}) P |D|^{-3},$$ with summation over $k,l = 0,1$ understood. We compute this expression using equation . First note that since the entries of $U$ are generators of ${\mathcal{A}}(SU_q(2))$, we see from and that $\rho(\delta^2(U_{kl})) = \rho(U_{kl})$, a relation that simplifies the above formula. We compute the degree $0$ part of $\rho(U_{kl}^*\,\delta(U_{lk}))$ with respect to the grading coming from ${\gamma}(t)$ –the only part that contributes to the trace– using the algebra relations of ${\mathcal{A}}(D_{q\pm}^2)$, $$\rho(U_{kl}^*\,\delta(U_{lk}))^0 = 2(1 - q^2)\,1 {\otimes}r_-(b)^2.$$ Using the basic equalities $$\tau_1(1) = 1, \quad \tau_1(r_\pm(b)^n) = 0, \quad
\tau_0^{{\mathord{\uparrow}}}(1) = -\tau_0^{{\mathord{\downarrow}}}(1) = -{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}, \quad
\tau_0^{{\mathord{\uparrow}}}(r_\pm(b)^n) = \tau_0^{{\mathord{\downarrow}}}(r_\pm(b)^n)
= \frac{(\pm 1)^n}{1 - q^n},$$ we find that $$\psi_1(U^{-1},U)
= 2(1 - q^2) (2\tau_0^{{\mathord{\uparrow}}}{\otimes}\tau_0^{{\mathord{\downarrow}}}+ \frac{2}{3}\tau_1 {\otimes}\tau_1)
\bigl(1 {\otimes}r_-(b)^2 \bigr)
- (\tau_1 {\otimes}\tau_0^{{\mathord{\downarrow}}}+ \tau_0^{{\mathord{\uparrow}}}{\otimes}\tau_1)
\bigl(1 {\otimes}1 \bigr) = -2.$$ Taking the proper coefficients, we finally obtain $$\operatorname{Index}(PUP) = -{{\mathchoice{{{\tfrac{1}{2}}}}{{{\tfrac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}{{{\scriptstyle\frac{1}{2}}}}}}\psi_1(U^{-1}, U) = 1.$$
Pseudodifferential calculus and cyclic cohomology {#sec:pdc}
=================================================
\[sec:cy-co\]
Recall [@CareyPRS; @ConnesMIndex; @Polaris] that a spectral triple $({\mathcal{A}},{\mathcal{H}},D)$ is *regular* (or *smooth*, or $QC^\infty$) if the algebra generated by ${\mathcal{A}}$ and $[D,{\mathcal{A}}]$ lies within the smooth domain $\bigcap_{n=0}^\infty \operatorname{Dom}\delta^n$ of the operator derivation $\delta(T) := |D|T - T|D|$. This condition permits to introduce the analogue of Sobolev spaces ${\mathcal{H}}^s := \operatorname{Dom}(1 + D^2)^{s/2}$ for $s \in {\mathbb{R}}$. Let ${\mathcal{H}}^\infty := \bigcap_{s\geq 0} H^s$, which is a core for $|D|$. Then $T{\colon}{\mathcal{H}}^\infty \to {\mathcal{H}}^\infty$ has *analytic order $\leq k$* if $T$ extends to a bounded operator from ${\mathcal{H}}^{k+s}$ to ${\mathcal{H}}^s$ for all $s \geq 0$. It turns out that ${\mathcal{A}}({\mathcal{H}}^\infty) \subset {\mathcal{H}}^\infty$.
Assume that $|D|$ is invertible –which is a generic case of the $D$ used in this paper (for a careful treatment of the noninvertible case, see [@CareyPRS]). The space ${\mathrm{OP}}^{\alpha}$ of *operators of order* $\leq{\alpha}$ consists of those $T {\colon}{\mathcal{H}}^\infty \to {\mathcal{H}}^\infty$ such that $$|D|^{-{\alpha}} T \in \bigcap_{n=1}^\infty \operatorname{Dom}\delta^n.$$ (Operators of order ${\alpha}$ have analytic order ${\alpha}$). In particular, ${\mathrm{OP}}^0 = \bigcap_{n=1}^\infty \operatorname{Dom}\delta^n$, the algebra of operators of order $\leq 0$ includes $A \cup [D,{\mathcal{A}}]$ and their iterated commutators with $|D|$. Moreover, $[D^2, {\mathrm{OP}}^{\alpha}] \subset {\mathrm{OP}}^{{\alpha}+1}$ and ${\mathrm{OP}}^{-\infty} := \bigcap_{{\alpha}\leq 0} {\mathrm{OP}}^{\alpha}$ is a two-sided ideal in ${\mathrm{OP}}^0$.
The algebra structure can be read off in terms of an *asymptotic expansion*: $T \sim \sum_{j=0}^\infty T_j$ whenever $T$ and each $T_j$ are operators from ${\mathcal{H}}^\infty$ to ${\mathcal{H}}^\infty$; and for each $m \in {\mathbb{Z}}$, there exists $N$ such that for all $M > N$, the operator $T - \sum_{j=1}^M T_j$ has analytic order $\leq m$. For instance, for complex powers of $|D|$ (defined by the Cauchy formula) there is a binomial expansion: $$[|D|^z, T] \sim \sum_{k=1}^\infty \binom{z}{k} \delta^k(T)\,|D|^{z-k} .$$
Thus far, we have employed finitely generated algebras ${\mathcal{A}}(X)$, where $X = SU_q(2)$, $D^2_{q\pm}$, ${\mathbb{S}}^1$ or ${\mathbb{S}}_q^2$. In each case, we can enlarge them to algebras $C^\infty(X)$ by replacing polynomials in the generators (given in a prescribed order) by series with coefficients of rapid decay: this is clear when $X = {\mathbb{S}}^1$, where smooth functions have rapidly decaying Fourier series. Using the symbol maps , and together with Lemma 2 of [@ConnesCIME], we can check that each such $C^\infty(X)$ is closed under holomorphic functional calculus. The foregoing results apply, mutatis mutandis, to the regular spectral triple $(C^\infty(SU_q(2)),{\mathcal{H}},D)$.
For convenience, we also summarize here the cyclic cohomology of the algebra ${\mathcal{A}}(SU_q(2))$. A cyclic $n$-cochain on an algebra ${\mathcal{A}}$ is an element $\varphi \in {C^{n}_\lambda(\mathcal{A})}$, the collection of $(n+1)$-linear functionals on ${\mathcal{A}}$ which in addition are cyclic, $\lambda \varphi = \varphi$, with $$\lambda\varphi(a_0, a_1,\dots, a_n)
= (-1)^n \varphi(a_n, a_0,\dots, a_{n-1}).$$ There is a cochain complex $(C_{\lambda}^{\bullet}({\mathcal{A}}) = \bigoplus_n {C^{n}_\lambda(\mathcal{A})},\,b)$ with (Hochschild) coboundary operator $b {\colon}{C^{n}(\mathcal{A})} \to {C^{n+1}(\mathcal{A})}$ defined by $$b\varphi(a_0, a_1,\dots, a_{n+1})
:= \sum_{j=0}^n (-1)^j \varphi(a_0,\dots, a_j a_{j+1},\dots, a_{n+1})
+ (-1)^{n+1} \varphi(a_{n+1} a_0, a_1,\dots, a_n) .$$ The cyclic cohomology $HC^{\bullet}({\mathcal{A}})$ of the algebra ${\mathcal{A}}$ is the cohomology of this complex, $$HC^n({\mathcal{A}}) := H^n(C_{\lambda}^{\bullet}({\mathcal{A}}),b).$$ Equivalently, $HC^{\bullet}({\mathcal{A}})$ can be described [@Book; @Polaris] by using the second filtration of a $(b,B)$ bicomplex of arbitrary (i.e., noncyclic) cochains on ${\mathcal{A}}$. Here the operator $B$ decreases the degree $B{\colon}{C^{n}(\mathcal{A})} \to {C^{n-1}(\mathcal{A})}$, and is defined as $B = N B_0$, with $$\begin{aligned}
& (B_0 \varphi)(a_0,\dots, a_{n-1})
:= \varphi(1,a_0,\dots, a_{n-1}) - (-1)^n \varphi(a_0,\dots,a_{n-1},1)
\\
& (N\psi)(a_0,\dots, a_{n-1}) := \sum_{j=0}^{n-1}
(-1)^{(n-1)j} \psi(a_j,\dots,a_{n-1}, a_0,\dots,a_{j-1}).\end{aligned}$$ It is straightforward to check that $B^2 = 0$ and that $b B + B b = 0$; thus $(b + B)^2 = 0$. By putting together these two operators, one gets a bicomplex $(C^{\bullet}({\mathcal{A}}), b, B)$ with ${C^{p-q}(\mathcal{A})}$ in bidegree $(p,q)$. To a cyclic $n$-cocycle one associates the $(b,B)$ cocycle $\varphi$, $(b + B)\varphi = 0$, having only one nonvanishing component $\varphi_{n,0}$ given by $\varphi_{n,0} := (-1)^{{\lfloorn/2\rfloor}} \psi$.
The cyclic cohomology of the algebra ${\mathcal{A}}(SU_q(2))$ was computed in [@MasudaNW]. The even components vanish while the odd ones were found to be one-dimensional and generated by the cyclic $1$-cocycle $\tau_{{\mathrm{odd}}}\in HC^1({\mathcal{A}}(SU_q(2)))$ which was obtained as a character of a $1$-summable Fredholm module, $$\tau_{{\mathrm{odd}}}(a^l b^m (b^*)^n, a^{l'} b^{m'} (b^*)^{n'}) = (n-m) \dfrac{q^{l(m'+n')} \prod_{i=1}^l (1 - q^{2i})}
{\prod_{i=0}^l (1- q^{2i+2n+2n'})} \,\delta_{n+n', m+m'} \delta_{l,-l'}$$ where we use the notation $a^{-l} = (a^*)^l$ for $l > 0$. Since $HC^1({\mathcal{A}}(SU_q(2)))$ is one-dimensional, the characters of the 1-summable Fredholm modules found in [@ConnesSUq] and in this paper, are all cohomologous to this cyclic cocycle.
Acknowledgments {#acknowledgments .unnumbered}
---------------
We thank Alain Connes for suggesting that we address this problem. GL and LD acknowledge partial support by the Italian Co-Fin Project SINTESI. Support from the Vicerrectoría de Investigación of the Universidad de Costa Rica is acknowledged by JCV.
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|
---
abstract: 'This letter considers a multi-source multi-relay network, in which relay nodes employ a coding scheme based on random linear network coding on source packets and generate coded packets. If a destination node collects enough coded packets, it can recover the packets of all source nodes. The links between source-to-relay nodes and relay-to-destination nodes are modeled as packet erasure channels. Improved bounds on the probability of decoding failure are presented, which are markedly close to simulation results and notably better than previous bounds. Examples demonstrate the tightness and usefulness of the new bounds over the old bounds.'
author:
- 'Amjad Saeed Khan and Ioannis Chatzigeorgiou [^1]'
bibliography:
- 'IEEEabrv.bib'
- 'IEEE\_CommLett\_Ref.bib'
title: '[Improved bounds on the decoding failure probability of network coding over multi-source multi-relay networks]{}'
---
=cmr12 at 22pt
Network coding, sparse random matrices, probability of decoding failure, linear dependence.
Introduction {#intro_section}
============
The exploitation of cooperative diversity and the inclusion of network coding in multi-source multi-relay networks, in order to achieve excellent performance and high diversity gain, has attracted the interest of the research community. For example, it has been demonstrated in [@Impact_NC] that the use of network coding combined with cooperative diversity not only increase network reliability but also improve network throughput. Moreover, it has been shown in [@CoCast_NC] and [@katti2008network] that they can be significantly useful for wireless networks with disruptive channel and connectivity conditions.
This letter considers linear network coding over a multi-source multi-relay network, where $N$ source nodes are supported by $M$ relay nodes for the delivery of packets over packet erasure channels. To the best of our knowledge, an exact expression for the probability of decoding failure at a destination is not available but an effort has been made in [@Seong_LTR], in which the author derives upper and lower bounds. However, the bounds presented in [@Seong_LTR] are tight only for a certain range of parameters, including erasure probabilities, the values of $N$, $M$ and the size of the finite field. As shown in Section \[sec:result\_sec\] of this letter, the existing upper bound is poor for a large number of source nodes and for large finite fields. Moreover, the existing lower bound is independent of the field size and is loose for small finite fields and low erasure probabilities.
The motivation for this work is to derive improved bounds on the probability of decoding failure. To this end, the main contributions of this letter can be summarized as follows: (i) we have reformulated the problem statement in order to associate the probability of decoding failure with the probability of attaining singular sparse random matrices, and (ii) we have revisited the expressions for upper and lower bounds in [@Seong_LTR] and obtained alternative expressions, which are better than the previous bounds for any field size and any value of $N$ and $M$. Furthermore, the proposed lower bound incorporates the effect of the field size in contrast to the previous lower bound.
System Model {#sec:system}
============
We consider a system with $N$ source nodes and $M$ relay nodes, $\{\mathrm{S}_1,\mathrm{S}_2,\hdots,\mathrm{S}_N\}$ and $\{\mathrm{R}_1,\mathrm{R}_2,\hdots,\mathrm{R}_M\}$, respectively, as shown in Fig. \[fig:system model\], where $M\geq N$. Each source node $\mathrm{S}_i$ has a packet $x_i$ to transmit to a destination $\mathrm{D}$ via $M$ relay nodes. No source-to-destination links are assumed. The links connecting source-to-relay and relay-to-destination nodes are modeled as independent packet erasure channels characterized by erasure probability $\epsilon_{\mathrm{SR}}$ and $\epsilon_{\mathrm{RD}}$, respectively.
The communication process is split into two phases. In the first phase, all the source nodes transmit their information packets simultaneously to the relay nodes over orthogonal broadcast channels. In the second phase, each relay node generates a coded packet by randomly combining the successfully received packets from the $N$ source nodes. The $M$ coded packets are then forwarded to the destination $\mathrm{D}$ over orthogonal channels. The coded packet $y_i$, which is transmitted by the $i^\mathrm{th}$ relay node, can be expressed as $y_i=\sum_{j=1}^{N}c_{i,j}x_j$, where $c_{i,j}$ is a coding coefficient selected independently at random over a finite field $F_q$ of size $q$. Because of the link condition $\epsilon_{\mathrm{SR}}$ between the source node $\mathrm{S}_j$ and the relay node $\mathrm{R}_i$, each relay node receives packets from different source nodes. In contrast to [@Multicast_capacity_2] where coding coefficients are chosen uniformly at random, our system model imposes that the zero coefficient is assigned to erased packets and the remaining $q-1$ non-zero coefficients are selected uniformly at random by each relay for successfully received packets. Consequently, the coding coefficient distribution is given by $$\begin{split}
\label{eq:uniform_distribution}
P[c_{i,j}=t]&=\left\{
\begin{array}{ll}
\!\epsilon_{\mathrm{SR}},&\!\!\!\mbox{if }t = 0\\[0.5em]
\displaystyle\!\!\frac{1-\epsilon_{\mathrm{SR}}}{q-1}, &\!\!\!\mbox{if }t \in F_q\setminus\{0\}
\end{array}
\right.
\end{split}$$ where $0\leq \epsilon_{\mathrm{SR}}\leq1$. For a given relay node $i$, the sequence $c_{i,1},\hdots, c_{i,N}$ forms a row vector, which is known as the coding vector of the coded packet $y_i$. As is commonly assumed in network coding [@Fitzek_ICC], coding vectors are transmitted along with the corresponding coded packets. When the destination $\mathrm{D}$ receives $N$ linearly independent coded packets, the packets of all source nodes can be recovered. Transmission of source packets over erasure channels and random linear coding at relay nodes is analogous to sparse random linear Network Coding (NC), which uses sparse random matrices [@Blomer_1997; @C-Cooper]. Based on the work of Bl$\ddot{\text{o}}$mer [@Blomer_1997] and Cooper [@C-Cooper], this paper derives improved upper and lower bounds on the probability that the destination will fail to recover the source packets.
{width="0.59\columnwidth"}
\[fig:system model\]
Preliminary Results and Former Bounds on the Probability of Decoding Failure
============================================================================
Consider a matrix $\mathbf{A} \in F_q^{M \times N}$, whose elements are the coding coefficients $c_{i,j}$ such that the $i^\mathrm{th}$ row of $\mathbf{A}$ represents the coding vector associated with the $i^\mathrm{th}$ coded packet received by the destination $\mathrm{D}$. The destination can recover the packets of the $N$ source nodes if and only if $rank(\mathbf{A})=N$. Thus, the decoding failure probability at the destination $\mathrm{D}$ can be defined as $\resizebox{0.21\textwidth}{!}{$P_{\mathrm{fail}}\!:=\!\mathrm{Pr}\{rank(\mathbf{A})\!<\!N\}$}$. It is related to the linear dependence of the vectors of matrix $\mathbf{A}$ and is defined as:
The vectors of matrix $\mathbf{A}\in F_q^{M \times N}$ are said to be *linearly dependent* if and only if there exists a column vector such that $$\mathbf{A}\mathbf{x}=\mathbf{0}.$$
When there is no packet loss between the relay-to-destination channels, i.e., $\epsilon_{\mathrm{RD}}=0$, the probability that the elements of the $i^\mathrm{th}$ row of matrix $\mathbf{A}$ add up to zero, i.e., $c_{i,1}+c_{i,2}+\hdots+c_{i,N}=0$, is given by [@Blomer_1997] $$\gamma_N=q^{-1}+(1-q^{-1})(1-\frac{1-\epsilon_{\mathrm{SR}}}{1-q^{-1}})^N.$$ Taking into account that matrix $\mathbf{A}$ consists of $M$ rows, the probability $\mathrm{Pr}(\mathbf{A}\mathbf{x}=\mathbf{0})$ can be obtained as $$\resizebox{0.4376\textwidth}{!}{$
\displaystyle \mathrm{Pr}(\mathbf{A}\mathbf{x}=\mathbf{0})=\gamma_N^{M}=\big(q^{-1}+(1-q^{-1})(1-\frac{1-\epsilon_{\mathrm{SR}}}{1-q^{-1}})^N\big)^M.
$}$$ The expected number of decoding failures at the destination $\mathrm{D}$ is given by the following theorem, which is a straightforward adaptation of [@Blomer_1997 Theorem 3.3], [@C-Cooper Theorem 3] to the system model under consideration.
\[pro:theorem\_1\] For a linear network coding scheme over $N$ source nodes, $M\geq N$ relay nodes and a single or multiple destinations, which are interconnected by links characterized by packet erasure probabilities $0\leq\epsilon_{\mathrm{SR}}\leq 1$ and $\epsilon_{\mathrm{RD}}=0$, the expectation of the decoding failures can be obtained as $$\label{eq:theorem_eq_1}
\mu_0(N,M)\!=\!{E}(\mathbf{A}\mathbf{x}=\mathbf{0})\!=\!\frac{1}{q-1}\sum_{w=1}^{N}\binom{N}{w}(q-1)^w\gamma_w^{M}$$ where $\mathbf{A}\in F_q^{M\times N}$ is the coding matrix at a destination.
Following the same line of reasoning, a direct extension of for $\epsilon_{\mathrm{RD}}\geq 0$ has been made in [@Seong_LTR Theorem 1] and was used to upper bound the probability of decoding failure.
The probability of decoding failure at a destination is bounded from above as: $$\label{eq:seong_eq_1}
\displaystyle P_\mathrm{fail}\leq\!\frac{1}{q-1}\!\!\sum_{w=1}^{N}\!\!\binom{N}{w}\!(q-1)^w\!\big [\epsilon_{\mathrm{RD}}+(1-\epsilon_{\mathrm{RD}})\gamma_w\big ]^M$$ where $N$ is the number of source nodes, $M\geq N$ is the number of relay nodes and $\epsilon_{\mathrm{SR}}$, $\epsilon_{\mathrm{RD}}$ represent the packet erasure probabilities between the network nodes.
However, is only tight for limited values of erasures $\epsilon_{\mathrm{SR}}$ and $\epsilon_{\mathrm{RD}}$, depending on $N$, $M$ and $q$. In particular, the upper bound takes values greater than 1 when either the field size is big or the difference between the number of source and relay nodes is small. This disparity between the probability of decoding failure and the upper bound will be demonstrated in Section \[sec:result\_sec\]. In an effort to improve the tightness of , Seong *et al.* proposed the selection of the minimum value between the upper bound in and 1 [@Seong_TCOM]. A lower bound on the probability of decoding failure has also been obtained by Seong in [@Seong_LTR Theorem 2]:
\[pro:theorem\_2\] Consider a network comprising $N$ source nodes and $M\geq N$ relay nodes, assume that links are modeled as packet erasure channels with erasure probabilities $\epsilon_{\mathrm{SR}}$ and $\epsilon_{\mathrm{RD}}$, and let $\mathbf{A}\in F_q^{M\times N}$ be the coding matrix at a destination node. The probability of decoding failure $P_\mathrm{fail}$ is lower bounded by $$\label{eq:theorem_eq_2}
\begin{split}
P_\mathrm{fail}\geq&\sum_{k=1}^{N}\binom{N}{k}\big ((\epsilon_{\mathrm{SR}}+\epsilon_{\mathrm{RD}}-\epsilon_{\mathrm{SR}}\epsilon_{\mathrm{RD}})^{M}\big )^{k}\\
&\times(1-(\epsilon_{\mathrm{SR}}+\epsilon_{\mathrm{RD}}-\epsilon_{\mathrm{SR}}\epsilon_{\mathrm{RD}})^{M})^{N-k}.
\end{split}$$
The bounds in and are used in [@Seong_TCOM] and [@disaster_conf]. For example, is employed in [@disaster_conf] to evaluate the performance gains introduced by linear NC in a practical network architecture for emergency communications. However, the following section will derive new bounds, which are considerably tighter than the previous bounds and can significantly improve the quality and accuracy of results presented in the literature.
Improved Bounds on the Probability of Decoding Failure
======================================================
Upper Bound
-----------
For $\epsilon_{\mathrm{RD}}=0$, an upper bound on the decoding failure probability can be obtained by extending and adapting [@Blomer_1997 Theorem 6.3] as follows:
\[pro:lemma\_1\] Let $\mathbf{A}\in F_q^{M\times N}$ be the coding matrix at a destination node of a network consisting of $N$ source nodes and $M$ relay nodes. If the internode erasure probabilities are $0\leq \epsilon_{\mathrm{SR}}\leq 1$ and $\epsilon_{\mathrm{RD}}=0$, the probability of decoding failure is upper bounded by $$\label{eq:lemma_1_eq}
\eta_\mathrm{max}(N,M)=1-\prod_{i=1}^{N}(1-\beta_{\max}^{M-i+1})$$ where $\displaystyle\beta_{\max}=\max(\epsilon_{\mathrm{SR}},\frac{1-\epsilon_{\mathrm{SR}}}{q-1})$.\
Let us assume that the first $i-1$ columns of $\mathbf{A}$, denoted by $\mathbf{A}_1,\mathbf{A}_2,\hdots,\mathbf{A}_{i-1}$, are linearly independent. This implies that by using elementary column operations, matrix $\mathbf{A}$ can be transformed into a matrix that contains an identity matrix. Without loss of generality, let us assume that the first $i-1$ rows form the identity matrix. The columns of this matrix represent the basis for the vector space spanned by $\mathbf{A}_1,\mathbf{A}_2,\hdots,\mathbf{A}_{i-1}$. Therefore, the probability that $\mathbf{A}_i$ is linearly independent from $\mathbf{A}_1,\mathbf{A}_2,\hdots,\mathbf{A}_{i-1}$ depends only on the last $M\!-\!i\!+\!1$ elements of $\mathbf{A}_i$. This probability is lower bounded by $1-\beta_{\max}^{M-i+1}$, where $\beta_{\max}$ specifies the maximum probability of obtaining an element from $F_q$. Hence, matrix $\mathbf{A}$ contains an $N\times N$ non-singular matrix with probability at least $\resizebox{0.15\textwidth}{!}{$\prod_{i=1}^{N}(1-\beta_{\max}^{M-i+1})$}$. As a result, the probability that matrix $\mathbf{A}$ does not contain an invertible matrix and, consequently, a decoding failure will occur is upper bounded by subtracting this product from one, which completes the proof.
Lemma \[pro:lemma\_1\] will be used to obtain a tighter upper bound on $P_{\mathrm{fail}}$. Before we invoke it, we shall first revisit and rewrite it as: $$\label{eq:lemma_2_eq}
P_\mathrm{fail}\leq\sum_{r=0}^{M}\binom{M}{r}\epsilon_{\mathrm{RD}}^{M-r}(1-\epsilon_{\mathrm{RD}})^r{\mu_0}(N,r).$$ This change is possible if $\ [\epsilon_{\mathrm{RD}}+(1-\epsilon_{\mathrm{RD}})\gamma_w\ ]^M$ is expanded into a sum, as per the binomial theorem.
\[pro:theorem\_3\] For a network coding scheme over multi-source multi-relay networks, composed of $N$ source nodes and $M$ relay nodes with packet erasures $\epsilon_{\mathrm{SR}}$ and $\epsilon_{\mathrm{RD}}$, the probability of decoding failure is upper bounded by $$\label{eq:theorem_3_eq}
\resizebox{0.4366\textwidth}{!}
{$
\displaystyle
P_\mathrm{fail}\!\leq\!\sum_{r=0}^{M}\!\binom{M}{r}\epsilon_{\mathrm{RD}}^{M-r}(1-\epsilon_{\mathrm{RD}})^r\!\min\{\eta_\mathrm{max}(N,r),\mu_0(N,r)\}.
$}$$
As inferred from , the number of packet deliveries by the relays follows the binomial distribution. If we employ Theorem \[pro:theorem\_1\] and Lemma \[pro:lemma\_1\] on the number of received coded packets $r$, a tight upper bound can be obtained by taking the minimum of outcomes and multiply with the probability distribution of $r$. Summing the resultant quantity gives , which concludes the proof.
It is worth noting that the upper bound is not simply the minimum between two cumulative probability distributions (CDFs), that is, the right-hand of and the CDF of for *all* possible numbers of relay nodes. Instead, the right hand of has been rewritten in the form of , which enabled us to identify the minimum between $\mu_0$ and $\eta_\mathrm{max}$ for *each* possible number of relay nodes, and use it in the computation of the CDF shown in .
Lower Bound
-----------
The bound that was derived in [@Blomer_1997 Theorem 6.3] was extended to an upper bound on the probability that an $M\times N$ matrix $\mathbf{A}$ does not contain an invertible $N\times N$ matrix in Lemma \[pro:lemma\_1\]. The same approach can be followed to obtain a lower bound as follows:
\[pro:lemmaupdate\_2\] Let $\mathbf{A}\in F_q^{M\times N}$ be the coding matrix at a destination of a network consisting of $N$ source nodes and $M$ relay nodes. If the internode erasure probabilities are $0\leq \epsilon_{\mathrm{SR}}\leq 1$ and $\epsilon_{\mathrm{RD}}=0$, the probability of decoding failure is lower bounded by $$\label{eq:lower_bound_eq}
\eta_\mathrm{min}(N,M)=1-\prod_{i=1}^{N}(1-\beta_{\min}^{M-i+1})$$ where $\displaystyle\beta_{\min}=\min(\epsilon_{\mathrm{SR}},\frac{1-\epsilon_{\mathrm{SR}}}{q-1})$.
The proof follows exactly the same line of reasoning as that of Lemma \[pro:lemma\_1\].
An improved lower bound on $P_{\mathrm{fail}}$ can be obtained if the right-hand side of (8) is denoted by $P_{0}(N,M)$ for $\epsilon_{\mathrm{RD}}=0$, that is $$\label{eq:lemma_3_eq}
P_{0}(N,M)=\sum_{k=1}^{N}\binom{N}{k}(\epsilon_{\mathrm{SR}}^{M})^k(1-\epsilon_{\mathrm{SR}}^{M})^{N-k}\\$$ and then combined with in Lemma \[pro:lemmaupdate\_2\]. In particular:
\[pro:theorem\_4\] For a linear network coding scheme over $N$ source nodes and $M\geq N$ relay nodes, let $\epsilon_{\mathrm{SR}}$ and $\epsilon_{\mathrm{RD}}$ be the packet erasure probabilities of the internode links. The probability of decoding failure is lower bounded by $$\label{eq:theorem_4_eq}
\resizebox{0.4366\textwidth}{!}
{$
\displaystyle
P_\mathrm{fail}\!\geq\!\sum_{r=0}^{M}\binom{M}{r}\epsilon_{\mathrm{RD}}^{M-r}(1-\epsilon_{\mathrm{RD}})^r\!\max\{\eta_\mathrm{min}(N,r),P_{0}(N,r)\}.
$}$$
In contrast to Theorem \[pro:theorem\_3\], here we employ Lemma \[pro:lemmaupdate\_2\] and on the number of received coded packets $r$, and we select the maximum of outcomes. The rest of the proof follows the same reasoning as that presented in the proof of Theorem \[pro:theorem\_3\].
Results {#sec:result_sec}
=======
This section compares the analytical expressions of the proposed bounds to simulation results. In addition, the proposed upper bound and lower bound, which shall be referred to as and , are contrasted with the old bounds represented by and , which shall be referred to as and . To obtain simulation results, each scenario was run over $10^4$ realizations, failures by the destination to recover the packets of all source nodes were counted, and the decoding failure probability was measured.
Fig. \[fig:figure\_2\] shows numerical results of the upper bounds obtained from and and labeled UB-old and UB-new, respectively. We observe that, in contrast to , is significantly tighter to the simulated performance. When the number of source nodes and the number of relay nodes increase to $N=30$ and $M=35$, respectively, it can be clearly seen that the curve moves far away from the simulated curve but the proposed UB-new expression still provides a tight bound. This reveals the fact that produces a worse approximation error for large values of $N$.
![Comparison between simulation results and the theoretical upper bounds obtained from and for different values of $N$ and $M$, when $q=2$, $\epsilon_{\mathrm{RD}}=0.1$ and $\epsilon_{\mathrm{SR}} \in [0.1, 0.9]$.](khan_2.eps){width=".975\columnwidth"}
\[fig:figure\_2\]
![Effect of field size $q$ on network performance and comparison between the proposed bounds and the old bounds for $\epsilon_{\mathrm{SR}}\in[0.1, 0.9]$, when $N=20$, $M=25$ and $\epsilon_{\mathrm{RD}}=0.1$.](khan_3.eps){width=".96\columnwidth"}
\[fig:figure\_3\]
{width="0.959\columnwidth"}
\[fig:figure\_4\]
![Network performance and comparison between the proposed bounds and the old bounds for $N=10$, an increasing number of relays $M$, , $\epsilon_{\mathrm{RD}}=0.1$ and different field size $q$.[]{data-label="fig:figure_5"}](khan_5.eps){width=".972\columnwidth"}
Fig. \[fig:figure\_3\] evaluates the probability of decoding failure for $q=\{4,64\}$, and contrasts the proposed bounds ( and ) with the old bounds ( and ). The figure demonstrates that for $\epsilon_{\mathrm{SR}}\in [0.1,0.7]$, the network experiences only a small probability of decoding failure. Furthermore, the figure shows that and are very close to the simulated performance and outperform and , respectively. In particular, when $q=64$, and are markedly loose while and are very tight to the actual simulation results. The performance of the network deteriorates for values of $\epsilon_{\mathrm{SR}}$ greater than 0.75. Moreover it is interesting to notice that, for large values of $q$, the upper bounds deviate from the simulation results and the simulations can be better approximated by the lower bounds. Figs. \[fig:figure\_4\] and \[fig:figure\_5\] plot the probability of decoding failure versus the number of relays $M$ with $N\!\!=\!\!10$ and $q\!\!=\!\!\{2,4\}$. It is evident that the probability of decoding failure decreases with an increasing number of relays and field size. The figures also demonstrate that, when $M\!<\!2N$, and are close to the simulated outcomes, compared to and , respectively. It follows from that depends only on the erasures $\epsilon_{\mathrm{SR}}$ and $\epsilon_{\mathrm{RD}}$, and does not depend on the field size $q$, thus shows no improvement for . However, LB-new approaches the simulation results, when $q$ increases to 4. For example in Fig. \[fig:figure\_5\], when $q=4$ and $M\leq 14$, both and are very tight, while and are noticeably far from the simulated performance.
Conclusions
===========
We presented improved upper and lower bounds on the probability of decoding failure in a multi-source multi-relay network, which employs linear network coding. The proposed analysis for counting failures provided significantly tighter bounds, which outperform existing bounds, derived in [@Seong_LTR]. Several examples, which considered various numbers of source nodes and relay nodes, different field sizes and a range of erasure probabilities, established the shortcomings of the existing bounds and demonstrated the tightness of the proposed improved bounds. Finally, we assert that the proposed bounds can also be used to better estimate the performance of systems employing sparse random linear network coding schemes, presented in the literature.\
[^1]: A. S. Khan and I. Chatzigeorgiou are with the School of Computing and Communications, Lancaster University, Lancaster, United Kingdom (e-mail: {a.khan9, i.chatzigeorgiou}@lancaster.ac.uk).
|
---
abstract: 'We consider a drone-based communication network, where several drones hover above an area and serve as mobile remote radio heads for a large number of mobile users. We assume that the drones employ free space optical (FSO) links for fronthauling of the users’ data to a central unit. The main focus of this paper is to quantify the geometric loss of the FSO channel arising from random fluctuation of the position and orientation of the drones. In particular, we derive upper and lower bounds, corresponding approximate expressions, and a closed-form statistical model for the geometric loss. Simulation results validate our derivations and quantify the FSO channel quality as a function of the drone’s instability, i.e., the variation of its position and orientation.'
author:
- |
Marzieh Najafi$^{\dag}$, Hedieh Ajam$^{\dag}$, Vahid Jamali$^{\dag}$, Panagiotis D. Diamantoulakis$^{\ddag}$,\
George K. Karagiannidis$^{\ddag}$, and Robert Schober$^{\dag}$\
$^{\dag}$Friedrich-Alexander University of Erlangen-Nuremberg, Germany\
$^{\ddag}$Aristotle University of Thessaloniki, Greece
bibliography:
- 'My\_Citation\_19-10-2017.bib'
title: |
Statistical Modeling of FSO Fronthaul Channel\
for Drone-based Networks
---
Introduction
============
Recently, there has been a growing interest in unmanned aerial vehicles (UAVs) and drones for civil applications, such as delivering cellular and internet services to remote regions or areas where a large number of users is *temporarily* gathered, e.g., a football match or a live concert, where *permanent infrastructure* does not exist or is costly to deploy [@Alouini_Drone; @Data_Collection_UAV_FSO]. In particular, drones may hover above the desired area and operate as mobile remote radio heads to assist the communication between the users and a central unit (CU) [@Alouini_Drone].
For these applications, free space optical (FSO) systems have been considered as promising candidates for fronthauling of the data gathered by the drones to the CU [@Alouini_Drone]. FSO systems offer the large bandwidths needed for data fronthauling and FSO transceivers are cheap and easy to implement [@Optic_Tbit; @MyTCOM]. However, the main factor that deteriorates the quality of the FSO link between a hovering drone and the CU is the instability of the drone, i.e., the variation of its position and orientation. Therefore, an immediate question is: *How good does the drone have to be in maintaining its position and orientation in order to achieve a certain FSO link quality?* The goal of this paper is to answer this question for uplink transmission by characterizing the geometric loss[^1] caused by random fluctuations of the drone’s position and orientation.
We note that, even in conventional FSO links where both transceivers are mounted on top of buildings, random fluctuations of the transceivers’ positions occur due to building sway which leads to a random geometric loss. For this case, corresponding statistical models were developed in [@Steve_pointing_error] and [@Alouini_Pointing]. However, the geometric loss for the case when the transmitter is a drone requires a new statistical model due to following differences: *i)* Unlike building sway, where the buildings exhibit limited movement due to wind loads and thermal expansion, for drone-based FSO communication, both the position and the orientation of the drone may fluctuate over time and have to be modelled as random variables (RVs). *ii)* For conventional FSO links, it is assumed that the laser beam is orthogonal with respect to (w.r.t.) the photo-detector (PD) plane at the receiver. However, this assumption may not hold for drone-based FSO communication. For example, the PD at the CU may receive data from several drones having different positions. Hence, it is not possible that the laser beams of all drones are orthogonal to the PD plane. Also, the positions of the drones may change due to changing traffic needs and the CU may not be able to adapt the orientation of the PD due to limited mechanical capabilities.
Drones and UAVs with FSO links have already been considered in the literature [@Alouini_Drone; @UAV_FSO; @Data_Collection_UAV_FSO; @Inter-UAV_FSO]. In particular, [@Alouini_Drone] discussed the advantages and challenges of FSO fronthauling for drone-based networks. Moreover, [@UAV_FSO; @Data_Collection_UAV_FSO; @Inter-UAV_FSO] studied a system consisting of several drones that communicate with each other through FSO links. Specifically, the authors of [@UAV_FSO] focused on the derivation of a *deterministic* model for the geometric loss assuming that the laser beam is always *orthogonal* to the receiver’s PD plane. However, to the best of the authors’ knowledge, a statistical model for the geometric loss of drone-based FSO links, which takes into account the fluctuations of the drone’s position and orientation as well as the non-orthogonality of the laser beam w.r.t. the PD plane, has not been developed in the literature, yet.
In this paper, we model the geometric loss of the drone-based FSO fronthaul channel while taking into account the drone’s instability and the non-orthogonality of the laser beam w.r.t. the PD plane. To this end, we first model the position and orientation of the drone as RVs. Then, we derive the geometric loss for a given realization of these RVs. In particular, we derive upper and lower bounds as well as approximate expressions for the geometric loss which are simpler than the exact expression. Finally, we derive a statistical model for the geometric loss assuming the drone’s position and orientation follow Gaussian distributions. Our simulation results validate the accuracy of the proposed bounds, approximations, and statistical model and quantify the quality of the FSO channel as a function of the drone’s instability.
System and Channel Models
=========================
In this section, we present the considered system model and the FSO channel model.
System Model
------------
We consider a drone-based communication network, where an arbitrary number of drones hover above an area, where a large number of users are concentrated, and operate as mobile remote radio heads to assist the communication between the users and a CU, see Fig. \[Fig:SysMod\]. In particular, we consider an uplink scenario where mobile users send their data to the drones over a multiple-access link, e.g., using sub-6 GHz radio frequency (RF) bands, and the drones forward the data over fronthaul links to a CU for final processing. Forwarding the aggregated user data received at the drones to the CU requires a huge data rate for the fronthaul links. Hereby, we propose to establish FSO links between the drones and the CU as large bandwidths can be realized at optical frequencies. The main goal of this paper is to develop a mathematical model that captures the effect of the fluctuation of a hovering drone’s position and orientation on the FSO channel quality. To do so, we formally define the position and orientation of the drone and the CU in our system model in the following.
In order to characterize an object in three dimensions, we need *at most* six independent variables, namely three variables to specify the position of a reference point of the object and three variables to quantify its orientation. With this in mind, we characterize the position and orientation of the CU and the drone as follows.
### CU
The CU is a fixed node located on top of a building. Without loss of generality, we choose the center of the PD as the reference point, which is located in the origin of the Cartesian coordinate system $(x,y,z)=(0,0,0)$. This coordinate system is referred to as Coordinate System 1, cf. Fig. \[Fig:Nodes\]. Moreover, we assume a circular PD of radius $a$. Note that it suffices to characterize the plane in which the PD lies to specify its orientation. Here, without loss of generality, we assume the PD lies in the $y-z$ plane at $x=0$.
### Drone
For the communication system under consideration, the parameters that directly affect the FSO channel are the position of the laser source of the drone and the direction of the laser beam. Therefore, without loss of generality, we refer to the position of the laser source and the direction of the laser beam as the drone’s position and orientation, respectively. We assume that the drone is in the hovering state. For practical reasons, in the hovering state, the position and orientation of the drone cannot be perfectly constant. Therefore, we model them as RVs. In particular, let $\mathbf{r}=(r_x,r_y,r_z)$ denote the vector of random position variables of the drone. Furthermore, let $\boldsymbol{\omega} = (\theta,\phi,\gamma)$ denote the vector of random orientation variables. The value of $\boldsymbol{\omega}$ depends on the coordinate system which is used to quantify these variables. Without loss of generality, we hereby use the following coordinate system which simplifies our analysis. For a given $\mathbf{r}$, let us define a new coordinate system, denoted as Coordinate System 2, with $\mathbf{r}$ as its origin and axes $x'$, $y'$, and $z'$, which are parallel to the $x$, $y$, and $z$ axes, respectively, cf. Fig. \[Fig:Nodes\]. We use variables $\theta$ and $\phi$ to determine the direction of the laser beam in a spherical representation of Coordinate System 2. In particular, $\theta\in[0,2\pi]$ denotes the angle between axis $x'$ and the projection of the beam vector onto the $x'-y'$ plane and $\phi\in[0,\pi]$ represents the angle between the beam vector and the $z'$ axis. The third orientation variable $\gamma$ is used to quantify the rotation around the beam vector. The advantage of the aforementioned representation of the orientation variables is two-fold. First, variable $\boldsymbol{\omega}$ does not change if coordinate $\mathbf{r}$ changes, i.e., the position and orientation variables are independent. Second, a rotation around the beam line does not affect the signal at the PD assuming rotational beam symmetry. Therefore, the value of $\gamma$ is irrelevant for our analysis. Hence, for simplicity of presentation, we drop $\gamma$ and use $\boldsymbol{\omega} = (\theta,\phi)$ in the remainder of the paper.
FSO Channel Model
-----------------
We assume direct detection at the CU where the PD responds to changes in the received optical signal power [@FSO_Survey_Murat]. In general, the FSO channel coefficient, denoted by $h$, is affected by several factors and can be modelled as follows
[lll]{}\[Eq:channel\] h=h\_p h\_a h\_g,
where $\eta$ is the responsivity of the PD and $h_p$, $h_a$, and $h_g$ are the path loss, atmospheric turbulence loss, and geometric loss, respectively. In particular, path loss $h_p$ is deterministic and represents the power loss over a propagation path due to attenuation. Atmospheric turbulence loss $h_a$ is random and induced by inhomogeneities in the temperature and the pressure of the atmosphere and is typically modelled as log-normal or Gamma-Gamma distributed RV. Moreover, geometric loss $h_g$ is caused by the divergence of the optical beam between the transmitter and the PD and the misalignment of the laser beam line and the center of the PD [@FSO_Survey_Murat; @FSO_Vahid]. Fluctuations of the drone’s position and orientation lead to a random geometric loss $h_g$. Hence, in this paper, we develop a statistical model for the geometric loss. This model allows us to study the performance of the considered communication system and the impact of the system parameters, such as the ability of the drone to maintain its position and orientation, on the FSO channel quality.
Modeling of the Geometric Loss
==============================
In this section, we first derive a deterministic model for the geometric loss for a given position and orientation of the nodes, and then a statistical model, assuming that the drone’s position and orientation fluctuate in the hovering state.
Deterministic Model {#Sec:hg_det}
-------------------
Here, we derive the geometric loss for a given state of the drone, i.e., for given $\mathbf{r}$ and $\boldsymbol{\omega}$. To do so, we first find the center of the beam footprint and the power density on the PD plane. Using these results, we then derive the geometric loss.
### Center of Beam Footprint
The line of the beam can be represented in Cartesian Coordinate System 1 as follows
[lll]{} \[Eq:BeamLine\] (x,y,z) = (r\_x,r\_y,r\_z) + t (d\_x,d\_y,d\_z),
where $t$ is an arbitrary real number and $\mathbf{d}=(d_x,d_y,d_z)$ denotes the beam direction which can be found as a function of $\theta$ and $\phi$ as
[lll]{} \[Eq:d\_angle\] =(()(),()(),()).
The center of the beam footprint on the PD can be obtained as the intersection point of the line of the laser beam and the PD plane, $x=0$. Denoting the center of the footprint of the beam on the PD as $\mathbf{f}=(f_x,f_y,f_z)$, cf. Fig. \[Fig:Nodes\], we obtain
[lll]{} \[Eq:FootPrint\_Center\] = (0,r\_y-r\_x(),r\_z-r\_x).
### Power Density on the PD Plane
We assume a Gaussian beam which dictates that the power distribution in any plane perpendicular to the direction of the wave propagation follows a Gaussian distribution [@FSO_Survey_Murat]. In particular, let us consider a perpendicular plane where the distance between the center of the beam footprint on the plane and the laser source is denoted by $L$. Then, the power density for any point on this perpendicular plane with distance $l$ to the center of the beam footprint is given by [@Steve_pointing_error]
[lll]{} \[Eq:PowerOrthogonal\] I\^(L,l) = (-),
where $w(L)$ is the beam width and is obtained as
[lll]{} \[Eq:BeamWidth\] w(L) = w\_0.
In (\[Eq:BeamWidth\]), $w_0$ denotes beam waist radius and $\rho(L)=(0.55C_n^2k^2L)^{-3/5}$ is the coherence length, where $C_n^2$ is the index of refraction structure parameter (assumed to be constant along the propagation path), $k = 2\pi/\lambda$ is the optical wave-number, and $\lambda$ is the optical wavelength. As mentioned before, for the problem at hand, the plane of the PD is not necessarily orthogonal to the beam direction. For this case, the power density in the PD plane, denoted by $I(y,z)$, is given in the following lemma.
\[Lem:PowerDensity\] Under the mild conditions $\Vert \mathbf{r} \Vert \gg \Vert \mathbf{f} \Vert$ and $\Vert \mathbf{r} \Vert \gg \Vert (y,z) \Vert$, where $\|\cdot\|$ denotes the norm of a vector, the power density at point $(y,z)$ on the PD plane is given by
[lll]{} \[Eq:Intensity\] I(y,z)&=()I\^(L(), l(,(y,z)))\
&=((\_[y]{}\^2+\_[z]{}\^2 + 2\_[yz]{})),
where $\psi=\sin^{-1}(\sin(\phi) \cos(\theta))$, $L(\mathbf{r})=\Vert \mathbf{r} \Vert$, $l(\boldsymbol{\omega},(y,z))=\rho_{y}\tilde{y}^2+\rho_{z}\tilde{z}^2
+ 2\rho_{yz}\tilde{y}\tilde{z}$, $\tilde{y} = y-f_y$, $\tilde{z} =z-f_z$, and $I^{\mathrm{orth}}(\cdot,\cdot)$ is given by (\[Eq:PowerOrthogonal\]). Moreover, $\rho_{y} = \cos^2(\phi)+\sin^2(\phi)\cos^2(\theta)$, $\rho_{z} = \sin^2(\phi)$, and $\rho_{yz} = -\cos(\phi)\sin(\phi)\sin(\theta)$.
Please refer to Appendix \[App:Lem\_PowerDensity\].
Note that the conditions under which (\[Eq:Intensity\]) in Lemma \[Lem:PowerDensity\] holds are met in practice, as in typical FSO links, $\Vert \mathbf{r} \Vert$ is on the order of several hundred meters, whereas $\Vert \mathbf{f} \Vert$ and $\Vert (y,z) \Vert$ are on the order of a few centimeters.
### Geometric Loss
The fraction of power collected at the PD, denoted by $h_g(\mathbf{r},\boldsymbol{\omega})$, can be obtained by integrating the power density obtained in Lemma \[Lem:PowerDensity\] over the PD area. This leads to
[lll]{}\[Eq:PowerPhotoDetector\] h\_g(,) = I(y,z) yz,
where $I(y,z) $ is given in (\[Eq:Intensity\]) and $\mathcal{A}$ is the set of $(y,z)$ within the PD area, i.e., $\mathcal{A}=\{(y,z)|y^2+z^2\leq a^2\}$. Unfortunately, the exact value of $h_g(\mathbf{r},\boldsymbol{\omega})$ cannot be derived in closed form. Instead, in the following theorem, we provide an upper and a lower bound on $h_g(\mathbf{r},\boldsymbol{\omega})$ which are subsequently used to derive an approximate closed-form expression for $h_g(\mathbf{r},\boldsymbol{\omega})$.
\[Theo:Power\] Using Lemma \[Lem:PowerDensity\], the geometric loss $h_g(\mathbf{r},\boldsymbol{\omega})$ is lower bounded by $h_g^{\mathrm{low}}(\mathbf{r},\boldsymbol{\omega})$ and upper bounded by $h_g^{\mathrm{upp}}(\mathbf{r},\boldsymbol{\omega})$ where
[lll]{} \[Eq:upper\_lower\_bound\] h\_g\^(,) =\
(-((y-u)\^2+\^2)) yz\
h\_g\^(,) =\
(-((y-u)\^2+\^2)) yz.
Here, $u=\sqrt{f_y^2+f_z^2}$, $\rho_{\min}=\frac{2}{\rho_y+\rho_z+\sqrt{(\rho_y-\rho_z)^2+4\rho_{zy}^2}}$, and $\rho_{\max}=\frac{2}{\rho_y+\rho_z-\sqrt{(\rho_y-\rho_z)^2+4\rho_{zy}^2}}$. For the special case, where the beam line is orthogonal to the plane of the PD, we obtain $\rho_{\max}=\rho_{\min}=1$ and $\rho_{yz}=0$ and the upper and lower bounds coincide and become identical to $h_g(\mathbf{r},\boldsymbol{\omega})$.
Please refer to Appendix \[App:Integral\].
We use Fig. \[Fig:Contour\] to illustrate the basic idea behind the upper and lower bounds proposed in Theorem \[Theo:Power\]. In particular, unlike the case where the optical beam is orthogonal w.r.t. the PD plane and the power density contours are circles [@Steve_pointing_error], the case where the optical beam is non-orthogonal w.r.t. the PD plane leads to power density contours which are *rotated ellipses*, e.g., the black solid contour in Fig. \[Fig:Contour\]. We have derived the lower bound assuming a contour that is a rotated ellipse whose major axis is perpendicular to the line connecting the center of the footprint to the origin, e.g., the green dash-dotted contour in Fig. \[Fig:Contour\]. Moreover, for the upper bound, the contour is a rotated ellipse whose minor axis is perpendicular to the line connecting the center of the footprint to the origin, e.g., the red dotted contour in Fig. \[Fig:Contour\]. In the special case where the major (minor) axis of the *original* power density contour is perpendicular to the line connecting the center of the footprint to the origin, the upper (lower) bound matches the exact geometric loss.
\[Fig:Contour\]
We emphasize that even for the case when the beam line is orthogonal w.r.t. the PD plane, the exact value of $h_g(\mathbf{r},\boldsymbol{\omega})$ is cumbersome and provides little insight. Therefore, in [@Steve_pointing_error], the authors proposed an approximation which was shown to be very accurate for $w(L)/a\geq 6$ and has been widely used subsequently [@Steve_MISO_FSO; @FSO_Receivers_UAV; @George_Pointing_error; @Alouini_Pointing_Beckman_Hoyt]. The proposed bounds in Theorem \[Theo:Power\] have two main advantages. First, for the special case when the beam line is orthogonal to the PD plane, the upper and lower bounds coincide with the exact $h_g(\mathbf{r},\boldsymbol{\omega})$. Second, the form of the integrals in (\[Eq:upper\_lower\_bound\]) allows us to employ the same technique as in [@Steve_pointing_error Appendix] to obtain accurate approximations. In particular, $h_g^{\mathrm{low}}(\mathbf{r},\boldsymbol{\omega})$ and $h_g^{\mathrm{upp}}(\mathbf{r},\boldsymbol{\omega})$ in (\[Eq:upper\_lower\_bound\]) can be approximated by [@Steve_pointing_error]
[lll]{} \[Eq:upper\_lower\_cal\] \_g\^(,) = A\_0 (-)\
\_g\^(,) = A\_0 (-),
respectively, where $A_0=\mathrm{erf}(\nu_{\min})\mathrm{erf}(\nu_{\max})$, $k_{{\min}}=\frac{\sqrt{\pi}\rho_{\min}\mathrm{erf}(\nu_{\min})}{2\nu_{\min}\exp(-\nu_{\min}^2)}$, and $\nu_{\min}=\frac{a}{w(L)}\sqrt{\frac{\pi}{2\rho_{\min}}}$. Similarly, $k_{{\max}}$ and $\nu_{\max}$ are obtained by changing the index $\min$ to $\max$ in the relevant equations. Moreover, $\mathrm{erf}(x)=\frac{1}{\sqrt{\pi}}\int_{-x}^{x}\exp(-t^2)\mathrm{d}t$ is the error function. Note that the only difference between the approximate upper and lower bounds in (\[Eq:upper\_lower\_cal\]) are the terms $k_{\max}$ and $k_{\min}$, respectively. This motivates us to propose the following approximation of $h_g(\mathbf{r},\boldsymbol{\omega})$
[lll]{} \[Eq:hg\_mean\] \_g(,) = A\_0 (-),
where $k_{\mathrm{mean}}=\frac{k_{\min}+k_{\max}}{2}$. Therefore, instead of considering the approximate upper and lower bounds in (\[Eq:upper\_lower\_cal\]), in the following, we employ the approximation in (\[Eq:hg\_mean\]) for our statistical analysis. We show in Section \[Sec:Sim\] that this approximation is accurate for a range of simulation parameters.
Statistical Model
-----------------
In the previous subsection, we derived an approximation of the geometric loss $h_g(\mathbf{r},\boldsymbol{\omega})$ for a drone’s given state $\mathbf{r}$ and $\boldsymbol{\omega}$ in (\[Eq:hg\_mean\]). However, in practice, the position and orientation of a hovering drone fluctuates, and hence, $\mathbf{r}$ and $\boldsymbol{\omega}$ are RVs. In the following, we first discuss the means and variances of these RVs.
### Mean
Let $\boldsymbol{\mu}_{\mathbf{r}}=(\mu_x,\mu_y,\mu_z)$ and $\boldsymbol{\mu}_{\boldsymbol{\omega}}=(\mu_{\theta},\mu_{\phi})$ denote the means of RVs $\mathbf{r}$ and $\boldsymbol{\omega}$, respectively. Since the drone is supposed to hover above the users, the mean position $\boldsymbol{\mu}_{\mathbf{r}}$ depends on the location of the users as well as the desired operating height of the drone. Given $\boldsymbol{\mu}_{\mathbf{r}}$, the drone’s tracking system aims to determine $\boldsymbol{\mu}_{\boldsymbol{\omega}}$ such that the beam line intersects with the center of the PD, i.e., $(0,0,0)$. This leads to
[lll]{} \[Eq:Angles\] \_=
\^[-1]{}()&\_x>0\
+\^[-1]{}()&
\
\_=-\^[-1]{}().
Note that the values of $\mu_{\theta}$ and $\mu_{\phi}$ may deviate from the above results if there is a tracking error. Nevertheless, in this paper, we assume perfect tracking where for a given $\boldsymbol{\mu}_{\mathbf{r}}$, $\boldsymbol{\mu}_{\boldsymbol{\omega}}$ is obtained from (\[Eq:Angles\]).
### Variance
Let $\boldsymbol{\sigma}_{\mathbf{r}}=(\sigma_x,\sigma_y,\sigma_z)$ and $\boldsymbol{\sigma}_{\boldsymbol{\omega}}=(\sigma_{\theta},\sigma_{\phi})$ denote the standard deviations of RVs $\mathbf{r}$ and $\boldsymbol{\omega}$, respectively. The values of $\boldsymbol{\sigma}_{\mathbf{r}}$ and $\boldsymbol{\sigma}_{\boldsymbol{\omega}}$ depend on how well the drone is able to maintain its position and orientation around the mean values $\boldsymbol{\mu}_{\mathbf{r}}$ and $\boldsymbol{\mu}_{\boldsymbol{\omega}}$, respectively. The smaller the values of the elements of $\boldsymbol{\sigma}_{\mathbf{r}}$ and $\boldsymbol{\sigma}_{\boldsymbol{\omega}}$ are, the more stable the drone is. Hence, we consider $\boldsymbol{\sigma}_{\mathbf{r}}$ and $\boldsymbol{\sigma}_{\boldsymbol{\omega}}$ as the drone’s quality measure and evaluate the performance of the FSO fronthaul link in terms of this measure.
In this paper, we assume that all position and orientation variables are independent from each other and follow Gaussian distributions, i.e.,
[lll]{} \[Eq:Gaussian\] = (\_x+\_x, \_y+\_y,\_z+\_z)\
= (\_+\_, \_+\_),
where $\epsilon_s\sim\mathcal{N}(0,\sigma^2_s)$ denotes a zero-mean normal RV with variance $\sigma^2_{s},\,s\in\{x,y,z,\theta,\phi\}$. We emphasize that the exact distributions of $\mathbf{r}$ and $\boldsymbol{\omega}$ have to be found via experimental measurements. Nevertheless, the adopted Gaussian model is a reasonble choice as it takes into account the first and second order moments of the RVs. Moreover, our assumption is inline with the Gaussian assumption made for derivation of the statistical model for the geometric loss due to building sway [@Steve_pointing_error; @Alouini_Pointing]. Substituting normal RVs $\mathbf{r}$ and $\boldsymbol{\omega}$ into (\[Eq:PowerPhotoDetector\]) and (\[Eq:hg\_mean\]), we obtain RVs $h_g(\mathbf{r},\boldsymbol{\omega})$ and $\tilde{h}_g(\mathbf{r},\boldsymbol{\omega})$, respectively.
Note that in (\[Eq:hg\_mean\]), $A_0$, $k_{\mathrm{mean}}$, and $u$ are RVs since $A_0$ and $k_{\mathrm{mean}}$ depend on RV $\boldsymbol{\omega}$ and $u$ depends on both RVs $\mathbf{r}$ and $\boldsymbol{\omega}$. Nevertheless, we observed from our simulations that the variances of $A_0$ and $k_{\mathrm{mean}}$ are several orders of magnitude smaller than the variance of $u$. The reason for this behaviour is that a small variation in $\boldsymbol{\omega}$, e.g., on the order of mrad, has a significant effect on $u=\sqrt{f_y^2+f_z^2}$ since the impact of this variation on $f_y$ and $f_z$ in (\[Eq:FootPrint\_Center\]) is scaled by $r_x$ which typically has a very large value, i.e., on the order of several hundred meters. On the other hand, the impact of variations in $\boldsymbol{\omega}$ is not scaled by $r_x$. Therefore, the main reason for the fluctuation of the geometric loss is the variation of the center of the beam footprint on the PD plane, i.e., $u$. Hence, in the following, we assume that the values of $A_0$ and $k_{\mathrm{mean}}$ are approximately constant and obtained based on the mean position and mean orientation of the drone, i.e., $\boldsymbol{\mu}_{\mathbf{r}}$ and $\boldsymbol{\mu}_{\boldsymbol{\omega}}$. Under this assumption, we determine the distribution of $u$ in the following theorem.
\[Theo:PDF\_r\] Assuming $\sigma_s\to 0,\,s\in\{x,y,z,\theta,\phi\}$, the distance between the center of the beam and the center of the PD, $r$, follows a Hoyt (Nakagami-q) distribution $u\sim\mathcal{H}(q,\Omega)$ with parameters $q=\sqrt{\frac{\min\{\lambda_1,\lambda_2\}}{\max\{\lambda_1,\lambda_2\}}}$ and $\Omega=\lambda_1+\lambda_2$. Moreover, $\lambda_1$ and $\lambda_2$ are the eigenvalues of matrix $\boldsymbol{\Sigma}$ which is given by
[lll]{} \[Eq:Cov\] =
\^2\_y+c\_1\^2\^2\_x+c\_2\^2\^2\_ & c\_1c\_5\^2\_x + c\_2c\_4\^2\_\
c\_1c\_5\^2\_x + c\_2c\_4\^2\_ & \^2\_z+c\_3\^2\^2\_+c\_4\^2\^2\_+c\_5\^2\^2\_x
,
where $c_1=-\tan(\mu_{\theta})$, $c_2=-\frac{\mu_x}{\cos^2(\mu_{\theta})}$, $c_3=\frac{\mu_x}{\sin^2(\mu_{\phi})\cos(\mu_{\theta})}$, $c_4=-\frac{\mu_x\cot(\mu_{\phi})\tan(\mu_{\theta})}{\cos(\mu_{\theta})}$, and $c_5=-\frac{\cot(\mu_{\phi})}{\cos(\mu_{\theta})}$ are constants.
Please refer to Appendix \[App:Theo\_PDF\_r\].
Based on the distribution of $u$ in Theorem \[Theo:PDF\_r\], the probability density function (PDF) of $\tilde{h}_g(\mathbf{r},\boldsymbol{\omega})$ in (\[Eq:hg\_mean\]), denoted by $f_{\tilde{h}_g}(x)$, is obtained as
[lll]{} \[Eq:PDF\] f\_[\_g]{}(x) = & ()\^[-1]{}\
& I\_0(-()), 0x A\_0,
where $\varpi = \frac{(1+q^2)k_{\mathrm{mean}}w^2(L)}{4q\Omega}$ is a constant and $I_0(\cdot)$ is the zero-order modified Bessel function of the first kind.
Now, we consider the special case where the average position and orientation of the drone correspond to a beam which is orthogonal w.r.t. the PD plane. In other words, we have $\mu_y=\mu_z=0$, $\mu_{\theta}=\pi$, and $\mu_{\phi}=\pi/2$. This leads to the following simplified matrix $\boldsymbol{\Sigma}$
[lll]{} \[Eq:Cov\_Rayleigh\] =
\^2\_y+\_[x]{}\^2\^2\_ & 0\
0& \^2\_z+\_x\^2\^2\_
,
which has eigenvalues $\lambda_1=\sigma^2_y+\mu_{x}^2\sigma^2_{\theta}$ and $\lambda_2=\sigma^2_z+\mu_x^2\sigma^2_{\phi}$. Hereby, assuming $\sigma^2_y=\sigma^2_z\triangleq\sigma^2_{p}$ and $\sigma^2_{\theta} =\sigma^2_{\phi}\triangleq\sigma^2_{o}$, RV $u$ follows a Rayleigh distribution [@Steve_pointing_error] and $\tilde{h}_g(\mathbf{r},\boldsymbol{\omega})$ follows the following distribution
[lll]{} \[Eq:PDF\_Rayleigh\] f\_[\_g]{}(x) = & ()\^[-1]{}, 0x A\_0,
where $\varrho=\frac{k_{\mathrm{mean}} w^2(L)}{4(\sigma^2_p+\mu^2_x\sigma^2_o)}$.
\[Remk:Special\] Note that as the values of $\lambda_1$ and $\lambda_2$ increase, the quality of the channel deteriorates since the probability of small values of $\tilde{h}_g$ increases. The simplified matrix $\boldsymbol{\Sigma}$ in (\[Eq:Cov\_Rayleigh\]) provides the important insight that the geometric loss is much more sensitive to the variance of the orientation $\sigma^2_{a}$ than to the variance of the position $\sigma^2_{p}$ since the variance of the orientation $\sigma^2_{a}$ is scaled by the mean distance between the drone and the CU, i.e., $\|\mathbf{r}\|=\mu_x$.
Simulation Results {#Sec:Sim}
==================
Unless stated otherwise, the default values of the parameters used for simulation are given by $C_n^2=10^{-14}$ m$^{2/3}$, $\lambda=1550$ nm, $a=10$ cm, $L=1$ km, $w_0=1$ mm [@Steve_pointing_error; @Alouini_Pointing]. Moreover, we obtained the simulation results reported in Figs. \[Fig:Stat\] and \[Fig:PDF\] based on $10^5$ realizations of RVs $\mathbf{r}$ and $\boldsymbol{\omega}$. To better quantify the non-orthogonality of the beam w.r.t. the PD plane, we express the mean position of the drone, $\boldsymbol{\mu}_{\mathbf{r}}$, in spherical coordinates as $(R,\alpha,\beta)$, i.e., $r_x=R\sin(\beta)\cos(\alpha), r_y=R\sin(\beta)\sin(\alpha)$, and $r_z = R\cos(\beta)$. Recall that for given $\boldsymbol{\mu}_{\mathbf{r}}$, $\boldsymbol{\mu}_{\boldsymbol{\omega}}$ is obtained from (\[Eq:Angles\]). Moreover, we assume identical standard deviations for the position variables, i.e., $\sigma_x=\sigma_y=\sigma_z= \sigma_p$, and identical standard deviations for the orientation variables, i.e., $\sigma_{\theta}=\sigma_{\phi}= \sigma_o$.
\[Fig:Bound\]
First, we study the effect of the non-orthogonality of the beam w.r.t. the PD plane for the deterministic geometric loss and investigate the accuracy of the bounds proposed in Theorem \[Theo:Power\], their approximations in (\[Eq:upper\_lower\_cal\]), and the proposed approximation in (\[Eq:hg\_mean\]). In Fig. \[Fig:Bound\], we show the deterministic loss due to the geometric loss in decibel (dB), i.e., $-10\log_{10}(h_g(\mathbf{r},\boldsymbol{\omega}))$, vs. $\alpha$ for $\beta=\pi/2$, $\sigma_p=\sigma_o=0$, and different values for the center of the beam footprint $(f_y,f_z)$. At $\alpha=0$, we have the special case of an orthogonal beam w.r.t. the PD plane where the loss for $(f_y,f_z)= (0,0)$ is due to the geometric loss and the loss for $(f_y,f_z)\neq (0,0)$ is due to both geometric loss and misalignment loss, cf. Footnote \[Ftn:Loss\]. We observe from Fig. \[Fig:Bound\] that as $\alpha$ increases, the loss increases. Note that although the beam line of the laser may not be orthogonal w.r.t. the PD plane, i.e., $\alpha\neq0$, $\alpha$ will be small in practice, i.e., $|\alpha|\ll \pi/2$. From Fig. \[Fig:Bound\], we observe that for $\alpha < \pi/4$, the loss due to the non-orthogonality of the beam is small (less than $1.5$ dB). Finally, Fig. \[Fig:Bound\] reveals that the proposed bounds and approximations are accurate for practical values of $\alpha$, i.e., $\alpha<\pi/4$.
Next, we study the effect of random fluctuations of the position and orientation of the drone on the *average* geometric loss, i.e., $\mathsf{E}_{h_g}\{h_g(\mathbf{r},\boldsymbol{\omega})\}$ where $\mathsf{E}_{x}\{\cdot\}$ denotes expectation w.r.t. RV $x$. Here, $\mathsf{E}_{h_g}\{h_g(\mathbf{r},\boldsymbol{\omega})\}$ is evaluated via simulation. In Fig. \[Fig:Stat\], we show the average geometric loss in dB vs. $\sigma$ for $\alpha=\pi/8$, $\beta=5\pi/8$, and different distances between the drone and the CU $L\in\{800,1000,1500\}$ m. Here, $\sigma$ is the standard deviation of the position or orientation, i.e., $(\sigma_p,\sigma_o)=(0,\sigma\,(\mathrm{cm}))$ or $(\sigma_p,\sigma_o)=(\sigma\,(\mathrm{mrad}),0)$. We observe from Fig. \[Fig:Stat\] that the proposed approximation in (\[Eq:hg\_mean\]) closely approaches the exact value of the geometric loss in (\[Eq:PowerPhotoDetector\]). Also, Fig. \[Fig:Stat\] reveals that the power loss due to orientation fluctuations on the order of mrad is much more severe than that due to position fluctuations on the order of cm. Furthermore, as the distance between the drone and the CU increases, the average geometric loss increases, too.
\[Fig:Stat\]
In Fig. \[Fig:PDF\], the PDF of the geometric loss is plotted for $\sigma_p=0$, $\sigma_o\in\{0.1,0.2\}$ mrad, and $(\alpha,\beta)\in\{(0,\pi/2),(\pi/8,5\pi/8)\}$. As can be observed from Fig. \[Fig:PDF\], the analytical statistical model proposed in (\[Eq:PDF\]) is in perfect agreement with the histogram of (\[Eq:PowerPhotoDetector\]). Note that the PDFs for the case where the beam is orthogonal w.r.t. the PD plane, i.e., $(\alpha,\beta)=(0,\pi/2)$, assume non-zero values at larger $h_g(\mathbf{r},\boldsymbol{\omega})$ (smaller $-10\log_{10}(h_g(\mathbf{r},\boldsymbol{\omega}))$) compared to the case where the beam is non-orthogonal w.r.t. the PD plane, i.e., $(\alpha,\beta)=(\pi/4,5\pi/4)$, cf. (\[Eq:PDF\]). Moreover, as the standard deviation $\sigma_o$ increases, the probability of larger geometric losses increases and hence, the corresponding PDFs become more heavy tailed.
\[Fig:PDF\]
Conclusions
===========
In this paper, we considered the modelling of a drone-based FSO fronthaul channel by quantifying the geometric loss caused by random fluctuations of the position and the orientation of the drone. We derived upper and lower bounds, corresponding approximate expressions, and a closed-form statistical model for the geometric loss. Furthermore, we validated our derivations via simulations and quantified the impact of the drone’s instability on the quality of the FSO channel using the developed model for the geometric loss. In future work, the proposed analytical model can be exploited for performance analysis of the considered drone-based communication system in terms of e.g. the outage probability, average bit/symbol error rate, and average achievable rate.
Proof of Lemma \[Lem:PowerDensity\] {#App:Lem_PowerDensity}
===================================
Note that $I(y,z)\mathrm{d}y\mathrm{d}z$ determines the fraction of power collected in the infinitesimally small area $\mathrm{d}y\mathrm{d}z$, i.e., $\mathrm{d}y\to 0$ and $\mathrm{d}z \to 0$, around the point $(0,y,z)$. Recall that the power density in any perpendicular plane is given by (\[Eq:PowerOrthogonal\]). To exploit this knowledge, we use the fact that any point $(0,y,z)$ in the PD plane is also located in another plane which is perpendicular to the beam line. Therefore, the power $I(y,z)\mathrm{d}y\mathrm{d}z$ can be obtained as $I(y,z)\mathrm{d}y\mathrm{d}z=I^{\mathrm{orth}}(L,l)\sin(\psi)\mathrm{d}y\mathrm{d}z$, where $\psi=\sin^{-1}(\sin(\phi) \cos(\theta))$. Next, we find $L$ and $l$. The distance $l$ is the distance between point $(0,y,z)$ and the beam line in (\[Eq:BeamLine\]). In general, the distance between a point $\mathbf{p}$ and a line specified by direction vector $\mathbf{u}$ and a given point $\mathbf{q}$ on the line can be obtained as
[lll]{} \[Eq:DistanceDefine\] l = ,
where $\times$ denotes the cross product between two vectors. For the problem at hand, we choose $\mathbf{p}=(0,y,z)$, $\mathbf{u}=\mathbf{d}$, and $\mathbf{q}=\mathbf{f}$, which leads to
[lll]{} \[Eq:Distance\] l = ((0,y,z)-) =\
\
=\_y+\_z+2\_[yz]{},
where we exploited the fact that $\Vert \mathbf{d}\Vert=1$ and introduced $\tilde{y} = y-f_y$ and $\tilde{z} =z-f_z$ where $\rho_y$, $\rho_z$, and $\rho_{yz}$ are given in Lemma \[Lem:PowerDensity\]. Moreover, the distance between the perpendicular plane and the laser source can be bounded as
[lll]{} - - L - + ,
where the extreme cases occur if the the beam line is parallel to plane $x=0$. In particular, we can safely assume that $\Vert \mathbf{r} - \mathbf{f} \Vert \pm \sqrt{\tilde{y}^2+\tilde{z}^2} \approx \Vert \mathbf{r} \Vert$ holds since the distance between the drone and the CU is much larger than $\Vert \mathbf{f} \Vert$ and $\sqrt{\tilde{y}^2+\tilde{z}^2}$. Substituting these results in (\[Eq:PowerOrthogonal\]) leads to (\[Eq:Intensity\]) and completes the proof.
Proof of Theorem \[Theo:Power\] {#App:Integral}
===============================
Note that in the $(y,z)$ plane, the contours of power density $I(y,z)=\bar{I}$ form ellipsoids given by
[lll]{} \[Eq:ellipse\] \_y(y-f\_y)\^2+2\_[yz]{}(y-f\_y)(z-f\_z)+\_z(z-f\_z)\^2=d,
where $d=\frac{w^2(L)}{2}\log\left(\frac{2\sin(\psi)}{\pi w^2(L)\bar{I}}\right)$. These ellipsoids are centered at point $(f_y,f_z)$ and rotated by angle $ \gamma=\frac{1}{2}\tan^{-1}\big(\frac{2 \rho_{yz}}{\rho_{y} -\rho_{z}}\big)$ counterclockwise, and have minor and major axis lengths of $\sqrt{\rho_{\min}/d}$ and $\sqrt{\rho_{\max}/d}$, respectively, where $\rho_{\min}=\frac{2}{\rho_y+\rho_z+\sqrt{(\rho_y-\rho_z)^2+4\rho_{zy}^2}}$ and $\rho_{\max}=\frac{2}{\rho_y+\rho_z-\sqrt{(\rho_y-\rho_z)^2+4\rho_{zy}^2}}$.
In order to obtain the lower and upper bounds of $h_g(\mathbf{r},\boldsymbol{\omega})$, we substitute the contour in (\[Eq:ellipse\]) by two rotated elliptic contours which have the same axis lengths $\rho_{\min}$ and $\rho_{\max}$; however, their main axes are either parallel or perpendicular to the line connecting $(f_y,f_z)$ and the origin, see Fig. \[Fig:Contour\]. Moreover, without loss of generality, we can define a new coordinate system by rotating the $y$ and $z$ axes by angle $\tau=\tan^{-1}(\frac{f_z}{f_y})$ such that the center of the ellipsoid in (\[Eq:ellipse\]) lies on the rotated $y$ axis, i.e., the center becomes $\Big(\big(\sqrt{f_y^2+f_z^2}\big),0\Big)$ in the new coordinate system. Note that the circular PD has the same description in the new and the old coordinate systems. This leads to lower and upper bounds $h_g^{\mathrm{low}}(\mathbf{r},\boldsymbol{\omega})$ and $h_g^{\mathrm{upp}}(\mathbf{r},\boldsymbol{\omega})$, respectively, as given in Theorem 1. This completes the proof.
Proof of Theorem \[Theo:PDF\_r\] {#App:Theo_PDF_r}
================================
We use the Taylor series expansions of $\tan(\theta)$, $\cot(\phi)$, and $\frac{1}{\cos(\theta)}$ in the expressions for $f_y$ and $f_z$, i.e.,
[lll]{} \[Eq:Taylor\] () &= (\_)+(-\_)\
() &= (\_)-(-\_)\
&= +(-\_).
We note that from (\[Eq:Gaussian\]), we have $\epsilon_{\theta}= \theta-\mu_{\theta}$ and $\epsilon_{\phi}=\phi-\mu_{\phi}$. Substituting (\[Eq:Taylor\]) into (\[Eq:FootPrint\_Center\]) and simplifying the results assuming perfect beam tracking using (\[Eq:Angles\]), we obtain
[lll]{} \[Eq:fyfz\_norm\] f\_y = \_y+c\_1\_x+c\_2\_\
f\_z = \_z+c\_3\_+c\_4\_+c\_5\_x,
where constants $c_1$, $c_2$, $c_3$, $c_4$, and $c_5$ are given in Theorem \[Theo:PDF\_r\]. To obtain (\[Eq:fyfz\_norm\]), we dropped the terms with orders higher than one, e.g., $\epsilon_{\theta}\epsilon_{\phi}$. Since $f_y$ and $f_z$ in (\[Eq:fyfz\_norm\]) are sums of Gaussian RVs, they are Gaussian distributed, too. However, $f_y$ and $f_z$ are correlated since $\epsilon_{x}$ and $\epsilon_{\theta}$ appear in both of them. The joint distribution of $f_y$ and $f_z$ is a bivariate Gaussian distribution $(f_y,f_z)\triangleq \bar{\mathbf{f}}\sim\mathcal{N}(\mathbf{0},\boldsymbol{\Sigma})$ where $\mathbf{0}=(0,0)$ and $\boldsymbol{\Sigma}$ is given in (\[Eq:Cov\]). Let $\boldsymbol{\Sigma}=\mathbf{U}\boldsymbol{\Lambda}\mathbf{U}^\mathsf{T}$ be the eigenvalue decomposition of $\boldsymbol{\Sigma}$ where $\boldsymbol{\Lambda}$ is a diagonal matrix with elements $\lambda_1$ and $\lambda_2$, $\mathbf{U}$ is a unitary matrix, i.e., $\mathbf{U}^\mathsf{T}\mathbf{U}=\mathbf{I}$, where $\mathbf{I}$ is the identity matrix and $(\cdot)^{\mathsf{T}}$ denotes the transpose operation. Using these definitions, it is easy to show that $\bar{\mathbf{f}}\sim\mathbf{g}\mathbf{U}^\mathsf{T}$ where $\mathbf{g}=(g_y,g_z)\sim\mathcal{N}(\mathbf{0},\boldsymbol{\Lambda})$. Now, we can express $u$ in terms of $\mathbf{g}$ as
[lll]{} u == \~ = .
Since $g_y$ and $g_z$ are independent zero-mean RVs with non-identical variances, $u$ follows a Hoyt (Nakagami-q) distribution $u\sim\mathcal{H}(q,\Omega)$, where $q=\sqrt{\frac{\min\{\lambda_1,\lambda_2\}}{\max\{\lambda_1,\lambda_2\}}}$ and $\Omega=\lambda_1+\lambda_2$ [@Nakagami_Hoyt]. This completes the proof.
[^1]: \[Ftn:Loss\] The receiver can only capture that fraction of power that falls onto the area of its photo-detector. This phenomenon is known as geometric loss. On the other hand, pointing errors further increase the geometric loss. This phenomenon is also known as misalignment loss [@Steve_pointing_error]. For simplicity of presentation, in this paper, we refer to the combined effect of these impairments as geometric loss.
|
---
abstract: 'In this paper we consider a coupled system of pdes modelling the interaction between a two–dimensional incompressible viscous fluid and a one–dimensional elastic beam located on the upper part of the fluid domain boundary. We design a functional framework to define weak solutions in case of contact between the elastic beam and the bottom of the fluid cavity. We then prove that such solutions exist globally in time regardless a possible contact by approximating the beam equation by a damped beam and letting this additional viscosity vanishes.'
address:
- 'CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, PSL Research University, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France'
- 'Inria Paris, 75012 Paris, France & Sorbonne Université, UMR 7598 LJLL,75005 Paris, France'
- 'IMAG, Univ Montpellier, CNRS, Montpellier, France'
author:
- 'Jean-Jérôme Casanova'
- Céline Grandmont
- Matthieu Hillairet
title: '[On an existence theory for a fluid-beam problem encompassing possible contacts]{}'
---
Introduction
============
In this paper we consider a fluid–structure system coupling a $2$D homogeneous viscous incompressible fluid with a $1$D elastic structure. When the elastic structure is at rest, the fluid domain is of rectangular type and the structure is located on the upper part of the fluid domain boundary. The fluid is described by the Navier–Stokes equations set in an unknown domain depending on the structure displacement that is assumed to be only transverse and that satisfies a beam equation. Since the fluid is viscous it sticks to the boundaries so that the fluid and the structure velocities are equal at the interface. Finally, the fluid applies a surface force on the structure. Such coupled nonlinear models can be viewed as toy models to describe the blood flow through large arteries.
The existence of a solution to the Cauchy problem associated with this kind of systems has been intensively studied in the last years. In [@DaVeiga; @Lequeurre11; @Lequeurre13] existence and uniqueness of a strong solution locally in time is proved in case additional viscosity is added to the structure equation (so that the structure displacement satisfies a damped Euler–Bernoulli equation). When no viscosity is added and in case the dynamics of the structure displacement is governed by a membrane equation, existence and uniqueness of a local strong solution is obtained in [@Grandmont-Hillairet-Lequeurre]. The beam case with no additional viscosity is investigated in [@Badra-Takahashi], where existence of strong solution locally in time (or for small data) is proved but with a gap between the regularity of the initial conditions and the propagated regularity of the structure displacement. Existence of weak solutions is obtained in [@Chambolle-etal] for $3$D-$2$D coupling where the structure behaviour is described by a viscous plate equation and in [@Grandmont08; @Muha-Canic13] in the non-viscous case. Let us also mention weak existence results on fluid–shell models [@Lengeler; @Muha-Schwarzacher]. Note that these results are obtained as long as the structure does not touch the bottom of the fluid cavity (or, in case of shells, as long as there is no self contact). More recently, in [@Grandmont-Hillairet], the authors establish existence of a global-in-time strong solution in the $2$D-$1$D case when the structure is governed by a damped Euler–Bernoulli equation. This global-in-time result is a consequence of a no contact one: it is proven therein that, for any $T>0,$ the structure does not touch the bottom of the cavity. The proof of this latter result relies strongly on the additional viscosity in the beam equation and on the control of the curvature of the structure.
The question we address here is: can we prove existence of a global weak solution regardless of a possible contact (for an undamped beam)? We aim to take advantage of the existence of global strong solution for a viscous structure and let the additional viscosity tend to zero. Our scheme is inspired by the one developed in [@SanMartin-Starovoitov-Tucsnak] where the global existence of a weak solution is derived for a $2$D fluid–solid coupled problem. However, in [@SanMartin-Starovoitov-Tucsnak] the solids are viewed as inclusions whose viscosities is infinite. The fluid–solid problem is then approximated by a completely fluid problem with different viscosities in the inclusions and in the fluid. The viscosity of the inclusions is then sent to infinity. In contrast, in our case the parabolic–hyperbolic fluid–structure system is approximated by a parabolic–parabolic one by adding viscosity to the structure. We prove that, up to the extraction of a subsequence, the sequence of solutions of the damped system converges towards a weak solution (in a sense to be defined) of the undamped system. The main difficulties are to define functional and variational frameworks compatible with a possible contact and to prove the strong compactness of the velocities, also in case of a possible contact. Indeed the proof developed for instance in [@Grandmont08], where the vanishing viscosity limit is also studied, strongly relies on the fact that the elastic structure does not touch the bottom of the fluid cavity.
The fluid-structure model
-------------------------
We introduce now the damped coupled fluid–structure system. We refer to this system as $(FS)_{\gamma}$, where the subscript $\gamma$ is used to track the dependency with respect to the “viscosity” of the structure. The configuration “at rest" of the fluid–structure system is assumed to be of the form $(0, L)\times (0, 1)$ where the elastic structure occupies the part of the boundary $(0, L)\times \{1\}$. The deformed fluid set is denoted by $\mathcal{F}_{h_{\gamma}(t)}$. It depends on the structure vertical deformation $h_{\gamma}=1+\eta_\gamma$, where $\eta_\gamma$ denotes the elastic vertical displacement. Thus, the deformed fluid configuration reads: $$\label{fluid.domain}
\mathcal{F}_{h_{\gamma}(t)}=\{(x,y)\in\mathbb{R}^{2}\mid 0<x<L,\,0<y<h_{\gamma}(x,t)\}.$$ The deformed elastic configuration is denoted by $\Gamma_{h_{\gamma}(t)}=\{(x,y)\in{\mathbb{R}}^{2}\mid x\in(0,L),\,y=h_{\gamma}(x,t)\}$.
The fluid velocity ${\boldsymbol{u}}_{\gamma}$ and the fluid pressure $p_{\gamma}$ satisfy the $2$-D incompressible Navier–Stokes equations in the fluid domain: $$\label{Navier-Stokes}
\begin{aligned}
\rho_{f}(\partial_{t}{\boldsymbol{u}}_{\gamma} + ({\boldsymbol{u}}_{\gamma}\cdot\nabla){\boldsymbol{u}}_{\gamma}) - {\text{div }}{\sigma({\boldsymbol{u}}_{\gamma},p_{\gamma})} &{}= 0\text{ in }\mathcal{F}_{h_{\gamma}(t)},\\
{\text{div }}{{\boldsymbol{u}}_{\gamma}}&{}=0\text{ in }\mathcal{F}_{h_{\gamma}(t)},
\end{aligned}$$ where $\sigma({\boldsymbol{u}}_{\gamma},p_{\gamma})$ denotes the fluid stress tensor given by the Newton law: $$\sigma({\boldsymbol{u}}_{\gamma},p_{\gamma})=\mu(\nabla{\boldsymbol{u}}_{\gamma} +( \nabla{\boldsymbol{u}}_{\gamma})^{T}) -p_{\gamma}I_{2}.$$ In the previous equations $\rho_{f}>0$ and $\mu>0$ are respectively the fluid density and viscosity. The structure displacement $\eta_{\gamma}$ satisfies a damped Euler–Bernoulli beam equation: $$\label{Beam.equation}
\rho_{s}\partial_{tt}\eta_{\gamma}-\beta\partial_{xx}\eta_{\gamma}-\gamma\partial_{xx}\partial_{t}\eta_{\gamma}+\alpha\partial_{x}^4\eta_{\gamma}=\phi({\boldsymbol{u}}_{\gamma},p_{\gamma},\eta_{\gamma})\text{ on }(0,L).$$ The constant $\rho_{s}>0$ denotes the structure density and $\alpha,\beta,\gamma$ are non negative parameters. Through this paper we assume that $\alpha>0$. This restriction guarantees sufficient regularity of the structure deformation in the compactness argument. The reader shall note for instance that we need $H^{1+\kappa} \cap W^{1,\infty}$ regularity of the deformation in Lemma \[lem:projector\].
The source term $\phi$ in the right-hand side of the beam equation arises from the action–reaction principle between the fluid and the structure. It represents the force applied by the fluid on the structure. It can be defined by the variational identity $$\label{def:forcefluide}
\int_{0}^{L}\phi({\boldsymbol{u}}_{\gamma},p_{\gamma},\eta_{\gamma})\cdot \varphi(x,h_{\gamma}(x)){\boldsymbol{e}}_{2}{\mathrm{d}}x=\int_{\Gamma_{h_{\gamma}(t)}}\sigma({\boldsymbol{u}}_{\gamma},p_{\gamma}){\mathbf{n}}_{\gamma}\cdot \varphi_{\vert \Gamma_{h_{\gamma}(t)}}{\boldsymbol{e}}_{2}{\mathrm{d}}\Gamma_{h_{\gamma}(t)},$$ for any regular test function $\varphi$ and where $\displaystyle{\mathbf{n}}_{\gamma}$ denotes the unit exterior normal to the deformed interface: $${\mathbf{n}}_{\gamma}=\frac{1}{\sqrt{1+(\partial_{x}\eta_{\gamma})^{2}}}\begin{pmatrix}
-\partial_{x}\eta_{\gamma}\\
1
\end{pmatrix}.$$ Since the fluid is viscous the following kinematic condition holds true at the interface $$\label{interface.condition}
{\boldsymbol{u}}_{\gamma}(x,h_\gamma(x,t),t)=\partial_{t}\eta_{\gamma}(x,t){\boldsymbol{e}}_{2}\text{ on } (0, L)\times (0, T).$$ We complement the fluid and structure boundary conditions with $$\label{fluid.boundary.condition}
\begin{aligned}
&{\boldsymbol{u}}_{\gamma}=0\text{ on }(0,L)\times \{0\},\\
&\eta_{\gamma}\text{ and }{\boldsymbol{u}}_{\gamma}\text{ are }L\text{-periodic with respect to }x.
\end{aligned}$$ Note that the kinematic condition together with the incompressibility constraint of the fluid velocity imply that, by taking into account the boundary conditions , $$\label{compatibilite-vitesse-structure}
\int_0^L\partial_t \eta_\gamma(t,x) {\rm d}x =0.$$ This condition states that the volume of the fluid cavity is preserved. This condition implies that the pressure $p_{\gamma}$ is uniquely determined in contrast with classical fluid–solid interaction problems. Finally the fluid–structure system is completed with the following initial conditions $$\label{initial.conditions}
\begin{aligned}
&\eta_{\gamma}(0)=\eta^{0}_\gamma\text{ and }\partial_{t}\eta_{\gamma}(0)=\eta_\gamma^{1}\text{ in }(0,L),\\
&{\boldsymbol{u}}_{\gamma}(0)={\boldsymbol{u}}_\gamma^{0}\text{ in }\mathcal{F}_{h_\gamma^{0}}\text{ with }h_\gamma^{0}=1+\eta_\gamma^{0}.
\end{aligned}$$
\[rem:korn\] As already noted in [@Chambolle-etal], due to the incompressibility constraint and the only transverse displacement of the beam $$\left((\nabla {\boldsymbol{u}}_\gamma)^T\cdot{\mathbf{n}}_\gamma\right)_2=0, \text{ on }\Gamma_{h_{\gamma}(t)}.$$ It implies that the force applied by the fluid on the beam can be defined as follows $$\int_{0}^{L}\phi({\boldsymbol{u}}_{\gamma},p_{\gamma},\eta_{\gamma})\cdot \varphi(x,h_{\gamma}(x)){\boldsymbol{e}}_{2}{\mathrm{d}}x=\int_{\Gamma_{h_{\gamma}(t)}}(\nabla{\boldsymbol{u}}_{\gamma}-p_{\gamma}I_2){\mathbf{n}}_{\gamma}\cdot \varphi_{\vert\Gamma_{h_{\gamma}(t)}}{\boldsymbol{e}}_{2}{\mathrm{d}}\Gamma_{h_{\gamma}(t)}.$$ For the same reason, a Korn equality also holds true $$\int_{\mathcal{F}_{h_\gamma(t)}} \vert\nabla{\boldsymbol{u}}_{\gamma} +( \nabla{\boldsymbol{u}}_{\gamma})^{T}\vert^2 =2\int_{\mathcal{F}_{h_\gamma(t)}} \vert\nabla{\boldsymbol{u}}_{\gamma} \vert^2.$$
The fluid–structure system – is denoted by $(FS)_{\gamma}$ and $(FS)_0$ corresponds to the system with $\gamma=0$ for which we are going to prove the existence of a global weak solution. The case where $\gamma>0$ is the one considered in [@Grandmont-Hillairet]. It is proven therein that the structure does not touch the bottom of the fluid cavity, namely $\min_{x\in (0, L)}(1+\eta_\gamma(x, t)) >0$, for all $t$, implying the existence of a unique global strong solution. In the case $\gamma=0,$ it is proven in [@Grandmont08; @Muha-Canic13] that a weak solution exists as long as the structure does not touch the bottom of the fluid cavity. In this paper, we investigate the vanishing viscosity limit ([*i.e.*]{} $\gamma\rightarrow 0$) and prove the convergence, up to the extraction of a subsequence, of the sequence of strong solutions $({\boldsymbol{u}}_\gamma,\eta_\gamma)$ solutions of $(FS)_{\gamma}$ defined on any time interval $(0, T)$ towards $({\boldsymbol{u}}, \eta)$ a weak solution (to be properly defined later on) of $(FS)_0$. Note that at the limit we loose the no contact property and have only: $\min_{x\in (0, L)}(1+\eta(x, t))\geq 0$, for all $t$. One key issue is thus to define an appropriate framework in case of contact. Moreover, and as it is standard for this kind of fluid–structure coupled problem, another important difficulty comes from the obtention of strong compactness of the approximate velocities. Such a property is mandatory in order to pass to the limit in the convective terms.
To conclude this introductory part, we point out that we do not address here the uniqueness of solutions. One reason is the lack of contact dynamics that should be added in case of contact, Hence, it is likely that our definition of weak solution below allows various rebounds of the elastic structure in case of contact (and consequently various solutions), one (or several) of these solutions coming from the construction process we consider herein. This issue is now well–identified in the fluid–solid framework [@Starovoitov03].
The rest of the paper splits into two sections. In the next section, we introduce and analyze a functional setting, we give the definition of weak solutions and state the main result. The last section is devoted to the proof of the existence result following standard steps: construction of a sequence of approximate solutions, derivation of compactness properties, passage to the limit. In the appendix, detailed proofs of technical lemma are given.
Problem setting
===============
In this section we first recall the energy estimates satisfied by any regular enough solution of the coupled problem. We then construct functional spaces and introduce a notion of weak solution relying on these energy estimates and compatible with a contact. Finally, we provide the rigorous statement of our main result and some technical lemma necessary to the following analysis.
Energy estimates
----------------
Let $\gamma \geq 0 $ and assume that $({\boldsymbol{u}}_\gamma, \eta_\gamma)$ is a classical solution to $(FS)_{\gamma}.$ Let then multiply the first equation of Navier–Stokes system by the fluid velocity ${\boldsymbol{u}}_\gamma$ and integrate over $\mathcal{F}_{h_\gamma(t)}$. Let also multiply the beam equation with the structure velocity $\partial_t\eta_{\gamma}$ and integrate over $(0, L)$. By adding these two contributions, after integration by parts in space – and by taking into account the coupling conditions (definition of $\phi$ and the kinematic condition ), the boundary conditions together with the incompressibility constraint and Remark \[rem:korn\] – we obtain $$\label{energy.equality}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\left(\rho_{f}\int_{\mathcal{F}_{h_{\gamma}(t)}}\vert{\boldsymbol{u}}_{\gamma}(t, {\boldsymbol{x}}) \vert^{2} {\mathrm{d}}{\boldsymbol{x}}+ \rho_{s}\int_{0}^{L}\vert\partial_{t}\eta_{\gamma}(t, x) \vert^{2}{\mathrm{d}}x+\beta\int_{0}^{L}\vert\partial_{x}\eta_{\gamma}(t, x) \vert^{2}{\mathrm{d}}x+\alpha\int_{0}^{L}\vert\partial_{xx}\eta_{\gamma}(t, x)\vert^{2} {\mathrm{d}}x\right)\\
&+\gamma\int_{0}^{L}\vert\partial_{tx}\eta_{\gamma}(t, x)\vert^{2} {\mathrm{d}}x+ \mu\int_{\mathcal{F}_{h_{\gamma}(t)}}\vert \nabla{\boldsymbol{u}}_{\gamma}(t, {\boldsymbol{x}}) \vert^{2}{\mathrm{d}}{\boldsymbol{x}}=0.
\end{aligned}$$ Note that we have used here that the set ${\mathcal{F}_{h_{\gamma}(t)}}$ moves with the velocity field ${\boldsymbol{u}}_{\gamma}$ thanks to the equality of velocities at the interface , that implies $$\int_{\mathcal{F}_{h_{\gamma}(t)}}(\partial_t{\boldsymbol{u}}_\gamma(t, {\boldsymbol{x}})+({\boldsymbol{u}}_{\gamma}(t, {\boldsymbol{x}})\cdot\nabla ){\boldsymbol{u}}_{\gamma}(t, {\boldsymbol{x}}))\cdot {\boldsymbol{u}}_{\gamma}(t, {\boldsymbol{x}}){\mathrm{d}}{\boldsymbol{x}}= \frac{1}{2}\frac{d}{dt}\int_{\mathcal{F}_{h_{\gamma}(t)}}\vert{\boldsymbol{u}}_{\gamma}(t, {\boldsymbol{x}}) \vert^{2} {\mathrm{d}}{\boldsymbol{x}}.$$ For $t >0,$ integrating over $(0, t)$ leads to $$\label{energy.estimates}
\begin{aligned}
&\frac{1}{2}\left(\rho_{f}\int_{\mathcal{F}_{h_{\gamma}(t)}}\vert{\boldsymbol{u}}_{\gamma}(t, {\boldsymbol{x}}) \vert^{2} {\mathrm{d}}{\boldsymbol{x}}+ \rho_{s}\int_{0}^{L}\vert\partial_{t}\eta_{\gamma}(t, x) \vert^{2} {\mathrm{d}}x+\beta\int_{0}^{L}\vert\partial_{x}\eta_{\gamma}(t, x) \vert^{2} {\mathrm{d}}x+\alpha\int_{0}^{L}\vert\partial_{xx}\eta_{\gamma}(t, x) \vert^{2} {\mathrm{d}}x\right.\\
&\left.+\gamma\int_0^t\int_{0}^{L}\vert\partial_{tx}\eta_{\gamma}(s, x) \vert^{2} {\mathrm{d}}x {\mathrm{d}}s+ \mu\int_0^t\int_{\mathcal{F}_{h_{\gamma}(s)}}\vert \nabla{\boldsymbol{u}}_{\gamma}(s, {\boldsymbol{x}}) \vert^{2} {\mathrm{d}}{\boldsymbol{x}}{\mathrm{d}}s \right)=\\
&\frac{1}{2}\left(\rho_{f}\int_{\mathcal{F}_{h^0_{\gamma}}}\vert{\boldsymbol{u}}^0_{\gamma}\vert^{2} {\mathrm{d}}{\boldsymbol{x}}+ \rho_{s}\int_{0}^{L}\vert\eta^1_{\gamma}\vert^{2}{\mathrm{d}}x+\beta\int_{0}^{L}\vert\partial_{x}\eta^0_{\gamma}\vert^{2}{\mathrm{d}}x+\alpha\int_{0}^{L}\vert\partial_{xx}\eta^0_{\gamma}\vert^{2}{\mathrm{d}}x\right).
\end{aligned}$$
As a consequence, we observe that, if $({\boldsymbol{u}}^0_\gamma, \eta^1_\gamma, \eta^0_\gamma)$ are such that the right-hand side of is uniformly bounded with respect to the viscosity parameter $\gamma \geq 0$, we have in particular $$\eta_{\gamma}\text{ is uniformly bounded in }L^{\infty}(0,T;H^{2}_{\sharp}(0,L))\cap W^{1,\infty}(0,T;L_{\sharp}^{2}(0,L)),$$ where the subscript $\sharp$ denotes spaces of periodic functions with respect to $x$. Thus the associated interface displacements $(\eta_\gamma)_{\gamma \geq 0}$ are uniformly bounded at least in $\mathcal{C}^{0}([0,T];\mathcal{C}^{1}_{\sharp}(0,L))$ thanks to the compact embedding $$L^{\infty}(0,T;H^{2}_{\sharp}(0,L))\cap W^{1,\infty}(0,T;L_{\sharp}^{2}(0,L))\hookrightarrow \mathcal{C}^{0,1-s}([0,T];\mathcal{C}^{1,2s-\frac{3}{2}}_{\sharp}(0,L)), \qquad \forall \, \tfrac{3}{4}<s<1.
\label{inclusion-eta}$$ Then, there exists $M>0$ depending on the initial data and independent of $\gamma$ such that $$\label{borne.eta}
0\leq 1+\eta_\gamma (t, x) \leq M , \quad \forall (x, t) \in [0, L]\times [0, T], \quad \forall \, \gamma \geq 0.$$
Finally, to define our functional setting, we rely below on the assumption that the initial data $({\boldsymbol{u}}^0, \eta^1, \eta^0)$ associated to $(FS)_0$ do satisfy the assumption that the right-hand side of is finite (for $\gamma =0$). So that we have at-hand an upper bound $M>0$ for the structure deformation $h=1+\eta$ for any physically reasonable solution. The above computations show also that, up to a good choice of regularized initial data the same functional framework can be used to describe the solutions to the damped system $(FS)_{\gamma}$ (for $\gamma >0$).
Functional spaces
-----------------
We design now a functional framework compatible with possible contact between the structure and the bottom of the fluid cavity. The parameter $M > 0$ is fixed in the whole construction.
Given a non-negative function $h \in \mathcal{C}^{1}_{\sharp}(0,L)$ such that $0\leq h \leq M$ we recall that we denote: $$\mathcal{F}_{h}=\{(x,y)\in\mathbb{R}^{2}\mid 0<x<L,\,0<y<h(x)\}.$$
In case $h$ vanishes two crucial difficulties appear. First, the set $\mathcal{F}_{h}$ does not remain connected (see Figure \[fig\_cusp\], the domain below the graph splits into a connected component between the red dots and a connected component outside the red dots). In particular, if $h$ is the deformation of a structure associated with a solution $({\boldsymbol{u}},p,\eta)$ to $(FS)_0,$ we may expect that the condition must be satisfied on each time–dependent connected component of the subset $\{x \in (0, L) \text{ s.t. } h(x) >0\}$ and not only globally on $(0,L).$ Secondly, the boundaries of $\mathcal{F}_{h}$ contain at least one “cusp” so that it does not satisfy the cone property (see [@Adams]). As a consequence, one must be careful in order to define a trace operator on $H^{1}(\mathcal F_h)$.
(0,0)node \[below left\] [$0$]{}; (10,0)node \[below right\] [$L$]{}; (2.5,0) node [$\bullet$]{}; (8,0) node [$\bullet$]{}; (0,0) – (10,0); (10,0) – (10,5); (0,5) – (0,0); (5.5,0.25)node \[above\] [$h(x)$]{}; (0,5) .. controls (1,5) and (1.5,0.1) .. (2.5,0); (2.5,0) .. controls (5,0) and (5,2) .. (6,1); (6,1) .. controls (6.5,0.5) and (6.5,0.125) .. (8,0); (8,0) .. controls (10.5,0.25) and (8.5,5) .. (10,5);
To overcome the second difficulty, we adapt the construction done in the context of fluid–solid problems in [@SanMartin-Starovoitov-Tucsnak]. Namely, we extend the fluid velocity fields – by taking into account their trace on the structure – on a time–independent domain whose regularity does not suffer from possible contacts.
First, let us make precise some specific notations for the various domains used in the analysis. We introduce a virtual container $\Omega = (0,L) \times (-1,2M).$ This set contains a part of the substrate ($(x,y) \in (0,L) \times (-1,0)$), the fluid film ($(x,y) \in \mathcal F_h$) and a virtual medium containing an extension of the structure (what remains of $\Omega$). Correspondingly, we also introduce three kinds of subsets of $\Omega.$ Given a continuous positive function $h$ we define first a subgraph domain (containing the substrate and the fluid film) $$\mathcal{F}^-_{h}=\{(x,y)\in{\mathbb{R}}^{2}\mid 0<x<L,\,-1<y<h(x)\},$$ then the epigraph domain (corresponding to the virtual elastic medium) $$\mathcal{S}_{h}=\{(x,y)\in{\mathbb{R}}^{2}\mid 0<x<L,\,h(x)<y<2M\}.$$ Finally, for the analysis, we need also more general sets. Given $a,b:(0,L)\rightarrow {\mathbb{R}}$ such that $a \leq b$, we also define the set $$\mathcal{C}_{a}^{b}=\{(x,y)\in{\mathbb{R}}^{2}\mid 0<x<L,\,a(x)<y<b(x)\}.$$ We emphasize that there is some overlap between these notations. In particular, ${\Omega}, \mathcal{F}_{h}, \mathcal{F}^-_{h}, \mathcal{S}_{h}$ can be seen as particular cases of sets of the form $\mathcal{C}_{a}^{b}$.
For the study of non cylindrical time–dependent problems, we also need notations for space–time domains. We use the convention that notations for time–independent domains extend to the time–dependent case by adding a hat. More precisely, we denote ${\widehat{\Omega}}={\Omega}\times(0,T)$ and $$\begin{aligned}
&\widehat{\mathcal{F}}_{h}=\bigcup_{t\in(0, T)}\mathcal{F}_{h(t)}\times\{t\}, &\widehat{\mathcal{F}}_{h}^{-}=\bigcup_{t\in(0, T)}\mathcal{F}_{h(t)}^{-}\times\{t\},\\
&\widehat{\mathcal{S}}_{h}=\bigcup_{t\in(0, T)}\mathcal{S}_{h(t)}\times\{t\}, &\widehat{\mathcal{C}}_{a}^{b}=\bigcup_{t\in(0, T)}\mathcal{C}_{a(t)}^{b(t)}\times\{t\},
\end{aligned}$$ where $h,a,b:(0,L)\times(0,T)\rightarrow {\mathbb{R}}$ are such that $h(x,t)\geq 0$ and $a(x,t)\leq b(x,t)$ for all $(x,t)\in (0,L)\times(0,T)$. We will also denote by ${\boldsymbol{n}}_h$ the vector $${\mathbf{n}}_{h}=\frac{1}{\sqrt{1+(\partial_{x}h)^{2}}}\begin{pmatrix}
-\partial_{x}h\\
1
\end{pmatrix}.$$ With these notations for the different sets, we introduce functional spaces to which our weak solutions will belong. The definition of these spaces is based on the following construction. Let us first introduce an extension operator:
\[def:bar\] Assume that $h\in \mathcal{C}^0_\sharp(0, L)$ with $0 \leq h \leq M.$ Let ${\boldsymbol{v}}\in {\boldsymbol{L}}^2_{\sharp}(\mathcal{F}_h)$ and $d\in L^2_\sharp(0, L)$, we define the extension operator by $$\overline {\boldsymbol{v}}= \left\{
\begin{array}{ll}
d {\boldsymbol{e}}_{2}, &\text{ in } \mathcal{S}_h,\\
{\boldsymbol{v}},&\text{ in } \mathcal{F}_h,\\
{\boldsymbol{0}},& \text{ in } \mathcal{C}_{-1}^0.
\end{array}
\right.$$
By construction, this extension operator defines a vector field $\overline {\boldsymbol{v}}\in {\boldsymbol{L}}^2_\sharp({\Omega}).$ In the previous definition the used symbol $\overline{{\boldsymbol{v}}}$ involves only ${\boldsymbol{v}}$ while the construction depends also on $d$. In what follows, this choice is justified as we consider functions ${\boldsymbol{v}}$ and $d$ satisfying the relation ${\boldsymbol{v}}_{|_{y=h}}= d{\boldsymbol{e}}_{2}$, where ${\boldsymbol{v}}_{|_{y=h}}$ denotes the function $x\mapsto {\boldsymbol{v}}(x, h(x))$ on $(0, L)$.
More precisely, when there is no contact this extension operator enjoys the following properties:
\[lem:bar\] Assume that $h\in W^{1, \infty}_\sharp(0, L)$ with $0<h(x)\leq M$ for $x\in [0, L]$ and let $s \in (0,1).$
1. If $s > 1/2$ and ${\boldsymbol{v}}\in {\boldsymbol{H}}^s_{\sharp}(\mathcal{F}_h)$ is divergence free with ${\boldsymbol{v}}_{|_{y=0}}=0$, and ${\boldsymbol{v}}_{|_{y=h}}=d {\boldsymbol{e}}_{2}$ with $d\in H^{s}_\sharp(0, L),$ we have that $$\begin{aligned}
& \overline{{\boldsymbol{v}}} \in {\boldsymbol{H}}^s(\Omega)\,, \qquad {\rm div }\, \overline{{\boldsymbol{v}}} = 0 \text{ in $\Omega$}, \qquad \overline{\boldsymbol{v}}\cdot{\boldsymbol{e}}_1=0 \text{ in $\mathcal{S}_h$}.\end{aligned}$$
2. If $0\leq s<1/2$ and ${\boldsymbol{v}}\in {\boldsymbol{H}}^s_{\sharp}(\mathcal{F}_h)$ is divergence free with ${\boldsymbol{v}}\cdot{\boldsymbol{e}}_2=0$ on $y=0$, and ${\boldsymbol{v}}_{|_{y=h}} \cdot{\boldsymbol{n}}_h=(0, d)^T \cdot{\boldsymbol{n}}_h$ on $(0,L)$ with $d\in H^s_\sharp(0, L),$ we have that: $$\begin{aligned}
& \overline{{\boldsymbol{v}}} \in {\boldsymbol{H}}^s(\Omega)\,, \qquad {\rm div }\, \overline{{\boldsymbol{v}}} = 0 \text{ in $\Omega$}, \qquad \overline{\boldsymbol{v}}\cdot{\boldsymbol{e}}_1=0 \text{ in $\mathcal{S}_h$}.\end{aligned}$$
3. In both cases $0\leq s < 1/2$ and $s>1/2$ the extension operator is a bounded linear mapping of its arguments whose norm can be bounded w.r.t $M$ only: $$\|\overline{{\boldsymbol{v}}}\|_{{\boldsymbol{H}}^s(\Omega)} \leq C(M)\left( \|d\|_{H^s_{\sharp}(0,L)} + \|{\boldsymbol{v}}\|_{H^s(\mathcal F_h)} \right).$$
We note that in both cases $\bar{{\boldsymbol{v}}}$ is by construction piecewisely divergence free and belongs to ${\boldsymbol{H}}^s$ (in the sets $ \mathcal{S}_h$, $ \mathcal{F}_h,$ $ \mathcal{C}_{-1}^0$). Consequently, in case (2) the extension is straightforwardly in ${\boldsymbol{H}}^s(\Omega).$ Only the continuity of normal traces is required to yield a global divergence free vector field. In Case (1) we require continuity of the full trace to obtain an ${\boldsymbol{H}}^s(\Omega)$ vector field.
\[rem:domaine\]
(i) In the case $\min_{x\in [0, L]} h(x)\geq 0$, we may extend vector fields defined on $\mathcal{F}^-_h$ with a similar bar-operator. Then, similar results for this extension operator hold true.
(ii) In Lemma \[lem:bar\] and in what follows, in order to avoid to denote the trace by the classical symbol $\gamma$, which is reserved here to the added viscosity on the structure, we denote by ${\boldsymbol v}_{|_{y=h}}$ the trace of ${\boldsymbol{v}}$ defined as ${\boldsymbol v}_{|_{y=h}}(x)={\boldsymbol{v}}(x, h(x))$. We note that when $h(x)>0$ for all $x\in [0, L]$, the associated linear trace operator is well defined from $H_\sharp^1(\mathcal{F}_h)$ into $H_\sharp^{\frac{1}{2}}(0, L)$. In the case where $h(x)\geq0$ for all $x\in [0, L]$ it is well defined from $H_\sharp^1(\mathcal{F}^-_h)$ into $H_\sharp^{\frac{1}{2}}(0, L)$. It is easy to verify that $$\label{est:trace}
\|{\boldsymbol{v}}_{|_{y=h}}\|_{H^\frac{1}{2}_\sharp(0, L)}\leq C(\|h\|_{W_\sharp^{1, \infty}(0, L)}) \|{\boldsymbol{v}}\|_{H^1_\sharp(\mathcal{F}^-_h)}.$$
Consequently, for a $W^{1, \infty}_{\sharp}(0,L)$–function $h$ satisfying $0 \leq h(x) \leq M$, for $x\in [0, L]$ and for $s \in (0,1),$ we set $$\begin{aligned}
&K^{s}[h]=\{{\boldsymbol{v}}\in {\boldsymbol{H}}^{s}_{\sharp}({\Omega})\mid {\text{div }}{{\boldsymbol{v}}}=0\text{ in }{\Omega},\,{\boldsymbol{v}}=0\text{ in }\mathcal{C}_{-1}^{0},\,\,{\boldsymbol{v}}\cdot{\boldsymbol{e}}_{1}=0\text{ in }\mathcal{S}_{h}\},\label{espace.Ks}\\
&X^{s}[h]=\{({\boldsymbol{w}},d)\in K^{s}[h]\times (H^{2s}_{\sharp}(0,L)\cap L_{\sharp, 0}^{2}(0,L))\mid {w_{2}}_{|_{(0, L)\times \{M\}}}=d\},\label{espace.Xs}\end{aligned}$$ where $$L_{\sharp, 0}^{2}(0,L)= \left\{d\in L_{\sharp}^{2}(0,L) \, s. t. \, \int_0^L d =0\right\}.$$ When $s=0$ we denote $K[h] = K^0[h]$ and $X[h] = X^0[h].$
Under the assumptions of Lemma \[lem:bar\] we have that $\bar{{\boldsymbol{v}}} \in K^s[h]$ and $(\bar{{\boldsymbol{v}}},d) \in X^s[h]$ in both cases $s \in (0,1/2)$ and $s \in (1/2,1)$. We emphasize that, for any ${\boldsymbol{v}}\in K[h]$, the divergence free condition implies that the trace on ${(0, L)\times \{M\}}$ of ${v_{2}}={\boldsymbol{v}}\cdot{\boldsymbol{n}}$ has a sense in $H^{-1/2}_\sharp(0, L)$. Similarly ${\boldsymbol{v}}_{|_{y=h}}\cdot {\boldsymbol{n}}_h$ also makes sense in $H^{-1/2}_\sharp(0, L)$. Following the construction of the extension operator above, one expects this trace to represent the structure velocity. Correspondingly, we introduce smooth variants of these functional spaces $\mathcal{K}[h]$ and $\mathcal{X}[h]$ defined by $$\begin{aligned}
&\mathcal{K}[h]=\{{\boldsymbol{w}}\in\mathcal{C}^{\infty}_{\sharp}({\Omega})\mid {\text{div }}{{\boldsymbol{w}}}=0\text{ in }{\Omega},\,{\boldsymbol{w}}=0\text{ in }\mathcal{V}(\mathcal{C}_{-1}^{0}),\,{\boldsymbol{w}}\cdot{\boldsymbol{e}}_{1}=0\text{ in } \mathcal V(\mathcal{S}_{h})\},\\
&\mathcal{X}[h]=\{({\boldsymbol{w}},d)\in \mathcal{K}[h]\times \mathcal{C}^{\infty}_{\sharp}(0,L)\mid {w_{2}}_{|_{(0, L)\times \{M\}}}=d\}.\end{aligned}$$ Here, we used“in $\mathcal{V}(\mathcal O)$” as a shortcut for the statement “in a neighbourhood of the open set $\mathcal O$".
Before defining the weak solutions, we now verify that the previous coupled spaces encode the fluid–structure nature of the problem and behave correctly (from an analytical standpoint). Once again, $h$ stands for a non–negative $W^{1, \infty}$–function satisfying $0\leq h \leq M$. The space $X[h]$ is endowed with the scalar product $$\label{def:produitscalaire}
\langle ({\boldsymbol{u}},\overset{\cdot}{\eta}),({\boldsymbol{w}},d)\rangle_{X[h]}:=\rho_{f}\int_{{\Omega}}{\boldsymbol{u}}\cdot{\boldsymbol{w}}+ \rho_{s}\int_{0}^{L}\overset{\cdot}{\eta}d,$$ and we endow the spaces $X^s[h]$ with a Hilbert structure associated with the norms $$\|({\boldsymbol{w}},d)\|_{X^s[h]}= \|{\boldsymbol{w}}\|_{H^s(\Omega)} + \|d\|_{H^{2s}(0,L)}.$$ For $s=0$ this Hilbert-norm does not correspond to the scalar product as defined in but the topologies are equivalent since $\rho_f$ and $\rho_s$ are both positive.
In order to prove the fluid–structure property, we show in the following lemma that, in the “virtual medium", the velocity–fields in $X[h]$ coincide with a structure velocity.
Let ${\boldsymbol{v}}\in K[h].$ There exists $d\in L_{\sharp}^{2}(0,L)$ such that ${\boldsymbol{v}}=d{\boldsymbol{e}}_{2}$ in $\mathcal{S}_{h}$.
By definition we have ${\boldsymbol{v}}=(0,v_{2})^{\top}$ in $\mathcal{S}_{h}$. Moreover the divergence condition ${\text{div }}{{\boldsymbol{v}}}=\partial_yv_{2}=0\text{ in }\mathcal{S}_{h},$ implies that $v_{2}(x,y)=v_{2}(x)$ in $\mathcal{S}_{h}$. Since ${\boldsymbol{v}}\in {\boldsymbol{L}}^2_\sharp({\Omega})$ and $0\leq h(x) \leq M, \forall x\in [0, L]$, we have $v_2\in L_\sharp^2(0, L)$.
Given a divergence free ${\boldsymbol{w}}\in {L}^2_{\sharp}(\Omega)$ it is classical that we can construct a stream function $\Psi \in {H}^1_{\sharp}(\Omega)$ such that ${\boldsymbol{w}}= \nabla^{\bot} \Psi.$ We show in the following lemma some additional properties satisfied by the stream function of an extended-field in $K[h]$:
\[lemma.X\[h\]\] Let $({\boldsymbol{w}}, d)\in X[h]$ and set $I=\{x\in[0,L]\mid h(x)>0\}$. There exists $\Psi \in {H}^1_\sharp(\Omega)$ such that ${\boldsymbol{w}}=\nabla^\perp\Psi$ which furthermore satisfies
- $\Psi(x,y)=b(x)$ in $\mathcal{S}_h$ with $b\in H^{1}_{\sharp}(0,L)$;
- $\Psi =0$ in $\mathcal{C}_{-1}^0$;
- $\Psi=0$ in $I^{c}\times(-1,2M)$.
We note that $\Psi$ is defined up to an additive constant. However, in $\mathcal{C}_{-1}^0$ we have ${\boldsymbol{w}}=0$, so that we fix this constant by choosing $\Psi=0$ in $\mathcal{C}_{-1}^0$. Then, due to the previous lemma ${\boldsymbol{w}}_{\vert_{(0, L)\times \{M\}}}= d{\boldsymbol{e}}_{2}$ and the identity ${\boldsymbol{w}}=d{\boldsymbol{e}}_{2}$ holds in $\mathcal{S}_{h}$. Thus $\partial_{x}\Psi=d$ in $\mathcal{S}_{h}$ and $\Psi(x,y)=b(x)$ in $\mathcal{S}_{h}$ where $b\in H^{1}_{\sharp}(0,L)$ satisfies $\partial_x b = d$. Remark that the $L$–periodicity of $b$ is ensured by $\int_{0}^{L}d(s){\mathrm{d}}s =0$.
Concerning the last point of the lemma, we emphasize that, since $\Psi \in {\boldsymbol{H}}^1_{\sharp}(\Omega)$, its trace is well defined on vertical lines $x=cst.$ Consequently, the value of $\Psi$ on $I^{c} \times (-1,2M)$ is well defined, whatever the topological properties of $I^c$ are. Now, given $a \in I^{c}$ (assuming $I^{c}$ is non-empty), we have $h(a)=0$ (by definition of $I^c$). The identity $\Psi(x,y)=b(x)$ in $ \mathcal{S}_{h}$, with $b\in H^{1}_{\sharp}(0,L)$, implies that $\Psi\in\mathcal{C}_\sharp^{0}(\mathcal{S}_{h})$. In particular, $\Psi$ is equal to a constant $b(a)$ on $\{a\}\times(-1,2M)$. Moreover, the function $\Psi$ is equal to $0$ on $\mathcal{C}_{-1}^{0}$. Finally, applying by a trace argument that $\Psi \in H^{1/2}(\{a\}\times (-1,2M))$ and using for example the following definition of the $H^{1/2}$-norm $${\left\Vert \Psi\right\Vert}_{H^{1/2}(\{a\}\times(-1,2M))}^{2}=\int_{\{a\}\times(-1,2M)}\Psi^{2} + \int_{\{a\}\times(-1,2M)}\int_{\{a\}\times(-1,2M)}\frac{\vert\Psi(a,x)-\Psi(a,y)\vert^{2}}{\vert x-y\vert^{2}},$$ we get that the trace of $\Psi$ cannot “jump” in $y=0.$ Therefore, we have $b(a)=0$ and $\Psi=0$ on $\{a\}\times(-1,2M)$.
We conclude this preliminary analysis of the space $X[h]$ by showing that we have density of smooth vector fields in $X[h]$. This is made precise in the following lemma:
\[lemma.density\]The embedding $X[h]\cap (\mathcal{C}_\sharp^{\infty}(\overline{{\Omega}})\times\mathcal{C}_\sharp^{\infty}(0, L))\subset X[h]$ is dense.
The difficulty of this proof is to deal with the case where $h$ has zeros. The main idea is to work with the stream function of the extended vector field. If we had $h(x)>0$, for all $x\in [0, L]$, a contraction in $y$ and a standard truncature and regularization argument on the stream function can be used. In the case where $h$ has zeros, one first cuts off the zeros of $h$ and then takes advantage of the better regularity of $\Psi$ on the structure. A detailed proof is given in the Appendix \[Annexe\].
Weak solutions and main result.
-------------------------------
In this section we introduce first our weak formulation of $(FS)_{\gamma}$.
We assume that the initial conditions $({\boldsymbol{u}}^{0},\eta^{0},\eta^{1})$ satisfy $$\begin{aligned}
&\eta^{0}\in H^{2}_{\sharp}(0,L) \text{ with }\displaystyle\min_{x\in [0,L]}(1+\eta^{0})>0,\label{CI1}\\
&({\boldsymbol{u}}^0,\eta^{1})\in {\boldsymbol{L}}_\sharp^{2}(\mathcal{F}_{h^{0}})\times L_{\sharp, 0}^{2}(0,L),\label{CI2}\\
& {\text{div }}{{\boldsymbol{u}}^{0}}=0 \text{ in }\mathcal{F}_{h^{0}},\label{CI3}\\
&{\boldsymbol{u}}^{0}\cdot {\boldsymbol{n}}^0 = 0 \text{ on } (0, L)\times \{0\} \text{ and } {\boldsymbol{u}}^{0}(\cdot, h_0(\cdot))\cdot {\boldsymbol{n}}^0= (0, \eta^{1}(\cdot) )^T\cdot {\boldsymbol{n}}^0\text{ on }(0, L).\label{CI4}\end{aligned}$$ We can then define $M>0$ by and construct the associated ${\Omega}.$ We have the following definition for a weak solution to $(FS)_{\gamma}$:
\[def.ws\] Let $({\boldsymbol{u}}^{0},\eta^{0},\eta^{1})$ satisfying – and $\gamma >0.$ We say that a pair $({\overline{{\boldsymbol{u}}}_\gamma},\eta_\gamma)$ is a weak solution to $(FS)_\gamma$ if it satisfies the following items:
i) $({\overline{{\boldsymbol{u}}}_\gamma},\eta_\gamma)\in L^{\infty}(0,T;{\boldsymbol{L}}_{\sharp}^{2}({\Omega}))\times \left(L^{\infty}(0,T;H^{2}_{\sharp, 0}(0,L))\cap W^{1,\infty}(0,T;L_\sharp^{2}(0,L))\right)$ with $$({\overline{{\boldsymbol{u}}}_\gamma}(t), \partial_t\eta_\gamma(t))\in X[h_\gamma(t)] \text{ for a.e. $t\in(0,T),$} \quad
\nabla \overline{{\boldsymbol{u}}}_\gamma \in L^{2}(\hat{\mathcal{F}}^{-}_{h_\gamma}),$$
ii) the kinematic condition $${\boldsymbol{u}}_\gamma(t, x, 1+\eta_\gamma(t, x))=\partial_t\eta_\gamma(t,x) {\bf{e}}_2 \quad \textrm{ on } (0, T) \times (0, L),$$
iii) For any $({\boldsymbol{w}}_\gamma, d_\gamma)\in \mathcal{C}_\sharp^{\infty}\left(\overline{{\widehat{\Omega}}}\right)\times\mathcal{C}_\sharp^{\infty}( [0, L]\times [0, T])$ such that $({\boldsymbol{w}}_\gamma(t), d_\gamma(t))\in\mathcal{X}[h_\gamma(t)]$ for all $t\in [0,T]$ we have for a.e. $t\in(0,T)$ $$\label{weak.formulation.FS}
\begin{aligned}
&\rho_{f}\int_{\mathcal{F}_{h_\gamma(t)}}{\boldsymbol{u}}_\gamma(t)\cdot{\boldsymbol{w}}_\gamma(t) -\rho_{f}\int_{0}^{t}\int_{\mathcal{F}_{h_\gamma(s)}}{\boldsymbol{u}}_\gamma\cdot\partial_{t}{\boldsymbol{w}}_\gamma + ({\boldsymbol{u}}_\gamma\cdot\nabla){\boldsymbol{w}}_\gamma\cdot{\boldsymbol{u}}_\gamma\\
&+\rho_{s}\int_{0}^{L}\partial_{t}\eta_\gamma(t)d_\gamma(t)-\rho_{s}\int_{0}^{t}\int_{0}^{L}\partial_{t}\eta_\gamma\partial_{t}d_\gamma+\mu\int_{0}^{t}\int_{\mathcal{F}_{h_\gamma(s)}} \nabla{\boldsymbol{u}}_\gamma:\nabla{\boldsymbol{w}}_\gamma\\
&+\beta\int_{0}^{t}\int_{0}^{L}\partial_{x}\eta_\gamma\partial_{x}d_\gamma + \alpha\int_{0}^{t}\int_{0}^{L}\partial_{xx}\eta_\gamma\partial_{xx}d_\gamma+\gamma\int_{0}^{t}\int_{0}^{L}\partial_{xt}\eta_\gamma\partial_{x}d_\gamma \\&
=\rho_{f}\int_{\mathcal{F}_{h_0}}{\boldsymbol{u}}^{0}\cdot{\boldsymbol{w}}_\gamma(0)+\rho_{s}\int_{0}^{L}\eta^{1}d_\gamma(0).
\end{aligned}$$ where ${\boldsymbol{u}}_\gamma={\overline{{\boldsymbol{u}}}_\gamma}_{\vert \widehat{\mathcal{F}}_{h_\gamma}}.$
The regularity statements in the first item of the definition comes from the energy estimate while the weak formulation is obtained classically by multiplying the fluid equation with ${\boldsymbol{w}}_{\gamma}$ and the beam equation with $d_{\gamma}$ and performing formal integration by parts. As usual for this type of fluid–structure problem, the test functions depend on the solution and thus on the parameter $\gamma,$ adding further nonlinearity to the system.
We recall that, from [@Chambolle-etal], [@Grandmont08], there exists a weak solution for $\gamma \geq 0$ as long as the beam does not touch the bottom of the fluid cavity. If $\gamma >0$ again and the initial data are smooth enough, it is also proved in [@Grandmont-Hillairet] that there exists a unique global in time strong solution such that $\min_{x\in [0, L]}h_\gamma(x, t)>0$ for all $t>0$. The main result of this paper is stated in the following theorem
\[main.theorem\] Suppose that $T>0$ and that the initial conditions $({\boldsymbol{u}}^{0},\eta^{0},\eta^{1})$ satisfy -.
Then $(FS)_0$ has a weak solution $({\boldsymbol{u}},\eta)$ on $(0,T).$ This solution satisfies furthermore for a.e. $t \in (0,T)$ $$\label{energy.estimates.thm}
\begin{aligned}
&\frac{1}{2}\left(\rho_{f}\int_{\mathcal{F}_{h(t)}}\vert{\boldsymbol{u}}(t, {\boldsymbol{x}}) \vert^{2} {\mathrm{d}}{\boldsymbol{x}}+ \rho_{s}\int_{0}^{L}\vert\partial_{t}\eta(t, x) \vert^{2} {\mathrm{d}}x+\beta\int_{0}^{L}\vert\partial_{x}\eta(t, x) \vert^{2} {\mathrm{d}}x+\alpha\int_{0}^{L}\vert\partial_{xx}\eta(t, x) \vert^{2} {\mathrm{d}}x\right)\\
& + \mu\int_0^t\int_{\mathcal{F}_{h(s)}}\vert \nabla{\boldsymbol{u}}(s, {\boldsymbol{x}}) \vert^{2} {\mathrm{d}}{\boldsymbol{x}}{\mathrm{d}}s \leq \\
&
.\qquad \qquad \frac{1}{2}\left(\rho_{f}\int_{\mathcal{F}_{h^0}}\vert{\boldsymbol{u}}^0\vert^{2} {\mathrm{d}}{\boldsymbol{x}}+ \rho_{s}\int_{0}^{L}\vert\eta^1\vert^{2}{\mathrm{d}}x+\beta\int_{0}^{L}\vert\partial_{x}\eta^0\vert^{2}{\mathrm{d}}x+\alpha\int_{0}^{L}\vert\partial_{xx}\eta^0\vert^{2}{\mathrm{d}}x\right).
\end{aligned}$$
Before detailing the proof of this result, we shall comment on the choice of test functions and the relations with a strong formulation of $(FS)_\gamma,$ in particular in the case where contacts occur. Since we focus on the construction of weak solutions, we stick here to a short description of formal arguments. Before contact ([*i.e.*]{} as long as $\min_{x \in [0,L]} h_\gamma(x,t) \geq \bar{\alpha}$ for some $\bar{\alpha} >0$), we claim that our definition coincides with the definition of [@Chambolle-etal] and that the solutions constructed in [@Grandmont-Hillairet] for smooth data match our definition also. In particular in this case we can choose test functions such that $({\boldsymbol{w}}_\gamma(t), d_\gamma(t))\in X^1[h_\gamma(t)]$. Before contact, we recover – from the weak formulation using a classical argument. First, we may take as test functions the vector fields ${\boldsymbol{w}}_\gamma \in \mathcal{C}^{\infty}_c(\hat \Omega)$ with $d = 0$ and we recover with a zero mean pressure $p$ by adapting an argument of de Rham. Assuming that $({\boldsymbol{u}}_\gamma,p_\gamma)$ is sufficiently smooth – to be able to define $\sigma({\boldsymbol{u}}_\gamma,p_\gamma) {\boldsymbol{n}}_{\gamma}$ on $\Gamma_{h_{\gamma}}$ – we can also recover the beam equation with the following construction that enables to extend structure test functions in the fluid domain
\[def:R\] Let $\lambda >0$ and $\zeta \in \mathcal{C}^{\infty}(\mathbb R)$ such that $\mathbf{1}_{[1,\infty)} \leq \zeta \leq \mathbf{1}_{[1/2,\infty)}. $ Given $d\in L_{\sharp,0}^2(0, L)$, we define $$\mathcal{R}_\lambda(d)(x,y)= \nabla^{\bot} (b(x) \zeta(y/\lambda)) \quad
\forall \, (x,y) \in \Omega.$$ where $b \in H^{1}_{\sharp}(0,L) \cap L^2_{\sharp,0}(0,L)$ satisfies $\partial_x b = d.$
We note that the above construction is well defined since $d$ is chosen to be mean free. We do not include the dependence on $\zeta$ in the name of our operator since it will be a given fixed function throughout the paper. The present lifting operator is a variant of the one introduced in [@Chambolle-etal]. It enjoys the following straightforward properties:
Let $\lambda >0$ and $h\in W^{1, \infty}_\sharp(0, L)$ satisfying $\lambda\leq h(x)\leq M, \forall x\in [0, L]$.
1. $\mathcal R_{\lambda}$ is a linear continuous mapping from $H^s_{\sharp}(0,L) \cap L^2_{\sharp}(0,L)$ into $K^s[h]$ for arbitrary $s \in [0,1].$
2. $\mathcal R_{\lambda}$ maps $\mathcal{C}^{\infty}_{\sharp}(0,L) \cap L^2_{\sharp,0}(0,L)$ into $\mathcal K[h].$
Consequently, before contact, for any arbitrary structure test function $d_{\gamma} \in \mathcal{C}^{\infty}_{\sharp}(0,L) \cap L^2_{\sharp,0}(0,L)$ we may consider $${\boldsymbol{w}}_{\gamma} := \mathcal R_{{\lambda}}[d_{\gamma}] \in \mathcal{C}^{\infty}_{\sharp}(\overline{\hat{\Omega}}),$$ so that $({\boldsymbol{w}}_{\gamma},d_{\gamma})$ is an admissible test function in our weak formulation. Classical integration by parts argument then enables to recover the structure equation multiplied by $d_{\gamma}.$ We note that, in this way, we recover up to a constant (indeed the test function $d_{\gamma}$ is mean free), but this constant mode corresponds to the choice of the constant normalizing the pressure in order to match the global volume preserving constraint .
When contact occurs, we recover a similar set of equations, assuming, once again, that the solution is sufficiently regular. Let consider for instance a simplified configuration such that, on some time interval $(T_0,T_1)$ there exist $\mathcal C^1$–functions $(a_k^{-},a_k^+): (T_0,T_1) \to \mathbb R^2$ ($k\in \mathbb N$) such that $$\{(t,x) \in (T_0,T_1 )\times(0,L) \ | \ h(t,x) >0 \} = \bigcup_{k\in\mathbb N} \bigcup_{t\in (T_0,T_1)} \{t\} \times (a_k^{-}(t),a_{k}^+(t))$$ In that case, we can reproduce similar arguments developed in the no contact case to recover the Navier–Stokes equations and the structure equations in each connected component of the fluid domain. More precisely, let introduce $$\begin{aligned}
& \hat{\mathcal F}_k := \{(t,x,y) \in (T_0,T_1) \times (0,L) \times (0,M) \text{ s.t. } a_k^{-}(t) < x < a_k^{+}(t) \quad 0 < y < h_{\gamma}(t,x) \}\,,
\\
& \hat{\Gamma}_k := \{(t,x) \in (T_0,T_1) \times (0,L) \text{ s.t. } a_k^{-}(t) < x < a_k^{+}(t) \}\,.\end{aligned}$$ First, by using that ${\boldsymbol{u}}_{\gamma}$ is divergence free on $\hat{\mathcal F}_{k}$ we obtain $$\int_{a_{k}^{-}(t)}^{a_k^{+}(t)} \partial_t \eta_{\gamma} = 0.$$ Second, by taking as a fluid test function a velocity field ${\boldsymbol{w}}_{\gamma}$ with compact support in $\hat{\mathcal F}_k$, we construct a pressure $p_{k,\gamma}$ on $\hat{\mathcal F_{k}}$ so that holds true. We recall that, at this point, $p_{k,\gamma}$ is defined up to a constant. The global pressure $p_{\gamma}$ is then constructed by concatenating all the $(p_{k,\gamma})_{k}$ to yield a pressure on $\hat{\mathcal F}_{h_{\gamma}}$ (that is defined up to a number of constants related to the number of parameters $k$). Third, we consider a mean free structure test function $d_{\gamma} \in \mathcal{C}^{\infty}_c(\hat{\Gamma}_{k}).$ Since $d_{\gamma}$ has compact support in the open set where $h_{\gamma} >0 $, $h_{\gamma}$ is bounded from below by some $\alpha_k>0$ on $\hat{\Gamma}_k$. So, instead of choosing the mean free anti-derivative $b_{\gamma}$ of the structure test function $d_{\gamma}$ in the definition of $\mathcal R_{{\alpha}_k}$, we choose the one that vanishes outside the support of $d_{\gamma}.$ In that way, we construct a test function ${\boldsymbol{w}}_{\gamma}$ such that $({\boldsymbol{w}}_{\gamma},d_{\gamma})$ is adapted to our weak formulation. So, we obtain on $\hat{\Gamma}_{k}$ up to a constant which is afterwards fixed by a suitable choice of the pressure on $\hat{\mathcal F}_{k}$. [Note that the structure equation is recovered on each component $\hat{\Gamma}_{k}$ and not on the whole interval $(0, L)$]{} and that the pressure is again uniquely defined but that there are more constants to fix than in the no contact case. To end up this remark, we emphasize that when $\min_{x\in [0, L]}h_\gamma(x, t)>0, \forall t\in [0, T]$, the test functions can be chosen in $X[h_\gamma(t)]$. Moreover if $\min_{x\in [0, L]}h_\gamma(x, t)>0$ the elastic test functions can be chosen independent of the solution and thus independent of the regularization parameter $\gamma$ (see [@Chambolle-etal]). It is not the case when a contact occurs since, as we saw in the previous construction, we require $d_\gamma=0$ in a neighbourhood of the contact points.
We end this part by giving a roadmap of our proof of Theorem \[main.theorem\]. To obtain a solution of the variational problem for $\gamma=0$ we consider the approximate fluid–structure system $(FS)_{\gamma}$ with a viscosity $\gamma>0$. From [@Grandmont-Hillairet-Lequeurre Theorem 1] this fluid–structure system $(FS)_{\gamma}$, completed with regularized initial conditions $({\boldsymbol{u}}^{0}_{\gamma},\eta^{0}_{\gamma},\eta^{1}_{\gamma})$, admits a unique strong solution $({\boldsymbol{u}}_{\gamma},p_{\gamma},\eta_{\gamma})$ such that $\min_{x\in [0, L]}h_\gamma(x, t)>0$. It ensures that the existence time interval does not depend on $\gamma$. Moreover, this solution satisfies the energy equality , and thus one can extract converging subsequences. We may then consider one cluster point of this sequence and show that this is a weak solution to $(FS)_0$. One key point here is that strong compactness of the approximate velocity fields is needed to pass to the limit in the convective nonlinear terms. The classical Aubin–Lions lemma does not apply directly because of the time-dependency of fluid domains and of the divergence free constraint. Many different strategies may be used to handle this difficulty [@Fujita-Sauer; @Grandmont08; @Lengeler; @Moussa]. Here we follow the line of [@SanMartin-Starovoitov-Tucsnak] where the existence of weak solutions for a fluid–solid problem beyond contact is proven. We first obtain compactness of a projection of the fluid and structure velocities. Roughly speaking the idea is to obtain compactness on fixed domains independent of time and of $\gamma$. So, we define an interface satisfying $0\leq\underline{h}\leq h_\gamma$, which is regular enough and “close" to $h_\gamma$ for all $\gamma$ small enough and we prove compactness of the projections of the velocity fields on the coupled space associated to $\underline{h}.$ We recover the compactness of the velocity fields by proving some continuity properties of the projection operators with respect to $\underline{h}.$ [In particular we prove that $X^{s}[\underline{h}]$ is a good approximation space of $X^{s}[h]$ in $H^s$ for some $s\geq 0$ whenever $\underline{h}$ is close to $h$.]{} This part of the proof is purely related to the definition of the spaces $X[h]$. Consequently we detail the arguments as preliminaries in the next subsection. We emphasize again that, since one may loose the no contact property at the limit, this study on the compactness of approximate velocity fields requires specific constructions. Once compactness is obtained, we pass finally to the limit in the weak formulation. Again, as contact may occur in the limit problem, we cannot follow [@Chambolle-etal] to construct a dense family of test functions independent of $\gamma$ to pass to the limit.
On the $h$-dependencies of the spaces $X^s[h]$ {#continuite_h}
----------------------------------------------
In this section, we analyze the continuity properties of the sets $X^s[h]$ with respect to the parameter $h.$ To start with, we remark that, given $h \in \mathcal{C}^0_{\sharp}(0,L)$ satisfying $0 \leq h \leq M$ the space $X^s[h]$ is a closed subspace of $$X^s := {\boldsymbol{H}}^s_{\sharp}(\Omega) \times H^{2s}_{\sharp}(0,L) .$$ We can then construct the projector $\mathbb P^s[h] : X^s \to X^s[h].$ We analyze in this section the continuity properties of these projectors with respect to the function $h.$ Our main result is the following lemma:
\[lem:projector\] Fix $0<\kappa<\frac{1}{2}$. Let $h$ and ${{\underline{h}}}$ belong to $H_\sharp^{1+\kappa}(0,L)\cap W_\sharp^{1,\infty}(0,L)$ with $0\leq {{\underline{h}}}\leq h\leq M$ and set $$\label{borne:A}{\left\Vert h\right\Vert}_{H^{1+\kappa}_{\sharp}(0,L)}+{\left\Vert h\right\Vert}_{W^{1,\infty}_{\sharp}(0,L)}+{\left\Vert {{\underline{h}}}\right\Vert}_{H^{1+\kappa}_{\sharp}(0,L)}+{\left\Vert {{\underline{h}}}\right\Vert}_{W^{1,\infty}_{\sharp}(0,L)}\leq A.$$ Let $s \in [0,\kappa/2)$ and $({\boldsymbol{w}},{\overset{.}{\eta}}) \in X^s[h]$ enjoying the further property $$\begin{aligned}
&{\boldsymbol{w}}_{\vert\mathcal{F}_{h}^{-}}\in {\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h}).\label{hyp-Ps2}\end{aligned}$$ Then, the following estimate holds true: $$\label{est.projector.main}
{\left\Vert \mathbb{P}^{s}[{{\underline{h}}}]({\boldsymbol{w}},{\overset{.}{\eta}}) -({\boldsymbol{w}},{\overset{.}{\eta}})\right\Vert}_{X^s}\leq C_{A}({\left\Vert {{\underline{h}}}-h \right\Vert}_{W^{1,\infty}_{\sharp}(0,L)}){\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})},$$ where $C_{A}(x)\underset{x\rightarrow 0}{\longrightarrow}0$.
The idea is to construct $({\boldsymbol{v}},b)\in X^{s}[{{\underline{h}}}]$ such that $$\label{est.projector}
{\left\Vert ({\boldsymbol{v}},d) -({\boldsymbol{w}},{\overset{.}{\eta}})\right\Vert}_{{\boldsymbol{H}}^{s}_{\sharp}({\Omega})\times H^{2s}_{\sharp}(0,L)}\leq C_{A}({\left\Vert {{\underline{h}}}-h \right\Vert}_{W^{1,\infty}_{\sharp}(0,L)}){\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})}.$$ The inequality then follows from the minimality property of the projection. The proof is divided in five steps. The two first ones are devoted to the construction of $({\boldsymbol{v}},d).$ The latter ones concern the derivation of .
[**Step 1. Geometrical preliminaries.**]{} Before going to the construction of a candidate $({\boldsymbol{v}},d)$ we define and analyze a change of variables $\chi$ that maps $\mathcal{F}^-_{{{\underline{h}}}}$ on $\mathcal{F}^-_{h}$. For $(x,y) \in \Omega,$ we set $$\chi(x,y) = \left(x,m(x)(y+1)-1\right), \; \text{ where } m(x) = \displaystyle\frac{h(x)+1}{{{\underline{h}}}(x)+1}$$ Clearly, $\chi$ realizes a one-to-one mapping between $\mathcal{F}^-_{{{\underline{h}}}}$ and $\mathcal{F}^-_{h}$. Thanks to the regularity assumptions on $h$ and ${{\underline{h}}}$, we remark that $m\in W_\sharp^{1,\infty}(0, L)\cap H_\sharp^{1+\kappa}(0, L)$ – since both spaces are algebras – and that $\chi \in {\boldsymbol{W}}_\sharp^{1, \infty}(\Omega).$
For the definition of ${\boldsymbol{v}},$ we shall transform ${\boldsymbol{w}}$ into a vector field ${\boldsymbol{w}}^{\chi}$ satisfying ${\boldsymbol{w}}^{\chi} \cdot {\boldsymbol{e}}_1 = 0$ on $\mathcal S_{{{\underline{h}}}}.$ To preserve simultaneously that ${\boldsymbol{w}}^{\chi}$ is divergence free, one multiplies, in a standard way, the vector field by the cofactor of $\nabla \chi$. So, we now analyze the multiplier properties of $\text{Cof}(\nabla\chi)^{\top}.$ First, we have $$\text{Cof}(\nabla\chi)^{\top} = \begin{pmatrix}
m(x)&0\\
-\partial_x m(x)(y+1)&1\\
\end{pmatrix} \in {\boldsymbol{L}}^\infty_\sharp(\Omega)\cap {\boldsymbol{H}}_\sharp^\kappa(\Omega).$$ Then, straightforward computations yield $$\label{prelim.h}
\|m-1\|_{W^{1,\infty}_{\sharp}(0,L)} \leq C_A \|h-\underline{h}\|_{W^{1,\infty}_{\sharp}(0,L)} ,$$ so that, $$\label{preli.C1}{\left\Vert \text{Cof}(\nabla\chi)^{\top}-\mathbb I_2\right\Vert}_{{\boldsymbol{L}}^{\infty}_{\sharp}({\Omega})}\leq C_A{\left\Vert h-{{\underline{h}}}\right\Vert}_{W^{1,\infty}_{\sharp}(0,L)},$$ with $C_A$ a constant depending only on the upper $A$ defined by . Finally we prove $H^{\sigma}$–estimates. To this end, we interpolate between $L^2$ and $H^\kappa$. Estimate implies $${\left\Vert \text{Cof}(\nabla\chi)^{T}-\mathbb I_2\right\Vert}_{{\boldsymbol{L}}^{2}_{\sharp}({\Omega})}\leq C_A{\left\Vert h-{{\underline{h}}}\right\Vert}_{W^{1,\infty}_{\sharp}(0,L)}.$$ Then, we remark that $${\left\Vert m-1\right\Vert}_{H^{1+\kappa}_{\sharp}(0,L)}\leq C_{A},$$ and that, for any given $f$, $a$ and $b$ regular functions defined on $(0,L),$ there holds $$\label{2D.1D.estimate}{\left\Vert f\right\Vert}_{H_\sharp^{\sigma}(\mathcal{C}_{a}^{b})}\leq {\left\Vert a-b\right\Vert}_{L_\sharp^{\infty}(0,L)}^{\frac{1}{2}}{\left\Vert f\right\Vert}_{H_\sharp^{\sigma}(0,L)},$$ for $0\leq \sigma\leq 1$. Consequently, we obtain also $${\left\Vert \text{Cof}(\nabla\chi)^{\top}-\mathbb I_2\right\Vert}_{{\boldsymbol{H}}^{\kappa}_{\sharp}({\Omega})}\leq C_A{\left\Vert m-1\right\Vert}_{H^{1+\kappa}_{\sharp}(0,L)}.$$ Using interpolation between the $L^{2}$ and the $H^{\kappa}$ estimates finally leads to $$\label{preli.Hs}{\left\Vert \text{Cof}(\nabla\chi)^{\top}-\mathbb I_2\right\Vert}_{{\boldsymbol{H}}^{\sigma}_{\sharp}({\Omega})}\leq C_A {\left\Vert h-{{\underline{h}}}\right\Vert}_{W^{1,\infty}_{\sharp}(0,L)}^{\frac{\kappa-\sigma}{\kappa}},$$ for $0\leq \sigma\leq \kappa$.
**[Step 2. Construction of $({\boldsymbol{v}},d)$.]{}** Let consider $({\boldsymbol{w}},{\overset{.}{\eta}}) \in X^s[h]$ enjoying the further property $$\begin{aligned}
&{\boldsymbol{w}}_{\vert\mathcal{F}_{h}^{-}}\in {\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h}).\label{hyp-Ps2-proof}\end{aligned}$$ As mentioned previously, to define $({\boldsymbol{v}},d)$ we first construct an intermediate vector field ${\boldsymbol{w}}^{\chi}$ obtained by the change of variables $\chi$ from ${\boldsymbol{w}}.$ However we note that $\chi$ does not map $\Omega$ into $\Omega$ so that we must at first extend ${\boldsymbol{w}}$ for $y\geq 2 M.$ Namely, we set $$\tilde{{\boldsymbol{w}}}(x,y) =
\left\{
\begin{array}{ll}
{\boldsymbol{w}}(x,y), & \text{ if $(x,y) \in \Omega$}, \\
{\overset{.}{\eta}}(x){\boldsymbol{e}}_{2}, & \text{ if $y >2M$}.
\end{array}
\right.$$ This extension preserves the divergence free constraint. We next define $${\boldsymbol{w}}^\chi = \text{Cof}(\nabla \chi)^{\top}\tilde{{\boldsymbol{w}}}\circ \chi.$$ The $\text{Cof}(\nabla \chi)^{\top}$ factor ensures that ${\boldsymbol{w}}^\chi$ is also divergence free. Next we define $({\boldsymbol{v}},d)$ as $$\label{def.v.b}
{\boldsymbol{v}}=\begin{cases}
\begin{array}{ll}
{\boldsymbol{w}}^\chi -{w_{2}^\chi}_{|_{y=0}}{\boldsymbol{e}}_{2}, & \text{ in } \mathcal{C}^{2M}_{0},\\
0, & \text{ in }\mathcal{C}^{0}_{-1},
\end{array}
\end{cases}
\qquad
d={\overset{.}{\eta}}-{w_{2}^\chi}_{|_{y=0}}.$$ The first step is to verify that $({\boldsymbol{v}}, d) \in X^s[{{\underline{h}}}]$, $\forall s<\kappa/2$. First, by taking into account assumption , since $\tilde{{\boldsymbol{w}}}$ coincides with ${\boldsymbol{w}}$ in $\mathcal F_{h}^{-}$, we have that $\tilde{{\boldsymbol{w}}}\in H_\sharp^1(\mathcal F_{h}^{-})$. Thus since the change of variables $\chi$ maps $\mathcal{F}^-_{{{\underline{h}}}}$ onto $\mathcal{F}^-_{h},$ the above analysis of the regularity of $\chi$ and of $ \text{Cof}(\nabla \chi)^{\top}$ implies that ${\boldsymbol{w}}^\chi\in {\boldsymbol{H}}^s_\sharp(\mathcal{F}^-_{{{\underline{h}}}})$, $\forall s<\kappa$ (see [@Grubsolo Proposition B.1] ). Moreover by the change of variables, the boundary $y=0$ is mapped to $y= m-1 $ which is lower than $h$ and strictly greater that $-1$. Hence, the trace of $\tilde{{\boldsymbol{w}}} \circ \chi$ on $y=0$ is well defined and belongs to $ H^{1/2}_{\sharp}(0,L)$. But by definition we have $${w_{2}^\chi}_{|_{y=0}} = -\partial_x m \ \tilde{w}_{1}\circ\chi_{|_{y=0}}+\tilde{w}_{2}\circ\chi_{|_{y=0}},$$ where $\partial_x m\in H^\kappa_\sharp(0, L).$ Classical multiplier arguments thus imply that ${w_{2}^\chi}_{|_{y=0}} \in H^{2s}_{\sharp}(0,L)$ for any $s < \kappa/2$ and that $$\label{eq.lapremieresurw2chi}
\|{w_{2}^\chi}_{|_{y=0}}\|_{H^{2s}_{\sharp}(0,L)} \leq C_A(\|h-{{\underline{h}}}\|_{W^{1,\infty}_{\sharp}(0,L)}) \|{\boldsymbol{w}}\|_{H^1(\mathcal F^{-}_h)},$$ where $C_{A}(x)\underset{x\rightarrow 0}{\longrightarrow}0$. Furthermore we have by construction that $$\text{
\qquad ${\boldsymbol{w}}^\chi= {\overset{.}{\eta}}{\boldsymbol{e}}_{2}$ in $\mathcal {S}_{{\underline{h}}}$.
}$$ Consequently, thanks to the regularity of ${\overset{.}{\eta}}$ and the one obtained on ${{w}^\chi_{2}}_{|_{y=0}}$, we deduce that ${\boldsymbol{v}}\in {\boldsymbol{H}}^{2s}(\mathcal {S}_{{\underline{h}}})\subset {\boldsymbol{H}}^{s}(\mathcal {S}_{{\underline{h}}})$. Finally $({\boldsymbol{v}}, d)\in {\boldsymbol{H}}^s(\Omega)\times H^{2s}(0, L)$, for $s<\kappa/2$. Let us now check the divergence free constraint and the fluid–structure velocity matching. We have by construction that $$\text{
$\text{div}\,{\boldsymbol{w}}^\chi =0$ in $\Omega$.}
$$ Thus ${\boldsymbol{v}}$ satisfies $$\text{
$\text{div}\,{\boldsymbol{v}}=0$ in $\mathcal{C}^{2M}_{0}$ and $\text{div}\,{\boldsymbol{v}}=0$ in $\mathcal{C}^{-1}_{0}$.
}$$ Since, by construction ${\boldsymbol{v}}_{|_{y=0}}=0$, we obtain $\text{div}\,{\boldsymbol{v}}=0$ in $\Omega$. Moreover ${\rm div}\, {\boldsymbol{w}}^{\chi} = 0$ on $\mathcal C_{-1}^0$ with ${\boldsymbol{w}}^{\chi} = 0$ on $y=-1.$ By integrating this divergence constraint we obtain the condition $$\int_{0}^L {w_{2}^\chi}_{|_{y=0}} = 0.$$ As a consequence, since $\dot\eta\in L^2_{\sharp,0}(0,L)$, we obtain $d \in L^2_{\sharp,0}(0,L).$
We now check the remaining compatibility conditions of $X^s[h].$ For $y \geq {{\underline{h}}},$ we have ${\boldsymbol{v}}= ( \dot{\eta} - {w_{2}^\chi}_{|_{y=0}}) {\boldsymbol{e}}_2$ so that ${\boldsymbol{v}}$ satisfies $${\boldsymbol{v}}\cdot {\boldsymbol{e}}_1 = 0\,, \text{ on $\mathcal S_{{{\underline{h}}}}$ }, \qquad {v_2}_{|_{y=M}} = d.$$ This ends the proof that $({\boldsymbol{v}},d) \in X^s[{{\underline{h}}}].$
[**Step 3. Splitting of $\|({\boldsymbol{w}},{\overset{.}{\eta}}) - ({\boldsymbol{v}},d)\|_{X^s}$.**]{} Let first remark that $${\boldsymbol{v}}- {\boldsymbol{w}}^\chi =\begin{cases}
\begin{array}{ll}
-{w_{2}^\chi}_{|_{y=0}}{\boldsymbol{e}}_{2},&\text{ in } \mathcal{C}^{2M}_{0},\\
-{\boldsymbol{w}}^\chi, & \text{ in }\mathcal{C}^{0}_{-1},
\end{array}
\end{cases}
\qquad
{\overset{.}{\eta}}-d ={w_{2}^\chi}_{|_{y=0}}.$$ Consequently we have $$\begin{aligned}
{\left\Vert ({\boldsymbol{w}},{\overset{.}{\eta}}) - ({\boldsymbol{v}},d)\right\Vert}_{X^s}&\leq {\left\Vert ({\boldsymbol{w}},{\overset{.}{\eta}}) - ({\boldsymbol{w}}^\chi,d)\right\Vert}_{X^s} +{\left\Vert {\boldsymbol{w}}^\chi - {\boldsymbol{v}}\right\Vert}_{{\boldsymbol{H}}^{s}_{\sharp}({\Omega})}\\
&\leq {\left\Vert {\boldsymbol{w}}-{\boldsymbol{w}}^\chi\right\Vert}_{{\boldsymbol{H}}^{s}_{\sharp}({\Omega})} + {\left\Vert {w_{2}^\chi}_{|_{y=0}}\right\Vert}_{H^{2s}_{\sharp}(0,L)}+ {\left\Vert {w_{2}^\chi}_{|_{y=0}}{\boldsymbol{e}}_{2}\right\Vert}_{{\boldsymbol{H}}^{s}_{\sharp}(\mathcal{C}^{2M}_{0})}+{\left\Vert {\boldsymbol{w}}^\chi\right\Vert}_{{\boldsymbol{H}}^{s}_{\sharp}(\mathcal{C}_{-1}^{0})}.
\end{aligned}$$ Recalling we obtain the bound $${\left\Vert {w_{2}^\chi}_{|_{y=0}}{\boldsymbol{e}}_{2}\right\Vert}_{{\boldsymbol{H}}_\sharp^{s}(\mathcal{C}_{0}^{2M})}\leq \sqrt{2M}{\left\Vert {w_{2}^\chi}_{|_{y=0}}{\boldsymbol{e}}_{2}\right\Vert}_{{\boldsymbol{H}}_\sharp^{s}(0,L)}\leq \sqrt{2M}{\left\Vert {w_{2}^\chi}_{|_{y=0}}\right\Vert}_{H_\sharp^{2s}(0,L)}.$$ Moreover, as ${\boldsymbol{w}}=0$ in $\mathcal{C}^{0}_{-1},$ we remark that an estimate on ${\left\Vert {\boldsymbol{w}}^\chi - {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{s}({\Omega})}$ implies an estimate on ${\left\Vert {\boldsymbol{w}}^\chi\right\Vert}_{{\boldsymbol{H}}^{s}(\mathcal{C}_{-1}^{0})}$. Finally is implied by the following estimate $$\label{est.projector.final}
{\left\Vert {\boldsymbol{w}}-{\boldsymbol{w}}^\chi\right\Vert}_{{\boldsymbol{H}}^{s}_{\sharp}({\Omega})}+ {\left\Vert {w_{2}^\chi}_{|_{y=0}}\right\Vert}_{H^{2s}_{\sharp}(0,L)}
\leq C_{A}({\left\Vert {{\underline{h}}}-h \right\Vert}_{W^{1,\infty}_{\sharp}(0,L)}){\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})}.$$ Thus we have to prove that ${\boldsymbol{w}}-{\boldsymbol{w}}^\chi={\boldsymbol{w}}-\text{Cof}(\nabla \chi)^{\top}{\boldsymbol{w}}\circ \chi $ and ${w_{2}^\chi}_{|_{y=0}}=-\partial_x m \ w_{1}\circ\chi_{|_{y=0}}+w_{2}\circ\chi_{|_{y=0}}$ can be estimated with respect to the difference $h-{{\underline{h}}}$. This is the aim of the two next steps respectively.
**Step 4. [Estimating ${\boldsymbol{w}}-{\boldsymbol{w}}^\chi.$]{}** We estimate the difference ${\boldsymbol{w}}- {\boldsymbol{w}}^{\chi}$ by considering successively each of the subdomains of $\Omega$: $\mathcal{S}_{h}$, $\mathcal{C}_{{{\underline{h}}}}^{h}$, $\mathcal{F}_{{{\underline{h}}}}$, $\mathcal{C}_{-1}^{0}.$
*Estimates in $\mathcal{S}_{h}$.* In $\mathcal{S}_{h}$, ${\boldsymbol{w}}={\overset{.}{\eta}}{\boldsymbol{e}}_2.$ By replacing in the definition of ${\boldsymbol{w}}^{\chi}$, we have also ${\boldsymbol{w}}^\chi = {\overset{.}{\eta}}{\boldsymbol{e}}_2$ in $\mathcal{S}_{{{\underline{h}}}}$ and since $h\geq {{\underline{h}}}$ we infer ${\boldsymbol{w}}- {\boldsymbol{w}}^{\chi} = 0$ in $\mathcal{S}_{h}$.\
[*Estimates in $\mathcal{C}^{h}_{{{\underline{h}}}}$.*]{} The identity ${\boldsymbol{w}}^\chi={\overset{.}{\eta}}{\boldsymbol{e}}_{2}$ still holds in $\mathcal{C}^{h}_{{{\underline{h}}}} \subset \mathcal S_{{{\underline{h}}}}$ which leads to $${\boldsymbol{w}}(x,y)- {\boldsymbol{w}}^{\chi}(x,y) = {\boldsymbol{w}}(x,y) - {\boldsymbol{w}}(x,h(x)) =\int_{y}^{h(x)}\partial_y{\boldsymbol{w}}(x,z){\mathrm{d}}z.$$ We obtain then $$\begin{aligned}
{\left\Vert {\boldsymbol{w}}^\chi - {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{L}}^{2}_{\sharp}(\mathcal{C}_{{{\underline{h}}}}^{h})}^{2}&{}=\int_{0}^{L}\int_{{{\underline{h}}}(x)}^{h(x)}\left\vert \int_{y}^{h(x)}\partial_y{\boldsymbol{w}}(x,z)dz\right\vert^{2}{\mathrm{d}}y{\mathrm{d}}x\\
&\leq \int_{0}^{L}\int_{{{\underline{h}}}(x)}^{h(x)}{\left\Vert h-{{\underline{h}}}\right\Vert}_{L^{\infty}_{\sharp}(0,L)}\int_{{{\underline{h}}}(x)}^{h(x)}\vert\partial_y{\boldsymbol{w}}(x,z)\vert^{2}{\mathrm{d}}z{\mathrm{d}}y{\mathrm{d}}x\\
&\leq {\left\Vert h-{{\underline{h}}}\right\Vert}_{L^{\infty}_{\sharp}(0,L)}^{2}{\left\Vert \nabla{\boldsymbol{w}}\right\Vert}_{{\boldsymbol{L}}^{2}_{\sharp}(\mathcal{F}^{-}_{h})}^{2}.
\end{aligned}$$ Moreover $${\left\Vert {\boldsymbol{w}}^\chi-{\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1/2}_{\sharp}(\mathcal{C}_{{{\underline{h}}}}^{h})}\leq {\left\Vert {\boldsymbol{w}}^\chi\right\Vert}_{{\boldsymbol{H}}^{1/2}_{\sharp}(\mathcal{C}_{{{\underline{h}}}}^{h})} + {\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1/2}_{\sharp}(\mathcal{C}_{{{\underline{h}}}}^{h})}\leq {\left\Vert {\boldsymbol{w}}^\chi\right\Vert}_{{\boldsymbol{H}}^{1/2}_{\sharp}(\mathcal{C}_{{{\underline{h}}}}^{h})} + {\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})}.$$ We have ${\boldsymbol{w}}^\chi={\overset{.}{\eta}}{\boldsymbol{e}}_2={\boldsymbol{w}}_{|_{y=h}}$. Hence, recalling the trace continuity estimate , we obtain $${\left\Vert {\boldsymbol{w}}^\chi\right\Vert}_{{\boldsymbol{H}}^{1/2}_{\sharp}(0,L)} \leq C_A{\left\Vert {\boldsymbol{w}}\right\Vert}_{H^{1}_{\sharp}(\mathcal{F}^{-}_{h})}.$$ We conclude that $${\left\Vert {\boldsymbol{w}}^\chi-{\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1/2}_{\sharp}(\mathcal{C}_{{{\underline{h}}}}^{h})}\leq C_A{\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})},$$ and by interpolation with the previous $L^2$-estimate, the following estimate in $H^s$ holds true $$\label{eq_enhaut}
{\left\Vert {\boldsymbol{w}}^\chi-{\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{s}_{\sharp}(\mathcal{C}^{h}_{{{\underline{h}}}})}\leq C_A{\left\Vert h-{{\underline{h}}}\right\Vert}_{W^{1,\infty}_{\sharp}(0,L)}^{1-2s}{\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})}.$$
[*Estimates in $\mathcal{F}_{{{\underline{h}}}}$:*]{} Consider the following splitting: $$\label{splitting.Du}
{\boldsymbol{w}}^\chi - {\boldsymbol{w}}= \text{Cof}(\nabla\chi)^{\top}({\boldsymbol{w}}\circ\chi - {\boldsymbol{w}}) +(\text{Cof}(\nabla\chi)^{\top}-\mathbb I_2){\boldsymbol{w}}.$$ Thanks to [@Grubsolo Proposition B.1] we obtain, for $s<s'\leq \kappa$ (see for the estimate of the $H^{s'}$-norm of the cofactor matrix): $$\begin{aligned}
{\left\Vert (\text{Cof}(\nabla\chi)^{\top}-\mathbb I_2){\boldsymbol{w}}\right\Vert}_{ {\boldsymbol{H}}^{s}_{\sharp}(\mathcal{F}_{{{\underline{h}}}})}&\leq C_{A} {\left\Vert (\text{Cof}(\nabla\chi)^{\top}-\mathbb I_2)\right\Vert}_{ {\boldsymbol{H}}^{s'}_{\sharp}({\Omega})}{\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})}\\
&\leq C_A {\left\Vert h-{{\underline{h}}}\right\Vert}_{W^{1,\infty}_{\sharp}(0,L)}^{\frac{\kappa-s'}{\kappa}}{\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})},\end{aligned}$$ Here we use the continuity of the multiplication $H^{s'}(\mathcal F^{-}_{{{\underline{h}}}}) \times H^{1}(\mathcal F^{-}_{h}) \to
H^{s}(\mathcal F^{-}_{{{\underline{h}}}}).$ The continuity constant of this mapping may depend on ${{\underline{h}}}.$ But, by a standard change of variables argument, we see that it depends increasingly on $\|{{\underline{h}}}\|_{W^{1,\infty}_{\sharp}(0,L)}$ only. This constant thus depends on $A$ only. We now take care of the first term of the right-hand side of . Let first note that we can bound the $L^2$–norm of ${\boldsymbol{w}}\circ \chi - {\boldsymbol{w}}$ as follows: $$\begin{aligned}
\int_{0}^{L}\int_{0}^{{{\underline{h}}}(x)}\vert{\boldsymbol{w}}(\chi(x,y)) - {\boldsymbol{w}}(x,y)\vert^{2}{\mathrm{d}}y{\mathrm{d}}x&{}=\int_{0}^{L}\int_{0}^{{{\underline{h}}}(x)}\vert {\boldsymbol{w}}(x,m(x)(y+1)-1)-{\boldsymbol{w}}(x,y)\vert^{2}{\mathrm{d}}y{\mathrm{d}}x\\
&\leq \int_{0}^{L}\int_{0}^{{{\underline{h}}}(x)}\left\vert \int_{y}^{m(x)(y+1)-1}\partial_y{\boldsymbol{w}}(x,z)dz\right\vert^{2}{\mathrm{d}}y{\mathrm{d}}x\\
&\leq \int_{0}^{L}\int_{0}^{{{\underline{h}}}(x)}(m(x)-1)(y+1)\int_{y}^{m(x)(y+1)-1}\vert \partial_y{\boldsymbol{w}}(x,z)\vert^{2}{\mathrm{d}}z{\mathrm{d}}y{\mathrm{d}}x\\
&\leq (M+1){\left\Vert m-1\right\Vert}_{L^{\infty}_{\sharp}(0,L)}{\left\Vert {{\underline{h}}}\right\Vert}_{L^{\infty}_{\sharp}(0,L)}{\left\Vert \nabla{\boldsymbol{w}}\right\Vert}_{{\boldsymbol{L}}^{2}_{\sharp}(\mathcal{F}^{-}_{h})}^{2}.
\end{aligned}$$ The previous estimate leads to $$\begin{aligned}
{\left\Vert {\boldsymbol{w}}\circ\chi-{\boldsymbol{w}}\right\Vert}_{{\boldsymbol{L}}^{2}_{\sharp}(\mathcal{F}_{{{\underline{h}}}})}&{}\leq \sqrt{(M+1)M}{\left\Vert m-1\right\Vert}_{L^{\infty}_{\sharp}(0,L)}^{1/2}{\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}_{h}^{-})}\nonumber\\
&\leq C_M{\left\Vert h-{{\underline{h}}}\right\Vert}_{W^{1,\infty}_{\sharp}(0,L)}^\frac{1}{2}{\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}_{h}^{-})}.
\end{aligned}$$ Finally, since ${\left\Vert \text{Cof}(\nabla\chi)^{\top}\right\Vert}_{L^\infty_\sharp(\mathcal{F}_{{{\underline{h}}}})}\leq C_A$, we deduce $$\label{est:L2:Fh}
{\left\Vert \text{Cof}(\nabla\chi)^{\top}({\boldsymbol{w}}\circ\chi-{\boldsymbol{w}})\right\Vert}_{{\boldsymbol{L}}^{2}_{\sharp}(\mathcal{F}_{{{\underline{h}}}})}\leq C_A{\left\Vert h-{{\underline{h}}}\right\Vert}_{W^{1,\infty}_{\sharp}(0,L)}^\frac{1}{2}{\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}_{h}^{-})}.$$ Next we remark that ${\boldsymbol{w}}\circ\chi - {\boldsymbol{w}}$ is bounded in ${\boldsymbol{H}}_{\sharp}^1(\mathcal{F}^{-}_{{{\underline{h}}}})$. Indeed ${\boldsymbol{w}}\in {\boldsymbol{H}}_{\sharp}^1(\mathcal{F}^-_{h})$ and thus ${\boldsymbol{w}}\in {\boldsymbol{H}}_{\sharp}^1(\mathcal{F}^{-}_{{{\underline{h}}}})$. It implies also that ${\boldsymbol{w}}\circ\chi \in {\boldsymbol{H}}_{\sharp}^1(\mathcal{F}_{{{\underline{h}}}})$ since $\chi$ belongs to $W_{\sharp}^{1, \infty}(\Omega)$ and maps $\mathcal{F}^-_{{{\underline{h}}}}$ in $\mathcal{F}^-_{h}$. Consequently, we have $${\left\Vert {\boldsymbol{w}}\circ\chi - {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}_{{{\underline{h}}}})}\leq {\left\Vert {\boldsymbol{w}}\circ\chi\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}_{{{\underline{h}}}})} + {\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}_{{{\underline{h}}}})} \leq C_{A}{\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})}.$$ Next, thanks to the fact that ${\left\Vert \text{Cof}(\nabla\chi)^{\top}\right\Vert}_{{\boldsymbol{H}}^\kappa_\sharp(\mathcal{F}_{{{\underline{h}}}})}\leq C_A$ (see ), we have, since $0\leq s<\kappa$ $${\left\Vert \text{Cof}(\nabla\chi)^{\top}({\boldsymbol{w}}\circ\chi - {\boldsymbol{w}})\right\Vert}_{{\boldsymbol{H}}^{s}_{\sharp}(\mathcal{F}_{{{\underline{h}}}})}\leq C_{A}{{\left\Vert \text{Cof}(\nabla\chi)^{\top}\right\Vert}_{{\boldsymbol{H}}^{\kappa}_{\sharp}(\Omega)}}{\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})}\leq C_{A}{\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})}.\label{est:H1:Fh}$$ By interpolating and , we obtain $${\left\Vert \text{Cof}(\nabla\chi)^{\top}({\boldsymbol{w}}\circ\chi - {\boldsymbol{w}})\right\Vert}_{{\boldsymbol{H}}^{s}_{\sharp}(\mathcal{F}_{{{\underline{h}}}})}\leq C_A({\left\Vert h-{{\underline{h}}}\right\Vert}_{W^{1,\infty}_{\sharp}(0,L)}){\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})}.$$ To summarize the estimates in $\mathcal{F}_{{{\underline{h}}}}$ we have proved that $$\label{eq.milieu}
{\left\Vert {\boldsymbol{w}}^\chi-{\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{s}_{\sharp}(\mathcal{F}_{{{\underline{h}}}})}\leq C_A({\left\Vert h-{{\underline{h}}}\right\Vert}_{W^{1,\infty}_{\sharp}(0,L)}){\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})}.$$
[*Estimates in $\mathcal{C}_{-1}^{0}$.*]{} The function ${\boldsymbol{w}}$ is equal to zero and we have to estimate only ${\boldsymbol{w}}^\chi$. As previously we obtain a first bound in $L^{2}$ involving $C_A({\left\Vert h-{{\underline{h}}}\right\Vert}_{W^{1,\infty}_{\sharp}(0,L))})$ and then we prove that ${\boldsymbol{w}}^\chi$ is bounded in some $H^{s}$ and we conclude using interpolation. For the $L^{2}$–norm we have $${\left\Vert {\boldsymbol{w}}^\chi\right\Vert}_{{\boldsymbol{L}}^{2}_{\sharp}(\mathcal{C}_{-1}^{0})}\leq {\left\Vert (\text{Cof}\nabla\chi)^{\top}\right\Vert}_{{\boldsymbol{L}}^{\infty}_{\sharp}(\mathcal{C}_{-1}^{0})}{\left\Vert {\boldsymbol{w}}\circ\chi\right\Vert}_{{\boldsymbol{L}}^{2}_{\sharp}(\mathcal{C}_{-1}^{0})},$$ and $$\begin{aligned}
{\left\Vert {\boldsymbol{w}}\circ\chi\right\Vert}^2_{{\boldsymbol{L}}^{2}_{\sharp}(\mathcal{C}_{-1}^{0})}=&\int_{0}^{L}\int_{-1}^{0}\vert {\boldsymbol{w}}(\chi(x,y))\vert^{2}{\mathrm{d}}y{\mathrm{d}}x&{}\\
&=\int_{0}^{L}\int_{0}^{m(x)-1}\vert {\boldsymbol{w}}(x,y)\vert^{2}\frac{dy}{m(x)}{\mathrm{d}}x\\
&\leq (1+M)\int_{0}^{L}\int_{0}^{m(x)-1}\left\vert\int_{0}^{y}\partial_y{\boldsymbol{w}}(x,z)dz\right\vert^{2}{\mathrm{d}}y{\mathrm{d}}x\\
&\leq (1+M)\int_{0}^{L}\int_{0}^{m(x)-1}y\int_{0}^{y}\vert\partial_z{\boldsymbol{w}}(x,z)\vert^{2}{\mathrm{d}}z{\mathrm{d}}y{\mathrm{d}}x\\
&\leq \frac{(1+M)}{2}{\left\Vert m-1\right\Vert}_{L^{\infty}_{\sharp}(0,L)}^{2}{\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})}^{2}.
\end{aligned}$$ Hence, using the estimates above with -, we conclude $${\left\Vert {\boldsymbol{w}}^\chi\right\Vert}_{{\boldsymbol{L}}^{2}_{\sharp}(\mathcal{C}_{-1}^{0})}\leq C_A{\left\Vert h-{{\underline{h}}}\right\Vert}_{W^{1,\infty}_{\sharp}(0,L)}{\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})}.$$ Moreover, for any $0\leq \sigma<\kappa$, we have $$\begin{aligned}
{\left\Vert {\boldsymbol{w}}^\chi\right\Vert}_{{\boldsymbol{H}}^{\sigma}_{\sharp}(\mathcal{C}_{-1}^{0})}&{}\leq C_{1}{\left\Vert (\text{Cof}\nabla\chi)^{\top}\right\Vert}_{{\boldsymbol{H}}^{\kappa}_{\sharp}(\mathcal{C}_{-1}^{0})}{\left\Vert {\boldsymbol{w}}\circ\chi\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{C}_{-1}^{0})}\\
&\leq C_A{\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})},
\end{aligned}$$ and using interpolation up to choose $\sigma \in (s,\kappa)$ $${\left\Vert {\boldsymbol{w}}^\chi\right\Vert}_{{\boldsymbol{H}}^{s}_{\sharp}(\mathcal{C}_{-1}^{0})}\leq C_A({\left\Vert h-{{\underline{h}}}\right\Vert}_{W^{1,\infty}_{\sharp}(0,L)}){\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})}.$$ To summarize we have proved that $$\label{eq.dessous}
{\left\Vert {\boldsymbol{w}}-{\boldsymbol{w}}^\chi\right\Vert}_{{\boldsymbol{H}}^{s}_{\sharp}(\mathcal C_{-1}^0)}\leq C_A({\left\Vert h-{{\underline{h}}}\right\Vert}_{W^{1,\infty}_{\sharp}(0,L)}){\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})}.$$ Finally, combining --, we obtain the expected estimate $$\label{eq.total}
{\left\Vert {\boldsymbol{w}}-{\boldsymbol{w}}^\chi\right\Vert}_{{\boldsymbol{H}}^{s}_{\sharp}({\Omega})}\leq C_A({\left\Vert h-{{\underline{h}}}\right\Vert}_{W^{1,\infty}_{\sharp}(0,L)}){\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})}.$$
**[Step 5. Estimating ${{w_{2}^\chi}_{|_{y=0}}}.$]{}** First, we recall that $${w_{2}^\chi}_{|_{y=0}}=-\partial_x m(\cdot)w_{1}(\cdot,m(\cdot)-1)+w_{2}(\cdot,m(\cdot)-1).$$ This term is first estimated in $L^2$, then in $H^{2\sigma}$ for $0<\sigma< \kappa/2$, and the final estimate is obtained by interpolation. First let estimate the $L^{2}$–norm $$\begin{aligned}
{\left\Vert -\partial_x m(\cdot)w_{1}(\cdot,m(\cdot)-1)\right\Vert}_{L^{2}(0,L)}^{2}&{}\leq {\left\Vert \partial_x m\right\Vert}_{L^{\infty}_{\sharp}(0,L)}^{2}\int_{0}^{L}\vert w_{1}(x,m(x)-1)\vert^{2}{\mathrm{d}}x\\
&\leq C_{A}\int_{0}^{L}(m(x)-1)\int_{0}^{m(x)-1}\vert\partial_yw_{1}(x,y)\vert^{2}{\mathrm{d}}y{\mathrm{d}}x\\
&\leq C_{A}{\left\Vert m-1\right\Vert}_{L^{\infty}_{\sharp}(0,L)}{\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})}\\
&\leq C_{A}{\left\Vert h-{{\underline{h}}}\right\Vert}_{L^{\infty}_{\sharp}(0,L)}{\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})}.
\end{aligned}$$ A similar estimate can be computed for ${\left\Vert w_{2}(\cdot,m(\cdot)-1)\right\Vert}_{L^{2}_{\sharp}(0,L)}$ so that, we obtain $${\left\Vert {w_{2}^\chi}_{|_{y=0}}\right\Vert}_{{L}^{2}_{\sharp}(0,L)}\leq C_A{\left\Vert h-{{\underline{h}}}\right\Vert}_{W^{1,\infty}_{\sharp}(0,L)}{\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})}.$$ For $0<\sigma< \kappa/2$ we obtain, similarly to $${\left\Vert {w_{2}^\chi}_{|_{y=0}}\right\Vert}_{{H}^{2\sigma}_{\sharp}(0,L)} \leq C_A({\left\Vert h-{{\underline{h}}}\right\Vert}_{W^{1,\infty}_{\sharp}(0,L)}){\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})}.$$ Using interpolation we finally obtain (up to choose $s \leq \sigma < \kappa/2$) that $${\left\Vert {w_{2}^\chi}_{|_{y=0}}\right\Vert}_{H^{2s}_{\sharp}(0,L)}\leq C_A({\left\Vert h-{{\underline{h}}}\right\Vert}_{W^{1,\infty}_{\sharp}(0,L)}){\left\Vert {\boldsymbol{w}}\right\Vert}_{{\boldsymbol{H}}^{1}_{\sharp}(\mathcal{F}^{-}_{h})},$$ which concludes the proof of and the proof of the lemma is completed.
Proof of Theorem \[main.theorem\]
=================================
This section is devoted to the proof of existence of weak solutions of $(FS)_0$. So, we fix $T>0$ and initial data $({\boldsymbol{u}}^0,\eta^0,\eta^1)$ satisfying –. We recall that the strategy is to approximate this problem by a sequence of viscous problems $(FS)_\gamma$, $\gamma>0$, for which existence results are available. The proof is divided into three steps. First, we analyze the Cauchy theory of $(FS)_{\gamma}$ when $\gamma >0$ and prove that the sequence of solutions converges, up to a subsequence, when $\gamma \to 0.$ We show in particular that possible weak limits are candidates to be weak solutions up to obtaining strong compactness of approximate velocities in $L^2.$ As explained in the introduction, this strong compactness property is the cornerstone of the analysis. Our proof builds on the projection/approximation argument provided by [@SanMartin-Starovoitov-Tucsnak] in the fluid–solid case. In our fluid–elastic setting, it requires to build a uniform bound by below ${{\underline{h}}}$ of the sequence of approximate structure deformations (in order to construct a fluid domain independent of $\gamma$ on which the Navier–Stokes equations are satisfied by the sequence of approximate solutions to be able to apply Aubin–Lions Lemma for projections of the velocities). The second step of the proof is devoted to the construction of ${{\underline{h}}}$ and the analysis of its properties. We then complete the proof of the $L^2$–strong compactness. This last step relies in particular on the continuity result obtained in subsection \[continuite\_h\].
Step 1. Construction of a candidate weak-solution. {#sec:proof1}
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Let us recall the strong existence result on $(0,T)$ stated in [@Grandmont-Hillairet Theorem 1]. Given $\gamma>0$ and initial data $({\boldsymbol{u}}_\gamma^{0},\eta_\gamma^{0},\eta_\gamma^{1})$ satisfying $$\begin{aligned}
&(\eta_\gamma^{0},\eta_\gamma^{1})\in H^{3}_{\sharp}(0,L)\times H^{1}_{\sharp}(0,L),\label{CI-gamma1}\\
&{\boldsymbol{u}}_\gamma^{0}\in {\boldsymbol{H}}_{\sharp}^{1}(\mathcal{F}_{h_{\gamma}^{0}}), {\text{div }}{{\boldsymbol{u}}_\gamma^{0}}=0\text{ in }\mathcal{F}_{h^{0}},\label{CI-gamma2}\\
&{\boldsymbol{u}}_\gamma^{0}(x,0)=0, \text{ and }{\boldsymbol{u}}_\gamma^{0}(x,h_\gamma^{0}(x))=\eta_\gamma^{1}(x){\boldsymbol{e}}_{2}, \forall x\in [0,L],\label{CI-gamma3}\\
&\min_{x\in[0,L]}h_\gamma^{0}(x)>0\text{ and }\int_{0}^{L}\eta_\gamma^{1}(x){\mathrm{d}}x=0,\label{CI-gamma4}\end{aligned}$$ the system $(FS)_{\gamma}$ admits a unique strong solution defined on $(0, T)$. This solution satisfies moreover $\min_{x\in [0, L]} h_\gamma(x, t) >0$ for all $t\in [0, T]$.
In order to apply this result we now explain the construction of a sequence of regular initial data $({\boldsymbol{u}}_\gamma^0,\eta_\gamma^0, \eta_\gamma^1)_{\gamma >0}$ approximating $({\boldsymbol{u}}^0, \eta^0, \eta^1)$. First we construct $\eta^0_\gamma\in H^{3}_{\sharp}(0,L)$ by a standard convolution of $\eta^0$ with a regularizing kernel. Since $\eta^0$ satisfies , this sequence is uniformly bounded in $H^2_\sharp(0,L)$ and satisfies $$\begin{aligned}
& \eta^0_\gamma\rightarrow \eta^0, \text{ in }H^2_\sharp(0,L),\\
& \| \eta^0_\gamma- \eta^0\|_{\mathcal{C}_\sharp^0([0, L])}\leq \gamma \|\eta^0\|_{H^2_\sharp(0, L)}\leq C\gamma.\end{aligned}$$ Since $\min_{x\in[0, L]}h^0(x)>0$, there exists $\lambda>0$ verifying $\min_{x\in[0, L]}h_\gamma^0(x)>\lambda >0$ for $\gamma$ small enough. We next construct $\eta^1_\gamma \in H^{1}_{\sharp}(0,L)\cap L^2_{\sharp, 0}(0, L)$. Since $\eta^1$ satisfies , this second sequence enjoys the following properties: $$\begin{aligned}
& \eta^1_\gamma\rightarrow \eta^1, \text{ in }L^2_\sharp(0,L),\\
& \| \eta^1_\gamma \|_{{L}^2_{\sharp,0}(0, L)}\leq \|\eta^1\|_{L^2_\sharp(0, L)}\leq C.\end{aligned}$$ We now build the approximate initial velocity fields ${\boldsymbol{u}}^0_{\gamma}.$ A key difficulty here is to match the continuity of velocity field at the structure interface together with preserving the divergence free condition, taking into account that the approximation is defined on an approximate domain depending on $\gamma.$ To handle this difficulty, we first define the extension of $\eta^1$ to the whole domain using the operator $\mathcal R_{\lambda}$ as defined in . Next we consider $\overline{{\boldsymbol{u}}^0}-\mathcal{R}_\lambda(\eta^1)$ which is in $K[h^0]$ and satisfies moreover $\overline{{\boldsymbol{u}}^{0}}-\mathcal{R}_\lambda(\eta^1)= {\boldsymbol{0}}$ in $\mathcal{S}_{h^0}\cup \mathcal{C}_{-1}^0$. Then we introduce the vertical contraction operator denoted by $${\boldsymbol{v}}\mapsto {\boldsymbol{v}}_\sigma(x, y)=(\sigma v_1(x, \sigma y), v_2(x, \sigma y)) \quad \forall \, \sigma >0.$$ We emphasize that this contraction operator preserves the divergence free constraint. By choosing $\sigma_{\gamma} = 1+ 2C\gamma/\lambda$ (with the constant $C$ above), we have $(\overline{{\boldsymbol{u}}^{0}}-\mathcal{R}_\lambda(\eta^1))_{\sigma_\gamma}= {\boldsymbol{0}}$ in $\mathcal{S}_{h_\gamma^0}\cup \mathcal{C}_{-1}^0$, and that $(\overline{{\boldsymbol{u}}^{0}}-\mathcal{R}_\lambda(\eta^1))_{\sigma_\gamma}$ converges to $\overline{{\boldsymbol{u}}^{0}}-\mathcal{R}_\lambda(\eta^1)$ in ${\boldsymbol{L}}_\sharp^2({\Omega})$ when $\gamma \to 0.$ Moreover $(\overline{{\boldsymbol{u}}^{0}}-\mathcal{R}_\lambda(\eta^1))_{\sigma_\gamma}$ belongs to ${\boldsymbol{L}}^2_{\sharp}(\mathcal{F}_{h_\gamma^0})$, is divergence free and satisfies $(\overline{{\boldsymbol{u}}^{0}}-\mathcal{R}_\lambda(\eta^1))_{\sigma_\gamma}\cdot{\boldsymbol{n}} =0$ on $\Gamma_{h^0_\gamma}$ and $(0, L)\times\{0\}$. Thus we approximate thanks to standard arguments (by truncation and regularization of the stream function for instance) this function by a divergence free function $(\overline{{\boldsymbol{u}}^{0}}-\mathcal{R}_\lambda(\eta^1))_{\gamma}$ in ${\boldsymbol{H}}^1_{\sharp}(\mathcal{F}_{h_\gamma^0})$ vanishing in a neighbourhood of $\Gamma_{h^0_\gamma}$ and $(0, L)\times\{0\}$. We may then set $${\boldsymbol{u}}_{\gamma}^0 = \left( \overline{(\overline{{\boldsymbol{u}}^{0}}-\mathcal{R}_\lambda(\eta^1))_{\gamma} }+\mathcal{R}_\lambda(\eta_\gamma^1)\right)_{|_{\mathcal{F}_{h_\gamma^0}}}.$$ Straightforward computations show that ${\boldsymbol{u}}_{\gamma}^0$ satisfies -. Moreover, remarking that the operator $\mathcal R_{\lambda}$ is continuous from $L^2_{\sharp,0}(0,L)$ into $L_\sharp^2(\Omega)$ we have as $\gamma$ goes to zero
$$\begin{aligned}
& \overline{{\boldsymbol{u}}^0_\gamma}\rightarrow \overline{{\boldsymbol{u}}^0}, \text{ in }L^2_\sharp(\Omega),\\
& \| {\boldsymbol{u}}^0_\gamma \|_{{L}^2(\mathcal F^0_{h})}\leq C\left( \|\eta^1\|_{L^2_\sharp(0, L)} + \|{\boldsymbol{u}}^0\|_{L^2(\mathcal F_h^0)} \right)\leq C,\end{aligned}$$
where $C$ does not depend on $\gamma$.
We now apply the result on existence of a strong solution for the viscous problem $(FS)_{\gamma>0}$. For fixed $\gamma >0$ the unique solution $({\boldsymbol{u}}_\gamma,\eta_\gamma)$ is global in time so that it exists on any time interval $(0,T)$. The first step is to verify that $\overline{{\boldsymbol{u}}}_{\gamma}$ as defined by $$\overline{{\boldsymbol{u}}}_{\gamma} =
\left\{
\begin{array}{ll}
\partial_t \eta_{\gamma} {\boldsymbol{e}}_2, & \text{ in $\mathcal S_{h_{\gamma}},$} \\
{\boldsymbol{u}}_{\gamma} , & \text{ in $\mathcal F_{h_{\gamma}},$} \\
{\boldsymbol{0}}, & \text{ in $\mathcal C_{-1}^0,$}
\end{array}\right.$$ together with $\eta_{\gamma}$ is a pair of weak solution to $(FS)_{\gamma}$ in the sense of Definition \[def.ws\]. First, we note that $({\boldsymbol{u}}_\gamma,\eta_\gamma)$ satisfies estimate so that we can define a constant $M$ involved in the definition of our weak solution framework. By construction we have that $$\eta_{\gamma} \in L^{\infty}(0,T;H^{2}_{\sharp}(0,L)) \cap W^{1,\infty}(0,T;L^2_{\sharp, 0}(0,L)).$$ Moreover, $$\overline{{\boldsymbol{u}}}_{\gamma} \in L^{\infty}(0,T ; L^2_{\sharp}(\Omega)), \quad
\nabla \overline{{\boldsymbol{u}}_{\gamma}}_{|_{\widehat{\mathcal F}_{h_{\gamma}}}} = \mathbf{1}_{\widehat{\mathcal F}_{h_{\gamma}}}\nabla {\boldsymbol{u}}_{\gamma} \in L^2(\widehat{\mathcal F}^{-}_{h_{\gamma}}),$$ and Lemma \[lem:bar\], implies that $$(\overline{{\boldsymbol{u}}}_{\gamma}(t),\eta_{\gamma})(t) \in X[h_{\gamma}(t)] \text{ for a.e. $t \in (0,T)$}
.$$ Thus the regularity statement $i)$ of Definition \[def.ws\] is satisfied. Moreover the solution satisfies the kinematic condition ${\boldsymbol{u}}_\gamma(t, x, 1+\eta_\gamma(t, x))=\partial_t\eta_\gamma(t,x){\boldsymbol{e}}_2$ is satisfied on $(0, L)$ which implies that $\partial_t\eta_\gamma\in L^2(0, T ; H^{1/2}_\sharp(0, L))$ as the trace of ${\overline{{\boldsymbol{u}}}_{\gamma}}_{|_{\widehat{\mathcal F}_{h_\gamma}}}$. Remember here that $\min_{x\in [0, L]} h_\gamma(x, t) >0$ for all $t\in [0, T]$ so that $\mathcal{F}_{h_\gamma}$ is a Lipschitz domain and consequently ${\overline{{\boldsymbol{u}}}_{\gamma}}_{\vert_{y=h_\gamma}}$ is well defined. Thus the second item $ii)$ of Definition \[def.ws\] holds true. Then, we note that, thanks to the regularity of solutions constructed in [@Grandmont-Hillairet Theorem 1], the system – is satisfied pointwise so that we can multiply the system with test functions $({\boldsymbol{w}}_{\gamma},d_{\gamma})$ for which the requirements in item $iii)$ of Definition \[def.ws\] are satisfied and obtain after integration by parts.
Moreover, we note that the solution $({\boldsymbol{u}}_{\gamma},\eta_{\gamma})$ satisfies the energy estimate with a right hand side that converges to $$\frac{1}{2}\left(\rho_{f}\int_{\mathcal{F}_{h^0}}\vert{\boldsymbol{u}}^0\vert^{2} {\mathrm{d}}{\boldsymbol{x}}+ \rho_{s}\int_{0}^{L}\vert\eta^1\vert^{2}{\mathrm{d}}x+\beta\int_{0}^{L}\vert\partial_{x}\eta^0\vert^{2}{\mathrm{d}}x+\alpha\int_{0}^{L}\vert\partial_{xx}\eta^0\vert^{2}{\mathrm{d}}x\right).$$ when $\gamma \to 0.$ Consequently, the sequence $(\overline{{\boldsymbol{u}}}_{\gamma},\eta_{\gamma})_{\gamma >0}$ satisfies the following bounds: $$\begin{aligned}
&\overline{{\boldsymbol{u}}}_{\gamma}\text{ is uniformly bounded in }\gamma\text{ in }L^{\infty}(0,T;{\boldsymbol{L}}_\sharp^{2}(\mathcal{F}_{h_{\gamma}(t)})),\label{bound-u-1}\\
&\|\nabla\overline{{\boldsymbol{u}}}_{\gamma}\|_{L^2(\widehat{\mathcal F}^{-}_{h_{\gamma}})} \text{ is uniformly bounded in }\gamma,
\label{bound-u-2}\\
&\eta_{\gamma}\text{ is uniformly bounded in }\gamma\text{ in }L^{\infty}(0,T;H^{2}_{\sharp}(0,L))\cap W^{1,\infty}(0,T;L_\sharp^{2}(0,L)).\label{est-eta}\end{aligned}$$
Furthemore the structure velocity $\partial_t\eta_\gamma$ is bounded uniformly with respect to $\gamma$ is $L^2(0, T ; H^{1/2}_\sharp(0, L))$ as the trace ${\overline{{\boldsymbol{u}}}_{\gamma}}_{\vert_{y=h_{\gamma}}}$.
Finally the sequence $(\overline{{\boldsymbol{u}}}_{\gamma})_{\gamma>0}$ satisfies additional uniform estimates that are summarized in the following lemma:
\[lem:extension-u\] The sequence $(\overline{{\boldsymbol{u}}}_{\gamma})_{\gamma >0}$ is uniformly bounded in $L^{4}({\widehat{\Omega}})$ and in $L^{2}(0,T;{\boldsymbol{H}}^{s}_{\sharp}({\Omega}))$ for arbitrary $s<1/2.$
The bound in $L^{2}(0,T;{\boldsymbol{H}}^{s}_{\sharp}({\Omega}))$ comes again from Lemma \[lem:bar\]. We next take care of the $L^{4}({\widehat{\Omega}})$ uniform bound. To prove it, it is sufficient to obtain independent uniform bounds for the restrictions of $\overline{{\boldsymbol{u}}}_{\gamma}$ to $\widehat{\mathcal S}_{h_{\gamma}}$ and to $\widehat{\mathcal F}^{-}_{h_{\gamma}}$ for the $L^4$–norm.
In $\widehat{\mathcal{S}}_{h_{\gamma}}$ we already know that $\overline{{\boldsymbol{u}}}_{\gamma}=\partial_{t}\eta_{\gamma}{\boldsymbol{e}}_{2}$ where $\partial_t{\eta}_{\gamma}$ is bounded in $L^{2}(0,T;H_\sharp^{1/2}(0,L))$. Moreover $\partial_{t}\eta_{\gamma}$ is also bounded in $L^{\infty}(0,T;L_\sharp^{2}(0,L)).$ By interpolation, we obtain that $\partial_{t}\eta_{\gamma}$ – and consequently resp. $\overline{{\boldsymbol{u}}}_{\gamma}$ – is uniformly bounded in $L^{4}(0,T;L_\sharp^{4}(0,L))$, resp. $L^4(\widehat{\mathcal S}_{h_{\gamma}})$. We would like to apply a similar interpolation argument on the domain $\mathcal F_{h_{\gamma}}^{-}$ using the uniform bounds on the restrictions of $\overline{{\boldsymbol{u}}}_{\gamma}$ and its gradient in $\mathcal F_{h_{\gamma}}^{-}$. To track the dependencies with respect to $h_{\gamma}$ to be able to ensure that the interpolation argument leads to uniform bounds, we use a change of variables. We denote by $$\tilde{{\boldsymbol{u}}}_{\gamma}(t,x,z) = \overline{{\boldsymbol{u}}}_{\gamma}(t,x,(h_{\gamma}(x) + 1)z - 1 ) \quad \forall \, (t,x,z) \in (0,T) \times \mathcal{C}_0^1.$$ Since $0 \leq h_{\gamma} \leq M,$ straightforward computations show that, for a.e. $t \in (0,T)$ we have $$\|\overline{{\boldsymbol{u}}}_{\gamma}(t)\|_{L_\sharp^4(\mathcal F_{h_{\gamma}(t)})} \leq (\|h_{\gamma}(t)\|_{L^{\infty}_{\sharp}(0,L)}+1)^{1/4} \|\tilde{{\boldsymbol{u}}}_{\gamma}(t)\|_{L_\sharp^4( \mathcal{C}_0^1)},$$ and $$\begin{aligned}
\|\tilde{{\boldsymbol{u}}}_{\gamma}(t)\|_{L_\sharp^2( \mathcal{C}_0^1)} & \leq \|\overline{{\boldsymbol{u}}}_{\gamma}(t)\|_{L_\sharp^2(\mathcal F_{h_{\gamma}(t)})}, \\
\|\nabla \tilde{{\boldsymbol{u}}}_{\gamma}(t)\|_{L_\sharp^2( \mathcal{C}_0^1)} & \leq (1+ \|h_{\gamma}(t)\|_{W^{1,\infty}_{\sharp}(0,L)}) \|\nabla \overline{{\boldsymbol{u}}}_{\gamma}\|_{L_\sharp^2(\mathcal F^-_{h_{\gamma}(t)})}.\end{aligned}$$ However, the following interpolation inequality holds true $$\|\tilde{{\boldsymbol{u}}}_{\gamma}(t)\|_{L_\sharp^4( \mathcal{C}_0^1)} \leq C \|\tilde{{\boldsymbol{u}}}_{\gamma}(t)\|^{1/2}_{L_\sharp^2( \mathcal{C}_0^1)}\|\nabla \tilde{{\boldsymbol{u}}}_{\gamma}(t)\|^{1/2}_{L_\sharp^2( \mathcal{C}_0^1)},$$ and thus $$\|\overline{{\boldsymbol{u}}}_{\gamma}(t)\|_{L_\sharp^4(\mathcal F^{-}_{h_{\gamma}(t)})} \leq C(1+ \|h_{\gamma}(t)\|_{W^{1,\infty}_{\sharp}(0,L)}) \|{{\boldsymbol{u}}}_{\gamma}(t)\|^{1/2}_{L_\sharp^2(\mathcal F^{-}_{h_{\gamma}(t)})}\|\nabla \tilde{{\boldsymbol{u}}}_{\gamma}(t)\|^{1/2}_{L_\sharp^2(\mathcal{C}_0^1)}.$$ Applying and together with the uniform bound for $h_{\gamma}$ in $L^{\infty}(0,T;W^{1,\infty}_{\sharp}(0,L))$ coming from , we obtain the desired bound on $\overline{{\boldsymbol{u}}}_{\gamma}$ which is uniformly bounded in $L^4(\mathcal F^{-}_{h_{\gamma}}).$
We now prove the existence of cluster points of the sequence $(\overline{{\boldsymbol{u}}}_{\gamma},\eta_{\gamma})_{\gamma >0}.$ First, thanks to , and to the compact embedding $$\begin{array}{ll}
\eta_{\gamma}\rightarrow\eta& \text{ uniformly in }\mathcal{C}^{0}([0,T];\mathcal{C}^{1}_{\sharp}(0,L)) ,\\
\eta_{\gamma}\rightharpoonup\eta& \text{ weakly} -\star \text{ in }W^{1,\infty}(0,T;L^2_{\sharp}(0,L)).
\end{array}$$
Next, using the energy estimates and Lemma \[lem:extension-u\], we may construct a divergence free function $\overline{{\boldsymbol{u}}}\in L^{\infty}(0,T;{\boldsymbol{L}}^{2}_{\sharp}({\Omega}))\cap L^{4}({\widehat{\Omega}})$, such that, up to a subsequence, the following convergences hold: $$\begin{array}{ll}
\overline{{\boldsymbol{u}}}_{\gamma}\rightharpoonup\overline{{\boldsymbol{u}}},& \text{ weakly}-\star \text{ in }L^{\infty}(0,T;{\boldsymbol{L}}^{2}_{\sharp}({\Omega})),\\
\overline{{\boldsymbol{u}}}_{\gamma}\rightharpoonup\overline{{\boldsymbol{u}}},&\text{ weakly in } L^{4}({\widehat{\Omega}}).
\end{array}$$
We now verify that any cluster point $(\overline{{\boldsymbol{u}}}, \eta)$ of the sequence $({\boldsymbol{u}}_{\gamma},\eta_{\gamma})_{\gamma >0}$ enjoys the properties of Definition \[def.ws\], which defines the weak solutions of the limit coupled system $(FS)_0.$ For simplicity, we do not relabel the sequence converging to $(\overline{{\boldsymbol{u}}}, \eta).$ We first note that, for fixed $\gamma >0,$ we have $$\int_0^L \partial_t\eta_{\gamma}=0.$$ This property is conserved in the weak limit so that $\partial_t\eta$ is mean free globally in time. Next, we verify that $(\overline{{\boldsymbol{u}}}(t),\partial_t \eta) \in X[h(t)]$ for a.e. $t \in (0,T).$ The divergence free condition is verified at the limit. Moreover, we note that, the property $\overline{{\boldsymbol{u}}}_{\gamma} = 0$ on $\widehat{\mathcal C}_{-1}^0$ is preserved in the weak limit. Furthermore, since $\eta_\gamma$ uniformly converges towards $\eta$, we easily obtain $\overline{{\boldsymbol{u}}} =\partial_{t}\eta{\boldsymbol{e}}_2,\text{ in }\widehat{\mathcal{S}}_{h}$, by testing the weak convergence of $ \overline{{\boldsymbol{u}}}_{\gamma}$ and $ \partial_{t}\eta_\gamma{\boldsymbol{e}}_2$, which are equal in $\widehat{\mathcal S}_{h_\gamma}$, against functions ${\boldsymbol{\varphi}} \in \mathcal C^{\infty}_c(\widehat{\mathcal S}_h)$.
Consequently, $(\overline{{\boldsymbol{u}}}(t),\partial_t \eta) \in X[h(t)].$ We now prove that $\overline{{\boldsymbol{u}}}$ has better regularity in $\widehat{\mathcal F}^-_{h}$ as stated in the following lemma:
\[lemma.gradient\] We have $\nabla \overline{{\boldsymbol{u}}} \in L^2(\widehat{\mathcal F}_h)$ and the sequence $\rho_\gamma^-\nabla \overline{{\boldsymbol{u}}}_{\gamma}$ converges to $\rho^-\nabla \overline{{\boldsymbol{u}}}$ weakly in $L^{2}({\widehat{\Omega}}).$
In the previous statement, we use the convention that if $\mathcal O \subset \widehat{\Omega}$ and $f \in L^2(\mathcal O)$ then $\mathbf{1}_{\mathcal O} f \in L^2(\widehat{\Omega})$ is the extension by $0$ of this $L^2(\mathcal O)$–function.
We remind that $\nabla \overline{{\boldsymbol{u}}}_{\gamma} \in {\boldsymbol{L}}^2(\widehat{\mathcal F}^{-}_{h_{\gamma}})$ so that $\rho_\gamma^-\nabla \overline{{\boldsymbol{u}}}_{\gamma}$ corresponds to the extension by ${\boldsymbol{0}}$ of this vector field. Because of , $\rho_\gamma^-\nabla \overline{{\boldsymbol{u}}}_{\gamma}$ is uniformly bounded in ${\boldsymbol{L}}^{2}(\widehat{\Omega})$. Thus $\rho^-_\gamma \nabla \overline{{\boldsymbol{u}}}_{\gamma}$ converges weakly to some $\bf{z}$ in $L^{2}(0,T;{\boldsymbol{L}}_\sharp^{2}({\Omega}))$. Thanks to the uniform convergence of $h_{\gamma}$ to $h$, we may then compute $\bf{z}$ by testing the weak convergence of $\rho^-_\gamma \nabla \overline{{\boldsymbol{u}}}_{\gamma}$ against functions ${\boldsymbol{\varphi}} \in \mathcal C^{\infty}_c(\widehat{\mathcal S}_h)$ and ${\boldsymbol{\varphi}} \in \mathcal C^{\infty}_c(\widehat{\mathcal F}_h^{-})$ respectively. This implies that ${\bf z}_{\vert\widehat{\mathcal{S}}_h}={\boldsymbol{0}}$ and ${\boldsymbol{z}}_{\vert\widehat{\mathcal{F}}^-_{h}}=(\nabla \overline{{\boldsymbol{u}}})_{\vert\widehat{\mathcal{F}}^-_{h}}$, which ends the proof.
The previous lemma gives the $H^1$ space regularity of $\overline{{\boldsymbol{u}}}_{|_{\mathcal{F}^-_{h(t)}}}$ (for a.e. $t \in (0,T)$). Since $\mathcal{F}^-_{h(t)}$ a Lipschitz domain, it enables us to define an $H^{1/2}$ trace of $\overline{{\boldsymbol{u}}}$ on ${\partial \mathcal F^-_{h(t)}}$.
This concludes the proof that $(\overline{{\boldsymbol{u}}},\eta)$ satisfies item $i)$ of Definition \[def.ws\]. [We also prove at first that the weak cluster point satisfies the expected energy estimate. Indeed, for any arbitrary small $\varepsilon >0$, thanks to the strong convergence of $\eta_{\gamma}$ to $\eta,$ we have that $h_{\gamma} > h- \varepsilon$ for $\gamma$ sufficiently small. We may apply then classical weak limit arguments to pass to the limit in the energy estimate satisfied by the $({{\boldsymbol{u}}}_{\gamma},\eta_{\gamma}).$ Consequently for almost every $t$ we have $$\begin{aligned}
&\frac{1}{2}\left(\rho_{f}\int_{\mathcal{F}^{-}_{h(t)-\varepsilon}}\vert{\boldsymbol{u}}(t, {\boldsymbol{x}}) \vert^{2} {\mathrm{d}}{\boldsymbol{x}}+ \rho_{s}\int_{0}^{L}\vert\partial_{t}\eta(t, x) \vert^{2} {\mathrm{d}}x+\beta\int_{0}^{L}\vert\partial_{x}\eta(t, x) \vert^{2} {\mathrm{d}}x+\alpha\int_{0}^{L}\vert\partial_{xx}\eta(t, x) \vert^{2} {\mathrm{d}}x\right)\\
& + \mu\int_0^t\int_{\mathcal{F}^{-}_{h(s)-\varepsilon}}\vert \nabla{\boldsymbol{u}}(s, {\boldsymbol{x}}) \vert^{2} {\mathrm{d}}{\boldsymbol{x}}{\mathrm{d}}s \\
& \leq
\liminf_{\gamma \to 0} \left[ \frac{1}{2}\left(\rho_{f}\int_{\mathcal{F}^{-}_{h(t)-\varepsilon}}\vert{\boldsymbol{u}}_{\gamma}(t, {\boldsymbol{x}}) \vert^{2} {\mathrm{d}}{\boldsymbol{x}}+ \rho_{s}\int_{0}^{L}\vert\partial_{t}\eta_{\gamma}(t, x) \vert^{2} {\mathrm{d}}x+\beta\int_{0}^{L}\vert\partial_{x}\eta_{\gamma}(t, x) \vert^{2} {\mathrm{d}}x \right.\right. \\
&\left. \left. +\alpha\int_{0}^{L}\vert\partial_{xx}\eta_{\gamma}(t, x) \vert^{2} {\mathrm{d}}x\right) + \mu\int_0^t\int_{\mathcal{F}^{-}_{h(s)-\varepsilon}}\vert \nabla{\boldsymbol{u}}_{\gamma}(s, {\boldsymbol{x}}) \vert^{2} {\mathrm{d}}{\boldsymbol{x}}{\mathrm{d}}s\right] \\
&
\leq \lim_{\gamma \to 0}
\frac{1}{2}\left(\rho_{f}\int_{\mathcal{F}_{h^0_{\gamma}}}\vert{\boldsymbol{u}}^0_{\gamma}\vert^{2} {\mathrm{d}}{\boldsymbol{x}}+ \rho_{s}\int_{0}^{L}\vert\eta^1_{\gamma}\vert^{2}{\mathrm{d}}x+\beta\int_{0}^{L}\vert\partial_{x}\eta^0_{\gamma}\vert^{2}{\mathrm{d}}x+\alpha\int_{0}^{L}\vert\partial_{xx}\eta^0_{\gamma}\vert^{2}{\mathrm{d}}x\right)
\\
&
=\frac{1}{2}\left(\rho_{f}\int_{\mathcal{F}_{h^0}}\vert{\boldsymbol{u}}^0\vert^{2} {\mathrm{d}}{\boldsymbol{x}}+ \rho_{s}\int_{0}^{L}\vert\eta^1\vert^{2}{\mathrm{d}}x+\beta\int_{0}^{L}\vert\partial_{x}\eta^0\vert^{2}{\mathrm{d}}x+\alpha\int_{0}^{L}\vert\partial_{xx}\eta^0\vert^{2}{\mathrm{d}}x\right).
\end{aligned}$$ Since $\varepsilon$ is arbitrary, we obtain the expected energy inequality. ]{}
We now show that any limit $(\overline{{\boldsymbol{u}}},\eta)$ also satisfies items $ii)$ and $iii)$ of Definition \[def.ws\] of weak solutions under the further assumption that the following lemma holds true:
\[lemma.convergence.L2\] Up to the extraction of a subsequence that we do not relabel, we have $(\rho_\gamma\overline{{\boldsymbol{u}}}_{\gamma},\partial_{t}\eta_{\gamma})\underset{\gamma\rightarrow 0}{\longrightarrow}(\rho\overline{{\boldsymbol{u}}},\partial_{t}\eta)$ strongly in $L^{2}({\widehat{\Omega}})\times L^{2}((0, T)\times (0,L))$.
So, fix $({\boldsymbol{w}}, d)\in\mathcal{C}^{\infty}({\widehat{\Omega}})\times \mathcal{C}^{\infty}(0, L) $ such that $({\boldsymbol{w}}(t), d(t))\in\mathcal{X}[h(t)]$ for all $t\in(0,T)$. Due to the uniform convergence of $h_\gamma$ and the special structure of $\mathcal{X}[h(t)]$ for which we require that ${\boldsymbol{w}}=0$ in the neighbourhood of $\mathcal{C}^0_{-1}$ and ${\boldsymbol{w}}\cdot{\boldsymbol{e}}_{1}=0$ in the neighbourhood of $\mathcal{S}_h(t)$, there exists $\gamma_{0}>0$ such that, for all $0<\gamma<\gamma_{0}$, $({\boldsymbol{w}}(t),d(t))\in X[h_{\gamma}(t)]$ for all $t\in(0,T)$. Hence, $({\boldsymbol{w}},d)$ is a test function for any $\gamma$ small enough and for a.e. $t\in(0,T)$, $(\overline{{\boldsymbol{u}}}_{\gamma},\eta_{\gamma})$ satisfies $$\label{weak.formulation.FS.gamma.limit}
\begin{aligned}
&\rho_{f}\int_{{\Omega}}\rho_{\gamma}\overline{{\boldsymbol{u}}}_{\gamma}(t)\cdot{\boldsymbol{w}}(t) -\rho_{f}\int_{0}^{t}\int_{{\Omega}}\rho_{\gamma}\overline{{\boldsymbol{u}}}_{\gamma}\cdot\partial_{t}{\boldsymbol{w}}+ (\rho_{\gamma}\overline{{\boldsymbol{u}}}_{\gamma}\cdot\nabla){\boldsymbol{w}}\cdot\rho_{\gamma}\overline{{\boldsymbol{u}}}_{\gamma}\\
&+\rho_{s}\int_{0}^{L}\partial_{t}\eta_{\gamma}(t)d(t)-\rho_{s}\int_{0}^{t}\int_{0}^{L}\partial_{t}\eta_{\gamma}\partial_{t}d+\mu\int_{0}^{t}\int_{{\Omega}}\rho_{\gamma} \nabla{\boldsymbol{u}}_{\gamma}:\nabla{\boldsymbol{w}}\\
&+\int_{0}^{t}\int_{0}^{L}\beta\partial_{x}\eta_{\gamma}\partial_{x}d + \alpha\partial_{xx}\eta_{\gamma}\partial_{xx}d+\gamma\int_{0}^{t}\int_{0}^{L}\partial_{tx}\eta_{\gamma}\partial_{x}d=\rho_{f}\int_{{\Omega}}\rho_{\gamma}\overline{{\boldsymbol{u}}}^{0}_{\gamma}\cdot{\boldsymbol{w}}(0)+\rho_{s}\int_{0}^{L}\eta^{1}_{\gamma}d(0).
\end{aligned}$$
Let us recall that, thanks to Lemma \[lemma.convergence.L2\], $\rho_\gamma {\boldsymbol{u}}_{\gamma}$ strongly converges in $L^2(\hat\Omega)$. The convergence of $\rho_{\gamma} \nabla{\boldsymbol{u}}_{\gamma}$ is proved in Lemma \[lemma.gradient\] and we can pass to the limit all the terms of . The pair $(\overline{{\boldsymbol{u}}},\eta)$ satisfies, for a.e. $t\in(0,T)$, $$\begin{aligned}
&\rho_{f}\int_{{\Omega}}\rho\overline{{\boldsymbol{u}}}(t)\cdot{\boldsymbol{w}}(t) -\rho_{f}\int_{0}^{t}\int_{{\Omega}}\rho\overline{{\boldsymbol{u}}}\cdot\partial_{t}{\boldsymbol{w}}+ (\rho\overline{{\boldsymbol{u}}}\cdot\nabla){\boldsymbol{w}}\cdot\rho\overline{{\boldsymbol{u}}}\\
&+\rho_{s}\int_{0}^{L}\partial_{t}\eta(t)d(t)-\rho_{s}\int_{0}^{t}\int_{0}^{L}\partial_{t}\eta\partial_{t}d+\mu\int_{0}^{t}\int_{{\Omega}}\rho {\boldsymbol{u}}:\nabla{\boldsymbol{w}}\\
&+\int_{0}^{t}\int_{0}^{L}\beta\partial_{x}\eta\partial_{x}d + \alpha\partial_{xx}\eta\partial_{xx}d=\rho_{f}\int_{{\Omega}}\rho\overline{{\boldsymbol{u}}}^{0}\cdot{\boldsymbol{w}}(0)+\rho_{s}\int_{0}^{L}\eta^{1}d(0),
\end{aligned}$$ which is a rewriting of the variational formulation . Thus the item $iii)$ of Definition \[def.ws\] is satisfied. The last point to verify is the kinematic condition at the fluid–structure interface. We know that $${\boldsymbol{u}}_\gamma(t, x, h_\gamma(t, x))= \partial_t \eta_\gamma(t, x) {\boldsymbol{e}}_2.$$ From Lemma \[lemma.convergence.L2\] the right hand side converges strongly in $L^2(0, T; L_\sharp^2(0, L))$ towards $\partial_t \eta$. It converges also weakly in $L^2(0, T; H_\sharp^{1/2}(0, L))$. The left hand side is the trace of the function $(x, z) \mapsto {\boldsymbol{u}}_\gamma(t, x, (1+h_\gamma(t, x))z + h_\gamma(t, x))$ on $z=1$. Thanks to the previous convergences this function converges strongly in $L^2(\widehat{{\mathcal C}_0^1})$ and weakly in $L^2(0, T; H^1_\sharp({\mathcal C}_0^1))$ towards ${\boldsymbol{u}}(t, x, (1+h(t, x))z + h(t, x))$. Hence by continuity of the trace we obtain $${\boldsymbol{u}}(t, x, h(t, x))= \partial_t \eta(t, x) {\boldsymbol{e}}_2,$$ so that item $ii)$ of Definition \[def.ws\], which completes the proof of Theorem \[main.theorem\].
It thus remains to prove Lemma \[lemma.convergence.L2\]. As is usual for fluid–structure problems, the sequence of domains is unknown and depends on time and here on the viscosity parameter, so that standard Aubin–Lions lemma cannot be applied directly to obtain compactness of the velocities. One key point is to build a piecewise in time, regular enough in space, interface, dealing with possible contact, close to the sequence of interfaces but always lower. The construction of this artificial interface is the aim of the next subsection. We then conclude the proof in the final subsection. Thanks to the variational formulation and to this well chosen interface “from below" we obtain bounds on the time derivative of an $L^2$ projection of the velocities, for which we are able to apply an adapted version of the Aubin–Lions lemma. It implies that the sequence of velocities is nearly compact. Next the key idea is to use that the velocities can be approximated, in $H^s$ for some $s>0$, by velocities associated to the interface “from below" so that we can “fill" the gap. This relies on the continuity properties of the $H^s$–projector operator obtained in Lemma \[lem:projector\].
Step 2. Construction of an interface from below.
------------------------------------------------
Before the construction of the interface from below, we analyze a simple method to approximate a given stationary deformation from below. Namely, given $h \in H^2_{\sharp}(0,L)$ satisfying $h \geq 0$ and $\mu >0,$ we denote $${{\underline{h}}}_{\mu} := [h-\mu]_{+}.$$ where the subscript $+$ denotes here the positive part of functions. The properties of this approximation process are summarized in the following lemma:
\[lemma.halpha\] Let $h \in H^{2}_{\sharp}(0,L)$ with $h \geq 0$ and $\mu >0.$ Then ${{\underline{h}}}_{\mu}\in \mathcal{C}_\sharp^{0}(0,L)$ and, given $\kappa \in (0,1/2),$ there exists a constant $C$ independent of $\mu$ and $h$ for which: $$\begin{aligned}
& {\left\Vert {{\underline{h}}}_{\mu}\right\Vert}_{H_\sharp^{1+\kappa}(0,L)} + {\left\Vert {{\underline{h}}}_{\mu}\right\Vert}_{W^{1,\infty}_\sharp(0,L)}\leq C{\left\Vert h\right\Vert}_{H_\sharp^{2}(0,L)}, \label{h1}\\
& {\left\Vert {{\underline{h}}}_{\mu}-h\right\Vert}_{W_\sharp^{1, \infty}(0,L)}\leq {\mu}+\displaystyle\sup_{\{x\in[0, L]\mid h(x)\leq \mu\}}\vert h'(x)\vert.\label{h2}
$$
Let $\mu>0$ and $h \in H^{2}_{\sharp}(0,L)$ non-negative. The first statement ${{\underline{h}}}_{\mu}\in \mathcal{C}_\sharp^{0}(0,L)$ is standard. We prove the two inequalities , successively.
[**Step 1: proof of inequality .**]{} Since $h \in {\mathcal C}^0_{\sharp}(0,L)$ the subset $\{x\in(0, L), h(x)>\mu\}$ is open. We may then construct at most denumerable sets $\{c_i, \; i \in \mathcal I_{\mu}\}$ and $\{ d_i, \; i \in \mathcal I_{\mu}\}$ such that $\{x\in(0, L), h(x)>\mu\}= \bigcup_{i\in\mathcal I_{\mu}}(c_{i},d_{i}).$ The successive derivatives of $[h-\mu]_{+}$ then read $$[h-\mu]'_{+}= h'\mathds{1}_{h>\mu}, \quad [h-\mu]{''}_{+}= h{''}\mathds{1}_{h>\mu}+\sum_{i\in \mathcal I_{\mu}}\delta_{c_{i} }h{'}(c_{i})- \delta_{d_{i}}h'(d_{i}),$$ To show that ${{\underline{h}}}_{\mu} \in H^{1+\kappa}_{\sharp}(0,L)$ and have a bound on its norm, we now prove that $[h- {\mu}]{''}_{+} \in H^{\kappa-1}_{\sharp}(0,L).$
Using the $H^{2}$-regularity of $h$ we obtain that $h{''}\mathds{1}_{h>\mu} \in L_\sharp^{2}(0,L) \subset H^{\kappa-1}_{\sharp}(0,L)$. Moreover for any test function $\varphi \in \mathcal{D}(0, L)$, we have: $$\vert\langle \delta_{c_{i}},\varphi\rangle\vert = \vert \varphi(c_{i})\vert\leq {\left\Vert \varphi\right\Vert}_{\mathcal{C}_\sharp(0,L)}\leq C{\left\Vert \varphi\right\Vert}_{H_\sharp^{1-\kappa}(0,L)},$$ where $C>0$ stands for the constant associated with the Sobolev embedding $H_\sharp^{1-\kappa}(0,L) \subset \mathcal{C}^0_\sharp(0,L)$. Hence $\delta_{c_{i}} \in H^{\kappa-1}(0,L)$, with a norm independent of $c_{i}$. To show that $[h- {\mu}]{''}_{+} \in H^{\kappa-1}_{\sharp}(0,L)$ it then remains to prove that the sums $$\sum_{i\in \mathcal I_{\mu}}\delta_{c_{i} }h{'}(c_{i}), \quad \sum_{i\in \mathcal I_{\mu}}\delta_{d_{i} }h{'}(d_{i}),$$ do converge normally in the Banach space $H_\sharp^{\kappa-1}(0,L).$
Let $i \in \mathcal I_{\mu},$ since $h(c_{i})=h(d_{i})=\mu,$ there exists $b_{i}\in(c_{i},d_{i})$ such that $h'(b_{i})=0$. This implies $$\vert h'(c_{i})\vert\leq \vert b_{i}-c_{i}\vert^{1/2}{\left\Vert h\right\Vert}_{H^{2}(c_{i},d_{i})} \leq \vert d_{i}-c_{i}\vert^{1/2}{\left\Vert h\right\Vert}_{H^{2}(c_{i},d_{i})},$$ and thus $$\label{regularity.second.derivative}
\begin{aligned}
\sum_{i \in \mathcal I_{\mu}} {\left\Vert h'(c_{i})\delta_{c_{i}}\right\Vert}_{H_\sharp^{\kappa-1}(0,L)}&{}\leq C\sum_{i \in \mathcal I_{\mu}}\vert d_{i}-c_{i}\vert^{1/2}{\left\Vert h\right\Vert}_{H^{2}(c_{i},d_{i})}\\
&\leq C\left(\sum_{i \in \mathcal I_{\mu}} \vert d_{i}-c_{i}\vert\right)^{1/2}\left(\sum_{i \in \mathcal I_{\mu}}^{N}{\left\Vert h\right\Vert}_{H^{2}(c_{i},d_{i})}^{2}\right)^{1/2}\\
&\leq CL^{1/2}{\left\Vert h\right\Vert}_{H_\sharp^{2}(0,L)}.
\end{aligned}$$ Consequently, we have $$\sum_{i\in \mathcal I_{\mu}}\delta_{c_{i} }h{'}(c_{i}) \in H^{\kappa -1}_{\sharp}(0,L) \textrm{ and } \|\sum_{i\in \mathcal I_{\mu}}\delta_{c_{i} }h{'}(c_{i})\|_{H^{\kappa -1}_{\sharp}(0,L) } \leq CL^{1/2}{\left\Vert h\right\Vert}_{H_\sharp^{2}(0,L)}.$$ The argument is similar for $\displaystyle\sum_{i\in \mathcal I_{\mu}}\delta_{d_{i}}h'(d_{i})$. This completes the proof of estimate .
[**Step 2: proof of estimate .**]{} Since $h\in H_\sharp^2(0, L)$, thanks to the continuous embedding of $H_\sharp^1(0, L)$ in $L_\sharp^\infty(0, L),$ we deduce that $[h-\mu]_{+}\in W_\sharp^{1,\infty}(0, L)$. Furthermore it is clear that, for any $ x\in [0, L]$, $$\vert [h-\mu]_{+}(x)-h(x)\vert \leq \mu, \qquad [h-\mu]'_{+}(x)-h'(x) = -h'(x) \mathds{1}_{h\leq\mu}(x).$$ This implies that is satisfied.
The next lemma ensures that the right-hand side of estimate goes to zero when $\mu$ goes to zero:
\[lemma.conv.zero\]Let $h \in \mathcal{C}_\sharp^{1}(0,L)$ with $h\geq 0$. The following limit holds $$\sup_{\{x\in [0, L]\mid h(x)\leq\mu\}}\vert h'(x)\vert \underset{\mu\rightarrow 0}{\longrightarrow} 0.$$
Since $ h \in \mathcal{C}^1_\sharp(0,L)$ we have that $\{ x \in [0,L]\mid h(x)\leq\mu\}$ is a compact subset of $[0,L]$ and that there exists $x_{\mu}\in [0,L]$ such that: $$\sup_{\{x\in [0, L]\mid h(x)\leq\mu\}} |h'(x)|= |h'(x_{\mu})|.$$ Note that, by construction, we have $h(x_{\mu})\leq \mu$.
Using the compactness of $[0,L]$ we have $x_{\mu}\rightarrow \bar x\in [0,L]$ as $\mu$ goes to zero (up to a subsequence). Using the continuity of $h$ and passing to the limit in the inequality $h(x_{\mu})\leq \mu$ we obtain $h(\bar x)=0$. Moreover, using that $h(x)\geq 0$ we deduce that $\bar x$ is a local minimum of $h$ and thus that $h'(\bar x)=0$. Finally the continuity of $h'$ ensures that $|h'(x_{\mu})|\underset{\mu\rightarrow 0}{\longrightarrow}|h'(\bar x)|=0$.
We recall that the sequence $(h_{\gamma})_{\gamma >0}$ we consider converges to $h$ strongly in $\mathcal{C}^{0}([0,T];\mathcal{C}_\sharp^{1}(0,L))$ with $h\in L^{\infty}(0,T;H_\sharp^{2}(0,L)) \cap W^{1,\infty}(0, T; L^2_\sharp(0, L))$. We now are in a position to build a family of approximating interfaces “from below" of any $h_\gamma$ for $\gamma$ small enough. Namely, given $\delta >0$ we construct a piecewise-constant (in time) function $\underline{h}_{\delta}$ such that there exists $\gamma_0 >0$ for which $$\begin{aligned}
& \|{{\underline{h}}}_{\delta}\|_{L^{\infty}(0,T;W^{1,\infty}_{\sharp}(0,L))} \leq C, \label{estimate.hb}\\
&{{\underline{h}}}_{\delta}\leq h_{\gamma}, && \,\forall \gamma\leq\gamma_{0}, \label{eq.interfacebelow}\\
&\| {{\underline{h}}}_{\delta}- h_\gamma\|_{L^\infty(0,T; W_\sharp^{1, \infty}(0 , L))}\leq \delta, && \,\forall \gamma\leq\gamma_{0}. \label{estimation-diffth}\end{aligned}$$ with $C$ depending only on initial data.
So, let fix $\delta >0$ and introduce parameters $\varepsilon >0,N\in \mathbb N$ to be made precise later on. We construct our piecewise approximation as follows. We consider the subdivision of the time interval $[0,T]=\cup_{0\leq k\leq N}I_{k}$ with $I_{k}=[k\Delta t,(k+1)\Delta t)$, $\Delta t=\frac{T}{N+1}$ and we fix $t_{k}\in I_{k}$ such that ${\left\Vert h(t_{k})\right\Vert}_{H^{2}_{\sharp}(0,L)}\leq {\left\Vert h\right\Vert}_{L^{\infty}(0,T;H^{2}_{\sharp}(0,L))}$. On each time interval $I_k$ we then define $$\label{def-sous-h}
{{\underline{h}}}_{\delta}(x, t)= [h-2\varepsilon]_+(x, t_k), \quad t\in I_k.$$
Estimate follows directly from Lemma \[lemma.halpha\] and estimate . Now, up to a good choice for the parameters $\varepsilon$ and $N,$ ${{\underline{h}}}_{\delta}$ satisfies the two properties and for $\gamma$ small enough. First, let us prove that ${{\underline{h}}}_{\delta}\leq h_{\gamma}$ for all $\gamma$ sufficiently small and $N$ sufficiently large (depending on $\varepsilon$). We recall that by interpolation, we have $h \in \mathcal{C}^{0,\theta}([0,T];\mathcal{C}^{1}_{\sharp}(0,L))$ for $\theta\in(0,\frac{1}{4})$ (see embedding ). Consequently, for any $k \leq N,$ we have $$\|h(x,t) - h(x,t_{k})\|_{L^{\infty}_{\sharp}(0,L)} \leq C \Delta t^{\theta}, \quad \forall \, t \in I_k.$$ Similarly, since $h_{\gamma}$ converges to $h$ in $\mathcal{C}^{0}([0,T];\mathcal{C}_\sharp^{1}(0,L))$ we can find $\gamma_0 >0$ such that, for $\gamma \leq \gamma_0,$ $$\|h(x,t) - h_{\gamma}(x,t)\|_{W^{1,\infty}_{\sharp}(0,L)} \leq \varepsilon, \quad \forall \, t \in (0,T).$$ Assuming that $N$ is chosen such that (with the above constant $C$) $$C \Delta t^{\theta}=C \left(\dfrac T{N+1} \right)^{\theta} < \varepsilon,$$ we have, for any $k \leq N$, $$h_{\gamma}(x,t)\geq h(x,t)-\varepsilon\geq h(x,t_{k})-2\varepsilon,\, \qquad \forall(x,t)\in(0,L)\times I_{k},\,\forall \gamma\leq \gamma_{0}.$$ Taking the positive part in the previous inequality (recall that $h_{\gamma}\geq 0$) we obtain .
We now estimate the difference between ${{\underline{h}}}_{\delta}$ and $h_{\gamma}$ for $\gamma \leq \gamma_0.$ Given $k \leq N$ we have, for all $\gamma \leq \gamma_0$ $$\begin{aligned}
\| {{\underline{h}}}_{\delta}- h_\gamma\|_{L^\infty(I_k; W_\sharp^{1, \infty}(0 , L))}&\leq \| {{\underline{h}}}_{\delta}- h\|_{L^\infty(I_k; W_\sharp^{1, \infty}(0, L))}
+ \| h- h_\gamma\|_{L^\infty(I_k; W_\sharp^{1, \infty}(0, L))},\\
&\leq \| [h-2\varepsilon]_+(t_k)- h(t_k)\|_{W_\sharp^{1, \infty}(0, L)}
+\| h({t_k})- h\|_{L^\infty(I_k; W_\sharp^{1, \infty}(0, L))}
+ \varepsilon,\\
&\leq \| [h-2\varepsilon]_+(t_k)- h(t_k)\|_{W_\sharp^{1, \infty}(0, L)}
+2\varepsilon.
\end{aligned}$$ Applying Lemma \[lemma.halpha\], this entails $$\| [h-2\varepsilon]_+(t_k)- h(t_k)\|_{W_\sharp^{1, \infty}(0, L)}\leq 2\varepsilon +\sup_{\{x\in [0, L], h(x,t_k)\leq 2\varepsilon\}}\vert\partial_x h(x,t_k)\vert.$$ Finally we obtain $$\| {{\underline{h}}}_{\delta}- h_\gamma\|_{L^\infty((0,T); W_\sharp^{1, \infty}(0 , L))}\leq 4 \varepsilon + \sup_{\{x\in [0, L], h(x, t_k)\leq 2\varepsilon\}}\vert\partial_x h(x, t_k)\vert .$$ Consequently, applying Lemma \[lemma.conv.zero\] and choosing $\varepsilon>0$ sufficiently small with the corresponding $N \in{\mathbb{N}}$ and $\gamma_{0}>0$ we obtain that the interface ${{\underline{h}}}_{\delta}$ satisfies .
Step 3. $L^2$-compactness of the velocities.
--------------------------------------------
In this section we study the $L^{2}$-convergence of the pair $(\rho_{\gamma}\overline{{\boldsymbol{u}}}_{\gamma},\partial_{t}\eta_{\gamma})$ stated in Lemma \[lemma.convergence.L2\]. We know that, up to a subsequence that we do not relabel, $\rho_{\gamma}\overline{{\boldsymbol{u}}}_{\gamma}\rightharpoonup\rho\overline{{\boldsymbol{u}}}$ weakly in $L^{2}(0,T;{\boldsymbol{L}}_\sharp^{2}({\Omega}))$, and $\partial_t\eta_\gamma \rightharpoonup \partial_{t}\eta$ in $L^2(0,T;L^2_{\sharp}(0,L))$. To prove the strong convergence of the sequence $(\rho_{\gamma}\overline{{\boldsymbol{u}}}_{\gamma},\partial_{t}\eta_{\gamma})$ to $(\rho\overline{{\boldsymbol{u}}},\partial_{t}\eta)$ it remains to show that the following convergence holds true: $$\label{convergence.int}
\rho_{f}\int_{0}^{T}\int_{{\Omega}}\vert \rho_{\gamma}\overline{{\boldsymbol{u}}}_{\gamma}\vert^{2}+\rho_{s}\int_{0}^{T}\int_{0}^{L}\vert\partial_{t}\eta_{\gamma}\vert^{2}\underset{\gamma\rightarrow 0}{\longrightarrow}\rho_{f}\int_{0}^{T}\int_{{\Omega}}\vert \rho\overline{{\boldsymbol{u}}}\vert^{2}+\rho_{s}\int_{0}^{T}\int_{0}^{L}\vert\partial_{t}\eta\vert^{2}.$$ We recall that we endow $X^0 := \mathbf{L}^2(\Omega) \times L^2_{\sharp}(0,L)$ with the scalar product: $$((\overline{{\boldsymbol{v}}},\dot{\eta}), (\overline{{\boldsymbol{w}}},d))_{X^0} = \rho_f \int_{\Omega} \overline{{\boldsymbol{v}}} \cdot \overline{{\boldsymbol{w}}} + \rho_s \int_{0}^L \dot{\eta} d.$$ In particular, with these notations, the right hand side of also reads: $$\int_0^T ((\rho_{\gamma}\overline{{\boldsymbol{u}}},\partial_t \eta_{\gamma}), (\overline{{\boldsymbol{u}}},\partial_t \eta_{\gamma}))_{X^0}.$$ By restriction, this bilinear form enables to consider any element of $X^0$ as an element of $(X^s)'$ via the formula $$\label{eq:embedX0}
\langle (\overline{{\boldsymbol{v}}},\dot{\eta}), (\overline{{\boldsymbol{w}}},d) \rangle_{(X^s)',X^s } = ((\overline{{\boldsymbol{v}}},\dot{\eta}), (\overline{{\boldsymbol{w}}},d))_{X^0}
\quad \forall \, ((\overline{{\boldsymbol{v}}},\dot{\eta}),(\overline{{\boldsymbol{w}}},d)) \in X^0 \times X^s.$$ In what follows we use this identification without mentioning it.
To obtain , we show actually that, up to extract again a denumerable times subsequences, we can prove that the error terms $$Err_{\gamma} := \rho_{f}\int_{0}^{T}\int_{{\Omega}}\vert \rho_{\gamma}\overline{{\boldsymbol{u}}}_{\gamma}\vert^{2}+\rho_{s}\int_{0}^{T}\int_{0}^{L}\vert\partial_{t}\eta_{\gamma}\vert^{2} - \left( \rho_{f}\int_{0}^{T}\int_{{\Omega}}\vert \rho\overline{{\boldsymbol{u}}}\vert^{2}+\rho_{s}\int_{0}^{T}\int_{0}^{L}\vert\partial_{t}\eta\vert^{2} \right) ,$$ satisfies $\limsup_{\gamma \to 0} |Err_{\gamma}| \leq \tilde\varepsilon$ for any arbitrary small $\tilde\varepsilon.$ We shall compute $\varepsilon$ with respect to the parameter $\delta >0$ fixing the interface from below ${{\underline{h}}}_{\delta}$ satisfying – as in the previous subsection. So let fix such a $\delta >0.$ We recall that the related interface ${{\underline{h}}}_{\delta}$ is constant on a family of intervals $(I_k)_{0 \leq k \leq N}$ covering $[0,T].$ Below, we denote ${{\underline{h}}}_{\delta,k}$ the value of ${{\underline{h}}}_{\delta}$ on $I_k.$ We then split the time integral and introduce the projector $\mathbb P^s[{{\underline{h}}}_{\delta,k}]$ for a given $s < 1/2.$ This yields: $$\begin{aligned}
Err_{\gamma} & = \sum_{k=0}^N \int_{I_{k}} ( (\rho_{\gamma} \overline{{\boldsymbol{u}}}_{\gamma},\partial_t \eta_{\gamma}),( \overline{{\boldsymbol{u}}}_{\gamma},\partial_t \eta_{\gamma}))_{X^0} - ( (\rho \overline{{\boldsymbol{u}}},\partial_t \eta),( \overline{{\boldsymbol{u}}},\partial_t \eta))_{X^0} \\
& = \sum_{k=0}^N \int_{I_{k}} ( (\rho_{\gamma} \overline{{\boldsymbol{u}}}_{\gamma},\partial_t\eta_{\gamma}),( \overline{{\boldsymbol{u}}}_{\gamma},\partial_t \eta_{\gamma})- \mathbb{P}^{s}[{{\underline{h}}}_{\delta, k}](\overline{{\boldsymbol{u}}}_{\gamma},\partial_{t}\eta_{\gamma}))_{X^0} \\
& \qquad + \int_{I_{k}} ( \mathbb{P}^{s}[{{\underline{h}}}_{\delta, k}](\overline{{\boldsymbol{u}}}_{\gamma},\partial_{t}\eta_{\gamma}),(\rho_{\gamma} \overline{{\boldsymbol{u}}}_{\gamma},\partial_t \eta_{\gamma}))_{X^0} \\
& \qquad - \int_{I_{k}} ( (\rho \overline{{\boldsymbol{u}}} ,\partial_t \eta),( \overline{{\boldsymbol{u}}},\partial_t \eta)- \mathbb{P}^{s}[{{\underline{h}}}_{\delta, k}](\overline{{\boldsymbol{u}}},\partial_{t}\eta),\partial_t \eta))_{X^0} \\
& \qquad - \int_{I_{k}} ( \mathbb{P}^{s}[{{\underline{h}}}_{\delta, k}](\overline{{\boldsymbol{u}}},\partial_{t}\eta),(\rho \overline{{\boldsymbol{u}}},\partial_t \eta))_{X^0} .\end{aligned}$$ Then, since $\mathbb P^s[{{\underline{h}}}_{\delta,k}](\overline{{\boldsymbol{u}}}_\gamma,\partial_t \eta_\gamma) \in X^s[{{\underline{h}}}_{\delta,k}]\subset X[{{\underline{h}}}_{\delta,k}] $ and thanks the identification , we write $$\begin{aligned}
( \mathbb{P}^{s}[{{\underline{h}}}_{\delta, k}](\overline{{\boldsymbol{u}}}_{\gamma},\partial_{t}\eta_{\gamma}),(\rho_{\gamma} \overline{{\boldsymbol{u}}}_{\gamma},\partial_t \eta_{\gamma}))_{X^0}
& = ( \mathbb{P}^{s}[{{\underline{h}}}_{\delta, k}](\overline{{\boldsymbol{u}}}_{\gamma},\partial_{t}\eta_{\gamma}),\mathbb P[{{\underline{h}}}_{\delta,k}](\rho_{\gamma} \overline{{\boldsymbol{u}}}_{\gamma},\partial_t \eta_{\gamma}))_{X^0} \\
& = \langle \mathbb P[{{\underline{h}}}_{\delta,k}](\rho_{\gamma} \overline{{\boldsymbol{u}}}_{\gamma},\partial_t \eta_{\gamma}) , \mathbb{P}^{s}[{{\underline{h}}}_{\delta, k}](\overline{{\boldsymbol{u}}}_{\gamma},\partial_{t}\eta_{\gamma}) \rangle_{(X^s)',X^s}.\end{aligned}$$ Proceeding similarly with the limit term, we obtain the following splitting $$Err_{\gamma} = \sum_{k=0}^N Err^{app}_{\gamma,k} + Err^{conv}_{\gamma,k} - Err^{app}_{k},$$ where, for arbitrary $k \leq N,$ we denote $$\begin{aligned}
Err^{app}_{\gamma,k} & = \int_{I_{k}} \Bigl( (\rho_{\gamma} \overline{{\boldsymbol{u}}}_{\gamma},\partial_t \eta_{\gamma}),( \overline{{\boldsymbol{u}}}_{\gamma},\partial_t \eta_{\gamma})- \mathbb{P}^{s}[{{\underline{h}}}_{\delta, k}](\overline{{\boldsymbol{u}}}_{\gamma},\partial_{t}\eta_{\gamma})\Bigr)_{X^0} \\
Err^{app}_{k} &= \int_{I_{k}} \Bigl( (\rho \overline{{\boldsymbol{u}}} ,\partial_t \eta),( \overline{{\boldsymbol{u}}},\partial_t \eta)- \mathbb{P}^{s}[{{\underline{h}}}_{\delta, k}](\overline{{\boldsymbol{u}}},\partial_{t}\eta),\partial_t \eta)\Bigr)_{X^0}\\
Err^{conv}_{\gamma,k} &= \int_{I_k} \langle \mathbb P[{{\underline{h}}}_{\delta,k}](\rho_{\gamma} \overline{{\boldsymbol{u}}}_{\gamma},\partial_t \eta_{\gamma}) , \mathbb{P}^{s}[{{\underline{h}}}_{\delta, k}](\overline{{\boldsymbol{u}}}_{\gamma},\partial_{t}\eta_{\gamma}) \rangle_{(X^s)',X^s}
- \langle \mathbb P[{{\underline{h}}}_{\delta,k}](\rho \overline{{\boldsymbol{u}}} ,\partial_t \eta) , \mathbb{P}^{s}[{{\underline{h}}}_{\delta, k}](\overline{{\boldsymbol{u}}},\partial_{t}\eta) \rangle_{(X^s)',X^s}.\end{aligned}$$
For the two first type of terms we use the fact that projection on $X^s[h_{\delta, k}]$ has good approximation properties. So, to estimate the error terms $Err^{app}_{\gamma,k}$, we use Lemma \[lem:projector\] for $\kappa=1/4$. Indeed, from the bound and Lemma , we know that $(\overline{{\boldsymbol{u}}}_{\gamma},\partial_t \eta_\gamma)$ satisfies, for a.e. $t \in I_k$, $(\overline{{\boldsymbol{u}}}_{\gamma}(t),\partial_t \eta_{\gamma}(t)) \in X^s[h_{\gamma}],$ for $s<1/2$, with $\overline{{\boldsymbol{u}}}_{\gamma} \in H_\sharp^1(\mathcal F^-_{h_{\gamma}}).$ Moreover we remark that both interfaces $h_{\gamma}$ and $h_\delta$ belong to $H^{1+\kappa}_{\sharp}(0,L) \cap W_\sharp^{1,\infty}(0,L)$ and that, thanks to the definition of $h_\delta$, there exists $A >0$ independent of $\gamma$ and $\delta$, such that $$\begin{aligned}
&{\left\Vert h_{\gamma}\right\Vert}_{L^{\infty}(0,T;H^{1+\kappa}_{\sharp}(0,L))}+{\left\Vert h_{\gamma}\right\Vert}_{L^{\infty}(0,T;W^{1,\infty}_{\sharp}(0,L))}+{\left\Vert {{\underline{h}}}_{\delta}\right\Vert}_{L^{\infty}(0,T;H^{1+\kappa}_{\sharp}(0,L))}+{\left\Vert {{\underline{h}}}_{\delta}\right\Vert}_{L^{\infty}(0,T;W^{1,\infty}_{\sharp}(0,L))}\leq A.\end{aligned}$$ Finally by construction $h_\delta$ and $h_\gamma$ are close in $W^{1, \infty}_\sharp(0, L)$ and is satisfied. Hence, Lemma \[lem:projector\] implies, for $s<1/8$ $$\begin{aligned}
{\left\Vert \mathbb{P}^{s}[{{\underline{h}}}_{\delta, k}](\overline{{\boldsymbol{u}}}_{\gamma}(\cdot,t),\partial_{t}\eta_{\gamma}(\cdot,t)) -(\overline{{\boldsymbol{u}}}_{\gamma}(\cdot,t),\partial_{t}\eta_{\gamma}(\cdot,t))\right\Vert}_{X^0}
&\leq{\left\Vert \mathbb{P}^{s}[{{\underline{h}}}_{\delta, k}](\overline{{\boldsymbol{u}}}_{\gamma}(\cdot,t),\partial_{t}\eta_{\gamma}(\cdot,t)) -(\overline{{\boldsymbol{u}}}_{\gamma}(\cdot,t),\partial_{t}\eta_{\gamma}(\cdot,t))\right\Vert}_{X^s}\\
&\leq C_{A}(\delta){\left\Vert \nabla \overline{{\boldsymbol{u}}}_{\gamma}(\cdot,t)\right\Vert}_{{\boldsymbol{L}}^{2}_{\sharp}(\mathcal{F}^{-}_{h_{\gamma}(t)})},\end{aligned}$$ with $\lim_{x\rightarrow 0} C_A(x)=0$. Using Cauchy–Schwartz inequality, we deduce from the previous estimate that $$\begin{aligned}
\sum_{k=0}^N |Err_{\gamma,k}^{app}| & \leq \sum_{k=0}^N \int_{I_k} {\left\Vert \mathbb{P}^{s}[{{\underline{h}}}_{\delta, k}](\overline{{\boldsymbol{u}}}_{\gamma}(\cdot,t),\partial_{t}\eta_{\gamma}(\cdot,t)) -(\overline{{\boldsymbol{u}}}_{\gamma}(\cdot,t),\partial_{t}\eta_{\gamma}(\cdot,t))\right\Vert}_{X^0}
\|(\overline{{\boldsymbol{u}}}_{\gamma},\partial_t \eta_{\gamma})\|_{X^0} \\
& \leq C_A({\delta})\|(\overline{{\boldsymbol{u}}}_{\gamma},\partial_t \eta_{\gamma})\|_{L^\infty(0, T;X^0)} \int_0^T \|\nabla \overline{{\boldsymbol{u}}}_{\gamma}\|_{L^2(\mathcal F_{h_{\gamma}(t)})} .\end{aligned}$$ Then we use the uniform estimates , to obtain that there exists a constant $C_1$ (depending only on initial data and $T$) such that, for $\gamma \leq \gamma_0$ $$\label{eq_Err1}
\sum_{k=0}^N |Err_{\gamma,k}^{app}| \leq C_1 C_A(\delta).$$ Similarly we have $$\label{eq_Err2}
\sum_{k=0}^N |Err_{k}^{app}| \leq C_1 C_A(\delta).$$ To complete the proof, the following term remains to be estimated $$\limsup_{\gamma \to 0} \sum_{k=0}^N |Err_{\gamma,k}^{conv}|.$$ At first, we prove that, for a fixed $k \leq N$ and up to a subsequence, $\mathbb{P}[{{\underline{h}}}_{\delta, k}](\rho_{\gamma}\overline{{\boldsymbol{u}}}_{\gamma},\partial_{t}\eta_{\gamma})$ converges strongly to $\mathbb{P}[{{\underline{h}}}_{\delta, k}](\rho\overline{{\boldsymbol{u}}},\partial_{t}\eta)$ in $L^{2}(I_{k};(X^{s}[{{\underline{h}}}_{\delta, k}])')$. Note that, since $(\rho_{\gamma}\overline{{\boldsymbol{u}}}_{\gamma},\partial_t \eta_{\gamma})$ converges weakly to $(\rho \overline{{\boldsymbol{u}}},\partial_t \eta)$ in $L^2(I_k;X^0)$ the only difficulty relies on showing that the sequence $\mathbb{P}[{{\underline{h}}}_{\delta, k}](\rho\overline{{\boldsymbol{u}}},\partial_{t}\eta)$ is relatively compact in $L^{2}(I_{k};(X^{s}[{{\underline{h}}}_{\delta, k}])').$ To do so we apply an adapted version of Aubin–Lions lemma that can be found in [@Fujita-Sauer Section 4.3] that reads
Let us consider three Hilbert spaces $M_i$, $i=1, 2,3 $ and two operators $T : M_0 \mapsto M_1$ and $S : M_0 \mapsto M_2$ satisfying
- $T$ and $S$ are two linear compact operators,
- $Su=0$ implies $Tu=0$.
If $(u_n)$ is bounded in $L^2(0, T; M_0)$ and $(\partial_t S u_n)$ is bounded in $L^2(0, T; M_2)$, then $T u_n$ is a compact set of $L^2(0, T; M_1)$.
We are going to use this version of Aubin–Lions lemma with the triplet $(X^0,(X^{s}[{{\underline{h}}}_{\delta, k}])',(X^{1}[{{\underline{h}}}_{\delta, k}])')$ and with $S= i_1\circ \mathbb{P}$, $T=i_s\circ \mathbb{P}$, where $i_l$ denotes the injection of $X[{{\underline{h}}}_{\delta, k}]$ into $(X^{l}[{{\underline{h}}}_{\delta, k}])'$. The mappings $i_l$ are indeed injective functions since Lemma \[lemma.density\] implies that the continuous embedding $X^l[{{\underline{h}}}_{\delta,k}] \subset X[{{\underline{h}}}_{\delta, k}]$ is dense, for $l>0$ and these densities imply that $X[{{\underline{h}}}_{\delta, k}] $ is continuously embedded in $(X^{l}[{{\underline{h}}}_{\delta, k}])'$, for $l>0$. Moreover, thanks to Rellich–Kondrachov theorem the embeddings $X^l[{{\underline{h}}}_{\delta,k}] \subset X[{{\underline{h}}}_{\delta, k}]$ are compact. The dual of a compact operator being compact, $X[{{\underline{h}}}_{\delta, k}] $ is compactly embedded in $(X^{l}[{{\underline{h}}}_{\delta, k}])'$ for $l>0$. Consequently $i_1\circ \mathbb{P}$ and $i_s\circ \mathbb{P}$ are compact linear operators. Moreover $i_1\circ \mathbb{P}({\boldsymbol{w}}, b)=0$ implies $\mathbb{P}({\boldsymbol{w}}, b)=0$ so that the second point is clearly satisfied.
Next applying , we have that $(\overline{{\boldsymbol{u}}}_{\gamma},\partial_{t}\eta_{\gamma})$ is uniformly bounded in $\gamma$ in $L^{2}(0,T;X^0)$. Thus the sequence $(\rho_{\gamma}\overline{{\boldsymbol{u}}}_{\gamma},\partial_{t}\eta_{\gamma})$ is bounded in $\gamma$ in $L^{2}(I_{k};X^0)$. We must now obtain a uniform bound for $\partial_{t}\mathbb{P}[{{\underline{h}}}_{\delta, k}](\rho_{\gamma}\overline{{\boldsymbol{u}}}_{\gamma},\partial_{t}\eta_{\gamma})$ in $L^p(I_k; (X^1[{{\underline{h}}}_{\delta,k}])').$ Precisely, we look for an estimate of the type $$\left| -\int_{I_{k}}\left( \mathbb{P}[{{\underline{h}}}_{\delta, k}](\rho_{\gamma}\overline{{\boldsymbol{u}}}_{\gamma},\partial_{t}\eta_{\gamma}), \partial_{t}({\boldsymbol{w}},b) \right)_{X^0}
\right| \leq C \int_{I_k} \|({\boldsymbol{w}}(t),d(t))\|_{X^1[{{\underline{h}}}_{\delta,k}]}^{2}
{\rm d}t,
\quad \forall \, ({\boldsymbol{w}},d) \in L^2(I_k; X^1[{{\underline{h}}}_{\delta,k}]).$$ To obtain such an estimate we use the variational formulation satisfied by $(\overline{{\boldsymbol{u}}}_{\gamma},\partial_{t}\eta_{\gamma})$. We consider $({\boldsymbol{w}},d) \in \mathcal C^{\infty}_c(I_k; X^1[{{\underline{h}}}_{\delta,k}])$. This is an admissible test function since ${{\underline{h}}}_{\delta}\leq h_\gamma$ and since, in the case where $\gamma>0$ for which $\min_{x\in [0, L]}h_\gamma(t, x)>0,\forall t\in [0, T]$, we can consider test functions in $C^{\infty}([0, T]; X^1[h_\gamma(t)])$. We obtain $$\begin{aligned}
-\int_{I_{k}}\left( \mathbb{P}[{{\underline{h}}}_{\delta, k}](\rho_{\gamma}\overline{{\boldsymbol{u}}}_{\gamma},\partial_{t}\eta_{\gamma}), \partial_{t}({\boldsymbol{w}},b) \right)_{X^0} &=-\rho_{f}\int_{I_{k}}\rho_{\gamma}\overline{{\boldsymbol{u}}}_{\gamma}\cdot\partial_{t}{\boldsymbol{w}}- \rho_{s}\int_{I_{k}}\partial_{t}\eta_{\gamma}\partial_{t}b\\
&=\rho_{f}\int_{I_{k}}\int_{\mathcal{F}_{h_\gamma(t)}}({{\boldsymbol{u}}}_{\gamma}\cdot\nabla){\boldsymbol{w}}\cdot{{\boldsymbol{u}}}_{\gamma}-\mu\int_{I_{k}}\int_{\mathcal{F}_{h_\gamma(t)}}\nabla{{\boldsymbol{u}}_\gamma}:\nabla {\boldsymbol{w}}\\
& \qquad -\beta\int_{I_{k}}\int_{0}^{L}\partial_{x}\eta_{\gamma}\partial_{x}b+ \alpha\int_{I_{k}}\int_{0}^{L}\partial_{xx}\eta_{\gamma}\partial_{xx}b \\
& \qquad +\gamma\int_{I_{k}}\int_{0}^{L}\partial_{tx}\eta_{\gamma}\partial_{x}b.
\end{aligned}$$ The nonlinear convection term is estimated using the $L^{4}$–regularity of $\overline{{\boldsymbol{u}}}_{\gamma}$ stated in Lemma \[lem:extension-u\], $$\begin{aligned}
\left\vert\rho_{f} \int_{I_{k}}\int_{{\Omega}}(\rho_{\gamma}\overline{{\boldsymbol{u}}}_{\gamma}\cdot\nabla){\boldsymbol{w}}\cdot\rho_{\gamma}\overline{{\boldsymbol{u}}}_{\gamma}\right\vert&{}\leq \rho_{f}\int_{I_{k}}{\left\Vert \overline{{\boldsymbol{u}}}_{\gamma}(t)\right\Vert}_{{\boldsymbol{L}}^{4}({\Omega})}^{2}{\left\Vert \nabla{\boldsymbol{w}}(t)\right\Vert}_{{\boldsymbol{L}}^{2}({\Omega})}{\mathrm{d}}t\\
&\leq \rho_{f}{\left\Vert \overline{{\boldsymbol{u}}}_{\gamma}\right\Vert}_{L^{4}({\widehat{\Omega}})}^{2}{\left\Vert \nabla{\boldsymbol{w}}\right\Vert}_{L^{2}(I_{k};{\boldsymbol{L}}^{2}({\Omega}))}.
\end{aligned}$$ The other terms are estimated directly and we obtain $$\left\vert \int_{I_{k}}\left( \mathbb{P}[{{\underline{h}}}_{\delta, k}](\rho_{\gamma}\overline{{\boldsymbol{u}}}_{\gamma},\partial_{t}\eta_{\gamma}), \partial_{t}({\boldsymbol{w}},b) \right)_{X^0}\right\vert \leq C{\left\Vert ({\boldsymbol{w}},b)\right\Vert}_{L^{2}(I_{k};X^{1}[{{\underline{h}}}_{\delta, k}])},$$ where $C$ depends only on the initial data. The previous inequality implies that $\partial_{t}\mathbb{P}[{{\underline{h}}}_{\delta, k}](\rho_{\gamma}\overline{{\boldsymbol{u}}}_{\gamma},\partial_{t}\eta_{\gamma})$ is bounded in $\gamma$ in $L^{2}(I_{k};(X^{1}[{{\underline{h}}}_{\delta, k}])')$. It then follows from the adapted version of Aubin–Lions lemma that $\mathbb{P}[{{\underline{h}}}_{\delta, k}](\rho_{\gamma}\overline{{\boldsymbol{u}}}_{\gamma},\partial_{t}\eta_{\gamma})$ is compact in $L^{2}(I_{k};(X^{s}[{{\underline{h}}}_{\delta, k}])')$. Moreover, since $(\overline{{\boldsymbol{u}}}_{\gamma},\partial_t \eta_{\gamma})$ converges weakly to $(\overline{{\boldsymbol{u}}},\partial_t \eta)$ in $L^2(I_k; X^s)$, for $s<1/2$ we also have that $\mathbb P^s[{{\underline{h}}}_{\delta,k}](\overline{{\boldsymbol{u}}}_{\gamma},\partial_t \eta_{\gamma})$ converges weakly to $\mathbb P^s[{{\underline{h}}}_{\delta,k}](\overline{{\boldsymbol{u}}},\partial_t \eta)$ in $L^2(I_k;X^s).$ Combining a strong and a weak convergence result leads to $$\label{eq_Err3}
\lim_{\gamma \to 0} Err_{\gamma,k}^{conv} = 0 \quad \forall \, k \leq N.$$ Finally, combining –, we conclud that $$\limsup_{\gamma \to 0} |Err_{\gamma} | \leq C_1C_A(\delta)$$ for arbitrary $\delta >0.$ We conclude the proof by remarking that $C_A(\delta) \to 0$ when $\delta \to 0.$ For completeness, we remark that in the computations of bounds for $Err_{\gamma}$ we only extract subsequences when we apply the Aubin–Lions lemma. Since we perform extraction a finite number of times for any value of the parameter $\delta$ that we can choose in a denumerable sets ([*i.e.*]{} a sequence converging to $0$), our proof induces indeed denumerable extractions of subsequences.
In the final weak formulation we consider fluid test functions that vanishes in the neighbourhood of the bottom of the fluid cavity and that are only transverse in the neighbourhood of the interface. Note that we could have also consider fluid test functions that vanishes in the neighbourhood any contact point. It imposes in particular the structure test function to be zero near each contact point so that they depend implicitly on the solution.
Proof of lemma \[lemma.density\] {#Annexe}
================================
This appendix is devoted to a density lemma in the space $X[h].$ We first recall the statement of the lemma to be proven and proceed to the proof.
\[lemma.density.app\] Let $h \in \mathcal C^0_{\sharp}(0,L)$ satisfy $0 \leq h(x) \leq M, \forall x\in [0, L]$ The embedding $X[h]\cap (\mathcal{C}^{\infty}(\overline{{\Omega}}) \times \mathcal C^{\infty}_{\sharp}(0,L)) \subset X[h]$ is dense.
First notice that the main difficulty here comes from the potential contact [*i.e.*]{} the points where $h$ is equal to zero. If there is no contact we may construct explicitly a smooth approximating sequence of any pair in $X[h]$ by adapting the arguments of [@Chambolle-etal], see also the construction of approximate initial data in Section \[sec:proof1\].
When $h$ vanishes, we propose an alternative proof: in this case we obtain that $(X[h]\cap \mathcal{C}^{\infty}(\overline{{\Omega}}) \times \mathcal C^{\infty}_{\sharp}(0,L))^{\perp}=\{({\boldsymbol{0}},0)\}$. So, let $({\boldsymbol{u}},\overset{\cdot}{\eta}) \in X[h]$ and assume it satisfies: $$\label{PS.1}\rho_{f}\int_{{\Omega}}{\boldsymbol{u}}\cdot{\boldsymbol{w}}+\rho_{s}\int_{0}^{L}\overset{\cdot}{\eta}d=0, \quad
\forall \, ({\boldsymbol{w}},d)\in X[h]\cap \mathcal{C}^{\infty}(\overline{{\Omega}}) \times \mathcal C^{\infty}_{\sharp}(0,L).$$ Using Lemma \[lemma.X\[h\]\] there exists $\Psi\in H^{1}_{\sharp}({\Omega})$ and $b\in H^{1}_{\sharp}(0,L)$ such that ${\boldsymbol{u}}=\nabla^{\bot}\Psi$ with $\Psi=b(x)$ in $\mathcal{S}_{h}$ and $\Psi=0$ in $\mathcal{C}^{0}_{-1} \cup I^{c}\times(-1,2M)$ where $\displaystyle I=\{x\in[0,L]\mid h(x)>0\}$. To complete the proof, we obtain that ${\boldsymbol{u}}$ vanishes in $I\times(-1,2M)$. Since $I$ is an open subset of $(0,L)$ we may construct an at most denumerable $\mathcal I$ such that $$I=\bigsqcup_{i\in\mathcal I}(a_{i},b_{i})$$ where the $(a_{i},b_{i})$ are the connected components of $I$. It is now sufficient to prove that, for arbitrary $i\in \mathcal I$ there holds ${\boldsymbol{u}}=0$ in ${\Omega}_{i}=(a_{i},b_{i})\times(-1,M)$. This is obtained by a suitable choice of functions $({\boldsymbol{w}},d)$ in .
Let fix $i \in \mathcal I$ and $\varepsilon>0$ small enough. Consider $\chi_{\varepsilon} \in \mathcal{C}^{\infty}_{c}((a_{i},b_{i}))$ such that $$\begin{aligned}
& \chi_{\varepsilon}=1 \text{ on $[a_{i}+\varepsilon,b_{i}-\varepsilon]$},
\quad
\text{supp}(\chi_{\varepsilon})\subset[a_{i}+\frac{\varepsilon}{2},b_{i}-\frac{\varepsilon}{2}], \\
& \|\chi_{\varepsilon}'\|_{L^{\infty}(\mathbb R)}<\frac{1}{\varepsilon}
\qquad \|\chi_{\varepsilon}''\|_{L^{\infty}(\mathbb R)}<\frac{1}{\varepsilon^{2}}.
\end{aligned}$$ Existence of such a truncation function is classical. We now introduce $${\boldsymbol{w}}_\varepsilon=\nabla^{\perp}(\chi_{\varepsilon}\Psi) \quad
d_{\varepsilon} = \partial_x (\chi_{\varepsilon}b).$$ It is straightforward that $({\boldsymbol{w}}_{\varepsilon},d_{\varepsilon}) \in {\boldsymbol{L}}^2((a_{i},b_{i}) \times (-1,M)) \times L^2_0((a_{i},b_{i}))$ and has support in $(a_i+\varepsilon,b_i-\varepsilon).$ On the other hand, there exists $\delta_{\varepsilon}>0$ such that $h(x)\geq \delta_{\varepsilon}$ on $(a_i+\varepsilon/2,b_i-\varepsilon/2)$. Setting $h_{\varepsilon} = \max(h,\delta_{\varepsilon})$ we have then that $h_{\varepsilon} \in \mathcal C^0_{\sharp}(0,L)$ does not vanish and $({\boldsymbol{w}}_{\varepsilon},d_{\varepsilon}) \in X[h_{\varepsilon}].$ Consequently, we may reproduce the arguments in the case of a non vanishing deformation to approximate $({\boldsymbol{w}}_{\varepsilon},d_{\varepsilon})$ by a sequence of pairs in $X[h_{\varepsilon}] \cap (\mathcal C^{\infty}(\overline{\Omega}) \times \mathcal C^{\infty}_{\sharp}(0,L)).$ Moreover, we emphasize that, by construction, this sequence has support in $(a_i+\varepsilon/2,b_i-\varepsilon/2)\times (-1,2M)$ also so that it is actually a sequence of $X[h] \cap (\mathcal C^{\infty}(\overline{\Omega}) \times \mathcal C^{\infty}_{\sharp}(0,L))$ that approximates $({\boldsymbol{w}}_{\varepsilon},d_{\varepsilon})$ in $X[h]$ also.
Consequently the identity holds true for $({\boldsymbol{w}}_{\varepsilon},d_{\varepsilon})$ also and we have $$\int_{{\Omega}_{i}}{\boldsymbol{u}}\cdot\nabla^{\perp}(\chi_{\varepsilon}\Psi) + \int_{a_{i}}^{b_{i}}\overset{\cdot}{\eta} \partial_x (\chi_{\varepsilon} b)=0.$$ But, recalling that $\nabla^{\bot} \Psi = {\boldsymbol{u}}$ and $\partial_x b = \dot{\eta}$ we may expand the differential operators to yield that: $$0 = \int_{{\Omega}_{i}}{\boldsymbol{u}}\cdot\nabla^{\perp}(\chi_{\varepsilon}\Psi) + \int_{a_{i}}^{b_{i}}\overset{\cdot}{\eta}z_\varepsilon=\int_{{\Omega}_{i}}\vert{\boldsymbol{u}}\vert^{2}\chi_\varepsilon + \int_{a_{i}}^{b_{i}}\vert\overset{\cdot}{\eta}\vert^{2}\chi_\varepsilon + \int_{{\Omega}_{i}}{\boldsymbol{u}}\cdot\nabla^{\perp}(\chi_{\varepsilon})\Psi + \int_{a_{i}}^{b_{i}}\overset{\cdot}{\eta}b{\chi}_{\varepsilon}'.$$ Since $\chi_{\varepsilon}$ depends on the $x$-variable only and $\chi'_{\varepsilon}$ vanishes on $\{a_i,a_i+ \varepsilon,b_i-\varepsilon,b_i\}$ , we have, by integrating by parts: $$\label{calcul-1}
-\int_{{\Omega}_{i}}{\boldsymbol{u}}\cdot\nabla^{\perp}(\chi_{\varepsilon})\Psi=\int_{a_i}^{a_i+\varepsilon} \int_{-1}^{2M} \frac{\Psi^{2}}{2}{\chi}_{\varepsilon}''
+ \int_{b_i-\varepsilon}^{b_i} \int_{-1}^{2M} \frac{\Psi^{2}}{2}{\chi}_{\varepsilon}''$$ Similarly we prove the following equality: $$\label{calcul-2}-\int_{a_{i}}^{b_{i}}\overset{\cdot}{\eta}\,b(x) {\chi}_{\varepsilon}'=
\int_{a_i}^{a_i+\varepsilon}\frac{b(x)^{2}}{2}{{\chi}_{\varepsilon}''}
+\int_{b_i-\varepsilon}^{b_i}\frac{b(x)^{2}}{2}{{\chi}_{\varepsilon}''}.$$ Since $\Psi=0$ on $\{a_{i}\}\times(-1,2M)$ a standard Poincaré inequalities entails that: $$\begin{aligned}
&\int_{a_{i}}^{a_i+\varepsilon} \int_{-1}^{2M} \frac{\Psi^{2}}{2}\leq \frac{\varepsilon^{2}}{4}\int_{a_{i}}^{a_i+\varepsilon} \int_{-1}^{2M}\vert\nabla\Psi\vert^{2}=\frac{\varepsilon^{2}}{4}\int_{a_{i}}^{a_i+\varepsilon}\vert{\boldsymbol{u}}\vert^{2},\\
&\int_{a_{i}}^{a_i+\varepsilon}\frac{b(x)^{2}}{2}\leq\frac{\varepsilon^{2}}{4}\int_{a_{i}}^{a_i+\varepsilon}\left|\partial_{x}b(x)\right|^{2}=\frac{\varepsilon^{2}}{4}\int_{a_{i}}^{a_i+\varepsilon}\vert\overset{\cdot}{\eta}\vert^{2},
\end{aligned}$$ We have a similar identity for integrals involving $(b_i-\varepsilon,b_i)$ by using that $\Psi = 0$ on $\{b_i\} \times \{-1,2M\}.$ Using finally that $L^{\infty}$-estimate on ${\chi}''_{\varepsilon}$ in - we conclude $$\label{calcul-3}\int_{a_i+\varepsilon}^{b_i-\varepsilon} \int_{-1}^{2M} \vert{\boldsymbol{u}}\vert^{2}+\int_{a_{i}+\varepsilon}^{b_{i}-\varepsilon}\vert\overset{\cdot}{\eta}\vert^{2}\leq \frac{1}{4}\left(\int_{a_i}^{a_i+\varepsilon} \int_{-1}^{2M} \vert{\boldsymbol{u}}\vert^{2}+\int_{a_i}^{a_i+\varepsilon}\vert\overset{\cdot}{\eta}\vert^{2}
+\int_{b_i-\varepsilon}^{b_i} \int_{-1}^{2M} \vert{\boldsymbol{u}}\vert^{2}+\int_{b_i-\varepsilon}^{b_i}\vert\overset{\cdot}{\eta}\vert^{2} \right).$$ Since $({\boldsymbol{u}},\dot{\eta})$ are both $L^2$-functions, the right-hand side of this identity vanishes when $\varepsilon \to 0.$ So, letting $\varepsilon\rightarrow 0$ we obtain $({\boldsymbol{u}},\overset{\cdot}{\eta})=({\boldsymbol{0}},0)$ in $\Omega_i$. This ends the proof.
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abstract: 'The circumgalactic medium (CGM) remains one of the least constrained components of galaxies and as such has significant potential for advancing galaxy formation theories. In this work, we vary the extragalactic ultraviolet background for a high-resolution cosmological simulation of a Milky Way-like galaxy and examine the effect on the absorption and emission properties of metals in the CGM. We find that a reduced quasar background brings the column density predictions into better agreement with recent data. Similarly, when the observationally derived physical properties of the gas are compared to the simulation, we find that the simulation gas is always at temperatures approximately 0.5 dex higher. Thus, similar column densities can be produced from fundamentally different gas. However, emission maps can provide complementary information to the line-of-sight column densities to better derive gas properties. From the simulations, we find that the brightest emission is less sensitive to the extragalactic background and that it closely follows the fundamental filamentary structure of the halo. This becomes increasingly true as the galaxy evolves from $z=1$ to $z=0$ and the majority of the gas transitions to a hotter, more diffuse phase. For the brightest ions (C[III]{}, C[IV]{}, O[VI]{}), detectable emission can extend as far as 120 kpc at $z=0$. Finally, resolution is a limiting factor for the conclusions we can draw from emission observations but with moderate resolution and reasonable detection limits, upcoming instrumentation should place constraints on the physical properties of the CGM.'
author:
- 'Lauren Corlies, David Schiminovich'
bibliography:
- 'library.bib'
title: 'Empirically Constrained Predictions for Metal-Line Emission from the Circumgalactic Medium'
---
Introduction
============
Perhaps the most basic process of galaxy formation, the flow of gas into and out of a galaxy, remains as one of the least understood. The key seems to lie in our lack of understanding of the circumgalactic medium (CGM). Roughly defined as the gas surrounding galaxies at 10 to 300 kpc, the CGM encompasses all gas in transition: gas falling onto the galaxy for the first time; gas that is being driven out by multiple feedback processes; gas that is being stripped from infalling satellite galaxies; and gas that is currently being recycled by the galaxy [see @Putman_review for review].
The structure of this gas halo depends on the mass and redshift of the galaxy in question. Currently, gas is thought to be accreted through two main modes - a “hot” mode where the gas is shock heated as it enters the halo and a “cold” mode where the gas remains in unshocked filamentary structures that can potentially penetrate all the way to the disk [@Keres_2005; @Fumagalli_2011]. Milky Way-like galaxies are thought to transition from the cold mode to the hot mode by the present day but the details of this transition are neither theoretically agreed upon nor well-constrained observationally [@Brooks_2009; @ryan; @Nelson_2015b].
In addition to these inflows, the outflow of gas from the galaxy is equally important in shaping the CGM [@Nelson_2015a; @Marasco_2015; @Suresh_2015]. Stellar feedback of some form is clearly needed to prevent the overcooling of gas and the formation of unrealistic stellar bulges in simulations [@Agertz_2011; @Brook_2011; @Hummels_2012]. It is also the most effective way of enriching the IGM to the non-pristine levels that are observed [@Oppenheimer_2008; @Wiersma_2010; @Barai_2013; @Ford_2013]. While such outflows are regularly seen, the exact physical process driving them and the extent of their influence is uncertain [@Turner_2015]. Multiple preferred forms of Type II supernovae feedback are implemented and recent work has begun to implement more detailed processes such as radiation pressure from supernovae [@Hopkins_2012; @Agertz_2013; @Ceverino_2014; @Trujillo_Gomez_2015], cosmic rays [@Booth_2013; @Salem_2014a; @Salem_2014b], AGN [@Sijacki_2007; @Booth_2009], and direct modeling of a kinetic energy component [@Simpson_2015] to name a few. In short, putting constraints on these many models is fundamental to furthering our understanding of galaxy formation.
In general, cosmological galaxy simulations are tuned to reproduce global and primarily *stellar* properties of galaxies such as the stellar mass function and the star formation rate density function [@Dave_2011; @schaye_eagle; @Nelson_2015a]. Another benchmark is the creation of thin, extended stellar disks [@Governato_2007]. The H[I]{} mass function is a constraining gas property but again looks at the total mass and not its distribution throughout the galaxy. [@Dave_2013] Recently, theoretical work has begun to compare the simulated CGM to column densities and equivalent width measurements as a function of impact parameter from the center of the galaxy [@hummels; @ford; @Liang_2015; @oppenheimer_2016]. The majority of the simulations have difficulty in matching the large amount and high covering fraction of O[VI]{} measurements, tracing the hottest gas phase (except recently for high-mass galaxies [@Suresh_2015b] and with cosmic ray feedback [@Salem_2015]). Their success varies when looking at cooler, less ionized lines (Mg[II]{}, C[III]{}, Si[IV]{} etc.) but in general, the data reveal large amounts of metal-enriched gas at large impact parameters that is hard to reproduce theoretically. In this way, measurements of the CGM can put strong restrictions on feedback models, independent of the global properties that are already used.
The most successful method of observing the CGM is in the absorption lines of quasar spectra. At higher redshifts, Lyman $\alpha$ and the ultraviolet metal lines of interest have shifted into the optical, making observations easier and successful [@Steidel_2010; @Simcoe_2004]. At low-redshift, several studies have begun pushing our knowledge of the more local CGM with measurements of Mg[II]{} [@Chen_2010] and O[VI]{} for a number of galaxies [@Prochaska_2011; @Thom_2008]. The recent installation of the Cosmic Origins Spectrograph (COS) on HST has enabled a new survey of the CGM of low-redshift ($z \approx 0.2$), massive, isolated galaxies. The COS Halos Survey has provided a large, uniformly measured sample of the H[I]{} column densities [@Tumlinson_HI], metal line absorption [@werk13], and O[VI]{} column densities [@tumlinson_OVI]. As accretion and outflows are expected to vary with redshift in addition to mass, low-redshift studies such as these are crucial as is the need to push to even lower redshifts.
A complementary approach is to observe the CGM $\emph{directly}$ in emission. Quasar spectra will always be limited by the small number of sight lines through each galaxy. An emission map has the potential to provide insight into the physical state of an entire galaxy halo. While promising, the low density of the gas has made this observation challenging. The most success has come from high-redshift surveys for Lyman $\alpha$ emitters [e.g. @Bridge_2013; @Gawiser_2007] and the more extended Lyman $\alpha$ blobs/halos [e.g. @Matsuda_2011; @Steidel_2011; @Steidel_2000] but metal-line emission has remained elusive [@Battaia_2015]. Recently, the development of new integral field units, MUSE and CWI (and its successor KCWI), now allows for a study of the kinematics of the gas. Early work has already suggested that the absorbers can be linked to global outflows [@Swinbank_2015] as well as filamentary inflows [@Martin_2014]. At low redshift, the upcoming FIREBall-2 is building upon its predecessor [@Milliard_2010] and pushing the boundaries of low surface brightness UV observations. This, in addition to any small or large near-future UV space telescope mean that direct UV observations of the CGM are closer than ever.
With these advancements in mind, this work looks to take advantage of new data while preparing for future observations. We take a high-resolution, cosmological, hydrodynamical simulation of a Milky Way-like galaxy and compare it to recent column density data. We then ask what emission we could presume to detect with upcoming facilities.
Previous studies of this same simulation provide a solid foundation for this work. @ximena demonstrated that in-falling satellites provide much of the cold, high metallicity gas found in the halo at $z=0$ whereas @ryan quantified how much gas of a given temperature is accreted at low-$z$. This existing physical insight allows us to better understand the evolution of the CGM and the contribution of different accretion modes.
In this paper, we look to build on this work when interpreting our emission predictions. In Section 2, basics of the simulation used and the photoionization model are summarized. In Section 3, the simulation is compared to column density observations to put empirical constraints on the interpretation of the simulation. In Section 4, the emission signatures of this gas and how they evolve are examined and its observational properties are explored. Finally, the broader context of the work is discussed in Section 5 and the results are summarized in Section 6.
Methodology
===========
Simulation Basics
-----------------
We analyze the cosmological, hydrodynamical simulation of @ryan performed with [enzo]{}, an Eulerian, adaptive mesh refinement, hydrodynamical code [@enzo]. A Milky Way-like halo was identified from within an initial low-resolution run with a periodic box of $L = 25\ h^{-1}$ Mpc comoving on a side with cosmological parameters consistent with WMAP5. This galaxy was centered in a box of length $\approx 5\ h^{-1}$ Mpc which was then resimulated with 10 levels of refinement. The selected galaxy has a final halo mass of $1.4 \times 10^{12} M_{\astrosun}$ and contains over 8.2 million dark matter particles within its virial radius, with $m_{\mathrm{DM}} = 1.7 \times 10^5 M_{\astrosun}$. The final stellar mass is $1.9 \times 10^{11} M_{\astrosun}$, placing the halo above the M$_{\mathrm(star)}$-M$_{\mathrm(halo)}$ relation as is common with simulations of this type [@guo_2010]. The maximum spatial resolution stays at 136-272 pc comoving or better at all times.
The simulation includes metallicity-dependent cooling, a metagalactic UV background, shielding of UV radiation by neutral hydrogen, and a diffuse form of photoelectric heating. The code simultaneously solves a complex chemical network involving multiple species (e.g. H[I]{}, H[II]{}, H$_2$, He[I]{}, He[II]{}, He[III]{}, $e^-$) and metal densities explicitly.
Star formation and stellar feedback are included in the simulation. Star particles have a minimum initial mass of $m_* = 1.0 \times 10^5 M_{\astrosun}$ and are created if $\rho > \rho_{\mathrm{SF}}$ and with a violation of the Truelove criterion. Supernovae feedback is modeled following @Cen_2005, with the fraction of the stellar rest-mass energy returned to the gas as thermal energy, $e_{\mathrm{SN}} = 10^{-5}$, consistent with the @Chabrier_2003 initial mass function. The metal yield from stars, assumed to be 0.025, represents metal production from supernovae of both Type Ia and Type II. This metallicity is traced as a single field and abundances are generated throughout the paper assuming the solar abundance. Feedback energy and ejected metals are distributed into 27 local cells centered at the star particle in question, weighted by the specific volume of the cell. The metals and thermal energy are released gradually, following the form: $f(t,t_i,t_*) = (1/t_*)[(t - t_i)/t_*] \exp[-(t - t_i)/t_*]$, where $t_i$ is the formation time of a given star particle, and $t_* = \mathrm{max}(t_{\mathrm{dyn}}, \ 3 \times 10^6 \mathrm{yr})$ where $t_{\mathrm{dyn}} = \sqrt{3\pi / (32 G \rho_{\mathrm{tot}})}$ is the dynamical time of the gas from which the star particle formed. The metal enrichment inside galaxies and in the IGM is followed self-consistently in a spatially resolved fashion. For details of these prescriptions, we direct the reader to @ryan.
Ionization Modeling
-------------------
To calculate the relevant ionization processes of interest, the simulation was post-processed with the photoionization code <span style="font-variant:small-caps;">cloudy</span> [version 10.0, last described in @cloudy] in conjunction with the cooling map generation code <span style="font-variant:small-caps;">roco</span> [@smith_2008] and the simulation analysis suite [yt]{} [@yt]. For each model discussed in the upcoming sections, the following procedure was used to produce the column density and emission predictions.
First, <span style="font-variant:small-caps;">cloudy</span> look-up tables of ion fractions and emissivity were constructed for a given ionization background as a function of temperature ($10^3 < T < 10^8, \ \Delta \log_{10} T = 0.1$) and hydrogen number density ($10^{-6} < n_{\mathrm{H}} < 10^2,\ \Delta \log_{10} n_{\mathrm{H}} = 0.5$). Each table assumes solar metallicity and abundances. The grid is then interpolated for every cell to the correct temperature and $n_{\mathrm{H}}$. Then, $n_{Xi}$, the number density of given ionization state of element $X$ (C[III]{}, Si[IV]{}, O[VI]{}, etc.) is calculated as: $$n_{Xi} = n_{\mathrm{H}}(n_X/n_{\mathrm{H}})(n_{Xi}/n_X)$$ where $(n_X/n_{\mathrm{H}})$ is the elemental abundance relative to hydrogen and $(n_{Xi}/n_X)$ is the ion fraction computed by <span style="font-variant:small-caps;">cloudy</span>. Here, the elemental abundance is given as the solar abundance scaled by the metallicity reported in the simulation. The emissivity is more straightforward as <span style="font-variant:small-caps;">cloudy</span> directly reports the emissivity at a given temperature and density that is then again scaled by the metallicity.
With these number densities and emissivities, producing the corresponding column density and surface brightness values is done as projections through the simulation with [yt]{}. Throughout the paper, we assume a box that is 320 kpc across and 500 kpc deep, ensuring the selection of gas associated with the galaxy. Each projection and radial profile is made with a resolution of 1 physical kpc, unless otherwise stated.
Finally, throughout the paper, the assumed ionization field, the extragalactic ultraviolet background (EUVB), is varied to examine the agreement of the simulation predictions with the column density measurements. To this end, we take the 2005 updated version of the @HM01 background of <span style="font-variant:small-caps;">cloudy</span> and split it into its two components - quasars and galaxies. Then, the intensity of each component can be varied and the changes in the predicted column densities studied. The quasar component dominates at short wavelengths and is responsible for the majority of the ionizing radiation in the calculations. In this way, varying the quasar component has more significant consequences than varying the galactic component. A more detailed discussion of the differences among these backgrounds is found in the Appendix.
Because this is a post-processing of the simulation, this technique is not fully self-consistent. It does not capture the underlying effects on the temperature and density that arise from changing the ionization background used in computing the cooling of the gas. However, the overall galaxy evolution and supernova feedback are thought to dominate the evolution of the gas density, temperature and metallicity more than the choice of EUVB and the ion fractions of interest here are less important in determining these large-scale properties. These limitations remain as part of the uncertainty in the following calculations but the overall conclusions should be robust.
However, the response of the simulation to the changes in the EUVB reflects the field’s true influence on the ionization state of the simulated gas, assuming ionization equilibrium. The field is a fundamental property of the physics of the calculation used in calculating the ion fractions and emissivity in <span style="font-variant:small-caps;">cloudy</span>. The density and temperature of the gas are not expected to vary much with the choice of EUVB, as discussed in the Appendix.
Absorption
==========
In this section, we look to place the simulation in the context of a set of current absorption-line observations. First, column density maps of a series of ions are generated and the resulting CGM structure is analyzed. Next, the reliability of the simulation is tested by examining its agreement with available observations, specifically the COS Halos survey [@werk13].\
![Column density maps of H[I]{}, Si[IV]{}, C[III]{}, and O[VI]{} respectively at $z=0.2$ with a resolution of 1 kpc. Note H[I]{} has a unique color bar. O[VI]{} has the largest covering factor with moderately high column densities extending uniformly over 100 kpc. C[III]{} has a smaller covering factor but reaches higher densities in visible filaments and stripped satellite material. Si[IV]{} is the weakest as its peak ionization temperature is slightly below the typical temperature of the halo gas. \[abs\_proj.fig\]](Figure1compressed.pdf){width="30.00000%"}
Column Density Maps
-------------------
In order to better visualize the CGM column density distribution of the simulated galaxy, we first examine Figure \[abs\_proj.fig\], which shows column density maps for four ions at $z=0.2$ with a resolution of 1 kpc. This set of ions allows us to probe from the coldest gas (H[I]{}) to the hottest gas (O[VI]{}) and the warm gas in between (Si[IV]{}, C[III]{}). What is first apparent is the intricate structures visible for all of these ions. The H[I]{} naturally has the largest column densities in the filaments that trace the high density structures within the gas.
The distribution of Si[IV]{} and C[III]{} closely follows that of the H[I]{}. The greater strength of C[III]{} is related to the fact that it is approximately 10 times more abundant than Si[IV]{} for a given metallicity. These low ionization ions are found mostly in the higher density gas because the high average temperature of the CGM outside these regions prefers higher ionization states. These trends are also true of other low ions, such as Si[III]{}, which show similar features. Conversely, in this map, although the O[VI]{} does retain traces of the same underlying structures seen as slightly enhanced column density regions, its higher ionization energy allows it to exist in hotter gas. In this way, the O[VI]{} has both the largest extent and obtains an appreciable column density value for almost the entire area of the map. This is consistent with @tumlinson_OVI who found O[VI]{} in all of their star forming galaxies, implying a high covering fraction as seen here.
Comparison to COS Halos Column Densities
----------------------------------------
With column density maps in hand, we now compare the simulation to the uniform, galaxy-selected, quasar sample of COS Halos data, which provides measurements of the column densities of multiple ions as a function of impact parameter to low sensitivities. The ions presented here (SiIII, SiIV, CIII, OVI) span a wide range in ionization energy while having a large number of observations in the COS Halos sample. CIV is excluded as observations are limited by the degraded sensitivity of COS for wavelengths $\lambda > 1500$ Å, necessary for this redshift sample [@werk13]. As we are considering a single galaxy, the simulation is not expected to reproduce every aspect of the larger population sampled by the survey. Because of this fact, coupled with the large number of upper and lower limits in the data, the comparison made here between simulated and observed column densities is visual. Every pixel in the column density map is shown so that the validity of the conclusions drawn here are easily confirmed.
{width="70.00000%"}
As a base case, we assume the standard HM05 background, labeled as g1q1 in figures. Using the method described in Section 2, the column densities are computed for each ion as a function of impact parameter from the galactic center at $z=0.2$, the approximate redshift of the data. The center panes of Figure \[coldens\_scatter.fig\] show the resulting average radial profiles of three projection angles as well as the value of each pixel for a single projection (shown in Figure \[abs\_proj.fig\]). Each pixel has a width of 1 kpc. The data points are detections and upper and lower limits respectively of the COS Halos data set. The color of the data point indicates whether the galaxy is considered star forming (with sSFR $> 10^{-11}$ yr$^{-1}$) or passive as in @werk13. At $z=0.2$, the simulated galaxy has a stellar mass of $1.9 \times 10^{11} M_{\astrosun}$ and star formation rate (SFR) of $6.22 \ M_{\astrosun} / yr$, making it a star-forming galaxy by this classification as expected. This stellar mass is typical of a COS Halos galaxy but this SFR leads to a rate at $z=0$ that is high compared to the actual Milky Way [noted by @ximena; @ryan]. However, in the COS Halos sample, 5 galaxies have this SFR or higher. These points show no special trends in the column densities [@werk13] so the comparison done here is valid.
These plots highlight both the average trends of the halo gas as well as the structures seen in the column density maps of Figure \[abs\_proj.fig\]. In general, the median column densities remain roughly constant with impact parameter. However, filamentary structures and satellite galaxies (the peak seen around 40 kpc) provide the possibility of a quasar sightline measuring higher than average column densities. Furthermore, the distribution of the pixel values does vary somewhat with the projection angle. In particular, a face-on projection reduces the scatter in the inner radii as the disk dominates the gas distribution. However, the average values are generally unaffected. Throughout the paper, we plot a mostly edge-on projection which allows for a better evaluation of how gas extends perpendicularly from the disk while lessening the influence of the disk itself.
It is apparent that for this simulation, this model is not a good fit to most of the data. The higher density filamentary structures bring many of the C[III]{} measurements into alignment with the data but the small covering fraction of these filaments makes it unlikely that they constitute a large fraction of the COS Halo absorber population. The Si[IV]{} data is composed of many upper limits which means the low predicted values may be more in-line with the simulation. However, the detections are still mostly too high to match the simulation at the larger impact parameters.
The O[VI]{} measurements, on the other hand, are comprised mostly of detections. Our inability to match the O[VI]{} prediction for the star-forming galaxy model highlights a true disagreement. The observed galaxies with similarly high star formation rates as our simulated galaxy have properties similar to the other star-forming galaxies while the simulation is more in agreement with the passive population. Taken together, this suggests that the details of the hot phase of the CGM are not being properly reproduced.
Nevertheless, it is encouraging that the simulated gas shows a roughly flat radial profile like the data, as this was not guaranteed a priori. Figure \[radial\_profiles.fig\] shows the radial profiles of the density, temperature, and metallicity at z=0.2 in orange. The density decreases much faster than temperature beyond the disk (excepting peaks which represent satellite galaxies) while the metallicity actually begins to rise beyond 50 kpc. The combination of warm/hot temperatures, falling density, and increasing metallicity combine to produce the roughly flat column densities seen here in Figure \[coldens\_scatter.fig\]. Yet, the points are not as tightly clustered as the data and are roughly two orders of magnitude too low.
It is tempting to simply increase the metallicity of the gas to increase the column densities but it is not clear that this would lessen the discrepancies with the observations. The metallicity and temperature of the gas are intimately linked. Increasing the metallicity of the gas may lower the temperature such that the simulation remains in disagreement with some of the data. For example, Si[III]{} and Si[IV]{} which prefer colder temperatures, would most likely benefit from added metallicity but O[VI]{} which prefers hotter temperatures may not. Furthermore, keeping the density and temperature fixed, the metallicity would need to be raised by approximately two orders of magnitude to bring better agreement with the data, assuming the fiducial HM05 EUVB. This would put much of the CGM at solar metallicity or above, in contrast to most expectations and the measurements of @werk14.
Instead, these discrepancies are an indication that the simulation which is tuned to reproduce bulk stellar properties of galaxies over time fails to do the same for these multi-phase CGM gas properties. However, this is not the first simulation to have such issues. @hummels also analyze an [enzo]{} simulation with a similar thermal feedback prescription but at lower resolution and report the same difficulties. Likewise, the SPH simulation of @ford also fails to reproduce the O[VI]{} densities even though they implement a non-thermal wind prescription for their feedback. One success is that of @Salem_2015, whose implementation of cosmic ray feedback successfully matches the data for all ions. A discussion of these different methods is found in Section 5.
Thus, the typical solution that is invoked and explored in these works and many others is a modification to the stellar feedback prescription, changing the density, temperature and metallicity of the simulated CGM [e.g. OWLS, EAGLE, FIRE described in @schaye_owlsim; @schaye_eagle; @hopkins_fire respectively]. This range of parameterizations can have an uncertain impact on the gas quantities such that new feedback solutions require the simulation to be re-run to capture the changes. We explore the role of feedback further in the Discussion Section.
However, the EUVB is also important in setting the ionization state of the gas and is not well constrained. Variations of the @HM96 background (e.g. HM96, HM01, HM05, HM12) are implemented in most simulations and <span style="font-variant:small-caps;">cloudy</span>. While the best models to date, there is still significant uncertainty in the exact strength and shape of the EUVB. To this end, we chose to explore the impact of this uncertainty by varying the intensity of the galaxy and quasar components of the HM05 background and examining the effects on the resulting column densities. In the following analysis, two bracketing cases of the quasar intensity are presented to examine the reasonable range of effects on the predicted column densities. At one end, the quasar intensity is 100 times less intense than standard (g1q01) and at the other, the quasar intensity is ten times more intense (g1q10). These properties are summarized in Table \[gq.tab\] for easy reference. We performed the same analysis for a range of quasar intensities spanning these two cases and the trends seen across the three values presented here are consistent with these results. Furthermore, these two cases bracket current estimates of the photoionization rate with high redshift Lyman $\alpha$ forest studies preferring higher backgrounds [@kollmeier_underproduction; @shull_2015] and with low-redshift H$\alpha$ upper limits preferring lower backgrounds [@adams_2011]. Further discussion of the EUVB and its uncertainty can be found in the Appendix. In addition, the galaxy intensity was varied in a similar way but with little to no effect on the predicted column densities as it provides less of the ionizing flux.
Name Galaxy Quasar
------- -------- --------
g1q01 1.0 0.01
g1q1 1.0 1.0
g1q10 1.0 10.0
: EUVB Model Summary \[gq.tab\]
The first and last columns of Figure \[coldens\_scatter.fig\] once again show the radial profiles of the column densities of our simulated galaxy but with these altered EUVBs. With g1q01 in the first column, it appears that lowering the quasar intensity to 0.01 times its normal value provides a much better fit to the low-ion data, Si[III]{}, Si[IV]{}, and C[III]{}. The majority of the pixels are now in better agreement with the data which is consistent with the idea that this softer spectrum is no longer over-ionizing the gas. Raising the quasar intensity as seen in the last column with the g1q10 models results in a much larger disagreement between the simulation and the data, which is thus consistent with the picture of over-ionization. Together, this suggests that photoionization is the dominant process in producing these low ions. O[VI]{}, on the other hand, is mostly unaffected, suggesting that the gas is predominantly collisionally ionized. Most of the halo volume is at about the same density and temperature, accounting for the small spread in O[VI]{} column density values. The scatter that is introduced is actually towards lower column densities with larger quasar intensity, consistent with the recombination rate of the lower density gas not being able to counterbalance the increased photoionization. This demonstrates that producing the correct amount of O[VI]{} is not a simple matter of increasing the photoionization of the CGM.
Observationally, there is support for the approach of varying the EUVB. @crighton allowed the power law slope of the @HM12 (HM12) background between 1-10 Rydbergs to vary and found that half the components in their absorption spectra preferred an altered slope. One component agrees with the findings here, preferring a softer background, but the others are better fit by a slightly harder spectrum. If variations of the EUVB are necessary to explain absorption components within the same sightline, it is reasonable to expect that the EUVB would vary amongst the many galaxies that comprise the COS Halos sample. Examining how this variation changes simulated predictions given otherwise identical physical conditions can thus provide insight into how to interpret such measurements.
On the other hand, the preference of this simulation for a weaker EUVB background is in fact in contrast both with these column density component measurements as well as the known limitation of the HM12 background failing to reproduce the column density distribution of Lyman $\alpha$ forest absorbers, known as the photon underproduction crisis [@kollmeier_underproduction; @shull_2015]. Solving this crisis calls for an increase in the photoionization rate of the HM12 background. However, the HM05 model used here is more consistent with the findings of @kollmeier_underproduction while it is less consistent with the log(N$_{\mathrm{H{\scriptsize I}}}$) $> 14.0$ distribution plotted in @shull_2015. This uncertainty in the low-redshift EUVB supports our decision to vary its intensity though in light of the on-going efforts in feedback and sub-grid physics, we acknowledge that this is likely not enough to bring full agreement between the simulation and observations. If a different feedback method allowed for the gas in the simulation to be cooler at late times, the EUVB would not have to be so low to produce the necessary amount of low ions such as Si[IV]{} and C[III]{}.
Overall, we find that column densities predicted by the simulation using the standard HM05 background do not provide the best match to recent observations. Instead, the simulation prefers a softer, reduced quasar intensity to produce the necessary large amount of low ionization gas. This demonstrates that simulation predictions are sensitive to the assumed EUVB and that this assumption should be considered in conjunction with efforts to vary feedback methods but that this preference of a reduced EUVB, in tension with certain observations, cannot solve the issue entirely.
Comparing to Derived Gas Properties
-----------------------------------
Just as <span style="font-variant:small-caps;">cloudy</span> can be used to predict column densities from simulated physical gas properties, measured column densities can be used to place constraints on the physical properties of the gas that is producing the absorption. Here, we compare the observationally derived gas density and temperature of @werk14 to those of the simulation.
![Hydrogen number density ($n_{\mathrm{H}}$) and temperature, weighted by the given ion number density along the line of sight within the g1q1 model. Colors correspond to the average column density of lines of sight contributing to each bin. Plotted square points are the values implied by the modeling of @werk14. The simulation and observations span the same range of densities while the simulation temperatures are universally higher. This also shows the O[VI]{} clearly in a different phase medium while data points are not included as the ion is explicitly not fit by @werk14. \[hden\_temp.fig\]](Figure3compressed.pdf){width="30.00000%"}
We choose to compare the modeled temperature and density from the data directly to the simulation as opposed to computing mock spectra from the simulation and comparing to the data directly. Both require [<span style="font-variant:small-caps;">c</span>loudy]{} modeling and assumptions about the EUVB, abundance patterns and ionization equilibrium and in this way, we do not need to re-analyze the spectra of the COS Halos team. Furthermore, the simulation densities and temperatures are weighted by the column density of the ion of interest, reflecting the preferential detection of higher column density features used in the modeling.
For each of the column density radial profiles shown in Figure \[coldens\_scatter.fig\], the hydrogen number density, $n_{\mathrm{H}}$, and temperature used in computing the column densities were projected along the same axis, weighted by the ion number density of interest within the g1q1 model. Figure \[hden\_temp.fig\] shows the resulting $n_{\mathrm{H}}$ and temperature for the four ions being discussed (Si[III]{}, Si[IV]{}, C[III]{}, O[VI]{}). The 2D histogram is colored to show the average column density of the lines of sight contributing to each bin.
The plotted points are derived from the modeling of the column density observations by @werk14. The reported values of the ionization parameter, log(U), are directly converted to $n_{\mathrm{H}}$, assuming the ionizing flux of HM05. For the temperature, <span style="font-variant:small-caps;">cloudy</span> models were generated using the adopted value of N$_{\mathrm{H}}$ and each permutation of the maximum and minimum values of log(U) and metallicity. For sight lines where there was an upper limit for the metallicity, a lower boundary of $[$Fe/H$] = -6$ was assumed. In this way, each absorber has four data points associated with it, the combinations of the maximum and minimum values for log(U) and metallicity, representing the range of acceptable values from the data. <span style="font-variant:small-caps;">cloudy</span> then reports the best equilibrium temperature of such a gas cloud, plotted here. This was done for the fiducial model, g1q1, which best matches the EUVB background used to derive the parameters in @werk14.
This plot more than any other shows that O[VI]{} is in a different phase from the other low-ionization species. It is found exclusively in the hottest gas and the fact that the column densities are uniform across the entire density range reflects the conditions necessary for its longer-lived existence instead of cooling immediately. Data points are not shown for the modeled COS-Halos data as O[VI]{} is explicitly left out of the <span style="font-variant:small-caps;">cloudy</span> modeling of @werk14, who are focused on cooler gas ($T < 10^5$ K).
More surprising is the discrepancy between the properties of the low ions in the simulation and the observations. For C[III]{}, Si[III]{}, and Si[IV]{}, the majority of the simulated points have log($n_{\mathrm{H}}) < -4$ but span a large range in temperature. Conversely, the observational points have a clear relation where the temperature decreases with increasing density. This trend is perhaps reproduced in C[III]{} in the simulation but at higher temperatures.
These plots show that gas with a measurable column density is found at low temperatures and high densities. Yet almost all of the gas in the simulation is at a higher temperature than those implied by @werk14. None of the gas within a radius of 100 kpc reaches this low of a temperature. Part of this may be due to observational selection; the majority of the simulated gas is at low density, below the detection threshold (see the second column of Figure \[hden\_temp\_evol.fig\]). Alternatively, there may be physical differences between the simulated gas and the modeled observations that can explain the discrepancy in the column densities. However, we have shown that this may also be alleviated by altering the assumed EUVB.
In short, although the simulated galaxy has column densities that can be brought into rough agreement with the data, the physical conditions of the gas producing such values is inconsistent with those derived from the data. However, the differences between these two approaches should be noted. <span style="font-variant:small-caps;">cloudy</span>, by design, constructs a cloud with uniform density and temperature in local ionization equilibrium. With only the EUVB fixed, it tends towards lower temperatures and produces a relationship between density and temperature seen as a track in the data plotted in Figure \[hden\_temp.fig\]. The simulation, on the other hand, is run with the intent of retaining the large-scale and complex structure of the CGM gas, with ionization fractions computed for individual cells using a fixed temperature and density. This work has shown that it is possible for this cosmological CGM to produce column densities in the range of those predicted by the idealized clouds intrinsic to <span style="font-variant:small-caps;">cloudy</span>, even with largely varying gas properties, if only the assumed EUVB is altered.
Emission
========
While the previous section demonstrates the power of absorption line studies, it also highlights some of their limitations. Absorption line measurements are extremely sensitive probes of low column density gas, but it is challenging to understand which physical structures in the halo we are probing with these data. Also, the limited number of sightlines per galaxy hinders any attempt at understanding the spatial extent or scale of the structures detected. If instead it was possible to image the entire galaxy in emission, such maps may begin to reveal coherent structures or asymmetries in a gaseous halo that could help place stronger limits on gas accretion and outflows from the galaxy.
Although the simulation fails to accurately predict the column density distribution of CGM gas, we feel that they have sufficient fidelity to obtain new estimates of CGM emission and to determine whether the emission signal is within reach of new observational capabilities. In particular, our analysis of CGM emission from a high resolution simulation of a galaxy at low redshift is distinct from most other recent studies that probe the overall, diffuse emission-line cooling from CGM halos at lower resolution. This work provides a benchmark for simulations of this type for future comparisons with different theoretical prescriptions and observations. In this section, we present a complementary set of emission maps and show how they vary with photoionizing background and redshift. We highlight the ways in which these maps provide a distinct and unique view of the morphology and evolution of the CGM. We also explore how emission varies with $n_{\mathrm{H}}$ and temperature in the context of the absorption discussed in the previous section. We also determine the detectability of such gas, thereby informing target selection and instrument design for upcoming missions.
Emission Maps
-------------
![Emission maps of Si[IV]{}, C[III]{}, C[IV]{} and O[VI]{} respectively at $z=0.2$ with a resolution of 1 kpc. As in Figure \[abs\_proj.fig\], O[VI]{} has the largest covering factor while the other ions more closely follow the underlying, cold gas structures. However, the surface brightness spans a wider range of values, demonstrating how sensitive the emission is to the density and temperature of the gas. \[emis\_proj.fig\]](Figure4compressed.pdf){width="30.00000%"}
Figure \[emis\_proj.fig\] shows the emission maps of four of the brightest lines at $z=0.2$, the same redshift as Figure \[abs\_proj.fig\] and with the same resolution. Throughout this section, we consider C[IV]{} in place of Si[III]{} as it emits more brightly and is an intermediate ion between C[III]{} and O[VI]{}, except within roughly the innermost 25 kpc (a slightly smaller radius than that seen by @vandeVoort_2013). In general, Si[III]{} follows the same trends as Si[IV]{}, which is shown. Furthermore, we do not present Lyman $\alpha$ emission (although it likely produces the strongest signal) because resonant scattering is known to change the extent and shape of the emission [e.g. @lake2015; @Dijkstra_2012; @Zheng_2011] and the required radiative transfer calculation for Lyman $\alpha$ and other lines is beyond the scope of this paper and deferred for future work. We assume that the impact of scattering for other emission lines is negligible.
Comparing Figures \[abs\_proj.fig\] and \[emis\_proj.fig\], it is visually apparent that emission and absorption trace the same high density structures. However, the emission surface brightness spans many more orders of magnitude making the relevant range much bigger than that of the column density measurements. A similarly large range was reported in earlier work by @Bertone_2012. Because much of the emission is expected to come from collisional excitation of the gas, the $n^2$ dependence of this process causes the large spread in values seen here and makes the higher density structures the brightest features and easiest to detect. Most of the gas far from the disk is at low emission levels that are undetectable, as discussed below. This suggests that even further from the galaxy, detections of the even lower density intergalactic medium will remain out of reach.
Furthermore, as with the column densities, the low ions (Si[IV]{}, C[III]{}) trace the filaments where the density is higher and, just as importantly, the temperature is lower. (The same is true for the not-shown Si[III]{} line, which is very similar to Si[IV]{}). C[III]{} is the strongest emitting line, consistent with previous work that has considered the line [@vandeVoort_2013; @Bertone_2013; @Bertone_2012]. As the peak of the emissivity curve for a given ion moves to higher temperatures, the emission becomes more prevalent for the majority of the diffuse halo which roughly has a temperature of $10^{5.5}$ K. For example, this is the peak temperature of O[VI]{} emissivity and as such, this line supplies strong emission throughout the entirety of the halo. Beyond $10^6$ K, however, the O[VI]{} emission will again become less volume filling as most of the CGM is not hotter than this. Such biasing of Si[IV]{} to higher density regions and O[VI]{} to less dense regions was also reported by @vandeVoort_2013 and @Bertone_2012 and appears to be a fundamental prediction of any simulation containing a warm/hot CGM halo.
{width="70.00000%"}
Photoionizing Background
------------------------
The results above assume the single, fiducial HM05 ionizing background. It was shown in the previous section that varying the intensity of this background can bring the simulation more in-line with the absorption observations but what effect does this have on the predicted emission? Figure \[emis\_scatter.fig\] shows the same radial profiles as Figure \[coldens\_scatter.fig\] but for the surface brightness values of each projected pixel. A comparison of these two plots shows that the median values of the emission is affected by the change in EUVB in mostly the same way as the column densities. To further quantify this notion, Table \[abs\_emis\_100kpc.tab\] shows the median value of the radial profiles of both the column density and surface brightness predictions at 100 kpc and $z=0.2$ for the three EUVBs considered in both sections. For Si[IV]{} and C[III]{}, the median values of the column density increase by almost two orders of magnitude in the g1q01 model relative to the fiducial model and the same is true for the emission. The O[VI]{} is more unaffected both in absorption and emission. When considering g1q10 relative to g1q1, both the column density and surface brightness medians decrease by the same orders of magnitude for Si[IV]{} and C[III]{}, and the O[VI]{} by 0.6 orders of magnitude. The ionizing background now over-ionizes the low-density gas as before, leading to larger changes in both absorption and emission of all ions, including O[VI]{}.
However, these median levels of emission are below the detection limits of upcoming surveys for all the models and thus the EUVB does not have a strong effect on what will realistically be possible to detect. In Figure \[emis\_scatter.fig\], the points are colored by rough detection probability cuts: green points are certain to be detected, blue points are likely to be detected and pink points are possible to detect. This coloring shows that the pixels with the highest surface brightnesses extend about equally far for all of the EUVBs. Instead, their distribution and number depends mostly on the density and temperature of the gas. Because a 2D image would capture all of these pixels, the number and brightness of these emission peaks can perhaps provide a more unambiguous look at these underlying gas properties. Such surface brightness effects are discussed in more detail in Section 4.4.1.
More importantly however, the pixels with the highest surface brightness have the highest possibility of detection and extend about equally far for all of the EUVBs. Their distribution and number depends mostly on the density and temperature of the gas. Because a 2D image would capture all of these pixels, the number and brightness of these emission peaks can perhaps provide a more unambiguous look at these underlying gas properties.
[ c c c c ]{}\
& g1q01 & g1q1 & g1q10\
Si[IV]{} & 11.46 & 9.45 & 7.00\
C[III]{} & 13.18 & 11.22 & 8.76\
O[VI]{} & 13.75 & 13.55 & 12.66\
\
\
\
\
& g1q01 & g1q1 & g1q10\
Si[IV]{} & -1.41 & -2.84 & -5.24\
C[III]{} & -0.47 & -1.96 & -4.23\
O[VI]{} & 0.50 & 0.33 & -0.46\
\
\
This variation of the emission radial profile with EUVB is opposed to previous studies which found no variation in their simulated emission profiles when the assumed background was increased by a factor of ten although neither explore lowering the EUVB [@Sravan_2015; @vandeVoort_2013]. Also, both average over a large number of halos over range of masses, which may smooth some of the changes seen here, particularly as we are presenting the median. However, creating the same radial profiles as Figure \[emis\_scatter.fig\] but for $z=1$, we also found little to no variation in the profiles. Yet the fiducial background is higher at $z=1$ than at $z=0.2$ and encompasses the regions covered by the g1q1 and g1q10 models at $z=0.2$. This suggests that the gas state is more dominated by collisional ionization at $z=1$, which is supported by the higher density of the gas at early times (see Section 4.3 and Figure \[hden\_temp\_evol.fig\] for more details).
We point out that the extremely low SB values predicted here are purely theoretical predictions from the gas itself. They do not include the EUVB photons as well as possible photon pumping and scattering from the host galaxy continuum. The UV continuum of a star forming galaxy typical of the COS Halos sample will be on the order of 500 photon s$^{-1}$ cm$^{-2}$ sr$^{-1}$, although this is dependent on the uncertain escape fraction of the ionizing photons [see Figure 13 of @werk14 for a plotted, typical galaxy SED]. With only a fraction of this flux contributing to the pumping or scattering of a specific line of interest, the floor set by these processes will be above the theoretical limit shown here but below any upcoming detectable limit. Furthermore, this will mostly affect the volume closer to the star-forming disk and not alter our predictions for the more distant CGM.
In addition to the projected simulation pixels, “observational” data has been generated from the physical parameters inferred from the <span style="font-variant:small-caps;">cloudy</span> modeling of @werk14. For each line of sight, the adopted N$_{\mathrm{H}}$ was paired with each combination of the maximum and minimum values of the metallicity and ionization parameter to compute the emission from such a cloud using <span style="font-variant:small-caps;">cloudy</span>. Then the maximum and minimum computed values of each sight line are plotted as connected points. The large acceptable range for a high faction of the sightlines is due to the degeneracy of the ionization parameter (which for this model is a proxy for $n_{\mathrm{H}}$) and the metallicity. As we previously noted, the measured column densities are not in agreement with the simulated values. However the emission predicted from the column density data using the method described above, bracket nearly all of the simulated emission values of Si[III]{}, Si[IV]{}, and C[III]{}. Data points are not shown for O[VI]{} since @werk14 explicitly model gas cooler than the gas seen in the simulation ($T < 10^5$ K).
The complex ionization structure in the CGM halo means that regions of strong absorption do not always produce significant associated emission, particularly for O[VI]{} which is known to have a low column. In part, this is because absorption-line measurements are typically probing gas in the ground state, allowing ionic absorption of incoming quasar photons. Emission, on the other hand, is generated as higher ions cool through the metal line of interest or through collisional excitation and cooling of lower ion gas; these processes tend to be more transitory in nature. We suggest here that the C[III]{} emission predictions may be the most reliable as the column density distribution is best reproduced by the simulation both for the fiducial EUVB and the modified, weaker EUVB. The column density of O[VI]{} is known to be underproduced and the temperature of the hot halo generating the O[VI]{} is contested amongst the simulations (discussed further in Section 5). However, we expect the overall trends seen here with O[VI]{} being found in the hotter, volume-filling gas to remain valid. C[IV]{} is likely intermediary but the lack of data in the COS Halos sample limits our conclusions.
Overall, the density structures revealed in the column density maps are also present in the emission but the emission values span a much larger dynamic range. This dynamic range reflects the emission’s biasing towards higher density and thus higher signal regions. However, this biasing and it’s unique dependence on density, temperature and metallicity can provide complementary constraints on these properties of the gas when combined with column density measurements. Another advantage of emission observations is that varying the ionizing background does not have a strong effect on the brightest emitting pixels as it does on the median column density that could be detected. Because the emission trends and detection possibilities do not vary with the EUVB, in the following sections, we present results using the fiducial background.
Redshift Evolution of the Emission
----------------------------------
Our simulations can also be used to predict the physical distribution of CGM emission over a a large range of redshift and imaging the CGM can be used to better understand what is driving the emission at each redshift. In particular, the nature of the CGM may change dramatically from $z=1$ to $z=0$ as the SFR declines on average and as galaxies in this mass range potentially transition from “cold mode” to “hot mode” accretion.
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![Covering fraction for the 4 lines of interest at two different surface brightness cutoffs: 10 and $10^2$ photons s$^{-1}$ cm$^{-2}$ sr$^{-1}$ respectively. This excludes the galactic disk. In general, the emission increases with redshift for all lines. At the lower level, O[VI]{} has the highest covering fraction at all redshifts but the highest one. At the higher surface brightness level, C[III]{} is the dominant ion except at $z=1$ where C[IV]{} increases rapidly. This shows the overall increasing emission with increasing redshift, the high covering fraction of low-SB O[VI]{}, and that the strongest emission is coming from ions with mid-ionization energies. (Here again the $(1+z)^4$ dimming is not accounted for - i.e. as if $z=0$ but this has no effect on the trends.) \[cover\_frac.fig\]](Figure7compressed.pdf){width="47.50000%"}
Figure \[emis\_theory\_zevol.fig\] shows the evolution of four emission lines for our simulated galaxy using the fiducial background. The proper physical size of the box is constant in each panel as well as the resolution of 1 physical kpc. Furthermore, in order to study the intrinsic evolution of the emission separate from the cosmic expansion, these plots show the true surface brightness of the object without accounting for the cosmological $(1+z)^4$ dimming - i.e. we set $z=0$. In this way, the brightness of the emission is directly related to the underlying density, temperature and metallicity and their evolution alone. We discuss the importance and effects of this dimming in the following section.
It is easy to see that as the redshift increases, the emission becomes brighter and extends further and more spherically from the disk. This is most striking for the low ions which emit at an appreciable level almost out to the virial radius at $z=1$ while they are limited to high-density features at $z=0$. However, the extended emission sphere surrounding the disk at $z=1$ in all four ions considered disappears by $z=0$, leaving only the filamentary structure behind. This increase in the bias of the emission towards high density regions at later times as compared to emission at $z=3$ was noted by @vandeVoort_2013, and we point out that this is true even when comparing $z=1$ to the present day.
These changes in the emission values are more easily seen when quantified as a covering fraction. Figure \[cover\_frac.fig\] shows the evolution of the fraction of pixels with an intrinsic emission above two different surface brightness limits \[10 and 100 photons s$^{-1}$ cm$^{-2}$ sr$^{-1}$\] for the four ions of interest within a square image that is 320 proper kpc per side with a resolution of 1 proper kpc. Disk pixels have been removed to emphasize the CGM. As redshift increases, the increasing brightness seen in Figure \[emis\_theory\_zevol.fig\] leads to higher covering fractions for the emission. For both limits, the fraction doubles between $z=0$ and $z=1$ for all the ions. O[VI]{} has the largest covering fraction at the lowest level considered except at $z=1$. C[III]{} and C[IV]{} have similar covering fractions for all the limits and are the dominant lines at the higher SB cutoff, seen in the right pane of Figure \[cover\_frac.fig\]. Most interestingly, by $z=1$, C[IV]{} has the largest covering fraction, overtaking C[III]{} and O[VI]{}. Even at the lowest surface brightness limit that we consider, the covering fraction is never higher than 0.25 for any ion as early as $z=1$, highlighting the difficulty of emission detections.
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To better understand these changes in the surface brightness maps, Figure \[hden\_temp\_evol.fig\] shows the density temperature diagram of the galaxy weighted by the emissivity of multiple ions for four different redshifts ($z=0,0.2,0.5,1.0$). For each of the ions, there is a clear trend towards lower densities and higher temperatures on average with decreasing redshift. The decrease in density causes a lower overall surface brightness across the majority of the halo. Simultaneously, the higher temperatures move the bulk of the gas away from the peak of the emissivity curve of the lower ions. In this way, the emission from most of the volume is reduced, leaving only the higher density and lower temperature filaments with an appreciable signal. Once again, O[VI]{} defies these trends as the higher temperatures at late times are more in line with its emissivity peak, accounting for its continued higher surface brightness throughout the area shown in Figure \[emis\_theory\_zevol.fig\].
![Projections of the density (left) and the density-weighted temperature (middle) and metallicity (right). The combined evolution of these quantities is what drives the changes predicted for the emission. Filaments are easily seen feeding the galaxy at $z=1$ in the density and as low metallicity regions and have weakened by $z=0$. The temperatures become higher and more uniform by $z=0$. \[dens\_temp\_projs.fig\]](Figure9compressed.pdf){width="50.00000%"}
The question then becomes: what is causing these systematic changes in the density and temperature of the CGM? While this is difficult to answer definitively, there are two dominant effects within the simulation: accretion and supernova feedback. The first effect arises from filaments feeding the galaxy, as seen in the density projections in the left column of Figure \[dens\_temp\_projs.fig\], showing the density evolution of the galaxy and its CGM. At $z=1$, there are three well-defined filaments penetrating the galactic halo down to the disk. As the redshift decreases, the galaxy mass increases, and cosmic expansion lowers the overall average density of the IGM, these features become broader and do not penetrate into the halo as deeply although they do supply additional gas along with stripped satellite material [@ryan; @ximena]. Instead, the gas density profile becomes more spherical and more extended as the galaxy evolves.
What’s surprising is that this change in CGM morphology is not reflected in the structure of the brighter emission. The fractured filaments exist as surface brightness peaks in the $z=1$ projection but the bright emission halo is more spherical whereas the brighter emission at $z=0$ is almost entirely contained in the remains of the filaments with no discernible symmetry.
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The explanation resides in the corresponding temperature projections, shown in the middle column of Figure \[dens\_temp\_projs.fig\]. At $z=1$, the halo has a complex temperature structure, with colder, denser gas contained in cold, in-falling gas and satellites. By $z=0$, this is replaced with a spherical hot halo, mimicking the density profile. Even the filaments are bringing in predominantly warm/hot gas [@ryan]. The weakening of the filaments, the slowing of their supply of cold gas, and their replacement with a more uniformly hot halo of lower density gas results in the loss of the large emission halos of $z=1$. By $z=0$, only the remaining cold, dense features are capable of producing a significant signal. Figure \[radial\_profiles.fig\] shows the radial profiles of the relevant quantities at various redshifts, which quantitatively demonstrates these trends.
In addition to this change in accretion mode, supernova-driven winds are also effective at creating low-density, high-temperature pressurized bubbles in their wake as they expand through the halo. At high redshift, it’s been suggested that galactic outflows are the dominant process powering the time-varying simulated emission [@Sravan_2015]. Looking again at the temperature projections of Figure \[dens\_temp\_projs.fig\], higher temperature regions extend perpendicular to the disk. Furthermore, multiple spherical plumes can be seen expanding away from the disk. These features are less visible in the density but still seen. This suggests that outside the filament regions, SN-driven outflows play a large role in shaping the density and temperature distribution of the halo. In particular, because the SN energy is injected as thermal feedback in this simulation, the temperature of the gas is efficiently raised to $10^{5.5}$K on average, much higher than the peak emissivity of low ion lines, such as C[III]{} and C[IV]{}.
In conclusion, at later times, the emission in the metal lines studied here is more structured, tracing the remaining high density structures, in contrast to the majority of the gas which becomes more spherically distributed and more uniformly hot. This emission may be an effective way to probe continued galactic accretion at low redshift. Additionally, the propensity for the gas to become hotter and more diffuse translates to a decrease in the magnitude and extent of the low-ion emission. Galactic winds coupled with a transition to hot mode accretion no longer resupplying cold gas likely explains the shift to low-density, high-temperature gas at late times.
Implications for Detection
--------------------------
Finally, with an emission signal this faint, understanding how these theoretical predictions relate to what can actually be detected is important for both furthering interpretations of future measurements as well as enabling fair comparisons of theory and observations. In this section, we examine how realistic surface brightness and angular resolution limitations can limit the conclusions that can be drawn about a galaxy’s CGM.
### Surface Brightness Limits
Recent work in CCD technology now allow us to reach extremely high quantum efficiencies in the UV [@Hamden_2012] leading to achieving unprecedented low surface brightness limits in the UV. In this section, we look to see how this translates into reaching levels where UV emission from the CGM of nearby galaxies can finally be detected.
We consider three regimes of detection possibility for the emission. Pixels with a surface brightness (SB) greater than $10^3$ photons s$^{-1}$ cm$^{-2}$ sr$^{-1}$ are certain to be detected by the specifications of any upcoming instrument and are colored green in the following plots. Those with $10^2 <$ SB $< 10^3$ photons s$^{-1}$ cm$^{-2}$ sr$^{-1}$ have a high probability of being detected and are plotted in blue. Finally, pixels with $10 <$ SB $< 10^2$ photons s$^{-1}$ cm$^{-2}$ sr$^{-1}$ have a possibility of being detected and are shown in pink. Exact confidence levels will vary for a given instrument and observing strategy but these are appropriate rules of thumb.
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Figure \[emis\_varyZ.fig\] again shows the surface brightness maps of Figure \[emis\_theory\_zevol.fig\] but now colored to show these detection probabilities. Unlike the theoretical projections, these maps take into consideration the $(1+z)^4$ dimming of the surface brightness due to the expansion of the universe. The brightest, easiest to detect emission (green) always comes from the galactic disk. Some pixels reach this brightness level into the filaments, especially in C[III]{}, but only at the lower redshifts.
Much more promising for CGM studies is the extent of the blue pixels, indicating regions that are likely to be detected. For the brightest ions (C[III]{}, C[IV]{},), these regions extend into a large portion of the filaments at $z=0$, out to as far as 100 kpc. The mid-level emission is less extended for the high ion, O[VI]{}, reaching a radius of only 50 kpc [similar to that found in previous work by @vandeVoort_2013; @Furlanetto_2004]. Thus, although Figure \[emis\_varyZ.fig\] shows that the extent of the emission decreases with redshift, the detectable emission is still an appreciable distance from the main galactic disk. At all redshifts it should be possible to detect emission from CGM gas beyond the galactic (star-forming) disk.
Finally, the lowest surface brightness limits naturally reveal the most extended structure and emphasize the importance of pushing the limits of future instrumentation. However, in considering a range of redshift, the combination of the surface-brightness dimming and the fixed surface brightness limits shapes the observable covering fraction. The radial profiles of Figure \[emis\_scatter.fig\] suggest how this is possible, where the colors of the simulated points correspond to the same limits as the projections of Figure \[emis\_varyZ.fig\]. At $z=0.2$, most of the points lie below any reasonable detection limit and the fraction at the lowest detection limit (pink) is greater than at the higher limits (blue, green) at all but the smallest radii. This trend is true at all redshifts. For O[VI]{} in particular, much of the emission is intrinsically emitted at the lowest limit considered here because it is mostly generated by the low-density, volume-filling gas. At $z=1$, most of this dim emission then falls below observational levels once the cosmological dimming is considered. In this way, the extent of possible O[VI]{} detections drops from 130 kpc at $z=0$ to 60 kpc at $z=1$.
For the low ions, the effect is less pronounced because they emit most brightly and significantly in the relatively over-dense filaments and this emission in fact increases with increasing redshift as seen in Figure \[emis\_theory\_zevol.fig\] [demonstrated also in @Bertone_2013]. The surface brightness dimming is thus offset by the inherent increased emissivity, seen as an increased theoretical covering fraction in Figure \[cover\_frac.fig\]. Thus, the decrease in detection extent is less steep yet still pronounced for C[III]{} (150 kpc to 100 kpc) and C[IV]{} (150 kpc to 90 kpc). Furthermore, most of this decrease is in place by $z=0.5$ and little change in the observable properties of the gas happens between $z=0.5$ and $z=1$. Work at higher redshifts indicates that these extents are decreasing slightly in physical scales but that they remain at the same fraction of the halo’s virial radius [@Sravan_2015; @vandeVoort_2013].
Thus, in order to make a clear detection of the CGM, it is necessary to reach a detection limit of at least 100 photons s$^{-1}$ cm$^{-2}$ sr$^{-1}$ to begin to probe extended emission in dense filamentary regions. Pushing down to 10 photons s$^{-1}$ cm$^{-2}$ sr$^{-1}$ provides the possibility of detecting emission from the diffuse, hot volume-filling phase of the CGM. Furthermore, observations close to $z=0$ increase the chances of detecting this phase as it is the first to drop out of range due to surface brightness dimming. However, between $z=0.5$ and $z=1$, the extent of the emission is relatively unchanged at these detection limits, enabling measurements across redshift that could trace the evolution of the CGM.
### Resolution Limitations
In addition to the surface brightness limits, the resolution of the image affects the types of conclusions that can be drawn from the observations. For all of the previous plots, the resolution of the projected grid has been set to 1 kpc physical such that the angular resolution varies with redshift. At this resolution, it is possible to see the filaments and streams feeding the galaxy. How does this change if the physical resolution is varied?
![C[III]{} emission at $z=0.2$ for four different resolutions - the fiducial 1 kpc, 5 kpc, 13kpc, and 25 kpc proper corresponding to angular resolutions of 0.3”, 1.5”, 4" and 7.6” respectively. The medium resolutions reproduce many of the features of the highest resolution and would allow for a more confident detection of filamentary CGM emission features. At the lowest resolution, it is possible to detect an elongation of the emission and the distances would make it identifiable as CGM material but filaments are less conclusive. \[res\_vary.fig\]](Figure12compressed.pdf){width="47.50000%"}
Figure \[res\_vary.fig\] shows the C[III]{} emission for the galaxy at $z=0.2$, with four different resolutions: the fiducial 1 kpc, 5 kpc, and 25 kpc. This corresponds to angular resolutions of 0.3”, 1.5”, 4" and 7.6” respectively. At the moderate resolution of 13 kpc, it is still possible to discern the filamentary features extending from the disk. At the lowest resolution, however, the CGM emission resembles an extended halo around the galaxy. Because of its physical extent, one can still associate this emission with the CGM but valuable information about the spatial distribution of the gas has been lost. The low resolution also makes it difficult to track the evolution of the CGM. At most, an elongation aligned with the disk could be seen developing, corresponding to the filaments feeding the disk but internal clumpy structures and features perpendicular to the disk are masked.
One optimistic consequence of these predictions derives from the fact that at higher redshift, the physical angular diameter of an object is almost unchanging due to cosmological expansion. Given the relatively constant physical extent of the emission, the resolution of any observation will not change much for galaxies between $z=0.5$ and $z=1$. In this range, the 25 kpc physical resolution corresponds to roughly 3”-4” and the 5 kpc resolution to 0.6”-0.8”. Thus, by ensuring the resolution is at least 4”, it should be possible to confirm emission from the CGM for $0.5<z<1$ and begin to resolve its structure at lower redshifts. At $z=0.2$, basic filamentary structure and stripped material should be distinguishable with this angular resolution. (See the 13 kpc panel of Figure \[res\_vary.fig\]. Within the next year, the balloon-borne FIREBall-2 will launch with this resolution and is expected to make a positive detection. A small, UV emission line explorer is currently being designed to have a comparable resolution but also to cover a much larger range in wavelength, providing complementary coverage to FIREBall-2. Finally, further in the future, a 12-meter class, UV/optical space telescope named the High-Definition Space Telescope (HDST) has been designed with the specification of 0.01” angular resolution between 100-500 nm. [^1] As shown in Figure \[res\_vary.fig\], this unprecedented resolution would allow for an evaluation of the predominance of filamentary accretion for the first time as well as structures created by galactic outflows.
Discussion
==========
It is only recently that comparing the CGM of simulations to data has become possible and that emission-line predictions from the simulations are relevant for upcoming observations. We are entering a new realm of detailed CGM studies for which it is necessary to understand the limitations of current simulations and to evaluate which conclusions we expect to remain robust.
In this paper, we have examined a single simulation of a Milky Way-like galaxy using one form of purely thermal supernova feedback in an [enzo]{} AMR simulation. Like @hummels who uses a similar prescription, we find that the simulation has difficulty in producing the necessary column densities for all the ions but especially O[VI]{}. @hummels found that implementing stronger feedback brought better agreement while we’ve found that lowering the assumed EUVB can be equally effective. On the contrary, the most common way to implement supernova feedback in SPH simulations is to give either a constant or physically-scaled velocity kick to a series of wind particles which carry away the SN energy in kinetic form. One such simulation by @ford found better agreement with the low-ions although they still fail to reproduce the O[VI]{} observations. This implies that the direct, thermal methods prevalent in AMR causes the gas to reach higher temperatures as opposed to the wind velocity approach of SPH simulations, which shock heat differently. Furthermore, the simple thermal feedback assumed in this simulation also leads to the well known over-cooling problem. The H[I]{} distribution of the galaxy is known to be too centrally concentrated at z=0 [@ximena] and too many stars are formed [@ryan].
However, new methods are emerging that incorporate additional components of the SN feedback. For example, @Liang_2015 included prescriptions for supernova pressure and momentum in addition to a thermal heating model in a series of RAMSES AMR simulations. Their fiducial model was also not a good fit to low-$z$ data and again, increasing the feedback and lowering the star formation efficiency led to greater agreement. Building on the wind velocity method, <span style="font-variant:small-caps;">arepo</span> and Illustris in particular include an implementation of AGN feedback with quasar and radio modes. @Suresh_2015b found that including the radio mode in particular is responsible for enriching the CGM in the simulations and reproduces the bimodality of star-forming and passive galaxies seen in O[VI]{} data. However, they still see a stronger mass dependence on the O[VI]{} distribution than what is observed and the radio mode feedback in Illustris is known to be too extreme, removing too much gas from the center of massive galaxies [@Suresh_2015b; @genel_2014]. @oppenheimer_2016 found that AGN feedback in their EAGLE SPH simulations does not have a large effect on the OVI column densities and that the lower values for passive galaxies is instead driven by their higher virial temperature. Additionally, the inclusion of non-equilibrium chemistry in their simulations also does not resolve the almost universal issue of producing too little O[VI]{} for star-forming galaxies and passive galaxies alike. Finally, @Salem_2015 found that implementing a two-fluid cosmic ray method resulted in a cosmic ray-driven wind that gradually accelerated the gas, allowing for a larger range of gas temperatures as well as a higher metallicity beyond 100 kpc. Both contribute to higher column densities for all of the ions considered in this paper. In particular, the simulation reproduced the O[VI]{} measurements of the COS Halos survey for star forming galaxies.
In short, reproducing both the stellar properties and the CGM properties of a given galaxy at low redshift is a major theoretical challenge and an important test of modern simulation methods. The majority have difficulties in capturing the multiphase medium required to produce such high levels of low and high ions in the data. Likely, a combination of these advanced feedback prescriptions will be necessary to remedy this. Thus, low-$z$ CGM absorption measurements are a powerful new way to constrain such prescriptions and further motivates the emission observations we are predicting in this work. Emission predictions can provide complementary constraints on these feedback processes. Understanding the role and effect of purely thermal supernova feedback in work such as this will allow us to estimate its importance in future work with more complex schemes.
In addition to the uncertainty in the feedback scheme, the resolution of the simulation potentially limits the conclusions that can be drawn from comparisons with the column density data. @werk14 found low number densities for the cool clouds causing the absorption in the COS Halos data, corresponding to cloud sizes of 0.1-2000 pc, the larger of which can be resolved by current zoom-in simulations. However, @crighton detected the presence of 8 smaller ($<$100-500 pc), higher density clouds in an QSO spectrum with a z=2.5 foreground galaxy. If these scales are the norm, simulations may not resolve these clouds which exist beyond the high density disk and which potentially contain a large fraction of the cool CGM gas. These resolution concerns and the possible existence of dense clouds relate back to the feedback mechanisms required to accurately reproduce the multiphase medium. Dense clumps will have cool, self-shielding cores that could explain the observations of low ions while the shells around them and the diffuse volume-filling gas could be responsible for the mid to high ions. Instead, lower resolution simulations may produce a mid-range, average temperature that does not precisely reflect the state of the gas. Further studies of the cooling of the gas in idealized simulations where the resolution can be much higher could tell us more about how cool gas forms and persists within the volume-filling hot phase of the galactic halo. In addition, if dense small clouds do exist, they should appear as bright points in emission studies as opposed to lower density clouds since the emission scales as the square of the density, offering a chance to probe the number and density of the emitting clouds if the angular resolution is high enough.
Finally, this is ultimately a simulation of a single Milky Way-like galaxy. The conclusions drawn here about the filamentary structure of the gas especially at low redshift might be specific to this particular galaxy. More than just changing the viewing angle is necessary for understanding the dependence of our predictions on the physical properties of the gas. Our exact expectations could change if the galaxy forms fewer stars; if the filaments are broader than expected here; if the environment of the galaxy increases the metallicity of the gas - to name a few examples. The cosmological study of emission sources by @Frank_2012 suggest that an appreciable number of sources will be detectable at the redshifts considered in this paper such that we can begin to measure how the emission varies with these physical parameters. Thus, it is only by conducting both future observational surveys and a larger range of cosmological simulations that we can begin to address the variance in emission signatures of the CGM.
Even with these limitations of the simulation, we expect the trends seen in our conclusions to be robust. Varying the EUVB can produce variations in the simulated column densities at low redshift and in future work this can be further explored in addition to alternative forms of feedback. Furthermore, there is seemingly a tension between the density and temperature of gas within simulations of this type and those modeled from the COS Halos measurements. The coexistence of large amounts of Si[IV]{}, C[III]{} and O[VI]{} and our failure to reproduce all three simultaneously indicates that there is a *more* complex temperature structure than what’s seen here. Because of this, details of the extent and shape of the emission may vary in future work but the dominance of the C[III]{}, C[IV]{}, and O[VI]{} lines has remained thus far and should persist. Similarly, low ions tracing higher density structures while high ions are more volume filling is a clear prediction of any warm/hot gaseous halo. In addition, altering feedback methods to capture a larger range of temperatures could lead to further structure in the emission at low redshift. Less clear is how numerous and how thin these features might become. In this way, the resolution of future observations could be the limiting factor in imaging the CGM structures, possibly even more so than surface brightness limits. However, the general conclusions of this paper regarding the required surface brightness limits and resolution limits should continue to reflect simulations of this type.
Summary and Conclusions
=======================
Observing the predicted gas halos of nearby galaxies has long been a goal of observations but the need to study this gas in the space ultraviolet coupled with the diffuse nature of the gas in question has made this challenging. Now, studies of the CGM at low redshift are entering an unprecedented age of sensitivity. Measurements can begin to constrain theoretical prescriptions in simulations as well as discriminate between them. In this work, we vary the EUVB in a high-resolution cosmological simulation of a Milky Way-like galaxy and examine its role in determining how the simulated column densities compare to recent data. We then predict the emission signal expected from such gas for upcoming instrumentation as well as how it varies with redshift and the physical properties of the gas itself.
Our main conclusions can be listed as follows:
1. Looking at column density maps at $z=0.2$, the largest values for the column densities of all the ions studied here are found in high density filamentary structures. The low-ions (H[I]{}, C[III]{}, Si[IV]{}) are found almost exclusively in these structures while O[VI]{} is found throughout the halo as its higher ionization energy allows it to exist in the volume-filling hot gas.
2. Varying the quasar component of the standard EUVB can significantly change the predicted column densities of the simulation. In particular, lowering this component by a factor of 100 brings the simulation values into much better agreement with the low-ion data of the COS Halos sample. The simulated O[VI]{} column densities remain too low at all impact parameters compared to the observed values for star-forming galaxies, even with the strongest EUVB.
3. Comparing the gas temperature and density in the simulation to that found through <span style="font-variant:small-caps;">cloudy</span> modeling of the COS Halos data shows that the simulation predicts higher temperatures than the data modeling. This demonstrates that it is possible to produce similar column densities from different gas distributions.
4. Examining the redshift evolution of the emission reveals that the emission becomes more structured at later times, tracing the remaining high density, low temperature features. This is in contrast to the majority of the gas which shifts to lower densities and higher temperatures from $z=1$ to $z=0$ due to the weakening of cold gas filaments and the progression of supernova-driven winds.
5. A surface brightness limit of 100 photons s$^{-1}$ cm$^{-2}$ sr$^{-1}$ should enable a clear detection of emission from the CGM with, C[III]{} emission extending as far as 100 kpc and O[VI]{} as far as 50 kpc at $z=0.2$. The predicted extent stays roughly constant for $0.5 < z< 1.0$ as the cosmological surface brightness dimming is balanced by an increasing intrinsic emissivity.
6. An angular resolution of 4” is necessary to begin to resolve the spatial distribution of the CGM out to $z=1$ and sub-arcsecond resolution is needed to resolve beyond a general elongation from the disk. At $z=0.2$, this same observations require an angular resolution of 7.6“ (for elongation) and 1.5” (for features) respectively.
These conclusions focus on results from the combination of predicted UV absorption and emission-line data from a simulated Milky Way-like galaxy, offering a physical explanation for the trends seen in observations for the existence and extent of multiple ions. Other studies have focused on varying feedback prescriptions to bring simulations into better agreement with recent data. However, this can also be reversed as simulation predictions can be extended to create true mock observations that can enable better interpretations of future data. To make more accurate predictions for observations, future work will have to include a number of details excluded here.
First, the low surface brightness of the emission in question means that the UV background can overpower the CGM signal. Including a model of the background signal and incorporating its subtraction will provide a better understanding of which CGM structures can be detected with confidence. Second, the continuum emission from the galaxy can also dominate the CGM emission-line signal close to the disk, especially at moderate to low resolution. The disk-halo interface is where the SN winds are being launched; understanding this transition is particularly important. Finally, the velocity structure of the gas has not been considered here, which can change the line profiles of the emission. @ryan examined the flow of gas into and out of the galaxy, finding that the majority of the accretion at low redshift was in the form of warm/hot gas. Associating emission with these flows is left for future work but will become crucial as integral field units that provide both spatial and spectral information are becoming commonplace. This kinematic information will provide the best observational evidence for both inflows and outflows of gas from galaxies.
We are grateful to M. Ryan Joung for generously sharing his simulation output and guidance. L.C. would also like to thank Yuan Li, Cameron Hummels, Munier Salem and Bruno Milliard for helpful discussions. L.C. and D.S. acknowledge support from NASA grant NNX12AF29G. L.C. would also like to acknowledge support from the Chateaubriand Fellowship.
Uncertainty in the Extragalactic Ultraviolet Background
=======================================================
The extragalactic ultraviolet background (EUVB) is an important component of any photoionization model implemented for both observations and simulations. However, the EUVB is historically not well constrained. Variations of the @HM96 background (e.g. HM96, HM01, HM05, HM12) are implemented in the majority of hydrodynamical simulations and in <span style="font-variant:small-caps;">cloudy</span>. Yet simulations attempting to match measurement of the low-redshift Lyman $\alpha$ forest find that the most recent HM12 model does not reproduce the observed column density distribution of absorbers. The simulations of @kollmeier_underproduction require a photoionization rate a factor of five higher than the HM12 model while those of @shull_2015 suggest a factor of 2-3 increase. However, unlike the previous work, @shull_2015 find that the HM12 model does in fact reproduce the distribution of absorbers with log(N$_{\mathrm{H{\scriptsize I}}}$) $> 14.0$ and that no single model reproduces the entire distribution. Both of these values are more consistent with HM96 and in-line with our high EUVB model, g1q10. Finally, at lower redshifts, constraints are even harder to place on the EUVB [@cooray_2016]. In contrast to both of these high redshift studies, @adams_2011 placed upper limits on the photoionization rate at $z=0$ from the non-detection of H$\alpha$ in UGC 1281 at roughly the values in the HM12 model and in UGC 7321 at roughly 10 times lower - corresponding to 10 and 100 times lower than the HM05 model assumed throughout this paper. This is in the range of our low model, g1q01. In this way, the models used here bracket the range of possible photoionization rates as they are known today. These theoretical and observational inconsistencies highlight the uncertainty in the shape and intensity of the EUVB.
In addition, these models depend on the escape fraction of ionizing photons from their host galaxy, a quantity that is expected to be low but has been argued to be anywhere from 0.01 to 0.3, depending on the galaxy mass and redshift [@dove_escape; @wise_escape2; @benson_escape; @roy_escape]. [see introduction of @wise_escape for an in-depth discussion.] Both of the above Lyman $\alpha$ studies agree that the likely source of the discrepancy is the prescription for the escape fraction which leads to a galactic contribution to the EUVB that is too low at low-redshift.
![Relevant EUVB backgrounds for this work: HM96 (assumed in the simulation) and HM05 (fiducial for <span style="font-variant:small-caps;">cloudy</span> modeling). The galaxy (red) and quasar (blue) components of HM05 are also plotted. The blue dashed lines represent the two quasar backgrounds assumed throughout the paper (100 times less intense and 10 times more intense respectively). The green vertical lines bracket the wavelengths of the emission examined in the paper. \[HM\_background.fig\]](Figure13compressed.pdf){width="47.50000%"}
Figure \[HM\_background.fig\] shows the EUVB relevant for this work. HM96 is the background assumed within the simulation and used for the chemical network that determined the cooling rate at each time step. HM05 is the background assumed and varied for all of the <span style="font-variant:small-caps;">cloudy</span> modeling discussed in the paper. Figure \[HM\_background.fig\] also shows how the HM05 background is broken into its components: the galactic (red) and the quasar (blue). Throughout the paper, this background is modified to either g1q01 (a quasar intensity 100 times smaller than fiducial) or g1q10 (a quasar intensity 10 times larger than fiducial). These models are plotted as blue dashed-dotted lines.
One concern is that the model assumed for the simulation (HM96) is not the same as the one assumed in the <span style="font-variant:small-caps;">cloudy</span> modeling done in this paper (HM05). At longer wavelengths, the HM96 and HM05 backgrounds are similar; however, the HM96 background is closer to the modified g1q10 background than to the fiducial HM05 at shorter wavelengths where the quasar component dominates. This suggests that the range of backgrounds being explored is reasonable. Furthermore, the only way the simulation directly depends on the EUVB is in the calculation of the heating and cooling. The heating will be dominated by physical processes such as supernova feedback. As for the cooling, for $T> 10^4$K, which is the case for all CGM gas considered, <span style="font-variant:small-caps;">cloudy</span> modeling shows that the cooling function assumed in the simulation varies somewhat with the ionization fraction at the low metallicities found in the CGM but is dominated by the overall metallicity. Thus pairing the simulated density and temperature with the varying EUVB in the <span style="font-variant:small-caps;">cloudy</span> modeling is not unreasonable.
The green vertical lines bracket the wavelength range of the emission lines considered here. They fall within the galaxy-dominated part of the spectrum. However, the quasar component contributes much more of the ionizing intensity and thus is more important in shaping the expected ion fractions for the column density and emission predictions. @werk14 examine how differences between HM01 and HM12 affect their measurements and find that repeating the analysis with HM12 lowers the gas ionization parameters by 0.1-0.4 dex, which must be accounted for in either the H[I]{} column density or the metallicity. The simulated column densities show larger variations with changes in EUVB because the density and temperature are fixed and only the ionizing intensity is changing. In the less constrained <span style="font-variant:small-caps;">cloudy</span> modeling, flexibility within setting the interdependent quantities of ionization parameter and metallicity can reduce the effect of the EUVB. The value of the simulation is that these gas properties are determined by the larger cosmological context instead of modeling an isolated cloud.
Finally, in addition to the uncertainty in the EUVB, ionizing photons from local stellar sources are expected to be the dominant source of photoionization in local star forming regions within the disk but again, the escape fraction of these photons into the halo is entirely uncertain. Similarly, star formation in the halo has been shown to change the extent and shape of a galaxy’s Lyman $\alpha$ emission but this triggered emission results in a greater predicted UV flux than what is currently measured. [@lake2015]. Because of these uncertainties and because we are focused on emission from gas further from the star-forming disk, including this ionization source is reserved for future work.
Investigating the Effects of Resolution
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Because the column density and surface brightness calculations depend so sensitively on the density, temperature, and metallicity of the gas, understanding how the resolution of the simulation affects these quantities is necessary to evaluate their robustness. On one hand, over-cooling can lead to large, artificially dense clumps and is known to leave a too centrally concentrated disk. On the other hand, observational evidence suggests that absorbing clouds can be small, high-density structures that would be under-resolved in the simulation [@crighton].
![Median and average column density and emission profiles of C[III]{} at $z=0.2$, binned for four different resolutions. 1 kpc is the resolution assumed throughout the paper and is roughly the underlying simulation resolution beyond the disk in the CGM. As the resolution increases, the median profile decreases as the gas structure is refined. At the very center of the disk and in the outer halo, the median profile of the simulation appears to be converging below 5 kpc. The exception is the disk-halo interface at roughly 20 kpc. Only at the highest resolution is the sharp transition from disk to halo captured. The average converges more quickly.[]{data-label="resolution.fig"}](Figure14compressed.pdf){width="55.00000%"}
The ideal solution would be to re-run the simulation at lower resolution and compare its output to the simulation analyzed here. Unfortunately, because of the length of time that has passed between this analysis and the original, high-resolution run, the exact initial conditions of this simulation can not be reproduced. However, we can attempt to address this issue by re-binning the high resolution output to lower resolutions. Figure \[resolution.fig\] shows the median and average profiles of the column density and emission of C[III]{} at $z=0.2$ for the volume considered in the above analysis. The average profiles of these quantities have converged except at the lowest resolution. On the other hand, at lower resolutions, the median profiles of the column density and the emission are both too high, corresponding to higher average density and fewer low-density regions. However, by a resolution of 5 kpc, the simulation seems to have converged on a median profile for the inner disk as well as the outer parts of the halo. The remaining region, around 20 kpc, corresponds to the edge of the disk-halo interface. It’s only at the highest resolution that this interface is properly resolved. At 5 kpc, this boundary is still blurred, not allowing for the sharp transition from high-density, cold gas to low-density, warm gas.
These profiles suggest that further resolving the outer halo should not greatly change the median predictions for the column density and emission. While higher densities clumps are mostly likely still not being resolved even with the maximum resolution, Figure \[resolution.fig\] suggests that these regions are small compared to the volume and won’t significantly alter the median profiles of either quantity.
[^1]: The report describing this proposed telescope can be found at <http://www.hdstvision.org/>
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abstract: 'Non-compact QED$_3$ with four-component fermion flavor content $N_f \geq 2$ is studied numerically near the chiral limit to understand its chiral symmetry breaking features. We monitor discretization and finite size effects on the chiral condensate by simulating the model at different values of the gauge coupling on lattices ranging in size from $10^3$ to $50^3$. Our upper bound for the dimensionless condensate $\beta^2\langle\bar\Psi\Psi\rangle$ in the $N_f=2$ case is $5 \times 10^{-5}$.'
address:
- 'Department of Physics, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, UK'
- 'Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801-3080, USA'
- 'Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, D-22603 Hamburg, Germany'
author:
- 'S.J. Hands , J.B. Kogut , L. Scorzato and C.G. Strouthos $^{\rm a}$'
title: 'The Chiral Limit of Non-compact QED$_3$'
---
Introduction
============
Over the last few years, QED$_3$ has attracted a lot of attention because of its potential applications to models of high $T_c$ superconductivity. It is believed to be confining and exhibit features such as dynamical mass generation when the number of fermion flavors $N_f$ is smaller than a critical value $N_{fc}$. It is therefore an interesting and challenging model and an ideal laboratory in which to study more complicated gauge field theories.
We are considering the four-component formulation of QED$_3$ where the Dirac algebra is represented by the $4 \times 4$ matrices $\gamma_0$, $\gamma_1$ and $\gamma_2$. This formulation preserves parity and gives each spinor a global $U(2)$ symmetry generated by $\bf1$, $\gamma_3, \gamma_5$ and $i\gamma_3 \gamma_5$; the full symmetry is then $U(2N_f)$. If the fermions acquire dynamical mass the $U(2N_f)$ symmetry is broken spontaneously to $U(N_f) \times U(N_f)$ and $2N_f^2$ Goldstone bosons appear in the particle spectrum.
At the present time the issue of spontaneous chiral symmetry breaking in QED$_3$ is not very well understood. Studies based on Schwinger-Dyson equations (SDEs) using the photon propagator derived from the leading order $1/N_f$ expansion suggested that for $N_f$ less than some critical value $N_{fc}$ the answer is positive with $N_{fc}\simeq3.2$ [@pisarski]. Apparently, for $N_f > N_{fc}$, the attactive interaction between a fermion and an antifermion due to photon exchange is overwhelmed by the fermion screening of the theory’s electric charge. More recent studies which treat the vertex consistently in both numerator and denominator of the SDEs have found a value for $N_{fc}$ either in agreement with the original study or slightly higher, with $N_{fc}\simeq4.3$ [@maris]. Finally an argument based on a thermodynamic inequality has predicted $N_{fc}\leq{3\over2}$ [@appelquist]. There have also been numerical attempts to resolve the issue via lattice simulations. The obvious advantage in this case is that one can study any $N_f$ without any assumption concerning the convergence of expansion methods. However, the principal obstruction to a definitive answer has been large finite volume effects resulting from the presence of a massless photon. Numerical studies of the quenched case have shown that chiral symmetry is broken [@quenched], whereas in the case of simulations with dynamical fermions opinions have divided on whether $N_{fc}$ is finite and $\approx 3$ or whether chiral symmetry is broken for all $N_f$ [@dynamical]. Recent studies of the $N_f=1$ model on small lattices appear in [@maris2002].
In our study we used the Hybrid Monte Carlo algorithm to simulate the non-compact version of lattice QED$_3$ with staggered fermions. In the continuum it corresponds to the four-component spinor formulation of the model [@burden]. We implement even-odd partitioning which implies that a single flavor of one-component staggered fermions can be simulated, which corresponds to $N_f=2$ in the continuum limit.
Results
=======
In this section we discuss the results of lattice simulations of QED$_3$ with $N_f \geq 2$. In our study we tried to detect and control the various drawbacks of the lattice method: (i) The lattice itself distorts continuum space-time physics considerably unless the lattice spacing $a$ can be chosen small compared to the relevant physical wavelengths in the system. (ii) the size of the lattice $L^3$ must be large compared to the dynamically generated correlations in the system; and (iii) the chiral limit can only be studied by simulating light fermions of mass $m_0$ in lattice units.
![Dimensionless condensate $\beta^2 \langle \bar\Psi\Psi \rangle$ vs. dimensionless bare mass $\beta m_0$ for $N_f=2,4,8,16$, $\beta=0.6$ on a $16^3$ lattice.[]{data-label="fig:all_Nf"}](all.condensates.eps)
In Fig.\[fig:all\_Nf\] we plot the dimensionless chiral condensate $\beta^2 \langle \bar\Psi\Psi \rangle$ vs. the dimensionless bare mass $\beta m_0$ (where $\beta\equiv \frac{1}{g^2a}$) for $N_f=2,4,8,16$. The coupling $\beta=0.6$ and the lattice volume is $16^3$. As $N_f$ increases the chiral condensate decreases (for $m_0 \geq 0$) because the interaction between the fermion and the antifermion is screened. However, as $m_0 \rightarrow 0$ all the curves tend to pass smoothly through the origin. This motivated us to study in more detail the pattern of chiral symmetry breaking at small $N_f$ on larger volumes near the chiral limit.
![Dimensionless condensate vs. dimensionless bare mass for $N_f=2$ at different values of the coupling $\beta$ and constant physical volume $(L/\beta)^3$.[]{data-label="fig:discret"}](discret.eps)
In order to check whether our lattice data are characteristic of the continuum limit we plot in Fig.\[fig:discret\] $\beta^2 \langle \bar\Psi\Psi \rangle$ vs. $\beta m_0$ for $N_f=2$ and coupling $\beta=0.45,0.60,0.75,0.90$. To disentangle the lattice discretization effects from the finite size effects we keep the volume in physical units $(L/\beta)^3$ constant. It can be inferred from the graph that discretization effects are small for $\beta \geq 0.60$, because the data almost fall on the same line within the resolution of our analysis.
In Fig.\[fig:cond\_b=0.6\] we present our results for the chiral condensate vs. bare mass for $N_f=2$ and $\beta=0.6$ on lattice sizes varying from $8^3$ to $48^3$. We infer that finite size effects become small for $L \geq 24$ and all the lines tend to pass smoothly through the origin. Our analysis of meson masses and susceptibilities in scalar and pseudoscalar channels showed that these quantities suffer from very strong finite size effects and therefore did not allow us to reach such a definitive conclusion [@qed3.2002].
Next we discuss the results from $N_f=2$ simulations at $\beta=0.75$. The $\beta=0.75$ data set is closer to the continuum limit than the data extracted at $\beta=0.60$. However, particular care is required because weak coupling data are very sensitive to finite size effects and accurate measurements require simulations on large lattices. In Fig.\[fig:cond\_b=0.75\] we present the results for the chiral condensate vs. fermion bare mass extracted from simulations with lattice sizes ranging from $10^3$ to $50^3$. These simulations were performed very close to the chiral limit, i.e. with $m_0 \leq 0.005$. We can see from the figure the finite size effects are under relatively good control and the data tend to pass smoothly through the origin. Therefore, we conclude that for $N_f=2$ $\beta^2 \langle \bar \Psi \Psi \rangle \leq 5 \times 10^{-5}$, which is a strong indication that QED$_3$ may be chirally symmetric for $N_f \geq 2$.
![Dimensionless condensate vs. dimensionless bare mass for $N_f=2$, $\beta=0.6$ and lattice sizes $8^3, 16^3, 24^3, 32^3, 48^3$.[]{data-label="fig:cond_b=0.6"}](cond_b=0.6.eps)
![Condensate vs. bare mass for $N_f=2$, $\beta=0.75$ and lattice sizes $10^3,
20^3, 30^3, 40^3$ and $50^3$.[]{data-label="fig:cond_b=0.75"}](cond.b=0.75.eps)
Conclusions
===========
In our study of QED$_3$ with $N_f \geq 2$ we attempted to establish whether chiral symmetry is broken or not by studying the behavior of the chiral condensate close to the continuum limit $g \rightarrow 0$, on different volumes in order to detect and control finite size effects and near the chiral limit $m_0 \rightarrow 0$.
Our upper bound for the condensate in the $N_f=2$ case is $\beta^2\langle \bar \Psi \Psi \rangle \leq 5 \times 10^{-5}$ and all the lines of $\beta^2 \langle \bar \Psi \Psi \rangle$ vs. $\beta m_0$ tend to pass smoothly through the origin which may imply that chiral symmetry is restored for $N_f \geq 2$. We are continuing this study to check if chiral symmetry is broken in the case of $N_f=1$.
Acknowledgements {#acknowledgements .unnumbered}
================
SJH and CGS were supported by the Leverhulme Trust. JBK was supported in part by NSF grant PHY-0102409. The computer simulations were done on the Cray SV1’s at NERSC, the IBM-SP at NPACI, and on the SGI Origin 2000 at the University of Wales Swansea.
[9]{}
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abstract: 'Given a filtration of a commutative monoid $A$ in a symmetric monoidal stable model category $\mathcal{C}$, we construct a spectral sequence analogous to the May spectral sequence whose input is the higher order topological Hochschild homology of the associated graded commutative monoid of $A$, and whose output is the higher order topological Hochschild homology of $A$. We then construct examples of such filtrations and derive some consequences: for example, given a connective commutative graded ring $R$, we get an upper bound on the size of the $THH$-groups of $E_{\infty}$-ring spectra $A$ such that $\pi_*(A) \cong R$.'
author:
- 'G. Angelini-Knoll'
- 'A. Salch'
bibliography:
- '/Users/gabrielangelini-knoll/Library/texmf/bibtex/bib/salch.bib'
title: 'A May-type spectral sequence for higher topological Hochschild homology'
---
Introduction.
=============
Suppose $A = F_0A \supseteq F_1A \supseteq F_2A\supseteq \dots$ is a filtered augmented $k$-algebra. In J. P. May’s 1964 Ph.D. thesis, [@MR2614527], May sets up a spectral sequence with input $\Ext^{*,*}_{E^*_0A}(k,k)$ and which converges to $\Ext^{*}_{A}(k,k)$. Here $E^*_0A = \oplus_{n\geq 0} F_nA/F_{n+1}A$ is the associated graded algebra of $A$.
In the present paper, we do the same thing for topological Hochschild homology and its “higher order” generalizations (as in [@MR1755114]). Given a filtered $E_{\infty}$-ring spectrum $A$, we construct a spectral sequence $$\label{ss 2304} E^1_{*,*} \cong THH_{*,*}(E^*_0A) \Rightarrow THH_*(A).$$ Here $E^*_0A$ is the [*associated graded $E_{\infty}$-ring spectrum of $A$*]{}; part of our work in this paper is to define this “associated graded $E_{\infty}$-ring spectrum,” and prove that it has good formal properties and useful examples (e.g. Whitehead towers; see \[whitehead tower 3408\], below).
More generally: given any generalized homology theory $H$, and given any simplicial finite set $X_{\bullet}$, we construct a spectral sequence $$\label{ss 2305} E^1_{*,*} \cong E_{*,*}(X_{\bullet}\otimes E^*_0A) \Rightarrow E_*(X_{\bullet}\otimes A).$$ We recover spectral sequence \[ss 2304\] as a special case of \[ss 2305\] by letting $E_* = \pi_*$ and letting $X_{\bullet}$ be a simplicial model for the circle $S^1$.
A major part of the work we do in this paper is to formulate a definition (see Definition \[def of dec filt obj\]) of a “filtered $E_{\infty}$-ring spectrum” which is sufficiently well-behaved that we can actually construct a spectral sequence of the form \[ss 2305\], identify its $E^1$- and $E^{\infty}$-terms and prove its multiplicativity and good convergence properties. Actually our constructions and results work in a somewhat wider level of generality than commutative ring spectra: we fix a symmetric monoidal stable model category $\mathcal{C}$ satisfying some mild hypotheses (spelled out in Running Assumptions \[ra:1\] and \[ra:2\]), and we work with filtered commutative monoid objects in $\mathcal{C}$. In the special case where $\mathcal{C}$ is the category of symmetric spectra in simplicial sets, in the sense of [@MR1695653] and [@schwedebook], the commutative monoid objects are equivalent to $E_{\infty}$-ring spectra. Our framework is sufficiently general that an interested reader could potentially also apply it to monoidal model categories of equivariant, motivic, and/or parametrized spectra.
In the appendix, we construct a version of spectral sequence \[ss 2305\] with coefficients in a filtered symmetric $A$-bimodule $M$: $$\label{ss 2305a} E^1_{*,*} \cong E_{*,*}(X_{\bullet}\otimes (E^*_0A, E^*_0M)) \Rightarrow E_*(X_{\bullet}\otimes (A,M)),$$ and as a special case, $$\label{ss 2304a} E^1_{*,*} \cong E_{*,*}THH(E^*_0A, E^*_0M) \Rightarrow E_*THH(A,M).$$
Some of the most important cases of filtered commutative ring spectra, or filtered commutative monoid objects in general, are those which arise from Whitehead towers: given a cofibrant connective commutative monoid in $\mathcal{C}$, we construct a filtered commutative monoid $$\label{whitehead tower 3408} A = \tau_{\geq 0}A \supseteq \tau_{\geq 1} A \supseteq \tau_{\geq 2} A \supseteq \dots$$ where the induced map $\pi_n(\tau_{\geq m} A) \rightarrow \pi_n(\tau_{\geq m-1} A)$ is an isomorphism if $n\geq m$, and $\pi_n(\tau_{\geq m}A) \cong 0$ if $n<m$. While the homotopy type of $\tau_{\geq m}A$ is very easy to construct, it takes us some work to construct a sufficiently rigid multiplicative model for the Whitehead tower \[whitehead tower 3408\]; this is the content of Theorem \[post filt\].
If $\mathcal{C}$ is the category of symmetric spectra in simplicial sets, then the associated graded ring spectrum of the Whitehead tower \[whitehead tower 3408\] is the generalized Eilenberg-Mac Lane ring spectrum $H\pi_*(A)$ of the graded ring $\pi_*(A)$. Consequently we get a spectral sequence $$\label{ss 2305b} E^1_{*,*} \cong E_{*,*}(X_{\bullet}\otimes H\pi_*A) \Rightarrow E_*(X_{\bullet}\otimes A),$$ and as a special case, $$\label{ss 2304b} E^1_{*,*} \cong THH_{*,*}(H\pi_*A) \Rightarrow THH_*(A).$$
Many explicit computations are possible using spectral sequence \[ss 2304b\] and its generalizations with coefficients in a bimodule (defined in Definition \[def of may filt\], and basic properties proven in Theorems \[thm on fund thm 2\] and \[thh-may coeff\]). For example, in [@K1localsphere], G. Angelini-Knoll uses these spectral sequences to compute the topological Hochschild homology of the algebraic $K$-theory spectra of a large class of finite fields. In lieu of explicit computations using our new spectral sequences, we point out that the mere existence of these spectral sequences implies an upper bound on the size of the topological Hochschild homology groups of a ring spectrum: namely, if $R$ is a graded-commutative ring and $X_{\bullet}$ is a simplicial finite set and $E_*$ is a generalized homology theory, then for any $E_{\infty}$-ring spectrum $A$ such that $\pi_*(A) \cong R$, $E_*(X_{\bullet}\otimes A)$ is a subquotient of $E_*(X_{\bullet} \otimes HR)$. Here we write $HR$ for the generalized Eilenberg-Maclane spectrum with $\pi_*(HR)\cong R$ as graded rings.
Consequently, in Theorem \[upper bound thm\] we arrive at the slogan: [*among all the $E_{\infty}$-ring spectra $A$ such that $\pi_*(A) \cong R$, the topological Hochschild homology of $A$ is bounded above by the topological Hochschild homology of $H\pi_*(R)$.*]{} This lets us extract a lot of information about the topological Hochschild homology of $E_{\infty}$-ring spectra $A$ from information depending [*only on the ring $\pi_*(A)$ of homotopy groups of $A$.*]{} We demonstrate how to apply this idea in Theorem \[polynomial case 120\] and its corollaries, by working out the special case where $R = \hat{\mathbb{Z}}_p[x]$ for some prime $p$, with $x$ in positive grading degree $2n$. We get, for example, that for any $E_{\infty}$-ring spectrum $A$ such that $\pi_*(A) \cong \hat{\mathbb{Z}}_p[x]$, the Poincaré series of the mod $p$ topological Hochschild homology $(S/p)_*(THH(A))$ satisfies the inequality $$\sum_{i\geq 0} \left( \dim_{\mathbb{F}_p} (S/p)_*(THH(A))\right) t^i \leq
\frac{(1 + (2p-1)t)(1 + (2n+1)t)}{(1 - 2nt)(1 - 2pt)},$$ and:
- If $p$ does not divide $n$, then $THH_{2i}(A) \cong 0$ for all $i$ congruent to $-p$ modulo $n$ such that $i\leq pn-p-n$, and $THH_{2i}(A) \cong 0$ for all $i$ congruent to $-n$ modulo $p$ such that $i\leq pn-p-n$. In particular, $THH_{2(pn-p-n)}(A) \cong 0$.
- If $p$ divides $n$, then $THH_{i}(A)\cong 0$, unless $i$ is congruent to $-1,0,$ or $1$ modulo $2p$.
There is some precedent for spectral sequence \[ss 2304\]: when $A$ is a filtered commutative ring (rather than a filtered commutative ring spectrum), M. Brun constructed a spectral sequence of the form \[ss 2304\] in the paper [@MR1750729]. In Theorem 2.9 of the preprint [@angeltveitpreprint], V. Angeltveit remarks that a version of spectral sequence \[ss 2304\] exists for commutative ring spectra by virtue of a lemma in [@MR1750729] on associated graded FSPs of filtered FSPs; filling in the details to make this spectral sequence have the correct $E^1$-term, $E^{\infty}$-term, convergence properties, and multiplicativity properties takes a lot of work, and even aside from the substantially greater level of generality of the results in the present paper (allowing $X_{\bullet}\otimes A$ and not just $S^1\otimes A$, working with commutative monoids in symmetric monoidal model categories rather than any particular model for ring spectra, working with coefficient bimodules as in \[ss 2305b\]), we think it is valuable to add these very nontrivial details to the literature.
We are grateful to C. Ogle and Ohio State University for their hospitality in hosting us during a visit to talk about this project and A. Blumberg for a timely and useful observation (that the Reedy model structure on inverse sequences may not have certain desired properties). The first author would like to thank Ayelet Lindenstrauss, Teena Gerhardt, and Cary Malkiewich for helpful conversations on the material in this paper and for hosting him at their respective universities to discuss this work.
Conventions and running assumptions
===================================
\[cof assump\] By convention, the “cofiber of $f:X\lra Y$" will mean that $f$ is a cofibration and we are forming the pushout $Y\coprod_X 0$ in the given pointed model category. By convention we will write $Y/X$ for $Y\coprod_X 0$ when $f:X\lra Y$ is a cofibration.
We will write $\Comm(\mathcal{C})$ for the category of commutative monoid objects in a symmetric monoidal category $\mathcal{C}$, we will write $s\mathcal{C}$ for the category of simplicial objects in $\mathcal{C}$, and we will write $\smash$ for the symmetric monoidal product in a symmetric monoidal category $\mathcal{C}$ since the main example we have in mind is the category of symmetric spectra where the symmetric monoidal product is the smash product.
\[ra:1\] Throughout, let $\mathcal{C}$ be a complete, co-complete left proper stable model category equipped with the structure of a symmetric monoidal model category in the sense of [@MR1734325], satisfying the following axioms:
- The unit object $\mathbbm{1}$ of $\mathcal{C}$ is cofibrant.
- A model structure (necessarily unique) on $\Comm(\mathcal{C})$ exists in which weak equivalences and fibrations are created by the forgetful functor $\Comm(\mathcal{C})\rightarrow\mathcal{C}$.
- The forgetful functor $\Comm(\mathcal{C})\rightarrow \mathcal{C}$ commutes with geometric realization of simplicial objects.
- Geometric realization of simplicial cofibrant objects in $\mathcal{C}$ commutes with the monoidal product, i.e., if $X_{\bullet},Y_{\bullet}$ are simplicial cofibrant objects of $\mathcal{C}$, then the canonical comparison map $$\left| X_{\bullet}\smash Y_{\bullet}\right| \rightarrow
\left| X_{\bullet}\right| \smash \left| Y_{\bullet}\right|$$ is a weak equivalence in $\mathcal{C}$.
Here are a few immediate consequences of these assumptions about $\mathcal{C}$:
1. Since being cofibrantly generated is part of the definition of a monoidal model category in [@MR1734325], $\mathcal{C}$ is cofibrantly generated and hence can be equipped with functorial factorization systems. We assume that a choice of functorial factorization has been made and we will use it implicitly whenever a cofibration-acyclic-fibration or acyclic-cofibration-fibration factorization is necessary.
2. Smashing with any given object preserves colimits. Smashing with any given cofibrant object preserves cofibrations and weak equivalences.
3. Axioms (TC1)-(TC5) of May’s paper [@MR1867203] are satisfied, so the constructions and conclusions of [@MR1867203] hold for $\mathcal{C}$. In particular, we have a natural filtration on any finite smash power of a filtered object in $\mathcal{C}$, which we say more about below.
4. \[item4ra1\] Since $\mathcal{C}$ is assumed left proper, a homotopy cofiber of any map $f: X \rightarrow Y$ between cofibrant objects in $\mathcal{C}$ can be computed by factoring $f$ as $f = f_2\circ f_1$ with $f_1: X \rightarrow \tilde{Y}$ a cofibration and $f_2: \tilde{Y}\rightarrow Y$ an acyclic fibration, and then taking the pushout of the square $$\xymatrix{ X\ar[r]^{f_1} \ar[d] & \tilde{Y} \\ 0. & }$$
5. \[item\] In particular, if $f$ is already a cofibration, the pushout map $Y \rightarrow Y\coprod_X 0$ is a homotopy cofiber of $f$.
\[ra:2\] In addition to Running Assumption \[ra:1\], we assume our model category $\mathcal{C}$ satisfies the following condition: a map $X_{\bullet}\rightarrow Y_{\bullet}$ in the category of simplicial objects in $\mathcal{C}$ is a Reedy cofibration between Reedy cofibrant objects whenever the following all hold:
1. \[it1\]Each object $X_n$ and $Y_n$ of $\mathcal{C}$ is cofibrant.
2. \[it2\] Each degeneracy map $s_i:X_n\rightarrow X_{n+1}$ and $s_i:Y_n\rightarrow Y_{n+1}$ is a cofibration in $\mathcal{C}$
3. \[it3\] Each map $X_n \rightarrow Y_n$ is a cofibration in $\mathcal{C}$.
A consequence of this assumption is that the geometric realization of a map of simplicial objects in $\mathcal{C}$ satisfying Item \[it1\], Item \[it2\], and Item \[it3\] is a cofibration.
The main motivating example of such a category $\mathcal{C}$ satisfying Running Assumption \[ra:1\] is the category of symmetric spectra in a pointed simplicial model category $\mathcal{D}$, denoted $Sp_{\mathcal{D}}$, as in [@schwedebook]. In the case when $\mathcal{C}$ is the category $Sp_{\mathcal{D}}$, then $\Comm(\mathcal{C})$ is the category of commutative ring spectra and it is known to be equivalent to the category of $E_{\infty}$-ring spectra. The existence of the desired model structure on $\Comm(\mathcal{C})$ is proven in Theorem 4.1 of [@MR1734325]. We ask that $\mathcal{D}$ admits the mixed $\Sigma$-equivariant model structure of [@schwedebook Thm. 3.8], so that $Sp_{\mathcal{D}}$ may be equipped with the stable positive flat model structure. The fact that the forgetful functor $\Comm(\mathcal{C})\rightarrow\mathcal{C}$ commutes with geometric realization in the stable positive flat model structure on $\mathcal{C}$ is a consequence of [@MR2580430 Thm. 1.6].
Under the additional hypothesis that $\mathcal{D}$ is a *graded concrete category* (see [@asReedy Def. 3.1]), the category $Sp_{\mathcal{D}}$ satisfies Running Assumption \[ra:2\], as the authors prove in [@asReedy]. In fact, in this setting Item \[it2\] of Running Assumption \[ra:2\] can be weakened to: each degeneracy map $s_i:X_n\rightarrow X_{n+1}$ is levelwise a cofibration in $\mathcal{D}$. The main example of a category $\mathcal{D}$ that satisfies all of these conditions is the category of pointed simplicial sets. *Consequently, the category of symmetric spectra in simplicial sets equipped with the stable positive flat model structure satisfies all of our running assumptions.*
Construction of the spectral sequence. {#construction of ss section}
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Filtered commutative monoids and their associated graded commutative monoids
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\[def of dec filt obj\] By a [*cofibrant decreasingly filtered object in $\mathcal{C}$*]{} we mean a sequence of cofibrations in $\mathcal{C}$ $$\dots \stackrel{f_3}{\longrightarrow} I_2 \stackrel{f_2}{\longrightarrow} I_1 \stackrel{f_1}{\longrightarrow} I_0,$$ such that each object $I_i$ is cofibrant.
\[def of dec filt comm mon\] By a [*cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$*]{} we mean:
- a cofibrant decreasingly filtered object $$\dots \stackrel{f_3}{\longrightarrow} I_2 \stackrel{f_2}{\longrightarrow} I_1 \stackrel{f_1}{\longrightarrow} I_0$$ in $\mathcal{C}$, and
- for every pair of natural numbers $i,j\in\mathbb{N}$, a map in $\mathcal{C}$ $$\rho_{i,j}: I_i\smash I_j \rightarrow I_{i+j},$$ and
- a map $\eta: \mathbbm{1}\rightarrow I_0$,
satisfying the axioms listed below. For the sake of listing the axioms concisely, it will be useful to have the following notation: if $i^{\prime}\leq i$, we will write $f_i^{i^{\prime}}: I_i\rightarrow I_{i^{\prime}}$ for the composite $$f_i^{i^{\prime}} = f_{i^{\prime}+1} \circ f_{i^{\prime}+2} \circ \dots \circ f_{i-1}\circ f_i.$$
Here are the axioms we require:
- [**(Compatibility.)**]{} For all $i,j,i^{\prime},j^{\prime}\in \mathbb{N}$ with $i^{\prime}\leq i$ and $j^{\prime}\leq j$, the diagram $$\xymatrix{
I_i \smash I_j \ar[d]_{f_i^{i^{\prime}}\smash f_j^{j^{\prime}}} \ar[r]^{\rho_{i,j}} & I_{i+j}\ar[d]^{f_{i+j}^{i^{\prime}+j^{\prime}}} \\
I_{i^{\prime}}\smash I_{j^{\prime}} \ar[r]^{\rho_{i^{\prime} , j^{\prime}}} & I_{i^{\prime} + j^{\prime}}}$$ commutes.
- [**(Commutativity.)**]{} For all $i,j\in \mathbb{N}$, the diagram $$\xymatrix{ I_i\smash I_j \ar[d]_{\chi_{I_i,I_j}}\ar[rd]^{\rho_{i,j}} & \\
I_j\smash I_i \ar[r]_{\rho_{j,i}} & I_{i+j} }$$ commutes, where $\chi_{I_i,I_j}: I_i\smash I_j \stackrel{\cong}{\longrightarrow} I_j\smash I_i$ is the symmetry isomorphism in $\mathcal{C}$.
- [**(Associativity.)**]{} For all $i,j,k\in\mathbb{N}$, the diagram $$\xymatrix{ I_i\smash I_j\smash I_k \ar[r]^{\id_{I_i}\smash \rho_{j,k}} \ar[d]_{\rho_{i,j}\smash \id_{I_k}} & I_i\smash I_{j+k}\ar[d]^{\rho_{i,j+k}} \\
I_{i+j}\smash I_k\ar[r]_{\rho_{i+j,k}} & I_{i+j+k}}$$ commutes.
- [**(Unitality.)**]{} For all $i\in \mathbb{N}$, the diagram $$\xymatrix{ \mathbbm{1}\smash I_i\ar[rd]^{\cong} \ar[d]_{\eta\smash \id_{I_i}} & \\ I_0\smash I_i\ar[r]_{\rho_{0,i}} & I_i }$$ commutes, where the map marked $\cong$ is the (left-)unitality isomorphism in $\mathcal{C}$.
We will sometimes write $I_{\bullet}$ as shorthand for this entire structure.
Note that, if $I_{\bullet}$ is a cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$, then $I_0$ really is a commutative monoid in $\mathcal{C}$, with multiplication map $\rho_{0,0}: I_0\smash I_0\rightarrow I_0$ and unit map $\eta: \mathbbm{1}\rightarrow I_0$. The objects $I_i$ for $i>0$ do not receive commutative monoid structures from the structure of $I_{\bullet}$, but instead play a role analogous to that of the nested sequence of powers of an ideal in a commutative ring.
\[def of Hausdorff filt\] Suppose $I_{\bullet}$ is a cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$.
- We shall say that $I_{\bullet}$ is [*Hausdorff*]{} if the homotopy limit of the $I_n$ is weakly equivalent to the zero object: $\holim_n I_n\simeq 0$.
- We shall say that $I_{\bullet}$ is [*finite*]{} if there exists some $n\in\mathbb{N}$ such that $f_{m}: I_m\rightarrow I_{m-1}$ is a weak equivalence for all $m>n$.
\[rem filt comm mon\] Definition \[def of dec filt comm mon\] has the advantage of concreteness, but there is an equivalent, more concise definition of a cofibrant decreasingly filtered commutative monoid. Observe that the the data of a decreasingly filtered commutative monoid, without the cofibrancy conditions, is the same as the data of a lax symmetric monoidal functor $$I_{\bullet} : (\mathbb{N}^{\op}, +, 0 ) \lra (\mathcal{C},\smash, \mathbbm{1}).$$ Recall that due to Day [@Day], the category of lax symmetric monoidal functors in $\mathcal{C}^{\mathbb{N}^{\op}}$ is equivalent to the category $\Comm \mathcal{C}^{\mathbb{N}^{\op}}$ of commutative monoid objects in the symmetric monoidal category $(\mathcal{C}^{\mathbb{N}^{\op}}, \otimes_{\text{Day}}, \mathbbm{1}_{\text{Day}})$ where $\otimes_{\text{Day}}$ is the Day convolution symmetric monoidal product also constructed in [@Day] and $\mathbbm{1}_{\text{Day}}$ is a cofibrant replacement for the unit of this symmetric monoidal product. (See [@MR3427193] for a modern treatment of this in the setting of quasi-categories.)
In particular, a decreasingly filtered commutative monoid is therefore equivalent to an object in $\Comm \mathcal{C}^{\mathbb{N}^{\op}}$. Now we claim that $\mathcal{C}^{\mathbb{N}^{\op}}$ with the projective model structure is cofibrantly generated and it is a monoidal model category satisfying the monoid axiom in the sense of Schwede-Shipley [@MR1734325]. The fact that a functor category with the projective model structure is cofibrantly generated follows from [@HTT A.2.8.3]. Therefore, whenever $\mathcal{C}$ is cofibrantly generated, as we assume in Running Assumption \[ra:1\], then $\mathcal{C}^{\mathbb{N}^{\op}}$ admits the projective model structure and it is cofibrantly generated. The fact that $\mathcal{C}^{\mathbb{N}^{\op}}$ is a closed symmetric monoidal model category satisfying the pushout product axiom follows by Propositions 2.2.15 and 2.2.16 of the thesis of Isaacson [@MR2713397]. To apply the theorem of Isaacson, we need to enrich $\mathbb{N}^{\op}$ in $\mathcal{C}$ by letting $$\mathbb{N}^{\op}(n,m)\cong \left \{ \begin{array}{l} \mathbbm{1} \text{ if } n\ge m \\ 0 \text{ otherwise,} \end{array} \right.$$ where $\mathbbm{1}$ is a cofibrant model for the unit of the symmetric monoidal product on $\mathcal{C}$. This ensures that all morphisms are “virtually cofibrant" in the sense of Isaacson [@MR2713397]. We claim that if $I_{\bullet}$ is a cofibrant object in the projective model structure on $\mathcal{C}^{\mathbb{N}^{\op}}$, then it is a sequence $$\xymatrix{ \dots \ar[r]^{f_3} & I_2 \ar[r]^{f_2}& I_1 \ar[r]^{f_1} & I_0}$$ such that each map $f_i$ is a cofibration and each object $I_i$ is cofibrant, and we will prove this in Lemma \[lem cof proj\], which follows. We do not prove the converse statement that all cofibrant objects in the projective model structure on $\mathcal{C}^{\mathbb{N}^{\op}}$ are of this form and we make no claim to its validity.
Due to Schwede-Shipley [@MR1734325 Thm 4.1], if we equip the category $\Comm \mathcal{C}^{\mathbb{N}^{\op}}$ with the model structure created by the forgetful functor $U: \Comm \mathcal{C}^{\mathbb{N}^{\op}}\rightarrow \mathcal{C}^{\mathbb{N}^{\op}}$, then cofibrant objects in $\Comm \mathcal{C}^{\mathbb{N}^{\op}}$ forget to cofibrant objects in $\mathcal{C}^{\mathbb{N}^{\op}}$ since $\mathcal{C}^{\mathbb{N}^{\op}}$ is cofibrantly generated, closed symmetric monoidal, and satisfies the pushout product axiom. Therefore cofibrant objects in $\Comm \mathcal{C}^{\mathbb{N}^{\op}}$ are cofibrant decreasingly filtered commutative monoids in $\mathcal{C}$ as defined in Definition \[def of dec filt obj\].
\[lem cof proj\] Let $\mathcal{D}$ be a cofibrantly generated model category, and let $\mathcal{D}^{\mathbb{N}^{\op}}_{proj}$ be the category of inverse sequences in $\mathcal{D}$, i.e., functors $\mathbb{N}^{\op}\rightarrow \mathcal{D}$, equipped with the projective model structure. (Recall that this is the model structure in which a map $F: X \rightarrow Y$ is a weak equivalence, respectively fibration, if $F(n): X(n) \rightarrow Y(n)$ is a weak equivalence, respectively fibration, for all $n\in\mathbb{N}$.) Let $P$ be a cofibrant object in $\mathcal{D}^{\mathbb{N}^{\op}}_{proj}$. Then, for all $n\in\mathbb{N}$, the object $P(n)$ of $\mathcal{D}$ is cofibrant, and the morphism $P(n+1) \rightarrow P(n)$ is a cofibration in $\mathcal{D}$.
First, a quick definition:
- given a morphism $f: Y \rightarrow Z$ in $\mathcal{D}$ and a nonnegative integer $n$, let $b_nf: \mathbb{N}^{\op}\rightarrow \mathcal{D}$ be the functor given by letting $b_nf(m) = Y$ if $m\geq n$, letting $b_{n}f(n-1) = Z$, and letting $b_nf(m) = 1$ if $m<n-1$. Here we are writing $1$ for the terminal object of $\mathcal{D}$. We let $b_nf(m+1) \rightarrow b_nf(m)$ be the identity map on $Y$ if $m\geq n$, we let $b_nf(n) \rightarrow b_nf(n-1)$ be the map $f: Y \rightarrow Z$, and we let $b_nf(m+1) \rightarrow b_nf(m)$ be the projection to the terminal object if $m<n-1$.
- Given an object $Y$ of $\mathcal{D}$ and a nonnegative integer $n$, we write $c_nY$ for $b_n\pi$ where $\pi$ is the projection $Y \rightarrow 1$ to the terminal object. It is easy to see that $c_n: \mathcal{D} \rightarrow \mathcal{D}^{\mathbb{N}^{\op}}$ is a functor by letting $c_nf(m): c_nY(m) \rightarrow c_nZ(m)$ be $f: Y\rightarrow Z$ if $m\geq n$ and $c_nf(m) = \id_1$ if $m<n$.
Fix some $n\in\mathbb{N}$, let $\phi: P(n)\rightarrow W$ be a map in $\mathcal{D}$, and let $g: V \rightarrow W$ be an acyclic fibration in $\mathcal{D}$. We have a commutative diagram in $\mathcal{D}^{\mathbb{N}^{\op}}_{proj}$: $$\label{lifting diagram 043285394} \xymatrix{
& c_nV \ar[d]^{c_ng} \\ P \ar[r]^h & c_nW }$$ where $h(n): P(n)\rightarrow c_nW(n)$ is $\phi$ if $m=n$, where $h(m)$ is the projection to the terminal object if $m<n$, and where, if $m>n$, then $h(m)$ is the composite $$P(m) \rightarrow P(m-1) \rightarrow \dots \rightarrow P(n+1) \rightarrow P(n) \stackrel{\phi}{\longrightarrow} W = c_nW(n).$$ Since $\phi$ and $\id_1$ are both acyclic fibrations, the map $c_ng$ is an acyclic fibration in the projective model structure on $\mathcal{D}^{\mathbb{N}^{\op}}$. So there exists a map $\ell: P \rightarrow c_nV$ filling in diagram \[lifting diagram 043285394\] and making it commute. Evaluating at $n$, we get that $\ell(n): P(n) \rightarrow c_nV(n) = V$ is a map satisfying $g\circ \ell(n) = \phi$. So $0 \rightarrow P(n)$ lifts over every acyclic fibration in $\mathcal{D}$, so $P(n)$ is cofibrant in $\mathcal{D}$.
Now suppose the map $P(n\leq n+1): P(n+1) \rightarrow P(n)$ fits into a commutative diagram $$\label{comm diag 45035834}\xymatrix{
P(n+1) \ar[d]_{P(n\leq n+1)} \ar[r]^{\psi} & V\ar[d]^t \\
P(n) \ar[r]_{\phi} & W
}$$ in $\mathcal{D}$, in which $t$ is an acyclic fibration. We have a commutative diagram in $\mathcal{D}^{\mathbb{N}^{\op}}_{proj}$: $$\label{lifting diagram 043285395} \xymatrix{
& c_nV \ar[d]^{i} \\ P \ar[r]^j & b_{n+1}t }$$ where $i(n): V\rightarrow W$ is $t$, where $i(m) = \id_V$ if $m>n$, and where $i(m) = \id_1$ if $m<n$; and where $j(n): P(n) \rightarrow W$ is $\phi$, where $j(n+1): P(n+1) \rightarrow V$ is $\psi$, where $j(m)$ is projection to the terminal object if $m<n$, and where, if $m>n+1$, then $j(m)$ is the composite $$P(m) \rightarrow P(m-1) \rightarrow \dots \rightarrow P(n+1) \rightarrow P(n) \stackrel{\phi}{\longrightarrow} W = b_nt(n).$$ Since $\id_V$ and $t$ and $\id_1$ are all acyclic fibrations, we know that $i$ is an acyclic fibration in the projective model structure on $\mathcal{D}^{\mathbb{N}^{\op}}$. So there exists a map $\ell: P \rightarrow c_nV$ filling in diagram \[lifting diagram 043285395\] and making it commute. Evaluating $\ell$ at $n$ yields a map $$\ell(n): P(n) \rightarrow c_nV(n) = V$$ such that $$\begin{aligned}
\nonumber t\circ \ell(n)
\nonumber &= i(n) \circ \ell(n) \\
\nonumber &= j(n) \\
\label{A} &= \phi, \mbox{\ \ \ and} \\
\nonumber \ell(n+1)
\nonumber &= \id_V\circ \ell(n+1) \\
\nonumber &= i(n+1) \circ \ell(n+1) \\
\nonumber &= j(n+1) \\
\nonumber &= \psi, \mbox{\ \ \ and} \\
\nonumber \ell(n) \circ P(n\leq n+1)
\nonumber &= c_nV(n\leq n+1) \circ \ell(n+1) \\
\label{B} &= \id_V \circ \psi.\end{aligned}$$ Equations \[A\] and \[B\] express exactly that the map $\ell(n)$ fills in the diagonal of diagram \[comm diag 45035834\], making it commute. So $P(n\leq n+1)$ lifts over every acyclic fibration. So $P(n\leq n+1)$ is a cofibration in $\mathcal{D}$.
[**(The associated graded monoid.)**]{} Let $I_{\bullet}$ be a cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$. By $E_0^*I_{\bullet}$, the [*associated graded commutative monoid of $I_{\bullet}$*]{}, we mean the graded commutative monoid object in $\mathcal{C}$ defined as follows:
- “additively,” that is, as an object of $\mathcal{C}$, $$E_0^*I_{\bullet} \cong \coprod_{n\in\mathbb{N}} I_n/I_{n+1}.$$
- The unit map $\mathbbm{1}\rightarrow E_0^*I_{\bullet}$ is the composite $$\mathbbm{1}\stackrel{\eta}{\longrightarrow} I_0 \rightarrow I_0/I_1\hookrightarrow E_0^*I_{\bullet} .$$ (Note that $E_0^*I_{\bullet}$ is constructed as an $I_0/I_1$-algebra).
- The multiplication on $E_0^*I_{\bullet}$ is given as follows. Since the smash product commutes with colimits, hence with coproducts, to specify a map $$E_0^*I_{\bullet} \smash E_0^*I_{\bullet}\rightarrow E_0^*I_{\bullet}$$ it suffices to specify a component map $$\nabla_{i,j}: I_i/I_{i+1}\smash I_j/I_{j+1}\rightarrow E_0^*I_{\bullet}$$ for every $i,j\in\mathbb{N}$. We define such a map $\nabla_{i,j}$ as follows: first, we have the commutative square $$\xymatrix{ I_{i+1}\smash I_j \ar[r]^{\rho_{i+1,j}} \ar[d]_{f_{i+1}\smash \id_{I_j}} & I_{i+j+1}\ar[d]^{f_{i+j+1}} \\
I_i\smash I_j \ar[r]^{\rho_{i,j}} & I_{i+j} }$$ so, since the vertical maps are cofibrations by Definition \[def of dec filt comm mon\], we can take vertical cofibers to get a map $$\tilde{\nabla}_{i,j}: I_i/I_{i+1}\smash I_j\rightarrow I_{i+j}/I_{i+j+1},$$ which is well-defined by Running Assumption \[ra:1\].
Now we have the commutative diagram $$\xymatrix{ I_{i+1}\smash I_{j+1}\ar[rd]^{\id_{I_{i+1}}\smash f_{j+1}}\ar[dd]^{f_{i+1}\smash \id_{I_{j+1}}}\ar[rr]^{\rho_{i+1,j+1}} & & I_{i+j+2}\ar[rd]^{f_{i+j+2}}\ar[dd]^(.7){f_{i+j+2}} & \\
& I_{i+1}\smash I_j \ar[dd]^(.7){f_{i+1}\smash \id_{I_j}} \ar[rr]^(.35){\rho_{i+1,j}} & & I_{i+j+1}\ar[dd]^{f_{i+j+1}} \\
I_i\smash I_{j+1}\ar[rd]^{\id_{I_i}\smash f_{j+1}} \ar[rr]^(.4){\rho_{i,j+1}} \ar[dd] & & I_{i+j+1}\ar[rd]^{f_{i+j+1}}\ar[dd] & \\
& I_i\smash I_j\ar[rr]^(.4){\rho_{i,j}} \ar[dd] & & I_{i+j} \ar[dd] \\
I_i/I_{i+1}\smash I_{j+1} \ar[rd]_{\id_{I_i/I_{i+1}}\smash f_{j+1}} \ar[rr]^(.4){\tilde{\nabla}_{i,j+1}} & & I_{i+j+1}/I_{i+j+2}\ar[rd]^{0} & \\
& I_i/I_{i+1}\smash I_j \ar[rr]^{\tilde{\nabla}_{i,j}} & & I_{i+j}/I_{i+j+1} }$$ in which the columns are cofiber sequences. So we have a factorization of the composite map $\tilde{\nabla}_{i,j}\circ \left( \id_{I_i/I_{i+1}}\smash f_{j+1}\right)$ through the zero object by Running Assumption \[ra:1\] Item \[item4ra1\]. Thus, we have the commutative square $$\xymatrix{ I_i/I_{i+1}\smash I_{j+1}\ar[d]^{\id_{I_i/I_{i+1}}\smash f_{j+1}} \ar[r] & 0 \ar[d] \\
I_i/I_{i+1}\smash I_j \ar[r]^{\tilde{\nabla}_{i,j}} & I_{i+j}/I_{i+j+1} }$$ and, taking vertical cofibers, a map $$I_i/I_{i+1}\smash I_j/I_{j+1}\rightarrow I_{i+j}/I_{i+j+1},$$ which we compose with the inclusion map $I_{i+j}/I_{i+j+1}\hookrightarrow E_0^*I_{\bullet}$ to produce our desired map $\nabla_{i,j}: I_i/I_{i+1}\smash I_j/I_{j+1}\rightarrow E_0^*I_{\bullet}$. (Note that all these maps are defined in the model category $\mathcal{C}$, not just in Ho($\mathcal{C}$).)
Filtered coefficient bimodules.
-------------------------------
Now we lay out all the same definitions as in the previous subsection, but with extra data: we assume that, along with our filtered commutative monoid object $I_{\bullet}$, we also have a choice of filtered bimodule object $M_{\bullet}$. These definitions are necessary in order to get a THH-May spectral sequence with coefficients, something that has already proven important in practical computations (e.g. of $THH(K(\mathbb{F}_q))$, in a paper by G. Angelini-Knoll [@K1localsphere]), but these definitions involve some repetition of those in the previous subsection, so we try to present them concisely.
\[def of dec filt bimodule\] Suppose we have a cofibrant decreasingly filtered commutative monoid $I_{\bullet}$ of a commutative monoid $I_0$ in $\mathcal{C}$. We therefore have structure maps $\rho_{i,j}:I_i\wedge I_j \lra I_{i+j}$ and maps $f_i:I_{i+1}\lra I_i$ for each integer $i$ and $j$. Let the sequence $$\xymatrix{\dots \ar[r]^{g_{n+1}} & M_n \ar[r]^{g_{n}}& \dots \ar[r]^{g_2} & M_1\ar[r]^{g_1} & M_0}$$ be a cofibrant decreasingly filtered object in $\mathcal{C}$ in the sense of \[def of dec filt obj\]. We call the sequence a [*cofibrant decreasingly filtered $I_{\bullet}$-bimodule*]{} if we have maps $$\psi_{i,j}^r : M_i \wedge I_j \lra M_{i+j}$$ $$\psi_{i,j}^{\ell} : I_i \wedge M_j \lra M_{i+j}$$ satisfying the following axioms. To write the axioms it will be helpful to use the notation, $$g_n^m=g_{m+1} \circ g_{m+2}\circ ... \circ g_n \text{ \hspace{.03in} for } n\ge m.$$
1. (Associativity.) The relations, $$\psi_{i+j,k}^r \circ (\psi_{i,j}^r \wedge \id_{I_k}) = \psi_{i,j+k}^r \circ ( \id_{M_i} \wedge \rho_{j,k})$$ $$\psi_{i+j,k}^r \circ ( \psi_{i,j}^{\ell} \wedge \id_{I_k} ) = \psi_{i,j+k}^{\ell} ( \id_{I_i} \wedge \psi_{j,k}^r )$$ $$\psi_{i,j+k}^{\ell} \circ (\id_{I_i} \wedge \psi_{j,k}^{\ell}) = \psi_{i+j,k}^{\ell} (\rho_{i,j} \wedge \id_{M_k} )$$ hold for all integers $i$, $j$, and $k$.
2. (Compatibility.) For $i,j, i', j'$ integers such that $i>i'$, $j>j'$, the following relations hold: $$g_{i+j}^{i'+j'} \circ \psi_{i,j}^r = \psi_{i',j'}^r \circ ( g_{i}^{i'} \wedge f_{j}^{j'})$$ $$g_{i+j}^{i'+j'} \circ \psi_{i,j}^{\ell}= \psi_{i',j'}^{\ell} \circ ( f_{i}^{i'} \wedge g_{j}^{j'}).$$
3. (Unitality.) The diagrams, $$\xymatrix{ M_0 & \ar[l]_{\simeq} M_0\smash S \ar[d]^{\text{id}_{M_0}\smash \eta} & S \wedge M_0 \ar[r]^{\simeq} \ar[d]_{\eta \wedge \id_{M_0}}& M_0 \\
& \ar[ul]^{\psi_{0,0}^r} M_0\smash I_0 & I_0\wedge M_0 \ar[ur]_{\psi_{0,0}^{\ell}} & \\
}$$ commute.
We say that the cofibrant decreasingly filtered bimodule $M_{\bullet}$ is [*symmetric*]{} if the factor-swap isomorphism $$\chi_{i,j}:M_i\wedge I_j \lra I_j\wedge M_i \text{ } \text{ satisfies } \text{ }\psi_{i,j}^{r}= \psi_{j,i}^{\ell}\circ \chi_{i,j} .$$
As in Definition \[def of Hausdorff filt\], we will say that a cofibrant decreasingly filtered $I_{\bullet}$-bimodule $M_{\bullet}$ is *Hausdorff* if $\holim_n M_n$ is weakly equivalent to the zero object, and we will say that $M_{\bullet}$ is *finite* if there exists $n\in\mathbb{N}$ such that $f_{m}: M_m\rightarrow M_{m-1}$ is a weak equivalence for all $m>n$.
Just as a cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$ can be considered as a cofibrant object in $\Comm \mathcal{C}^{\mathbb{N}^{\op}}$ (See Remark \[rem filt comm mon\]), we can define a cofibrant decreasingly filtered $I_{\bullet}$-bimodule as an cofibrant symmetric $I_{\bullet}$-bimodule in the category of functors $\mathcal{C}^{\mathbb{N}^{\op}}$.
[**(The associated graded bimodule.)**]{} Let $I_{\bullet}$ be a cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$, and let $M_{\bullet}$ be a cofibrant decreasingly filtered $I_{\bullet}$-bimodule. By $E_0^*M_{\bullet}$, the [*associated graded bimodule of $M_{\bullet}$*]{}, we mean the graded $E_0^*I_{\bullet}$-bimodule object in $\mathcal{C}$ defined as follows:
- “additively,” that is, as an object of $\mathcal{C}$, $$E_0^*M_{\bullet} \cong \coprod_{n\in\mathbb{N}} M_n/M_{n+1}.$$
- The left action map $E_0^*I_{\bullet} \smash E_0^*M_{\bullet} \rightarrow E_0^*M_{\bullet}$ is defined as follows. Using the fact that $f_j$ and $g_{i+j+1}$ are cofibrations in the diagram, $$\xymatrix{
I_{j+1} \wedge M_{i} \ar[d]_{f_j \wedge 1_{M_i}} \ar[r]^{\psi^{\ell}_{j+1,i}} & M_{i+j+1} \ar[d]^{g_{i+j+1}} \\
I_{j} \wedge M_{i} \ar[r]_{\psi^{\ell}_{j,i}} & M_{i+j} \\
}$$ we get a map $\tilde{\Psi} _{i,j}: I_j/I_{j+1}\smash M_i \lra M_{i+j}/M_{i+j+1} $. We then observe that the diagram, $$\xymatrix{ I_{j+1}\smash M_{i+1}\ar[rd]^{\id_{I_{j+1}}\smash g_{i+1}}\ar[dd]^{f_{j+1}\smash \id_{M_{i+1}}}\ar[rr]^{\psi^{\ell}_{j+1,i+1}} & & M_{i+j+2}\ar[rd]^{g_{i+j+2}}\ar[dd]^(.65){g_{i+j+2}} & \\
& I_{j+1}\smash M_i \ar[dd]^(.7){f_{j+1}\smash \id_{M_i}} \ar[rr]^(.35){\psi^{\ell}_{j+1,i}} & & M_{i+j+1}\ar[dd]^{f_{i+j+1}} \\
I_j\smash M_{i+1}\ar[rd]^{\id_{I_j}\smash g_{i+1}} \ar[rr]^(.35){\psi^{\ell}_{j,i+1}} \ar[dd] & & M_{i+j+1}\ar[rd]^{g_{i+j+1}}\ar[dd] & \\
& I_j\smash M_i\ar[rr]^(.35){\psi^{\ell}_{j,i}} \ar[dd] & & M_{i+j} \ar[dd] \\
I_j/I_{j+1}\smash M_{i+1} \ar[rd]_{\id_{I_j/I_{j+1}}\smash g_{i+1}} \ar[rr]^(.35){\tilde{\Psi}_{j,i+1}} & & M_{i+j+1}/M_{i+j+2}\ar[rd]^{0} & \\
& I_j/I_{j+1}\smash M_i \ar[rr]^{\tilde{\Psi}_{j,i}} & & M_{i+j}/M_{i+j+1} },$$ produces a factorization of the the composite $ (\id_{I_{j}/I_{j+1}}\wedge g_{i+1})\circ \tilde{\Psi}_{i,j}$ through the zero object. There is a commutative diagram $$\xymatrix{
I_j/I_{j+1} \wedge M_{i+1} \ar[r] \ar[d]_{\id_{I_{j}/I_{j+1}}\wedge g_i} & 0 \ar[d] \\
I_j/I_{j+1} \wedge M_i \ar[r]^{\tilde{\Psi}_{i,j}} & M_{i+j}/ M_{i+j+1} \\ }$$ which, since the maps $f_i$ and $g_i$ are cofibrations by Definition \[def of dec filt bimodule\], produces a map, $$\bar{\Psi}_{i,j}: I_{j}/I_{j+1} \wedge M_{i}/M_{i+1} \lra M_{i+j} /M_{i+j+1}.$$ We therefore have a module map, $$\Psi: \coprod_{j\in\mathbb{N}} I_j/I_{j+1} \wedge \coprod_{i\in\mathbb{N}} M_i/M_{i+1} \lra \coprod_{k\in\mathbb{N}} M_k / M_{k+1},$$ given on each wedge factor by the composite, $$\xymatrix{ I_{j} / I_{j+1} \wedge M_{i}/M_{i+1} \ar[r]^{\bar{\Psi}_{i,j}} & M_{i+j} /M_{i+j+1} \ar[r]^{ \iota_{i+j} } & \coprod_{k\in \mathbb{N} } M_k / M_{k+1}, }$$ where $\iota_{i+j}$ is the inclusion.
- The right action map $E_0^*M_{\bullet} \smash E_0^*I_{\bullet} \rightarrow E_0^*M_{\bullet}$ is defined in the same way as above and the symmetry of $M_{\bullet}$ along with these maps gives $E_0^*M_{\bullet}$ the structure of a symmetric bimodule over $E_0^*I_\bullet$.
Tensoring and pretensoring over simplicial sets.
------------------------------------------------
We will write $fSet$ for the category of finite sets. First we introduce the [*pretensor product*]{}, which is merely a convenient notation for the well-known “Loday construction” of [@MR981743]:
\[def of tensoring\] We define a functor $$- \tilde{\otimes} -: sf\Sets \times \Comm\mathcal{C}\rightarrow
s\Comm\mathcal{C},$$ which we call the [*pretensor product*]{}, as follows. If $X_{\bullet}$ is a simplicial finite set and $A$ a commutative monoid in $\mathcal{C}$, the simplicial commutative monoid $X_{\bullet}\tilde{\otimes} A$ is given by:
- For all $n\in\mathbb{N}$, the $n$-simplex object $$(X_{\bullet} \tilde{\otimes} A)_n =
\coprod_{x\in X_n} A$$ is a coproduct, [*taken in $\Comm(\mathcal{C})$*]{}, of copies of $A$, with one copy for each $n$-simplex $x\in X_n$. Recall that the categorical coproduct in $\Comm(\mathcal{C})$ is the smash product $\smash$.
- For all positive $n\in\mathbb{N}$ and all $0\leq i\leq n$, the $i$th face map $$d_i : (X_{\bullet} \tilde{\otimes} A)_n \rightarrow
(X_{\bullet} \tilde{\otimes} A)_{n-1}$$ is given on the component corresponding to an $n$-simplex $x\in X_n$ by the map $$A \rightarrow \coprod_{y\in X_{n-1}} A$$ which is inclusion of the coproduct factor corresponding to the $(n-1)$-simplex $\delta_i(x)$.
- For all positive $n\in\mathbb{N}$ and all $0\leq i\leq n$, the $i$th degeneracy map $$s_i : (X_{\bullet} \tilde{\otimes} A)_n \rightarrow
(X_{\bullet} \tilde{\otimes} A)_{n+1}$$ is given on the component corresponding to an $n$-simplex $x\in X_n$ by the map $$A \rightarrow \coprod_{y\in X_{n+1}} A$$ which is inclusion of the coproduct factor corresponding to the $(n+1)$-simplex $\sigma_i(x)$.
We have defined the pretensor product on objects; it is then defined on morphisms in the evident way.
We define the [*tensor product*]{} $$- \otimes -: sf\Sets \times \Comm\mathcal{C}\rightarrow
\Comm\mathcal{C}$$ as the geometric realization of the pretensor product: $$X_{\bullet}\otimes A = \left| X_{\bullet}\tilde{\otimes} A\right| .$$
It is easy to check that $X_{\bullet}\tilde{\otimes} A$ is indeed a simplicial object in $\Comm(\mathcal{C})$.
In the case that $\mathcal{C}$ is the category of symmetric spectra, the tensor product $X_{\bullet}\otimes A$ agrees with the tensoring of commutative ring spectra over simplicial sets. (This is proven in [@MR1473888], although using (an early incarnation of) $S$-modules, rather than symmetric spectra; the symmetric monoidal Quillen equivalence of $S$-modules and symmetric spectra, as in [@MR1819881], then gives us the same result in symmetric spectra.) The same is true when $E$ is a commutative $S$-algebra and $\mathcal{C}$ is the category of $E$-modules. In fact, the tensor product defined in \[def of tensoring\] agrees with the tensoring over simplicial sets in every case of a symmetric monoidal model category whose category of commutative monoids is tensored over simplicial sets that is known to the authors.
In particular, when $X_{\bullet}$ is the minimal simplicial model for the circle, i.e., $X_{\bullet} = (\Delta[1]/d\Delta[1])_{\bullet}$, then $X_{\bullet}\tilde{\otimes} A$ is the cyclic bar construction on $A$, and hence (by the main result of [@MR1473888]) $X_{\bullet}\otimes A$ agrees with the topological Hochschild homology ring spectrum $THH(A,A)$.
For other simplicial sets, $X_{\bullet}\otimes A$ is regarded as a generalization of topological Hochschild homology, e.g. as “higher order Hochschild homology” in [@MR1755114]. For the definition of tensoring a simplicial finite set with a commutative monoid with coefficients in a bimodule, see the appendix.
The fundamental theorem of the May filtration.
----------------------------------------------
The fundamental theorem of the May filtration relies on the following lemma. This lemma also occurs as Lemma 4.7 in May’s [@MR1867203]. (May’s treatment of this particular lemma addresses the question of the compatibility of cofiber sequence \[useful cof seq\] with other cofiber sequences that arise naturally in this context, but this is omitted from our treatment.)
\[n=2 case of hard lemma\] Suppose $I,J$ are objects of $\mathcal{C}$ and $I^{\prime}\rightarrow I$ and $J^{\prime}\rightarrow J$ are cofibrations. Suppose $I,J,I^{\prime},J^{\prime}$ are cofibrant. Let $P$ denote the pushout (which, by Running Assumption \[ra:1\], is a homotopy pushout) of the diagram $$\xymatrix{ I^{\prime}\smash J^{\prime}\ar[r]\ar[d] & I^{\prime}\smash J \\
I\smash J^{\prime} & } .$$ Let $f: P \rightarrow I\smash J$ denote the canonical map given by the universal property of the pushout. Then $f$ is a cofibration by the pushout product axiom in the definition of a monoidal model category, as in [@MR1734325]. Then the cofiber of $f$ is isomorphic to $(I/I^{\prime})\smash (J/J^{\prime})$, where $I/I^{\prime}$ and $J/J^{\prime}$ denote the cofibers of $I^{\prime}\rightarrow I$ and $J^{\prime}\rightarrow J$, respectively. So $$\label{useful cof seq} P \stackrel{f}{\longrightarrow} I\smash J \rightarrow (I/I^{\prime})\smash (J/J^{\prime})$$ is a cofiber sequence.
We define three diagrams in $\mathcal{C}$: $$\begin{aligned}
X_1 & = & \left( \vcenter{\xymatrix{
I^{\prime}\smash J^{\prime}\ar[r]\ar[d] & I\smash J^{\prime} \\
I^{\prime}\smash J & }}\right) \\
X_2 & = & \left( \vcenter{\xymatrix{
I^{\prime}\smash J\ar[r]\ar[d] & I\smash J \\
I^{\prime}\smash J & }}\right) \\
X_3 & = & \left( \vcenter{\xymatrix{
I^{\prime}\smash (J/J^{\prime})\ar[r]\ar[d] & I\smash (J/J^{\prime}) \\
0 & }}\right) .\end{aligned}$$ The obvious maps of diagrams $$X_1 \rightarrow X_2\rightarrow X_3$$ are trivially seen to be levelwise cofiber sequences. Since colimits commute with colimits, we then have a commutative diagram where each row is a cofiber sequence: $$\xymatrix{
\colim X_1 \ar[r]\ar[d]^{\cong} & \colim X_2 \ar[r]\ar[d]^{\cong} & \colim X_3\ar[d]^{\cong} \\
P \ar[r] & I\smash J \ar[r] & (I/I^{\prime})\smash (J/J^{\prime}) .}$$
\[def of may filt 0 1\] [**(Some important colimit diagrams I.)**]{}
- If $S$ is a finite set, we will equip the set $\mathbb{N}^S$ of functions from $S$ to $\mathbb{N}$ with the strict direct product order, that is, $x \leq y$ in $\mathbb{N}^S$ if and only if $x(s)\leq y(s)$ for all $s\in S$.
- If $T \stackrel{f}{\longrightarrow} S$ is a function between finite sets, let $\mathbb{N}^T \stackrel{\mathbb{N}^f}{\longrightarrow} \mathbb{N}^S$ be the function of partially-ordered sets defined by $$\left(\mathbb{N}^f(x)\right)(s) = \sum_{\{ t\in T: f(t) = s\}} x(t) .$$ One checks easily that this defines a functor $$\mathbb{N}^{-}: f\Sets \rightarrow \POSets$$ from the category of finite sets to the category of partially-ordered sets.
- For each finite set $S$, we also equip $\mathbb{N}^S$ with the $L^1$-norm: $$\begin{aligned}
\left| - \right|: \mathbb{N}^S & \rightarrow & \mathbb{N} \\
\left| x\right| & = & \sum_{s\in S} x(s).\end{aligned}$$ One checks easily that, if $T \stackrel{f}{\longrightarrow} S$ is a function between finite sets, the induced map $\mathbb{N}^f$ preserves the $L^1$ norm, that is, $$\left| x\right| = \left| \mathbb{N}^f(x)\right|$$ for all $x\in \mathbb{N}^T$.
\[def of may filt 0 2\] [**(Some important colimit diagrams II.)**]{}
- - If $S$ is a finite set, for each $n\in\mathbb{N}$ we will let $\mathcal{D}^S_n$ be the sub-poset of $\mathbb{N}^S$ consisting of all functions $x\in\mathbb{N}^S$ such that $\left| x \right| \geq n$.
- If $T \stackrel{f}{\longrightarrow} S$ is a function between finite sets, let $\mathcal{D}^T_n \stackrel{\mathcal{D}^f_n}{\longrightarrow} \mathcal{D}^S_n$ be the function of partially-ordered sets defined by restricting $\mathbb{N}^f$ to $\mathcal{D}^T_n$. One checks easily that, this assignment $\xymatrix{ f \ar@{|->}[r]& D_{n;x}^f}$ respects composition of functions in the variable $f$.
- - For each $x\in \mathbb{N}^S$ and each $n\in \mathbb{N}$, let $\mathcal{D}^S_{n; x}$ denote the following sub-poset of $\mathbb{N}^S$: $$\mathcal{D}^S_{n; x} = \left\{ y \in \mathbb{N}^S: y \geq x, \mbox{\ and\ } \left| y\right| \geq n + \left| x\right| \right\} .$$ So, for example, $\mathcal{D}^S_{n; \vec{0}} = \mathcal{D}^S_n$, where $\vec{0}$ is the constant zero function.
- If $T \stackrel{f}{\longrightarrow} S$ is a function between finite sets and $x\in \mathbb{N}^T$ and $n\in\mathbb{N}$, let $\mathcal{D}^T_{n;x} \stackrel{\mathcal{D}^f_{n;x}}{\longrightarrow} \mathcal{D}^S_{n;\mathcal{D}^f_n(x)}$ be the function of partially-ordered sets defined by restricting $\mathbb{N}^f$ to $\mathcal{D}^T_{n;x}$. One checks easily that, for each $n\in\mathbb{N}$ and each $x\in\mathbb{N}^T$, this defines a functor $$\mathcal{D}^{-}_{n;x}: f\Sets \rightarrow \POSets$$ from the category of finite sets to the category of partially-ordered sets.
\[def of may filt 0 3\] [**(Some important colimit diagrams III.)**]{}
- Let $S$ be a finite set and let $n$ be a nonnegative integer. We write $\mathcal{E}^S_n$ for the set $$\mathcal{E}^S_{n,k} = \left\{ x\in \{ 0,1, \dots , n\}^{S} : \sum_{s\in S} x(s) \geq k \right\}$$ where $k\ge n$. We partially-order $\mathcal{E}^S_{n,k}$ by the strict direct product order, i.e., $x^{\prime}\leq x$ if and only if $x^{\prime}(s)\leq x(s)$ for all $s \in S$. When $n=k$, we simply write $\mathcal{E}^S_{n}$ for this poset.
- The definition of $\mathcal{E}^S_n$ is natural in $S$ in the following sense: if $T\stackrel{f}{\longrightarrow} S$ is a injective map of finite sets, then $\mathbb{N}^f$ naturally restricts to a function $\mathcal{E}^T_n\rightarrow \mathcal{E}^S_n$.
\[def of may filt 0 4\] [**(Some important colimit diagrams IV.)**]{}
- Suppose $I_{\bullet}$ is a cofibrant decreasingly filtered object in $\mathcal{C}$ and suppose $S$ is a finite set. We will let $$\mathcal{F}^S(I_{\bullet}): \left(\mathbb{N}^S\right)^{\op}\rightarrow \mathcal{C}$$ be the functor sending $x$ to the smash product $$\smash_{s\in S} I_{x(s)},$$ and defined on morphisms in the apparent way, and let $$\mathcal{F}^S_n(I_{\bullet}): \left(\mathcal{D}^S_n\right)^{\op}\rightarrow \mathcal{C}$$ be the functor which is the composite of $\mathcal{F}^S(I_{\bullet})$ with the inclusion of $\mathcal{D}^S_n$ into $\mathbb{N}^S$: $$\left(\mathcal{D}^S_n\right)^{\op} \hookrightarrow
\left(\mathbb{N}^S\right)^{\op}\stackrel{\mathcal{F}^S(I_{\bullet})}{\longrightarrow}
\mathcal{C}.$$
- If $x\in \mathcal{D}^S_n$, we will write $\mathcal{F}^S_{n; x}(I_{\bullet})$ for the restriction of the diagram $\mathcal{F}^S(I_{\bullet})$ to $\mathcal{D}^S_{n; x}$, i.e., $\mathcal{F}^S_{n; x}(I_{\bullet})$ is the composite $$(\mathcal{D}^S_{n; x})^{\op} \hookrightarrow \left(\mathbb{N}^S\right)^{\op}\stackrel{\mathcal{F}^S(I_{\bullet})}{\longrightarrow}
\mathcal{C}.$$
- Finally, let $\mathcal{M}^S_n(I_{\bullet})$ denote the colimit $$\mathcal{M}^S_n(I_{\bullet}) = \colim \left(\mathcal{F}^S_n(I_{\bullet})\right)$$ in $\mathcal{C}$. By the natural inclusion of $\mathcal{D}^S_n$ into $\mathcal{D}^S_{n-1}$, we now have a sequence in $\mathcal{C}$: $$\label{may filt diagram} \dots \rightarrow \mathcal{M}^S_3(I_{\bullet}) \rightarrow \mathcal{M}^S_2(I_{\bullet})
\rightarrow \mathcal{M}^S_1(I_{\bullet}) \rightarrow \mathcal{M}^S_0(I_{\bullet}) \cong \smash_{s\in S} I_0.$$ We refer to the functor $\mathbb{N}^{\op}\rightarrow \mathcal{C}$ given by sending $n$ to $\mathcal{M}^S_n(I_{\bullet})$ as the [*May filtration on $\smash_{s\in S}I_0$*]{}.
- The May filtration is functorial in $S$ in the following sense: if $T\stackrel{f}{\longrightarrow} S$ is a map of finite sets, we have a functor $$\begin{aligned}
\mathcal{D}^f_n: \mathcal{D}^{T}_n & \rightarrow & \mathcal{D}^S_n \\
\left(\mathcal{D}^f_n(x)\right)(s) & \mapsto & \sum_{\left\{ t\in T: f(t) = s\right\}} x(t)\end{aligned}$$ and a map of diagrams from $\mathcal{F}^T_n(I_{\bullet})$ to $\mathcal{F}^S_n(I_{\bullet})$ given by sending the object $$\mathcal{F}^T_n(I_{\bullet})(x) = \smash_{t\in T}I_{x(t)}$$ to the object $$\mathcal{F}^S_n(I_{\bullet})(\mathcal{D}^f_n(x)) = \smash_{s\in S} I_{\Sigma_{\{ t\in T: f(t) = s\}} x(t)}$$ by the map $$\smash_{t\in T}I_{x(t)} \rightarrow \smash_{s\in S} I_{\Sigma_{\{ t\in T: f(t) = s\}} x(t)}$$ given as the smash product, across all $s\in S$, of the maps $$\smash_{\{ t\in T: f(t) = s\}} I_{x(t)} \rightarrow I_{\Sigma_{\{ t\in T: f(t) = s\}} x(t)}$$ given by multiplication via the maps $\rho$ of Definition \[def of dec filt comm mon\].
To really make Definition \[def of may filt 0 4\] precise, we should say in which order we multiply the factors $I_{x(t)}$ using the maps $\rho$; but the purpose of the associativity and commutativity axioms in Definition \[def of dec filt comm mon\] is that any two such choices commute, so any choice of order of multiplication will do.
\[def of may filt\] [**(Definition of the May filtration.)**]{} If $I_{\bullet}$ is a cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$ and $X_{\bullet}$ a simplicial finite set, by the [*May filtration on $X_{\bullet}\tilde{\otimes} I_0$*]{} we mean the functor $\mathcal{M}^{X_{\bullet}}(I_{\bullet}): \mathbb{N}^{\op} \rightarrow \mathcal{C}^{\Delta^{\op}}$ given by sending a natural number $n$ to the simplicial object of $\mathcal{C}$ $$\xymatrix{ \mathcal{M}^{X_0}_n(I_{\bullet}) \ar[r] &
\mathcal{M}^{X_1}_n(I_{\bullet}) \ar@<1ex>[l] \ar@<-1ex>[l] \ar@<1ex>[r] \ar@<-1ex>[r] &
\mathcal{M}^{X_2}_n(I_{\bullet}) \ar@<2ex>[l] \ar@<-2ex>[l] \ar[l] \ar@<2ex>[r] \ar@<-2ex>[r] \ar[r] &
\dots \ar@<-3ex>[l] \ar@<-1ex>[l] \ar@<1ex>[l] \ar@<3ex>[l] ,}$$ with $\mathcal{M}^{X_i}_n(I_{\bullet})$ defined as in Definition \[def of may filt 0 4\], and with face and degeneracy maps defined as follows:
- The face map $$d_i: \mathcal{M}^{X_j}_n(I_{\bullet}) \rightarrow \mathcal{M}^{X_{j-1}}_n(I_{\bullet})$$ is the colimit of the map of diagrams $$\mathcal{F}^{X_j}_n(I_{\bullet}) \rightarrow \mathcal{F}^{X_{j-1}}_n(I_{\bullet})$$ induced, as in Definition \[def of may filt 0 4\], by $\delta_i: X_j\rightarrow X_{j-1}$.
- The degeneracy map $$s_i: \mathcal{M}^{X_j}_n(I_{\bullet}) \rightarrow \mathcal{M}^{X_{j+1}}_n(I_{\bullet})$$ is the colimit of the map of diagrams $$\mathcal{F}^{X_j}_n(I_{\bullet}) \rightarrow \mathcal{F}^{X_{j+1}}_n(I_{\bullet})$$ induced, as in Definition \[def of may filt 0 4\], by $\sigma_i: X_j\rightarrow X_{j+1}$.
\[rem on structure\] Note that the associative, commutative, and unital multiplications on the objects $I_i$, via the maps $\rho$ of Definition \[def of dec filt comm mon\], also yield (by taking smash products of the maps $\rho$) associative, commutative, and unital multiplication natural transformations $$\label{action maps on F} \mathcal{F}^S_m(I_{\bullet})\smash \mathcal{F}^S_n(I_{\bullet})\rightarrow\mathcal{F}^S_{m+n}(I_{\bullet}),$$ hence, on taking colimits, associative, commutative, and unital multiplication maps $$\mathcal{M}^S_m(I_{\bullet})\smash \mathcal{M}^S_n(I_{\bullet})\rightarrow\mathcal{M}^S_{m+n}(I_{\bullet}),$$ i.e., the functor $$\begin{aligned}
\mathbb{N}^{\op} & \rightarrow & \mathcal{C} \\
n & \mapsto & \mathcal{M}^S_n(I_{\bullet}) \end{aligned}$$ is a cofibrant decreasingly filtered commutative monoid, in the sense of Definition \[def of dec filt comm mon\]. Note furthermore that, if $f: T\rightarrow S$ is a map of finite sets, then the induced maps $\mathcal{F}^T_m(I_{\bullet}) \rightarrow \mathcal{F}^S_m(I_{\bullet})$ commute with the multiplication maps \[action maps on F\], and so $\mathcal{M}^T_{\bullet}(I_{\bullet})\rightarrow\mathcal{M}^S_{\bullet}(I_{\bullet})$ is a map of cofibrant decreasingly filtered commutative monoids.
Consequently, for any simplicial finite set $X_{\bullet}$, we have that $\mathcal{M}_{\bullet}^{X_{\bullet}}(I_{\bullet})$ is a simplicial object in the category of cofibrant decreasingly filtered commutative monoids in $\mathcal{C}$. Since geometric realization commutes with the monoidal product in $\mathcal{C}$ by our running assumptions on $\mathcal{C}$, this in turn implies that the geometric realization $\left| \mathcal{M}_{\bullet}^{X_{\bullet}}(I_{\bullet})\right|$ of $\mathcal{M}_{\bullet}^{X_{\bullet}}(I_{\bullet})$ is a cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$ by Running Assumption \[ra:2\]. It can easily be shown that $\mathcal{M}_n^{X_{\bullet}}(I_{\bullet})$ satisfies Running Assumption \[ra:2\]’s Item \[it2\] for each $n$ whenever $I_0$ is cofibrant as an object in $\Comm \mathcal{C}$. Running Assumption \[ra:2\]’s Item \[it1\] and Item \[it3\] are satisfied by definition of $\mathcal{M}_n^{X_{\bullet}}(I_{\bullet})$ and by definition of the maps $$\mathcal{M}_n^{X_{\bullet}}(I_{\bullet})\rightarrow \mathcal{M}_{n-1}^{X_{\bullet}}(I_{\bullet}).$$ Therefore, the decreasingly filtered commutative monoid $\left| \mathcal{M}_{\bullet}^{X_{\bullet}}(I_{\bullet})\right|$ is a cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$. Recall that, by the main theorem of the authors’ paper [@asReedy], there is a symmetric monoidal model structure on symmetric spectra in simplicial sets in which Running Assumption \[ra:2\] holds.
\[cofiber computation lemma\] Let $I_{\bullet}$ be a cofibrant decreasingly filtered object in $\mathcal{C}$ and let $S$ be a finite set. Suppose $n\in \mathbb{N}$. We have the canonical inclusion of categories $\iota: \mathcal{D}^S_{n+1} \rightarrow \mathcal{D}^S_n$. Let $\tilde{\mathcal{F}}_{n+1}^S(I_{\bullet})$ be the left Kan extension of $\mathcal{F}_{n+1}^S(I_{\bullet}): (\mathcal{D}^S_{n+1})^{\op}\rightarrow \mathcal{C}$ along $\iota^{\op}$, i.e., if we write $$Kan: \mathcal{C}^{(\mathcal{D}_{n+1}^S)^{\op}} \rightarrow \mathcal{C}^{(\mathcal{D}_{n}^S)^{\op}}$$ for the left adjoint of the map $\mathcal{C}^{(\mathcal{D}_{n}^S)^{\op}} \rightarrow \mathcal{C}^{(\mathcal{D}_{n+1}^S)^{\op}}$ induced by $\iota$, then $$\tilde{\mathcal{F}}_{n+1}^S(I_{\bullet}) = Kan(\mathcal{F}_{n+1}^S(I_{\bullet})).$$ By the universal property of this Kan extension, we have a canonical map $c: \tilde{\mathcal{F}}_{n+1}^S(I_{\bullet}) \rightarrow \mathcal{F}_n^S(I_{\bullet})$.
Then the cofiber of the map $$\colim \left( \tilde{\mathcal{F}}_{n+1}^S(I_{\bullet})\right) \stackrel{\colim c}{\longrightarrow} \colim \left(\mathcal{F}_n^S(I_{\bullet})\right),$$ computed in $\mathcal{C}$, is isomorphic to the coproduct in $\mathcal{C}$ $$\coprod_{\{ x\in \mathbb{N}^S: \left| x\right| = n\}} \left( \left( \smash_{s\in S} I_{x(s)}\right)/\left( \colim \mathcal{F}_{1; x}^S(I_{\bullet})\right) \right) .$$ This isomorphism is natural in the variable $S$.
(It is a elementary exercise in combinatorics to show that there are $\binom{n+\hash(S)}{n}$ summands in this coproduct, where $\hash(S)$ is the cardinality of $S$.)
One knows that the left Kan extension of $\mathcal{F}_{n+1}^S(I_{\bullet})$ agrees with $\mathcal{F}_{n+1}^S(I_{\bullet})$ where both are defined, so $$\begin{aligned}
\tilde{\mathcal{F}}_{n+1}^S\left(I_{\bullet}\right)\left( x\right) & \cong & \mathcal{F}_{n+1}^S\left(I_{\bullet}\right)\left( x\right) \\
& = & \smash_{s\in S} I_{x(s)}\end{aligned}$$ for all $x\in \mathcal{D}_{n+1}^S \subseteq \mathcal{D}_{n}^S$. The elements of $\mathcal{D}_{n}^S$ which are [*not*]{} in $\mathcal{D}_{n+1}^S$ are those $x$ such that $\left| x\right| = n$, and by the usual basic results (see e.g. [@MR1712872]) on Kan extensions one knows that, for all $x$ such that $\left| x\right| = n$, we have an isomorphism of $\tilde{\mathcal{F}}_{n+1}^S(I_{\bullet})\left( x\right)$ with the colimit of the values of $\mathcal{F}_{n+1}^S(I_{\bullet})$ over those elements of $(\mathcal{D}_{n+1}^S)^{\op}$ which map to $x$, i.e., the colimit of the values of $\mathcal{F}_{n+1}^S(I_{\bullet})$ over $(\mathcal{D}_{1, x}^S)^{\op}\subseteq (\mathcal{D}_{n+1}^S)^{\op}$, i.e., $\colim \left(\mathcal{F}_{1, x}^S(I_{\bullet})\right)$.
The map $c$ can be shown to be a cofibration by iterated use of the pushout product axiom, so the cofiber of $c$ is a homotopy cofiber. By the previous paragraph the levelwise cofiber $\cof c: (\mathcal{D}_{n}^S)^{\op} \rightarrow\mathcal{C}$ of the natural transformation $c$ is given as follows: $$(\cof c)\left( x \right) \cong \left\{ \begin{array}{lll} 0 & \mbox{\ if\ } & \left| x\right| > n \\
\left(\mathcal{F}_n^S(I_{\bullet})\right)/\left(\colim \left(\mathcal{F}_{1, x}^S(I_{\bullet})\right)\right) & \mbox{\ if\ } &
\left| x\right| = n.\end{array}\right.$$ Hence, on taking colimits, we have $$\begin{aligned}
\cof \colim c & \cong & \colim \cof c \\
& = & \coprod_{\{ x\in\mathbb{N}^S: \left| x\right| = n\}} \left( \left( \smash_{s\in S} I_{x(s)}\right)/\left( \colim \mathcal{F}_{1; x}^S(I_{\bullet})\right) \right) ,\end{aligned}$$ as claimed.
\[rather hard lemma\] Suppose $S$ is a finite set and suppose $Z_{s,1} \rightarrow Z_{s,0}$ is a cofibration for each $s\in S$. Suppose the objects $Z_{s,1},Z_{s,0}$ are all cofibrant. Let $\mathcal{G}_S: (\mathcal{E}^S_1)^{\op}\rightarrow \mathcal{C}$ be the functor given on objects by $$\mathcal{G}_S(x) = \smash_{s\in S} Z_{s,x(s)},$$ and given on morphisms in the obvious way.
Then the smash product $$\smash_{s\in S} Z_{s,0} \rightarrow \smash_{s\in S} \left( Z_{s,0}/Z_{s,1}\right)$$ of the cofiber projections $Z_{s,0} \rightarrow Z_{s,0}/Z_{s,1}$ fits into a cofiber sequence: $$\colim \mathcal{G}_S \rightarrow \smash_{s\in S} Z_{s,0} \rightarrow \smash_{s\in S} \left( Z_{s,0}/Z_{s,1}\right).$$
If the cardinality of $S$ is one, the statement of the lemma is true by the definition of a cofiber.
The case in which the cardinality of $S$ is two is precisely Lemma \[n=2 case of hard lemma\], already proven.
For the case in which the cardinality of $S$ is greater than two, we introduce a notation we will need to use: let $\mathcal{PO}$ denote the category indexing pushout diagrams, i.e., $$\mathcal{PO} = \left( \vcenter{\xymatrix{ & [1^{\prime}] \ar[ld]\ar[rd] & \\ [1] & & [0] }}\right) ,$$ the symbols $[1^{\prime}],[1],[0]$ each representing an object, and the arrows each representing a morphism. We observe that $\mathcal{PO}$ and $\mathcal{E}^S_1$ are not arbitrary small categories but are in fact partially-ordered sets; this simplifies some of the arguments we will give in the rest of the proof.
Suppose the cardinality of $S$ is greater than two. Choose an element $s_0\in S$. We will write $S^{\prime}$ for the complement $$S^{\prime} = \{ s\in S: s\neq s_0\}$$ of $s_0$ in $S$. Define objects $X_1^{\prime},X_2^{\prime},Y_1^{\prime},Y_2^{\prime}$ in $\mathcal{C}$ as follows: $$\begin{aligned}
Y^{\prime}_1 & = & \colim \mathcal{G}_{S^{\prime}} \\
X^{\prime}_1 & = & \smash_{s\in S^{\prime}} Z_{s,0} \\
Y^{\prime}_2 & = & Z_{s_0,1} \\
X^{\prime}_2 & = & Z_{s_0,0}.\end{aligned}$$ Now we apply the statement of the lemma, in the (already proven, above) case $S = \{ 1,2\}$ and using $X_1^{\prime},X_2^{\prime},Y_1^{\prime},Y_2^{\prime}$ in place of $Z_{1,0},Z_{2,0},Z_{1,1},Z_{2,1}$ to obtain a cofiber sequence $$\label{preliminary cof seq} \colim \mathcal{B} \rightarrow \smash_{s\in S} Z_{s,0} \rightarrow \smash_{s\in S} \left( Z_{s,0}/Z_{s,1}\right) ,$$ where $\mathcal{B}$ is the functor $\mathcal{PO}\rightarrow \mathcal{C}$ given by: $$\begin{aligned}
\mathcal{B}\left( [1^{\prime}] \right) & = & \left( \colim \mathcal{G}_{S^{\prime}}\right) \smash Z_{s_0,1} \\
\mathcal{B}\left( [1] \right) & = & \left( \smash_{s\in S^{\prime}} Z_{s_0,0} \right) \smash Z_{s_0,1} \\
\mathcal{B}\left( [0] \right) & = & \left( \colim \mathcal{G}_{S^{\prime}} \right) \smash Z_{s_0,0} .\end{aligned}$$
By Lemma \[n=2 case of hard lemma\], we know that the map $\colim G_{S^{\prime}}\lra \smash_{s\in S'} Z_{s,0}$ is a cofibration in the case $S=\{1,2\}$. Since $\colim \mathcal{B}$ is constructed as a pushout, the pushout product axiom ensures that the map $\colim \mathcal{B}\lra \smash_{s\in S} Z_{s,0}$ is also a cofibration as long as $\colim G_{S^{\prime}}\lra \smash_{s\in S'} Z_{s,0}$ is a cofibration. It suffices to show that $\colim\mathcal{B}\cong \colim \mathcal{G}_S$. This will show that the map $\colim G_{S}\lra \smash_{s\in S}Z_{s,0}$ is a cofibration and allow us to identify the cofiber, thus completing the induction on the cardinality of the set $S$. We can reindex, to describe $\colim\mathcal{B}$ as the colimit of a larger diagram $\mathcal{H}$: $$\begin{aligned}
\mathcal{H}: (\mathcal{E}_1^{S^{\prime}})^{\op}\times \mathcal{PO} & \rightarrow & \mathcal{C} \\
\left( x,[1^{\prime}] \right) & \mapsto & \left( \smash_{s\in S^{\prime}} Z_{s,x(s)}\right) \smash Z_{s_0,1} \\
\left( x,[1] \right) & \mapsto & \left( \smash_{s\in S^{\prime}} Z_{s,0}\right) \smash Z_{s_0,1} \\
\left( x,[0] \right) & \mapsto & \left( \smash_{s\in S^{\prime}} Z_{s,x(s)}\right) \smash Z_{s_0,0} \end{aligned}$$
We have a functor $$\begin{aligned}
P: (\mathcal{E}_1^{S^{\prime}})^{\op}\times \mathcal{PO} & \rightarrow & (\mathcal{E}_1^S)^{\op} \\
\left( x,[1^{\prime}] \right)(s) & = & \left\{ \begin{array}{lll} x(s) & \mbox{\ if\ } & s\neq s_0 \\ 1 & \mbox{\ if\ } & s = s_0 \end{array}\right. \\
\left( x,[1] \right)(s) & = & \left\{ \begin{array}{lll} 0 & \mbox{\ if\ } & s\neq s_0 \\ 1 & \mbox{\ if\ } & s = s_0 \end{array}\right. \\
\left( x,[0] \right)(s) & = & \left\{ \begin{array}{lll} x(s) & \mbox{\ if\ } & s\neq s_0 \\ 0 & \mbox{\ if\ } & s = s_0 \end{array}\right. .\end{aligned}$$
Now we claim that the canonical map $\colim\mathcal{H}\rightarrow\colim\mathcal{G}_S$ given by $P$ is an isomorphism in $\mathcal{C}$. We define a functor $$\begin{aligned}
I: (\mathcal{E}_1^S)^{\op} & \rightarrow & (\mathcal{E}_1^{S^{\prime}})^{\op}\times \mathcal{PO} \\
x & \mapsto & \left\{ \begin{array}{lll}
\left(I^{\prime}(x),[1^{\prime}]\right) & \mbox{\ if\ } & x(s_0)= 1 \\
\left(I^{\prime}(x),[0]\right) & \mbox{\ if\ } & x(s_0)= 0 \end{array}\right. \end{aligned}$$ where $I^{\prime}: (\mathcal{E}_1^S)^{\op}\rightarrow (\mathcal{E}_1^{S^{\prime}})^{\op}$ is the functor given by restriction, i.e., $I^{\prime}(x)(s) = x(s)$ for $s\in S'$. Now we observe some convenient identities: $$\begin{aligned}
(\mathcal{G}_S\circ P)(x,j) & = & \left\{ \begin{array}{lll}
\left( \smash_{s\in S^{\prime}} Z_{s,x(s)}\right) \smash Z_{s_0,1} & \mbox{\ if\ } & j = [1^{\prime}] \\
\left( \smash_{s\in S^{\prime}} Z_{s,x(s)}\right) \smash Z_{s_0,0} & \mbox{\ if\ } & j = [0] \\
\left( \smash_{s\in S^{\prime}} Z_{s,0}\right) \smash Z_{s_0,1} & \mbox{\ if\ } & j = [1] \end{array}\right. \\
& = & \mathcal{H}(x,j), \mbox{\ and} \\
(\mathcal{H}\circ I)(x) & = & \left\{ \begin{array}{lll}
\left( \smash_{s^{\prime}} Z_{s,x(s)}\right) \smash Z_{s_0,1} & \mbox{\ if\ } & x_{s_0} = 1 \\
\left( \smash_{s^{\prime}} Z_{s,x(s)}\right) \smash Z_{s_0,0} & \mbox{\ if\ } & x_{s_0} = 0\end{array}\right. \\
& = & \mathcal{G}_S(x).\end{aligned}$$ We conclude that $P,I$ give mutually inverse maps between $\colim \mathcal{G}_S$ and $\colim \mathcal{H}$, i.e., $\colim \mathcal{G}_S\cong \colim\mathcal{H}$ and hence $\colim \mathcal{G}_S\cong \colim\mathcal{B}$, as desired. So from cofiber sequence \[preliminary cof seq\], we have a cofiber sequence $$\colim \mathcal{G}_S \rightarrow \smash_{s\in S} Z_{s,0} \rightarrow \smash_{s\in S} \left( Z_{s,0}/Z_{s,1}\right),$$ as desired.
From inspection of the colimit diagrams one sees that the cofiber sequence \[preliminary cof seq\] does not depend on the choice of $s_0\in S$, and naturality in $S$ follows.
\[cofinality lemma\] Let $S$ be a finite set, let $n$ be a positive integer, and let $x\in \mathbb{N}^S$. Let $\mathcal{E}_n^S$ and $\mathcal{D}_{n; x}^S$ be as in Definition \[def of may filt 0 2\] and Definition \[def of may filt 0 3\]. Let $J_{n;x}^S$ be the functor (i.e., morphism of partially-ordered sets) defined by $$\begin{aligned}
J_{n;x}^S: \mathcal{E}^S_n & \rightarrow & \mathcal{D}_{n; x}^S \\
(J_{n;x}(y))(s) & = & x(s) + y(s).\end{aligned}$$ Then $J_{n;x}$ has a right adjoint. Consequently $J_{n;x}$ is a cofinal functor, i.e., for any functor $F$ defined on $\mathcal{D}_{n; x}^S$ such that the limit $\lim F$ exists, the limit $\lim (F\circ J_{n;x}^S)$ also exists, and the canonical map $\lim (F\circ J_{n;x}^S) \rightarrow \lim F$ is an isomorphism.
We construct the right adjoint explicitly. Let $K_{n;x}^S$ be the functor defined by $$\begin{aligned}
K_{n;x}^S: \mathcal{D}_{n; x}^S & \rightarrow & \mathcal{E}^S_n \\
(K_{n;x}^S(y))(s) & = & \min\{ n, y(s) - x(s)\}. \end{aligned}$$ (We remind the reader that every element $y\in \mathcal{D}_{n; x}^S$ has the property that $y(s) \geq x(s)$ for all $s\in S$, so $y(s) - x(s)$ will always be nonnegative.)
Now suppose $z\in \mathcal{E}_n^S$ and $y\in \mathcal{D}_{n; x}^S$. Then:
- $z\leq K_{n;x}^S(y)$ if and only if $z(s) \leq K_{n;x}^S(y)(s)$ for all $s\in S$,
- i.e., $z\leq K_{n;x}^S(y)$ if and only if $z(s)\leq \min\{ n, y(s) - x(s)\}$ for all $s\in S$.
- By the definition of $\mathcal{E}_n^S$, $z(s)\leq n$ for all $s\in S$. Hence $z\leq K_{n;x}^S(y)$ if and only if $z(s) \leq y(s) - x(s)$ for all $s\in S$,
- i.e., $z\leq K_{n;x}^S(y)$ if and only if $x(s) + z(s) \leq y(s)$ for all $s\in S$,
- i.e., $z\leq K_{n;x}^S(y)$ if and only if $J_{n;x}^S(z) \leq y$.
Hence $\hom_{\mathcal{E}_n^S}(z, K_{n;x}^S(y))$ is nonempty if and only if $\hom_{\mathcal{D}_{n;x}^S}(J_{n;x}^S(z),y)$ is nonempty. Since $\mathcal{E}_n^S$ and $\mathcal{D}_{n;x}^S$ are partially-ordered sets and hence their hom-sets are either nonempty or have only a single element, we now have a (natural) bijection $$\hom_{\mathcal{E}_n^S}(z, K_{n;x}^S(y)) \cong \hom_{\mathcal{D}_{n;x}^S}(J_{n;x}^S(z),y)$$ which is exactly what we are looking for: $J_{n;x}^S$ is left adjoint to $K_{n;x}^S$.
For the fact that having a right adjoint implies cofinality, see section IX.3 of Mac Lane’s [@MR1712872]. (Mac Lane handles the equivalent dual case.)
[**(Fundamental theorem of the May filtration.)**]{} \[thm on fund theorem\] Let $I_{\bullet}$ be a cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$, and let $X_{\bullet}$ be a simplicial finite set. Then the associated graded commutative monoid $E_0^*\left| \mathcal{M}^{X_{\bullet}}(I_{\bullet})\right|$ of the geometric realization of the May filtration is weakly equivalent, as a commutative graded monoid, to the tensoring $X_{\bullet}\otimes E_0^* I_{\bullet}$ of $X_{\bullet}$ with the associated graded commutative monoid of $I_{\bullet}$: $$E_0^*\left| \mathcal{M}^{X_{\bullet}}(I_{\bullet})\right| \simeq
X_{\bullet}\otimes E_0^* I_{\bullet}.$$
We must compute the filtration quotients $$\left| \mathcal{M}^{X_{\bullet}}_n(I_{\bullet})\right|/\left| \mathcal{M}^{X_{\bullet}}_{n+1}(I_{\bullet})\right|
\cong
\left| \mathcal{M}^{X_{\bullet}}_n(I_{\bullet})/\mathcal{M}^{X_{\bullet}}_{n+1}(I_{\bullet})\right| .$$ We handle this as follows. First, we claim that there exists, for any finite set $S$ and for all $n\in \mathbb{N}$, a cofiber sequence $$\label{putative cof seq 1} \colim \left(\mathcal{F}^S_{n+1}(I_{\bullet})\right)\rightarrow \colim \left(\mathcal{F}^S_{n}(I_{\bullet})\right) \rightarrow \coprod_{x\in \mathbb{N}^S: \left| x\right| = n}
\left( \smash_{s\in S}\left( I_{x(s)}/I_{1+x(s)} \right)\right)$$ in $\mathcal{C}$, natural in $S$. We have already defined (in Definition \[def of may filt 0 4\]) how $\mathcal{F}^S_n$ is natural, i.e., functorial in $S$; by taking the obvious coproduct of quotients, this naturality in $S$ induces a naturality in $S$ on the terms $\coprod_{x\in \mathbb{N}^S: \left| x\right| = n}
\left( \smash_{s\in S}\left( I_{x(s)}/I_{1+x(s)} \right)\right)$ appearing in \[putative cof seq 1\]. The claim that \[putative cof seq 1\] is a cofiber sequence implies that $$\label{important cofiber}
|\mathcal{M}_n^{X_k}(I_{\bullet})| / |\mathcal{M}_{n+1}^{X_k}(I_{\bullet})|\cong \coprod_{x\in \mathbb{N}^{X_k}: \left| x\right| = n}
\left( \smash_{s\in X_k}\left( I_{x(s)}/I_{1+x(s)} \right)\right),$$ and naturally implies the necessary naturality with respect to the face and degeneracy maps.
We now show that the cofiber sequence \[putative cof seq 1\] exists. First, by the universal property of the Kan extension, the map of diagrams $\mathcal{F}^S_{n+1}(I_{\bullet}) \rightarrow \mathcal{F}^S_{n}(I_{\bullet})$ factors uniquely through the map to the left Kan extension $\mathcal{F}^S_{n+1}(I_{\bullet})\rightarrow \tilde{\mathcal{F}}_{n+1}^S(I_{\bullet})$ from Lemma \[cofiber computation lemma\], and the cofiber of the map $\colim (\mathcal{F}^S_{n+1}(I_{\bullet}) )\rightarrow \colim (\mathcal{F}^S_{n}(I_{\bullet}) )$ agrees with the cofiber of the map $\colim (\tilde{\mathcal{F}}^S_{n+1}(I_{\bullet}))\rightarrow \colim (\mathcal{F}^S_{n}(I_{\bullet}) )$. By Lemma \[cofiber computation lemma\], this cofiber is the coproduct $$\coprod_{\{ x\in \mathbb{N}^S: \left| x \right| = n\}} \left( \left( \smash_{s\in S} I_{x(s)}\right)/\left( \colim \mathcal{F}_{1; x}^S(I_{\bullet})\right) \right) .$$ In Lemma \[cofinality lemma\], we showed that the functor $J_{1;x}$ is cofinal, hence that the comparison map of colimits $$\label{consequence of cofinality lemma} \colim \left( \mathcal{F}_{1; x}(I_{\bullet})\circ J_{1;x}\right) \rightarrow \colim \left(\mathcal{F}_{1; x}(I_{\bullet})\right)$$ is an isomorphism. (We here have a colimit, not a limit as in the statement of Lemma \[cofinality lemma\], since $\mathcal{F}_{1; x}(I_{\bullet})$ is a [*contravariant*]{} functor on $\mathcal{D}_{1; x}^S$. Of course Lemma \[cofinality lemma\] still holds in this dual form.)
Now Lemma \[rather hard lemma\] identifies the cofiber $$\left(\smash_{s\in S} I_{x(s)}\right)/\left( \colim \left(\mathcal{F}_{1; x}(I_{\bullet})\circ J_{x}\right) \right)$$ with $\smash_{s\in S}\left( I_{x(s)}/I_{1+x(s)} \right)$, as desired. So we have our cofiber sequence of the form \[putative cof seq 1\].
All isomorphisms in the lemmas we have invoked in this proof are natural in $S$, with the exception of the isomorphisms from Lemma \[cofinality lemma\] and Lemma \[rather hard lemma\] which directly involve $\mathcal{E}_1^S$, only because we did not specify in Lemma \[rather hard lemma\] how $\mathcal{G}_S$ is functorial in $S$. In the present proof, $\mathcal{G}_S$ is $\mathcal{F}_{1; x}(I_{\bullet})\circ J_{1;x}$, and the cofinality of $J_{1;x}$ together with the fact that $K_{1;x} \circ J_{1;x} = \id_{\mathcal{E}_1^S}$ implies, on inspection of the colimit diagrams, that the isomorphism $$\begin{aligned}
\colim \mathcal{G}_S & = & \colim \left( \mathcal{F}_{1; x}(I_{\bullet})\circ J_{x}\right) \\
& \cong & \colim \left(\mathcal{F}_{1; x}(I_{\bullet})\right)\end{aligned}$$ is natural in $S$; details are routine and left to the reader. We conclude that the cofiber sequence \[putative cof seq 1\] is indeed natural in $S$.
Now we have the sequence of simplicial commutative monoids in $\mathcal{C}$: $$\xymatrix{
\vdots \ar[d] & \vdots \ar[d] & \vdots \ar[d] & \\
\mathcal{M}^{X_0}_2(I_{\bullet}) \ar[r]\ar[d] &
\mathcal{M}^{X_1}_2(I_{\bullet}) \ar@<1ex>[l] \ar@<-1ex>[l] \ar@<1ex>[r] \ar@<-1ex>[r]\ar[d] &
\mathcal{M}^{X_2}_2(I_{\bullet}) \ar@<2ex>[l] \ar@<-2ex>[l] \ar[l] \ar@<2ex>[r] \ar@<-2ex>[r] \ar[r]\ar[d] &
\dots \ar@<-3ex>[l] \ar@<-1ex>[l] \ar@<1ex>[l] \ar@<3ex>[l] \\
\mathcal{M}^{X_0}_1(I_{\bullet}) \ar[r]\ar[d] &
\mathcal{M}^{X_1}_1(I_{\bullet}) \ar@<1ex>[l] \ar@<-1ex>[l] \ar@<1ex>[r] \ar@<-1ex>[r]\ar[d] &
\mathcal{M}^{X_2}_1(I_{\bullet}) \ar@<2ex>[l] \ar@<-2ex>[l] \ar[l] \ar@<2ex>[r] \ar@<-2ex>[r] \ar[r]\ar[d] &
\dots \ar@<-3ex>[l] \ar@<-1ex>[l] \ar@<1ex>[l] \ar@<3ex>[l] \\
\mathcal{M}^{X_0}_0(I_{\bullet}) \ar[r] &
\mathcal{M}^{X_1}_0(I_{\bullet}) \ar@<1ex>[l] \ar@<-1ex>[l] \ar@<1ex>[r] \ar@<-1ex>[r] &
\mathcal{M}^{X_2}_0(I_{\bullet}) \ar@<2ex>[l] \ar@<-2ex>[l] \ar[l] \ar@<2ex>[r] \ar@<-2ex>[r] \ar[r] &
\dots \ar@<-3ex>[l] \ar@<-1ex>[l] \ar@<1ex>[l] \ar@<3ex>[l]
}$$ and geometric realization commuting with cofibers together with the isomorphism \[important cofiber\] implies that the comparison map $$\label{comparison map 1} X_{\bullet}\otimes E_0^* I_{\bullet}\rightarrow E_0^*\left| \mathcal{M}^{X_{\bullet}}(I_{\bullet})\right|$$ of objects in $\mathcal{C}$ is a weak equivalence. Hence the comparison map \[comparison map 1\] in $\Comm(\mathcal{C})$ must also be a weak equivalence, since the weak equivalences in $\Comm(\mathcal{C})$ are created by the forgetful functor $\Comm(\mathcal{C})\rightarrow\mathcal{C}$, by assumption.
Construction of the topological Hochschild-May spectral sequence.
-----------------------------------------------------------------
\[def of gen hom thy\] By a [*connective generalized homology theory on $\mathcal{C}$*]{} we shall mean the following data:
- for each integer $n$, a functor $H_n: \Ho(\mathcal{C}) \rightarrow \Ab$, and
- for each integer $n$ and each cofiber sequence $$X \rightarrow Y \rightarrow Z$$ in $\mathcal{C}$, a map $\delta_n^{X \rightarrow Y \rightarrow Z}: H_n(Z) \rightarrow H_{n-1}(X)$,
satisfying the axioms:
Exactness
: For each cofiber sequence $$X \stackrel{f}{\longrightarrow} Y \stackrel{g}{\longrightarrow} Z$$ in $\mathcal{C}$, the sequence of abelian groups $$\xymatrix{
\dots \ar[r] &
H_{n+1}(Y) \ar[r]^{H_{n+1}(g)} &
H_{n+1}(Z) \ar`r_l[ll] `l[dll]_{\delta_{n+1}^{X \rightarrow Y \rightarrow Z}} [dll] \\
H_n(X) \ar[r]^{H_n(f)} &
H_n(Y) \ar[r]^{H_n(g)} &
H_n(Z) \ar`r_l[ll] `l[dll]_{\delta_{n}^{X \rightarrow Y \rightarrow Z}} [dll] \\
H_{n-1}(X) \ar[r]^{H_{n-1}(f)} &
H_{n-1}(Y) \ar[r]^{H_{n-1}(g)} &
\dots }$$ is exact.
Additivity
: For each integer $n$ and each collection of objects $\{ X_i\}_{i\in I}$ in $\Ho(\mathcal{C})$, the canonical map of abelian groups $$\coprod_{i\in I} H_n(X_i) \rightarrow H_n(\coprod_{i\in I} X_i)$$ is an isomorphism.
Naturality of boundaries
: For each integer $n$ and each map of cofiber sequences $$\xymatrix{
X^{\prime} \ar[r] \ar[d]^g & Y^{\prime} \ar[r] \ar[d]^{f} & Z^{\prime} \ar[d]^{h} \\
X \ar[r] & Y \ar[r] & Z,
}$$ the square of abelian groups $$\xymatrix{
H_n(Z^{\prime}) \ar[d]_{H_n(h)} \ar[r]^{\delta^{X^{\prime}\rightarrow Y^{\prime}\rightarrow Z^{\prime}}} &
H_{n-1}(X^{\prime}) \ar[d]^{H_{n-1}(g)} \\
H_n(Z) \ar[r]^{\delta^{X\rightarrow Y\rightarrow Z}} &
H_{n-1}(X)
}$$ commutes.
Connectivity of the unit object
: We have $H_n(\mathbbm{1}) \cong 0$ for all $n<0$.
Connectivity of smash products
: Suppose that $X,Y$ are objects of $\mathcal{C}$, and that $A,B$ are nonnegative integers such that $H_n(X) \cong 0$ for all $n<A$, and $H_n(Y) \cong 0$ for all $n<B$. Then $H_n(X\smash Y) \cong 0$ for all $n<A+B$.
Clearly Definition \[def of gen hom thy\] is just a formulation, in a general pointed model category, of the Eilenberg-Steenrod axioms (from [@MR0012228]) for a generalized homology theory with connective (i.e., vanishing in negative degrees) coefficients. The “connectivity of smash products” axiom is easily proven anytime one has an $E$-homology Künneth spectral sequence in $\mathcal{C}$, which is the case in (for example) any of the usual models for the stable homotopy category.
\[def of thh-may ss\] If $I_{\bullet}$ is a cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$, $X_{\bullet}$ is a simplicial finite set, and $H_*$ is a connective generalized homology theory on $\mathcal{C}$, then by the [*topological Hochschild-May spectral sequence for $X_{\bullet}\tilde{\otimes} I_{\bullet}$*]{} we mean the spectral sequence in $\mathcal{A}$ obtained by applying $H_*$ to the tower of cofiber sequences in $\mathcal{C}$ $$\label{tower of sequences} \xymatrix{
\vdots \ar[d] & \\
\left| \mathcal{M}_2^{X_{\bullet}}(I_{\bullet})\right| \ar[r]\ar[d] & \left| \mathcal{M}_2^{X^{\bullet}}(I_{\bullet})\right|/\left| \mathcal{M}_3^{X^{\bullet}}(I_{\bullet})\right| \\
\left| \mathcal{M}_1^{X_{\bullet}}(I_{\bullet})\right| \ar[r]\ar[d] & \left| \mathcal{M}_1^{X^{\bullet}}(I_{\bullet})\right|/\left| \mathcal{M}_2^{X^{\bullet}}(I_{\bullet})\right| \\
\left| \mathcal{M}_0^{X_{\bullet}}(I_{\bullet})\right| \ar[r] & \left| \mathcal{M}_0^{X^{\bullet}}(I_{\bullet})\right|/\left| \mathcal{M}_1^{X^{\bullet}}(I_{\bullet})\right| .}$$ That is, it is the spectral sequence of the exact couple $$\xymatrix{
D^1_{*,*} \cong \bigoplus_{i,j} H_i \left| \mathcal{M}_j^{X_{\bullet}}(I_{\bullet})\right| \ar[r] & \bigoplus_{i,j} H_i \left| \mathcal{M}_j^{X_{\bullet}}(I_{\bullet})\right| \cong D^1_{*,*} \ar[d] \\
& E^1_{*,*} \cong \bigoplus_{i,j} H_i \left| \mathcal{M}_j^{X_{\bullet}}(I_{\bullet})\right|/\left| \mathcal{M}_{j+1}^{X_{\bullet}}(I_{\bullet})\right| \ar[ul] . }$$
[**(Connectivity conditions.)**]{}\[connectivity conditions lemma\] Let $H_*$ be a connective generalized homology theory on $\mathcal{C}$. Suppose $\mathcal{C}$ is stable, and suppose that there exist objects $Z,H$ of $\mathcal{C}$ such that $H_*(-)$ is naturally isomorphic to $[\Sigma^* Z, H \smash -]$.
- Let $$\label{sequence 13083} \dots \rightarrow Y_2 \rightarrow Y_1 \rightarrow Y_0$$ be a sequence in $\mathcal{C}$, and suppose that $H_n(Y_i) \cong 0$ for all $n<i$. Then $[\Sigma^n Z, \holim_i (H \smash Y_i)] \cong 0$ for all $n$.
- Suppose that $A$ is a nonnegative integer and that $$\label{simplicial object 1309}\xymatrix{
X_0
\ar[r] &
X_1 \ar@<1ex>[l]\ar@<-1ex>[l]\ar@<1ex>[r]\ar@<-1ex>[r] &
X_2 \ar@<2ex>[l]\ar@<-2ex>[l]\ar[l]
\ar@<2ex>[r]\ar@<-2ex>[r]\ar[r] &
\dots \ar@<3ex>[l]\ar@<-3ex>[l]\ar@<1ex>[l]\ar@<-1ex>[l] }$$ is a simplicial object of $\mathcal{C}$. Suppose that $H_n(X_i) \cong 0$ for all $n<A$ and all $i$. Then $H_n\left(\left| X_{\bullet}\right|\right) \cong 0$ for all $n<A$.
<!-- -->
- Since $\mathcal{C}$ is assumed stable, the homotopy limit $\holim_i Y_i$ is the homotopy fiber of the map $$\prod_{n\in \mathbb{N}} Y_n \stackrel{\id - T}{\longrightarrow} \prod_{n\in \mathbb{N}} Y_n$$ in $\Ho(\mathcal{C})$, where $T$ is the product of the maps $Y_n \rightarrow Y_{n-1}$ occuring in the sequence \[sequence 13083\]. For each object $Z$ of $\mathcal{C}$, we then have the long exact sequence $$\xymatrix{
\dots \ar[r] &
[\Sigma^j Z, \prod_{n\in \mathbb{N}} H \smash Y_n] \ar[r]^{\id - T} &
[\Sigma^j Z, \prod_{n\in \mathbb{N}} H \smash Y_n] \ar`r_l[ll] `l[dll] [dll] \\
[\Sigma^{j-1} Z, \holim_i H \smash Y_i] \ar[r] &
[\Sigma^{j-1} Z, \prod_{n\in \mathbb{N}} H \smash Y_n] \ar[r]^{\id - T} &
[\Sigma^{j-1} Z, \prod_{n\in \mathbb{N}} H \smash Y_n] \ar`r_l[ll] `l[dll] [dll] \\
[\Sigma^{j-2} Z, \holim_i H \smash Y_i] \ar[r] &
[\Sigma^{j-2} Z, \prod_{n\in \mathbb{N}} H \smash Y_n] \ar[r]^{\id - T} &
\dots }$$ hence the Milnor exact sequence $$0 \rightarrow R^1\lim_i [\Sigma^{j+1} Z, H\smash Y_i] \rightarrow [\Sigma^j Z, \holim_i H\smash Y_i] \rightarrow \lim_i [\Sigma^j Z, H\smash Y_i] \rightarrow 0 .$$ The assumption that $[\Sigma^j Z, H\smash Y_i] \cong 0$ for $j<i$ guarantees that the sequence $$\dots \rightarrow [\Sigma^j Z, H\smash Y_2] \rightarrow [\Sigma^j Z, H\smash Y_1] \rightarrow [\Sigma^j Z, H\smash Y_0]$$ is eventually constant and zero for all $j$, hence both its limit and $R^1\lim$ vanish for all $j$, hence $[\Sigma^j Z, \holim_i H\smash Y_i]\cong 0$ for all $j$.
- The Bousfield-Kan spectral sequence, i.e., the $H$-homology spectral sequence of the simplicial object \[simplicial object 1309\], has input $E^1_{s,t} \cong \pi_s(H\smash X_t)$ and converges to $H_{s+t}\left(\left| X_{\bullet}\right|\right)$, since $ \left [ \Sigma^* Z, \holim H\smash Y_i\right ]$ vanishes. The differential in this spectral sequence is of the form $d^r: E^r_{s,t} \rightarrow E^r_{s-r,t+r-1}$, hence this spectral sequences has a nondecreasing upper vanishing curve at $E^1$, hence converges strongly. Triviality of $E^1_{s,t}$ for $s<A$ and $t<0$ then gives us that $H_s\left(\left| X_{\bullet}\right|\right)$ vanishes for $s<A$.
\[conn lem 2\] Suppose $H_*$ is a connective generalized homology theory as defined in Definition \[def of gen hom thy\], and $\mathcal{M}_i^{S}(I_{\bullet})$ is the $i$-th degree of the May filtration for a finite set $S$ and a cofibrant decreasingly filtered commutative monoid $I_{\bullet}$ as defined in Definition \[def of dec filt comm mon\]. Then, if $H_m(I_i)\cong 0$ for all $m,i\in \mathbb{N}$ such that $m<i$, then $$H_m(\mathcal{M}_i^{S}(I_{\bullet}))\cong 0$$ for all $m,i\in \mathbb{N}$ such that $m< i$.
The proof is inductive on the cardinality of $S,$ which we denote $\hash (S)$. First recall that by Definition \[def of may filt 0 4\], $$\mathcal{M}_i^{S}(I_{\bullet})=\colim \mathcal{F}_i^{S}(I_{\bullet})$$ and $\mathcal{F}_i^S(I_{\bullet})$ is a functor $$\mathcal{F}_i^{S}(I_{\bullet}) : (\mathcal{D}_i^{S})^{\text{op}} \lra \mathcal{C},$$ where $(\mathcal{D}_i^{S})^{\text{op}}$ is the full sub-poset of $(\mathbb{N}^S)^{\text{op}}$ containing exactly the objects $x\in \mathbb{N}^S$ such that $|x|\ge i$. Also, recall the definition $$\mathcal{E}_{n,k}^S=\{ x\in \{0, ... ,n\}^{S} | \sum_{s\in S} x(s)\ge k \}$$ where $k\ge n$ and the convention of writing $\mathcal{E}_n^S$ when $n=k$. One can easily see that the diagram $(\mathcal{E}_n^S)^{\op} \lra \mathcal{C}$ is cofinal in in the diagram $(\mathcal{D}_n^S)^{\op} \lra \mathcal{C}$.
The case $\hash (S)=1$ is trivial since the constant diagram $I_i$ is cofinal in $\mathcal{F}_i^{S}(I_{\bullet})$. The claim follows by the assumption that $$H_m(I_{i})\cong 0$$ for all $m,i\in \mathbb{N}$ such that $m< i$.
The case $\hash (S)\ge 2$ and $i=0$ is trivial as well by the following argument. First, the constant diagram $\bigsmash_{s\in S} I_0$ is cofinal. Second, we assumed that $H_m(I_0)\cong 0$ for all $m,i\in \mathbb{N}$ such that $m<0$, and, by Definition \[def of gen hom thy\], a connective generalized homology theory, satisfies $H_m(X\smash Y)\cong 0$ for $m< m_1 + m_2$ whenever $H_{i}(X)\cong H_j(Y)\cong 0$ for all $i< m_1$ and all $j<m_2$.
Suppose $\hash(S)=2$. In the case $i=1$, the diagram $\mathcal{F}_i^{S}(I_{\bullet})$ contains the pushout diagram $$\xymatrix{ I_1\smash I_1 \ar[d] \ar[r] & I_0 \smash I_1 \\ I_1 \smash I_0 }$$ as a cofinal subdiagram. This colimit, which is $\mathcal{M}_1^S(I_{\bullet})$, is a homotopy pushout by Lemma \[n=2 case of hard lemma\] and can therefore be written as the cofiber in the cofiber sequence $$I_1\smash I_1 \lra I_0 \smash I_1 \vee I_1 \smash I_0 \lra (I_0 \smash I_1 \vee I_1 \smash I_0)/(I_1\smash I_1) \cong (I_0/I_1 \smash I_1) \vee (I_1\smash I_0/I_1) .$$ Since $H_m(I_1\smash I_1)\cong 0$ for $m<2$ and $H_m((I_0 \smash I_1) \vee (I_1 \smash I_0))\cong 0$ for $m<1$, by the long exact sequence in $H_*$, $H_m( \colim \mathcal{F}_i^{S}(I_{\bullet}))\cong 0$ for $m<1$. This proves the claim for $i=1$. When $i>1$, we consider the cofinal subdiagram $\mathcal{E}_i^S(I_{\bullet}):(\mathcal{E}_i^S)^{\op} \rightarrow \mathcal{C}$ of $\mathcal{F}_i^S(I_{\bullet})$. By filling in vertices with pushouts, which we denote $P_{(j,k)}$, we can write $\colim \mathcal{F}_i^{S}(I_{\bullet})\cong \colim \mathcal{E}_i^S(I_{\bullet})$ as an iterated pushout; for example, when $n=2$, $$\xymatrix{
&&\vdots \ar[d] & \vdots \ar[d] \\
&\dots \ar[r] \ar[d] & I_1 \smash I_2 \ar[r] \ar[d] & I_0\smash I_2 \ar[d] \\
\dots \ar[r] &I_2\smash I_1 \ar[r] \ar[d] & I_1 \smash I_1 \ar[d] \ar[r]& P_{(0,1)} \ar[d] \\
\dots \ar[r] & I_2 \smash I_0 \ar[r] & P_{(1,0)} \ar[r] & P_{(0,0)}. \\}$$ Note that the colimit of a pushout diagram agrees with the homotopy colimit of that diagram when the maps are all cofibrations and the objects are all cofibrant as is the case here.
On each of the objects $\mathcal{E}_i^S(I_{\bullet})(x)$ and the objects $P_{(j,k)}$ where $0\le j+k<i$, the functor $H_m(-)$ evaluates to zero whenever $m<i$. The same is true about $\colim \mathcal{F}_i^{S}(I_{\bullet})\cong P_{(0,0)}$, completing the case $\hash(S)=2$.
Now assume $$H_m(\mathcal{M}_i^{S}(I_{\bullet}))\cong 0$$ for all $m,i\in \mathbb{N}$ such that $m<i$, whenever $\hash (S)< n$. By the same method of filling vertices in cubes that we use in the case $\hash(S)=2$, we just need to prove the case $i=1$ and the case $i>1$ will follow by the fact that the colimit $\colim \mathcal{F}_i^S(I_{\bullet})= \mathcal{M}_i^{S}(I_{\bullet}))$ can be written as an iterated $n$-cube. It therefore remains to prove that $$H_m(\mathcal{M}_1^{S}(I_{\bullet}))\cong 0$$ for all $m\in \mathbb{Z}$ such that $m<1$, whenever $\hash (S)=n$. We consider a single directed $n$-cube missing a single terminal vertex, i.e. a functor $$\mathcal{E}_1^S(I_{\bullet}): (\mathcal{E}_1^S)^{\op} \lra \mathcal{C}$$ which is a cofinal diagram in $\mathcal{F}_1^S(I_{\bullet})$ so that $\colim \mathcal{E}_1^S(I_{\bullet})\cong \mathcal{M}_1^S(I_{\bullet})$.
We then consider the subdiagrams of $ \mathcal{E}_1^S(I_{\bullet})$ that are $(n-1)$-cube shaped diagrams of $\mathcal{C}$ containing the object $ \mathcal{E}_1^S(I_{\bullet})(x)$ such that $x(s)=1$ for all $s\in S$. We then remove the terminal vertex in each of these $n-1$-cubes and denote the $k$-th truncated $(n-1)$-cube $$\mathcal{E}_1^S(I_{\bullet})^{(k)}$$ where $k\in\{1,...,n\}$ runs over all truncated sub $(n-1)$-cubes of this type.
We then construct a diagram, which we call $\mathcal{B}$, which is the same as $\mathcal{E}_1^S$ except that all the vertexes removed as in the process above are replaced with the colimits $\colim \mathcal{E}_1^S(I_{\bullet})^{(k)}$. By universality of the colimit we get maps $$\mathcal{E}_1^S(I_{\bullet})^{(k)} \lra I_0\smash ... \smash I_0 \smash I_1 \smash I_0 \smash ... \smash I_0$$ for each $k$, where $I_1$ is in the $k$-th position, and by iterated use of the pushout product axiom, this map is a cofibration. We can therefore consider the map of diagrams $$\mathcal{B} \lra \mathcal{E}_1^S(I_{\bullet})$$ and take levelwise cofibers. We call the resulting diagram $\mathcal{Cof}$. By examination of the levelwise cofibers, we see that $$\colim \mathcal{Cof} \cong \bigvee_{k=1}^n \left ( I_{\delta_{1k}}\smash \dots \smash I_{\delta_{nk}}\right) / \mathcal{E}_1^S(I_{\bullet})^{(k)}$$ where $\delta_{jk}=0$ if $j\ne k$ and $\delta_{kk}=1$. We also observe that $\colim \mathcal{Cof}\cong \hocolim \mathcal{Cof}$ since it can be written as an iterated pushout of cofibrant objects along cofibrations [@dugco Prop. 13.10]. The object $\colim \mathcal{Cof}$ is therefore a model for the homotopy cofiber of the map $$\colim \mathcal{B} \lra \colim \mathcal{E}_1^S(I_{\bullet})$$ even though this map is not necessarily a cofibration. We therefore get a long exact sequence in $H_*$ so $H_m(\mathcal{F}_1^S(I_{\bullet}))\cong 0$ for $m<\text{min} \{m', m'' \}$ where $H_j( \colim \mathcal{B}) \cong 0$ for $j<m'$ and $H_j( \colim \mathcal{Cof} ) \cong 0$ for $k<m''$. We know $H_j( \colim \mathcal{Cof} ) \cong 0$ for all $j<1$ since for each $k\in \{1,\dots, n\}$ there is an isomorphism $H_j \left ( I_{\delta_{1k}}\smash \dots \smash I_{\delta_{nk}}\right )\cong 0 $ for all $j<1$, and by the inductive hypothesis there is an isomorphism $H_j(\mathcal{E}_1^S(I_{\bullet})^{(k) }) \cong 0 $ for all $j<1$.
To prove $H_j( \colim \mathcal{B}) \cong 0$ for $j<1$, first note that $\colim \mathcal{B}=\colim \mathcal{E}_{1,2}^S$ as defined. We observe that the object $\colim \mathcal{E}_{1,2}^S$ can be written as a colimit of $n$ truncated $(n-1)$-cubes whose pairwise intersections are $(n-2)$-cubes. We repeat the process and form $\mathcal{B}_1$ by eliminating terminal vertices in each $(n-1)$-cube and replacing each one with the colimit of that $(n-1)$-cube. Observe that $\colim \mathcal{B}_1\cong \colim \mathcal{E}_{1,3}^{S}$. We produce another sequence $$\colim \mathcal{B}_1 \lra \colim \mathcal{B} \lra \colim \mathcal{Cof}_1$$ and as before $H_j(\mathcal{Cof}_1)\cong 0$ for $j<1$. This begins an inductive procedure that ends with $B_{n-2}$ such that $\colim B_{n-2}\cong \colim \mathcal{E}_{1,n}^S$ and since $\hash (S)=n$, $\colim \mathcal{E}_{1,n}^S\cong \wedge _{s\in S} I_1$. Since $H_j( \smash _{s\in S} I_1)\cong 0$ for $j<1$ and $H_j(\mathcal{Cof}_m)\cong 0$ for $j<1$ and all $1\le m \le n-2$, we have shown that $H_j(\mathcal{B})\cong 0$ for all $j<1$.
The colimit $\colim \mathcal{E}_{1}^S(I_{\bullet})$ therefore has the property that $H_j( \colim \mathcal{E}_{1}^S(I_{\bullet}))\cong 0$ for $j<1$ and, thus, we have proven our claim.
\[subprop on thh-may ss\] Suppose $I_{\bullet}$ is a Hausdorff cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$, $X_{\bullet}$ is a simplicial finite set, and $H_*$ is a connective generalized homology theory on $\mathcal{C}$. Suppose $H_*(-)\cong [\Sigma^*Z,-\wedge H]$ for some objects $Z$ and $H$ in $\mathcal{C}$. Suppose the following connectivity axiom:
- [**(Connectivity axiom.)**]{} $H_m(I_n)\cong 0$ for all $m<n$.
Then the topological Hochschild-May spectral sequence is strongly convergent, its differential satisfies the graded Leibniz rule, and its input and output and differential are as follows: $$\begin{aligned}
E^1_{s,t}\cong H_{s,t}(X_{\bullet}\otimes E_0^*I_{\bullet}) & \Rightarrow & H_{s}(X_{\bullet}\otimes I_{0}) \\
d^r: E^r_{s,t} & \rightarrow & E^r_{s-1,t+r} \end{aligned}$$
It is standard (see e.g. the section on Adams spectral sequences in [@MR1718076]) that the $H$-homology spectral sequence of a tower of cofiber sequences of the form \[tower of sequences\] converges to $H_*\left(\left| \mathcal{M}_0^{X_{\bullet}}(I_{\bullet})\right|\right)$ as long as $\left[\Sigma^* Z, \holim_i \left( H\smash\left| \mathcal{M}_i^{X_{\bullet}}(I_{\bullet})\right| \right) \right]$ is trivial. By Lemma \[conn lem 2\] $H_m\left( \left| \mathcal{M}_i^{X_{\bullet}}(I_{\bullet})\right| \right) \cong 0$ for all $m<i$, so by Lemma \[connectivity conditions lemma\], $$\left[\Sigma^n Z, \holim_i \left(H \smash \left| \mathcal{M}_i^{X_{\bullet}}(I_{\bullet})\right|\right)\right] \cong 0$$ for all $n$, as desired. Hence the spectral sequence converges to $$H_*\left(\left| \mathcal{M}_0^{X_{\bullet}}(I_{\bullet})\right|\right) \cong H_*(X_{\bullet}\otimes I_0).$$ That the differential has the stated bidegree is a routine and easy computation in the spectral sequence of a tower of cofiber sequences.
The sequence $$\xymatrix{ \dots \ar[r] & \left | \mathcal{M}_2^{X_{\bullet}}(I_{\bullet}) \right | \ar[r] & \left | \mathcal{M}_1^{X_{\bullet}}(I_{\bullet}) \right | \ar[r] & \left | \mathcal{M}_0^{X_{\bullet}}(I_{\bullet}) \right | }$$ is a cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$ as observed in Remark \[rem on structure\] and therefore, in particular, it produces a “pairing of towers” in the sense of [@mult1] and therefore by Proposition 5.1 of [@mult1] the differentials in the spectral sequence satisfy a graded Leibniz rule.
The statements about convergence are also standard: the connectivity axiom and the “connectivity of smash products” axiom from Definition \[def of gen hom thy\] together imply that our spectral sequence has a nondecreasing upper vanishing curve already at the $E^1$-term, so the spectral sequence converges strongly.
Another construction of our $THH$-May spectral sequence $$\label{our ss 4308} E^1_{*,*} \cong H_{*}(X_{\bullet} \otimes E_0^{*}I_{\bullet}) \Rightarrow H_{*}(X_{\bullet} \otimes I_0))$$ is possible using the Day convolution product. This construction is conceptually cleaner, but it does not, to our knowledge, simplify the process of proving that the resulting spectral sequence has the correct input term, output term and convergence properties.
Recall from Remark \[rem filt comm mon\] that a cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$ is equivalent to a cofibrant object in $\Comm \mathcal{C}^{\mathbb{N}^{\op}}$ where $\Comm \mathcal{C}^{\mathbb{N}^{\op}}$ has the model structure created by the forgetful functor to $\mathcal{C}^{\mathbb{N}^{\op}}$ and the category $\mathcal{C}^{\mathbb{N}^{\op}}$ is equipped with the projective model structure.
Now fix a simplicial finite set $X_{\bullet}$. A cofibrant commutative monoid object $I$ in $\mathcal{C}^{\mathbb{N}^{\op}}$ is a cofibrant decreasingly filtered commutative monoid object $I_{\bullet}$ in $\mathcal{C}$, and we can form the pretensor product $X_{\bullet}\tilde{\otimes} I_{\bullet}$, a simplicial object in $\Comm \mathcal{C}^{\mathbb{N}^{\op}}$. For example, if $X_{\bullet}$ is the usual minimal simplicial model $(\Delta[1]/\delta\Delta[1])_{\bullet}$ for the circle, then $X_{\bullet}\tilde{\otimes} I_{\bullet}$ is the cyclic bar construction using the Day convolution as the tensor product: $$(\Delta[1]/\delta\Delta[1])_{\bullet} \tilde{\otimes} I =
\left(
\vcenter{ \xymatrix{I \ar[r] & I\otimes_{{\rm Day}} I \ar@<1ex>[l]\ar@<-1ex>[l]\ar@<1ex>[r]\ar@<-1ex>[r] & I\otimes_{{\rm Day}} I\otimes_{{\rm Day}} I \ar@<2ex>[l]\ar@<-2ex>[l]\ar[l] \ar@<2ex>[r]\ar@<-2ex>[r]\ar[r] & \dots \ar@<3ex>[l]\ar@<-3ex>[l]\ar@<1ex>[l]\ar@<-1ex>[l] } } \right )$$
Since $I$ is a functor $\mathbb{N}^{\op} \rightarrow \mathcal{C}$, we will write $I(n)$ for the evaluation of this functor at a nonnegative integer $n$. (If we instead think of $I$ as a decreasingly filtered commutative monoid, as in most of the rest of this paper, we would write $I_n$ instead of $I(n)$.) We write $\left( (\Delta[1]/\delta\Delta[1])_{\bullet}\tilde{\otimes} I \right ) (i)$ for the the simplicial object in $\mathcal{C}$ $$\left( (\Delta[1]/\delta\Delta[1])_{\bullet} \tilde{\otimes} I \right) (i) =
\left(
\vcenter{ \xymatrix{I (i) \ar[r] & (I\otimes_{{\rm Day}} I) (i) \ar@<1ex>[l]\ar@<-1ex>[l]\ar@<1ex>[r]\ar@<-1ex>[r] & (I\otimes_{{\rm Day}} I\otimes_{{\rm Day}} I )(i)\ar@<2ex>[l]\ar@<-2ex>[l]\ar[l] \ar@<2ex>[r]\ar@<-2ex>[r]\ar[r] & \dots \ar@<3ex>[l]\ar@<-3ex>[l]\ar@<1ex>[l]\ar@<-1ex>[l] } } \right )$$ Applying geometric realization to $\left ( (\Delta[1]/\delta\Delta[1])_{\bullet}\tilde{\otimes} I\right )(i)$, we get a cofibrant decreasingly filtered object in in $\mathcal{C}$ (assuming Running Assumption \[ra:2\]) $$\left | \left( (\Delta[1]/\delta\Delta[1])_{\bullet}\tilde{\otimes} I \right ) (0) \right | \leftarrow \left | \left( (\Delta[1]/\delta\Delta[1])_{\bullet}\tilde{\otimes} I\right ) (1) \right | \leftarrow \left| \left( (\Delta[1]/\delta\Delta[1])_{\bullet}\tilde{\otimes} I\right) (2) \right| \leftarrow \dots$$ and the spectral sequence obtained by applying a generalized homology theory $E_*$ to this cofibrant decreasingly filtered object in $\mathcal{C}$ is precisely the spectral sequence \[our ss 4308\], the spectral sequence constructed and considered throughout this paper. (It is an easy exercise in unwinding definitions to check that this spectral sequence agrees with the one constructed in Definition \[def of thh-may ss\], but to verify that the resulting spectral sequence has the expected input term, output term, and convergence properties amounts to exactly the same proofs already found in this paper which aren’t expressed in terms of Day convolution.)
Decreasingly filtered commutative ring spectra. {#section 2}
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Whitehead towers
----------------
Let $R$ be a cofibrant connective commutative monoid in $\mathcal{C}$. For this section and the next, let $\mathcal{C}$ be a model for the homotopy category of spectra such as symmetric spectra, $S$-modules, or orthogonal spectra. The goal of this section is to produce a cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$ as a specific multiplicative model for the Whitehead tower of a connective commutative monoid in $\mathcal{C}$. Part of the proof uses a Postnikov tower of a commutative ring spectrum constructed as a tower of square-zero extensions, so first we define square-zero extensions in this context.
\[sz\] By a square-zero extension in $\mathcal{C}$, we mean a fiber sequence $$I \lra \tilde{A} \lra A$$ where $\tilde{A}$ is the pullback in $\Comm \mathcal{C}$ of $$\xymatrix{
\tilde{A} \ar[r] \ar[d] & A \ar[d]^(.45){\epsilon} \\
A \ar[r]^(.4){d} & A \ltimes \Sigma I, }$$ the map $\epsilon$ is defined to be the inclusion of $A$ into $A\ltimes \Sigma I$ and $d$ represents a class $[d]\in TAQ_S^0(A,\Sigma I)$. (For a definition of $TAQ^*_S(A,\Sigma I)$, see [@MR1732625] or [@MR1990939].) Note that, a priori, $A$ must be a commutative monoid in $\mathcal{C}$ and $I$ must be a $A$-bimodule. By $A\ltimes \Sigma I$ we mean the trivial square-zero extension of $A$ by $\Sigma I$; that is, additively $A\ltimes \Sigma I:=A\vee \Sigma I$ and its multiplication is the map $$\mu: A\smash A \vee A\smash I \vee I\smash A \vee I\smash I \lra A\vee I$$ determined by the maps $$\mu_A : A\smash A \rightarrow A \hookrightarrow A\vee I$$ $$\psi^{\ell} :A\smash I \rightarrow I \hookrightarrow A\vee I$$ $$\psi^{r} : I\smash A \rightarrow I \hookrightarrow A\vee I$$ $$sq : I\smash I \rightarrow 0 \hookrightarrow A\vee I$$ where $\mu_A$ is the multiplication on $A$, $\psi^{r}$ and $\psi^{\ell}$ are the right and left action maps of $I$ as an $A$-bimodule and $sq$ is the usual map $I\smash I\rightarrow I\hookrightarrow A\vee I$, which in this case factors through the zero object.
\[post def\] Let $R$ be a connective commutative monoid in $\mathcal{C}$. By a *Postnikov tower of square-zero extensions* associated to $R$, we mean a tower $$\xymatrix{ \dots \ar[r] & \tau_{\le 3}R \ar[r] & \tau_{\le 2}R \ar[r]& \tau_{\le 1}R \ar[r] & \tau_{\le 0}R \\
&\Sigma^{3}H\pi_3R\ar[u]&\ar[u] \Sigma^{2}H\pi_2R & \Sigma^{1}H\pi_1R \ar[u]&\ar[u] H\pi_0 R }$$ of fiber sequences where $\pi_k(\tau_{\le n} R)= \pi_k(R) $ for $k\le n$ and $\pi_k(\tau_{\le n} R)=0$ for $k>n$, such that the fiber sequences $$\Sigma^n H\pi_nR \lra \tau_{\le n} R \lra \tau_{\le n-1} R$$ are square-zero extensions.
As defined it is not clear that such Postnikov towers of square-zero extensions actually exist for a given commutative monoid in $\mathcal{C}$, but it is a theorem that they do.
\[thm dfcm\] Let $R$ be a connective commutative monoid in $\mathcal{C}$. Then there exists a model for the Postnikov tower associated to $R$ which is a Postnikov tower of square-zero extensions.
See Theorem 4.3 and the comments after in [@MR2125041] and Theorem 8.1 in [@MR1732625]. Also, see Lurie’s Corollary 3.19 from [@0709.3091] for the result in the setting of quasi-categories.
Recall from Remark \[rem filt comm mon\] that a cofibrant object in the category $\Comm \mathcal{C}^{\mathbb{N}^{\op}}$ equipped with the projective model structure is a cofibrant decreasingly filtered commutative monoid. We may define certain $n$-truncated decreasingly filtered commutative monoids in the following way.
Let $J_n \subset \mathbb{N}$ be the sub-poset of the natural numbers consisting of all $i\in \mathbb{N}$ such that $i\le n$. We give this poset the structure of a symmetric monoidal category $(J_n, \dot{+}, 0)$ by letting $$i \dot{+} j = \text{min} \{ i+j, n\}.$$
We may consider lax symmetric monoidal functors in $(\mathcal{C}^{J_n^{\op}})$ for each $n$ again as a consequence of [@Day Ex. 3.2.2] these are equivalent to the commutative monoids in the functor category under a Day convolution symmetric monoidal product. We may also consider the model structure on $\Comm (\mathcal{C}^{J_n^{\op}})$ created by the forgetful functor to $(\mathcal{C}^{J_n^{\op}})$, where $(\mathcal{C}^{J_n^{\op}})$ has the projective model structure. In this model structure, it is an easy exercise to show that the cofibrant objects are objects $I^{\le n}$ in $(\mathcal{C}^{J_n^{\op}})$ such that each $I^{\le n}_i$ is cofibrant in $\mathcal{C}$ for $i\le n$ and each map $f_i :I^{\le n}_i\rightarrow I^{\le n}_{i-1}$ is a cofibration in $\mathcal{C}$ for each $i\le n$.
\[post filt\] Let $R$ be a cofibrant connective commutative monoid in $\mathcal{C}$, then there exists a cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$ $$\xymatrix{ \dots \ar[r] & \tau_{\ge 2} R \ar[r] & \tau_{\ge 1} R \ar[r] & \tau_{\ge 0}R }$$ with structure maps $$\rho_{i,j}: \tau_{\ge i } R \smash \tau_{\ge j} R \longrightarrow \tau_{\ge i+j}R$$ such that $\pi_k(\tau_{\ge n}R)\cong \pi_k(R)$ for $k\ge n$ and $\pi_k(\tau_{\ge n}R)\cong 0$ for $k<n$. This cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$ is denoted $\tau_{\ge \bullet}R$.
Let $R$ be a cofibrant connective commutative monoid in $\mathcal{C}$ and let $$\xymatrix{
\dots \ar[r] & \tau_{\le 2} R \ar[r] & \tau_{\le 1} R \ar[r] & \tau_{\le 0} R \\
& \Sigma^2 H\pi_2(R) \ar[u] & \Sigma H\pi_1(R) \ar[u] & H\pi_0(R) \ar[u] }$$ be a Postnikov tower of square-zero extensions of $R$ in the sense of Definition \[post def\]. To prove the theorem we need to do the following:
1. Construct $\tau_{\ge n} R$.
2. Construct maps $\rho_{i,j}:\tau_{\ge i} R \wedge \tau_{\ge j} R \lra \tau_{\ge i+j} R$.
3. Show that the maps $\rho_{i,j}$ satisfy associativity, commutativity, unitality and compatibility.
The procedure will be inductive. First, define $\tau_{\ge 0} R:=R$ where $R$ was assumed to be a cofibrant connective commutative monoid in $\mathcal{C}$ and is therefore an object in $\Comm \mathcal{C}^{J_0^{\op}}$. To construct $\tau_{\ge 1} R$, we consider the map of commutative ring spectra $R\lra H\pi_0R$. We can assume this map is a fibration, since if it wasn’t we could factor the map in commutative ring spectra into an acyclic cofibration and a fibration. We then define $\tau_{\ge 1}R$ to be the fiber of this map. By design, we have constructed an object $I^{\le 1}_{\bullet}$ in $\Comm \mathcal{C}^{J_1^{\op}}$. Commutativity, associativity and unitality follow by the definition of a symmetric $R$-bimodule action of $R$ on $\tau_{\ge 1}R$. This completes the base step in the induction.
Suppose we have a object $I^{\le n-1}_{\bullet}\in\ob \Comm \mathcal{C}^{{J_{n-1}}^{\op}}$ for an arbitrary $n\ge 1$. As before, we define $\tau_{\ge i}R$ to be $I^{\le n-1}_i$ for all $i\le n-1$. Define $$P_n := \colim_{\mathcal{D}_n} \tau_{\ge i}R \wedge \tau_{\ge j}R$$ where $\mathcal{D}_n$ is the full subcategory of $\mathbb{N}^{\op}\times \mathbb{N}^{\op}$ with objects $(i,j)$ such that $0<i\le j <n$ and $i+j\ge n$. Since $I^{\le n-1}_{\bullet}$ is in $\Comm \mathcal{C}^{{J_{n-1}}^{\op}}$, there is a unique map $P_n\lra \tau_{\ge n-1}R$.
The fact that the fiber sequence $\Sigma^kH\pi_kR \lra \tau_{\le k}R \lra \tau_{\le k-1}R$ is a square-zero extension for each $k$ implies that the natural maps $$\Sigma^iH\pi_iR \wedge \Sigma^j H\pi_jR \lra \Sigma^{n-1} H\pi_{i+j}R,$$ factor through $0$ for each $(i,j)\in \mathcal{D}_n$. We get an induced map on fibers by considering the diagrams $$\xymatrix{
\tau_{\ge k} R \ar[d] \ar[r] & R \ar[r] \ar[d] & \tau_{\le k-1}R \ar[d] \\
\Sigma^kH\pi_kR \ar[r]& \tau_{\le k}R \ar[r]& \tau_{\le k-1}R}$$ for $k<n$. There are therefore commutative diagrams $$\xymatrix{
\tau_{\ge i}R \wedge \tau_{\ge j}R \ar[r] \ar[d] & \Sigma^iH\pi_iR \wedge \Sigma^j H\pi_jR \ar[d]^{0} \\
\tau_{\ge n-1 } R \ar[r] & \Sigma^{n-1} H\pi_nR
}$$ for each $(i,j)\in \mathcal{D}_n$, hence, the map $$\tau_{\ge i}R \wedge \tau_{\ge j}R \lra \tau_{\ge n } R \lra \Sigma^{n-1} H\pi_nR$$ factors through zero for each $(i,j)\in \mathcal{D}_n$.
We need the map $\tau_{\ge n-1 } R \rightarrow \Sigma^{n-1} H\pi_nR $ to be a fibration, so we use the factorization $$\xymatrix{ \tau_{\ge n-1 } R \ar[dr] \ar[rr]&& \Sigma^{n-1} H\pi_nR \\
& \overline{\tau_{\ge n-1 } R} \ar[ur] & }$$ into a trivial cofibration followed by a fibration.
We can define $\tau_{\ge n}R$ to be the pullback, in the category of $R$-modules in ${\mathcal{C}}$, of the diagram $$\xymatrix{ \tau_{\ge n}R \ar[r] \ar[d]& 0 \ar[d] \\
\overline{\tau_{\ge n-1 }R} \ar[r]^f & \Sigma^{n-1} H\pi_nR. }$$
We then also need to replace $P_n$ by $\overline{P}_n$ where $\overline{P}_n$ is the same colimit as $P_n$ except that each instance of $\tau_{\ge n-1 }R$ is replaced by $\overline{\tau_{\ge n-1 }R}$. There is therefore a map $P_n\rightarrow \overline{P}_n$ and there is a map $\overline{P}_n\rightarrow \Sigma^{n-1}H\pi_{n-1}R$ that factors through the zero map by the same considerations as above.
Therefore, by the universal property of the pullback, there exists a unique map $g$ $$\xymatrix{
\overline{P}_n \ar[drr] \ar[ddr] \ar[dr]^g && \\
& \tau_{\ge n}R \ar[r] \ar[d] & 0 \ar[d] \\
& \overline{\tau_{\ge n-1}R} \ar[r] & \Sigma^{n-1}H\pi_{n-1}R.
}$$ By composing the maps $\tau_{\ge i} R \wedge \tau_{\ge j} R\rightarrow \overline{P}_n$ and $\overline{\tau_{\ge n-1}R}\wedge \tau_{\ge i}R\rightarrow \overline{P}_n$ with the map $g$, we produce the necessary maps $\rho_{i,j}: \tau_{\ge i} R \wedge \tau_{\ge j} R \lra \tau_{\ge \text{min}\{i+j,n\}}R$ where $0<i\le j<n$. This also proves, by construction, that they satisfy the compatibility axiom (that is, naturality of the lax symmetric monoidal functor $J_n^{\op} \rightarrow \mathcal{C}).$ The factor swap map produces all the maps $$\rho_{i,j}: \tau_{\ge i}R \wedge \tau_{\ge j}R \lra \tau_{\ge \text{min}\{i+j,n\}}R$$ where $i>j$ and the commutativity and compatibility necessary for those maps as well. The maps $\rho_{0,n}$ and $\rho_{n,0}$ are the $R$-module action maps that we produced by working in the category of $R$-modules and and again by construction these maps satisfy commutativity and compatibility with the other maps. Unitality is also easily satisfied for each $\rho_{i,j}$ with $i,j\in \{0,...,n\}$, since all these maps are $R$-module maps.
We just need to check associativity. By assumption, we have associativity for all the maps $\rho_{i,j}$ where $i,j<n$, we therefore just need to show that the associativity diagrams involving the maps $\rho_{i,j}$ for $i$ or $j$ equal to $n$. Since the symmetric monoidal product on $R$-modules is assumed to be associative, we know that, for $i,j,k\in\{0,n\}$, the diagrams $$\xymatrix{
\tau_{\ge i} R \wedge\tau_{\ge j} R \wedge \tau_{\ge k} R \ar[r] \ar[d]& \tau_{\ge i} R\wedge \tau_{\ge j+k } R \ar[d]\\
\tau_{\ge i+j} R\wedge \tau_{\ge k} R \ar[r] & \tau_{\ge i+j+k} R }$$ commute. We also know, by construction, that the diagram $$\label{eq A}
\xymatrix{
\tau_{\ge i} R \wedge \tau_{\ge j} R \ar[d] \ar[dr] & \\
\tau_{\ge n-1} R & \ar[l] \tau_{\ge n} R }$$ commutes for all $i+j\ge n$. The diagram $$\label{eq B} \xymatrix{
\tau_{\ge i} R \wedge \tau_{\ge n} R \ar[r] \ar[d] & \tau_{\ge n} R \ar[d] & \ar[l] \tau_{\ge n} R \wedge \tau_{\ge i} R \ar[d] \\
\tau_{\ge i} R \wedge \tau_{\ge n-1} R \ar[r] & \tau_{\ge n-1} R & \ar[l] \tau_{\ge n-1} R \wedge \tau_{\ge i }R }$$ also commutes by construction.
We need to show that for $i,j,k\in\{0,1,...,n\}$ with either $i$, $j$, or $k$ equal to $n$, then $$\label{eq C} \xymatrix{ \tau_{\ge i} R \wedge \tau_{\ge j} R \wedge \tau_{\ge k} R \ar[r] \ar[d] & \tau_{\ge i} R \wedge \tau_{\ge j+k} R \ar[d]\\ \tau_{\ge i+j} R \wedge \tau_{\ge k} R \ar[r] & \tau_{\ge n} R }$$ commutes. This follows by combining the commutativity of Diagram \[eq A\], Diagram \[eq B\], and the diagrams of the form of Diagram \[eq C\] when $i,j,k<n$, and using the fact that $\tau_{\ge n} R \rightarrow \tau_{\ge n-1} R $ is a monomorphism, since it is the pullback of a monomorphism in $\mathcal{C}$ by construction, and hence it is retractile.
We have therefore produced an object in $\Comm\mathcal{C}^{J_n^{\op}}$. By induction, we can therefore produce an object in $\Comm\mathcal{C}^{\mathbb{N}^{\op}}$ and then cofibrantly replace it to produce a cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$, denoted $\tau_{\ge \bullet}R$, as desired.
Since we have functorial factorizations of maps and functorial cofibrant replacement in our setting, the above theorem is entirely functorial, in other words, a map of connective commutative ring spectra $A\rightarrow B$ induces a map of Whitehead towers $\tau_{\ge n} A \rightarrow \tau_{\ge n} B$ compatible with the multiplication maps $\rho^{A}_{i,j}$ and $\rho_{i,j}^A$. This induces a map of associated graded commutative monoids in $\mathcal{C}$ $$E_0^{*}( \tau_{\ge \bullet} A) \longrightarrow E_0^{*}(\tau_{\ge \bullet} B).$$ and a map of THH-May spectral sequences $$\xymatrix{ H_*(X_{\bullet}\otimes E_0^{*} (\tau_{\ge \bullet} A)) \ar[d] \ar@{=>}[r] & H_*(X_{\bullet}\otimes A) \ar[d]\\
H_*(X_{\bullet}\otimes E_0^{*} (\tau_{\ge \bullet} B )) \ar@{=>}[r] & H_*(X_{\bullet} \otimes B). }$$
\[exm 2\] Let $R$ be a commutative ring spectrum with homotopy groups $\pi_k(R)\cong \hat{\mathbb{Z}}_p$ for $k=0,n$ and $\pi_k(R)\cong 0$ otherwise. Then one can build $$0 \lra \Sigma^n H\hat{\mathbb{Z}}_p \lra R$$ as a cofibrant decreasingly filtered commutative ring spectrum using Theorem \[post filt\]. Since one can construct a Postnikov truncation of a commutative ring spectrum as a commutative ring spectrum [@MR1732625], we can produce an example of this type by considering the truncation of the connective $p$-complete complex K-theory spectrum $$\Sigma^2H\pi_2ku_p \lra ku_p^{\le 2} \lra H\pi_0ku_p.$$
The results of this subsection naturally leads to the question of whether topological Hochschild homology commutes with Postnikov limit; i.e the question of whether the map $$THH(R) \rightarrow \holim THH(R^{\le n})$$ is an equivalence. In the following section, we prove this result in the more general case of tensoring with a simplicial set. One could therefore try to compute $THH(R)$ for some ring spectrum by computing $THH(R^{\le n})$ for each $n$ using the THH-May spectral sequence and then computing the limit. As an example, we carry this out in the case $R = \hat{ku}_p$ and $n = 2$ in the next subsection.
Tensoring with simplicial sets commutes with the Postnikov limit.
-----------------------------------------------------------------
\[finiteness lemma 9273\] Let $X_{\bullet}$ be a simplicial pointed finite set. Let $E$ be a spectrum and let $R$ be an $E_{\infty}$-ring spectrum. Suppose that $E_n(R)$ is finite for all integers $n$. Then $E_n(X_{\bullet}\otimes R)$ is finite for all integers $n$.
We will make use of the “pretensor product” $\tilde{\otimes}$ defined in Definition \[def of tensoring\]. The Bousfield-Kan-type spectral sequence obtained by applying $E_*$ to the simplicial object $X_{\bullet}\tilde{\otimes} R$ has $E^1$-term $E^1_{s,t} \cong \coprod_{X_s} E_t(R)$, differentials $d^r: E^r_{s,t} \rightarrow E^r_{s-r,t+r-1}$, and converges to $E_{s+t}(X_{\bullet}\otimes R)$. Consequently, this spectral sequence is half-plane with exiting differentials, in the sense of [@MR1718076]. Hence, the spectral sequence is strongly convergent, by Theorem 6.1 of [@MR1718076], and finiteness of $E^1_{s,t}$ for all $s,t$ such that $s+t = n$ implies finiteness of $E_n(X_{\bullet}\otimes R)$.
Let $R$ be an $E_\infty$-ring spectrum and let $n$ be an integer. We will write $R^{\leq n}$ for the $n$-th Postnikov truncation of $R$, that is, $R^{\leq n}$ is $R$ with $E_\infty$-cells attached to kill all the homotopy groups of $R$ above dimension $n$. Consequently we have a map of $E_{\infty}$-ring spectra $R \rightarrow R^{\leq n}$ which induces an isomorphism $\pi_i(R) \rightarrow \pi_i(R^{\leq n})$ for all $i\leq n$, and such that $\pi_i(R^{\leq n}) \cong 0$ if $i>n$.
In the statements of Theorems \[p-complete postnikov completion\] and \[rational postnikov completion\], the homotopy limit in \[comparison map 2308437\] and \[comparison map 2308438\] can be taken in $E_{\infty}$-ring spectra or in spectra; since the forgetful functor from $E_{\infty}$-ring spectra to spectra commutes with homotopy limits, the maps \[comparison map 2308437\] and \[comparison map 2308438\] are weak equivalences either way.
\[p-complete postnikov completion\] Let $R$ be a connective $E_\infty$-ring spectrum. Let $p$ be a prime number such that the $i$-th mod $p$ homotopy group $(S/p)_i(R)$ is finite for each integer $i$. Let $X_{\bullet}$ be a simplicial pointed finite set. Then the natural map of $E_\infty$-ring spectra $$\label{comparison map 2308437} (X_{\bullet} \tensor R)^{\hat{}}_p \rightarrow \left( \holim_n X_{\bullet}\otimes (R^{\leq n})\right)^{\hat{}}_p$$ is a weak equivalence.
Since we assumed that $(S/p)_i(R)$ is finite for each $i$ and since we assumed that $X_{\bullet}$ is a simplicial [*finite*]{} set, Lemma \[finiteness lemma 9273\] implies that $(S/p)_i(X_{\bullet}\otimes R^{\leq n})$ is finite for all $i$ and all $n$. So the first right-derived limit $\lim_n{}^1(S/p)_i(X_{\bullet}\otimes R^{\leq n})$ vanishes, by the well-known vanishing of $\lim{}^1$ for inverse sequences of finite abelian groups.
For each nonnegative integer $n$, the map of Bousfield-Kan-type spectral sequences $$\label{ss map 21038}\xymatrix{
E^1_{s,t} \cong \coprod_{X_s} (S/p)_t(R) \ar@{=>}[r] \ar[d] & (S/p)_{s+t}(X_{\bullet}\otimes R)\ar[d] \\
E^1_{s,t} \cong \coprod_{X_s} (S/p)_t(R^{\leq n}) \ar@{=>}[r] & (S/p)_{s+t}(X_{\bullet}\otimes R^{\leq n}) }$$ is an isomorphism on the portion of the $E^1$-page satisfying $t < n-1$. The differential in these Bousfield-Kan-type spectral sequences is of the form $d^r: E^r_{s,t} \rightarrow E^r_{s-r,t+r-1}$, and consequently both of these Bousfield-Kan-type spectral sequences are half-plane spectral sequences with exiting differentials, in the sense of [@MR1718076]. Consequently both spectral sequence are strongly convergent, by Theorem 6.1 of [@MR1718076].
Furthermore, since the map of spectral sequences \[ss map 21038\] is an isomorphism at $E^1$ in bidegrees $(s,t)$ satisfying $t< n-1$, and since elements in total degree $u$ can only interact, by supporting differentials or being hit by differentials, with elements in bidegrees $(s,t)$ such that $t\leq u+1$, the map of spectral sequences \[ss map 21038\] is an isomorphism of spectral sequences when restricted to total degrees $< n-1$. Hence the map of abelian groups $(S/p)_u(X_{\bullet} \otimes R) \rightarrow (S/p)_u(X_{\bullet}\otimes R^{\leq n})$ is an isomorphism when $u< n-1$. Hence the map of graded abelian groups $$(S/p)_*(X_{\bullet} \otimes R) \rightarrow \lim_n (S/p)_*(X_{\bullet}\otimes R^{\leq n})$$ is an isomorphism. Vanishing of $\lim{}^1$ then tells us that the map $$(S/p)_*(X_{\bullet} \otimes R) \rightarrow (S/p)_*(\holim_n X_{\bullet}\otimes R^{\leq n})$$ is an isomorphism, i.e., that $$\label{map 324098743} X_{\bullet} \otimes R \rightarrow \holim_n X_{\bullet}\otimes R^{\leq n}$$ is a $S/p$-local weak equivalence, i.e., that \[map 324098743\] is a weak equivalence after $p$-completion.
\[rational postnikov completion\] Let $R$ be a connective $E_\infty$-ring spectrum. Suppose that, for each integer $i$, the $\mathbb{Q}$-vector space $\pi_i(R) \otimes_{\mathbb{Z}}\mathbb{Q}$ is finite-dimensional. Let $X_{\bullet}$ be a simplicial pointed finite set. Then the natural map of $E_\infty$-ring spectra $$\label{comparison map 2308438} L_{H\mathbb{Q}}(X_{\bullet} \tensor R) \rightarrow L_{H\mathbb{Q}}\left( \holim_n X_{\bullet}\otimes (R^{\leq n})\right)$$ is a weak equivalence. (Here we are writing $L_{H\mathbb{Q}}$ for Bousfield localization at the Eilenberg-Mac Lane spectrum $H\mathbb{Q}$, i.e., $L_{H\mathbb{Q}}$ is rationalization.)
Essentially the same proof as that of Theorem \[p-complete postnikov completion\]; the only substantial difference is that, rather than $\lim{}^1$ vanishing being a consequence of finiteness of the mod $p$ homotopy groups, in the present situation we have vanishing of $$\lim_n{}^1 \left(\pi_*(X_{\bullet}\otimes R^{\leq n}))\otimes_{\mathbb{Z}}\mathbb{Q}\right)$$ due to the fact that $\lim{}^1$ vanishes on any inverse sequence of finite-dimensional vector spaces over a field; see [@MR1361910].
Let $p$ be a prime, and let $R$ be a $p$-local connective $E_\infty$-ring spectrum. Suppose that, for each integer $i$, the $\mathbb{Z}_{(p)}$-module $\pi_i(R)$ is finitely generated. Let $X_{\bullet}$ be a simplicial pointed finite set. Then the natural map of $E_\infty$-ring spectra $$\label{comparison map 2308439} X_{\bullet} \tensor R \rightarrow \holim_n X_{\bullet}\otimes (R^{\leq n})$$ is a weak equivalence.
It follows from the pullback square in rings $$\xymatrix{
\mathbb{Z}_{(p)} \ar[r]\ar[d] & \mathbb{Q} \ar[d] \\ \hat{\mathbb{Z}}_p \ar[r] & \mathbb{Q}_p
}$$ that a map of connective finite-type $p$-local spectra which is both a $p$-complete weak equivalence and a rational weak equivalence is also a weak equivalence.
Let $R$ be a connective $E_\infty$-ring spectrum. Suppose that, for each integer $i$, the abelian group $\pi_i(R)$ is finitely generated. Let $X_{\bullet}$ be a simplicial pointed finite set. Then the natural map of $E_\infty$-ring spectra $$X_{\bullet} \tensor R \rightarrow \holim_n X_{\bullet}\otimes (R^{\leq n})$$ is a weak equivalence.
Again, it is classical that a map of connective finite-type spectra which is a rational equivalence and a $p$-complete weak equivalence at each prime $p$ is also a weak equivalence.
Applications
============
We now present two calculations: first, we calculate $(S/p)_*THH(R)$ when $R$ has the property that $\pi_*(R)\cong \hat{\mathbb{Z}}_p[x]/x^2$ where $|x|>0$; second, we provide a bound on topological Hochschild homology of a connective commutative ring spectrum $R$ in terms of $THH(H\pi_*(R))$ and we give an explicit bound in the case $\pi_*(R)\cong \hat{\mathbb{Z}}_p[x]$ where $|x|=2n$ for $n>0$.
Topological Hochschild homology of Postnikov truncations
--------------------------------------------------------
Let $R$ be a commutative ring spectrum with the property that $\pi_*(R)\cong \hat{\mathbb{Z}}_p[x]/x^2$ with $|x|>0$. We will consider the THH-May spectral sequence $$(S/p)_*(THH(H\hat{\mathbb{Z}}_p \ltimes \Sigma^{n} H\hat{\mathbb{Z}}_p)) \Rightarrow (S/p)_*(THH(R))$$ produced using the short filtration of a commutative ring spectrum $R$ given in Example \[exm 2\]. First, we compute the input of the $S/p$-THH-May spectral sequence for this example.
Let $p$ be an odd prime, then $$(S/p)_*(THH(H\hat{\mathbb{Z}}_p\ltimes \Sigma^{n} H\hat{\mathbb{Z}}_p)) \cong E(\lambda_1)\otimes_{\mathbb{F}_p} P(\mu_1)\otimes_{\mathbb{F}_p} HH_*(E(x))$$ where $|x|=n$. The grading of $HH_*(E(x))$ is given by the sum of the internal and homological gradings.
Due to Bökstedt [@bok], there is an isomorphism $$\pi_*(S/p\smash THH(H\hat{\mathbb{Z}}_p))\cong E(\lambda_1)\otimes_{\mathbb{F}_p} P(\mu_1).$$ Let $S\ltimes \Sigma^{n}S$ be the trivial split square-zero extension of $S$ by $\Sigma^nS$. Then $H\mathbb{Z}$ and $S\ltimes \Sigma^{n}S$ are commutative $S$-algebras and $H\hat{\mathbb{Z}}_p \ltimes \Sigma^{n} H\hat{\mathbb{Z}}_p\simeq H\hat{\mathbb{Z}}_p \smash S\ltimes \Sigma^{n}S$. By [@MR1783629 Thm. 3.1], there are equivalences $$\begin{array}{rcl} THH(H\hat{\mathbb{Z}}_p\ltimes \Sigma^{n} H\hat{\mathbb{Z}}_p)) &\simeq & THH(H\hat{\mathbb{Z}}_p \smash(S\ltimes \Sigma^{n}S)) \\
& \simeq & THH(H\hat{\mathbb{Z}}_p )\smash THH(S\ltimes \Sigma^n S ) \\ \end{array}$$ of commutative ring spectra. Since $S/p\smash H\hat{\mathbb{Z}}_p\simeq H\mathbb{F}_p $ and the spectrum $THH(H\hat{\mathbb{Z}}_p )$ is a $H\hat{\mathbb{Z}}_p$-algebra, the spectrum $S/p\smash THH(H\hat{\mathbb{Z}}_p)$ naturally has the structure of a $H\mathbb{F}_p$-module. Hence, there are isomorphisms $$\begin{array}{rc} \pi_*(S/p\smash THH(H\hat{\mathbb{Z}}_p )\smash THH(S\ltimes \Sigma^nS )) &\cong \\
\pi_*(S/p\smash THH(H\hat{\mathbb{Z}}_p )\smash_{H\mathbb{F}_p} H\mathbb{F}_p\smash THH(S\ltimes \Sigma^nS )) &\cong \\
\pi_*(S/p\smash THH(H\hat{\mathbb{Z}}_p ))\otimes_{\mathbb{F}_p} {H\mathbb{F}_p}_*(THH(S\ltimes \Sigma^{n}S)). &\\ \end{array}$$ Now, we apply the Bökstedt spectral sequence $$HH_*( {H\mathbb{F}_p}_*(S\ltimes \Sigma^n S) ) \Rightarrow H_*(THH(S\ltimes \Sigma^nS ))$$ where the input is $HH_*(E(x)).$ If $|x|$ is odd, then $HH_*(E(x))\cong E(x)\otimes_{\mathbb{F}_p} \Gamma(\sigma x)$, which can be seen from the standard fact that $\Tor_*^{E(x)}(\mathbb{F}_p,\mathbb{F}_p)\cong \Gamma(\sigma x)$ and a change of rings argument, for example see [@MR1209233]. If $|x|$ is even, then one easily computes $$HH_n(E(x)) \cong \left \{ \begin{array}{ll} E(x) & *=0 \\ \Sigma^{|x|(2i-1)} k\{1\} & n=2i-1 \\ \Sigma^{|x|(2i+1)} k\{x\}& n=2i \end{array} \right.$$ for $i\ge 1$. There is an isomorphism of bigraded rings $$HH_{*,*}(E(x))\cong E(x)[x_i,y_j: i\ge 1, j\ge 0 ]/\sim$$ where the degrees are given by $|x_i|=(2i,2|x|i+|x|)$ and $|y_j|=(2j+1,2j|x|+|x|)$, and the equivalence relation is the one that makes all products zero. The representatives in the cyclic bar complex for $x_i$ and $y_j$ are $x^{\otimes 2i+1}$ and $1\otimes x^{\otimes 2j+1}$ respectively. Whether $|x|$ is even or odd, the Bökstedt spectral sequence collapses for bi-degree reasons. (Also see [@MR2183525 Prop. 3.3] for the more general calculation of $HH_*(\mathbb{F}_p[x]/x^n)$ when $|x|=2n$ and $n>0$ and $p\nmid n$.)
(Rigidity of $S/p\wedge THH$ for Postnikov truncations ) \[rigid square-zero\] Let $R$ be a connective $E_{\infty}$-ring spectrum with $\pi_*(R)\cong \hat{\mathbb{Z}}_p[x]/x^2$, $\pi_i(R)\cong 0$ for $i\neq0,k$. Suppose that $$p \not \equiv k+1 \text{ mod } 2k+1.$$ Then $\pi_*(S/p\wedge THH(R))$ depends only on $\pi_*(R^{\le 2k})$; i.e only on $p$ and $k$.
The THH-May spectral sequence $$(S/p)_{*,*}(THH(H\hat{\mathbb{Z}}_p\ltimes \Sigma^{2k} H\hat{\mathbb{Z}}_p)) \Rightarrow (S/p)_*(THH(R))$$ collapses since there are no possible differentials for bidegree reasons under the assumptions on $k$ with respect to $p$.
Corollary \[rigid square-zero\] can be considered a rigidity theorem in the sense that $S/p\wedge THH$ does not see the first Postnikov $k$-invariant in the cases given by the congruences above.
Let $p$ be a prime such that $p\not \equiv 2 \text{ mod }3$, then $$\pi_*(S/p\smash THH(ku_p^{\le 2}))\cong E(\lambda_1)\otimes_{\mathbb{F}_p} P(\mu_1)\otimes_{\mathbb{F}_p} HH_*(E(x)),$$ where $|x|=2$ and the degree of $HH_*(E(x))$ in $\pi_*$ is given by the sum of the internal and homological degree.
Upper bounds on the size of $THH$.
----------------------------------
Many explicit computations are possible using the $THH$-May spectral sequence; for example, G. Angelini-Knoll’s computations of topological Hochschild homology of the algebraic $K$-theory of finite fields, in [@K1localsphere]. These computations are sufficiently lengthy that they merit their own separate paper.
In lieu of explicit computations using the $THH$-May spectral sequence, we point out that the mere existence of the $THH$-May spectral sequence implies an upper bound on the size of the topological Hochschild homology groups of a ring spectrum: namely, if $R$ is a graded-commutative ring and $X_{\bullet}$ is a simplicial finite set and $E_*$ is a generalized homology theory, then for any $E_{\infty}$-ring spectrum $A$ such that $\pi_*(A) \cong R$, $E_*(X_{\bullet}\otimes A))$ is a subquotient of $E_*(X_{\bullet} \otimes HR)$. Here $HR$ is the generalized Eilenberg-Mac Lane spectrum of the graded ring $R$.
In particular:
\[upper bound thm\] For all integers $n$ and all connective $E_{\infty}$-ring spectra $A$, the cardinality of $THH_n(A)$ is always less than or equal to the cardinality of $THH_n(H\pi_*(A))$.
Below are more details in a more restricted class of examples, namely, the $E_{\infty}$ ring spectra $A$ such that $\pi_*(A) \cong \hat{\mathbb{Z}}_p[x]$.
We put a partial ordering on power series with integer coefficients as follows: given $f,g\in \mathbb{Z}[[t]]$, we write $f\leq g$ if and only if, for all nonnegative integers $n$, the coefficient of $t^n$ in $f$ is less than or equal to the coefficient of $t^n$ in $g$.
Lemma \[finite generation of THH lemma\] is surely not a new result:
\[finite generation of THH lemma\] Suppose that $A$ is a connective $E_{\infty}$-ring spectrum such that the abelian group $\pi_n(A)$ is finitely generated for all $n$. Suppose that $X_{\bullet}$ is a simplicial finite set. Then $\pi_n(X_{\bullet}\otimes A)$ is finitely generated for all $n$.
First, a quick induction: if we have already shown that the abelian group $\pi_n(A^{\smash m})$ is finitely generated for all $n$, then the Künneth spectral sequence $$\Tor^{\pi_*(S)}_{*,*}(\pi_*A, \pi_n(A^{\smash m})) \Rightarrow \pi_*(A^{\smash m+1})$$ is finitely generated in each bidegree and is a first-quadrant spectral sequence (with differentials according to the Serre convention), hence strongly convergent and has $E_{\infty}$-page a finitely generated abelian group in each total degree. So the abelian group $\pi_n(A^{\smash m+1})$ is also finitely generated for each $n$.
Consequently in the Bousfield-Kan-type spectral sequence $$\begin{aligned}
E^1_{s,t} \cong \pi_t(A^{\smash \#(X_s)})
& \Rightarrow \pi_*(X_{\bullet}\otimes A) \\
d^r: E^r_{s,t}
& \rightarrow E^r_{s-r,t+r-1}\end{aligned}$$ obtained by applying $\pi_*$ to the simplicial ring spectrum $X_{\bullet}\tilde{\otimes} A$ (here we are using the pretensor product, of Definition \[def of tensoring\]), each bidegree is a finitely generated abelian group, and the spectral sequence is half-plane with exiting differentials, hence also strongly convergent by Theorem 6.1 of [@MR1718076]. Consequently $\pi_n(X_{\bullet}\otimes A)$ is a finitely generated abelian group for each integer $n$.
\[polynomial case 120\] Let $n$ be a positive integer, $p$ a prime number, and let $E$ be an $E_{\infty}$-ring spectrum such that $\pi_*(E) \cong \hat{\mathbb{Z}}_p[x]$, with $x$ in grading degree $2n$. Then the Poincaré series of the mod $p$ topological Hochschild homology $(S/p)_*(THH(E))$ satisfies the inequality $$\sum_{i\geq 0} \left( \dim_{\mathbb{F}_p} (S/p)_*(THH(E))\right) t^i \leq
\frac{(1 + (2p-1)t)(1 + (2n+1)t)}{(1 - 2nt)(1 - 2pt)}.$$
It is a classical computation of Bökstedt (see [@bok]) that $$(S/p)_*(THH(H\hat{\mathbb{Z}}_p)) \cong E(\lambda_1)\otimes_{\mathbb{F}_p} P(\mu_1),$$ with $\lambda_1$ and $\mu_1$ in grading degrees $2p-1$ and $2p$ respectively. Now we use the splitting theorem of Schwänzl, Vogt, and Waldhausen, Lemma 3.1 of [@MR1783629]: if $K$ is a commutative ring, and $W$ is a $q$-cofibrant $S$-algebra (i.e., up to equivalence, an $A_{\infty}$-ring spectrum), then there exists a weak equivalence of $S$-modules (not necessarily a weak equivalence of $S$-algebras!): $$THH(W\smash HK)\simeq THH(W)\smash THH(HK) \simeq THH(W)\smash HK\smash_{HK} THH(HK).$$
In our case, $W$ is the free $A_{\infty}$-algebra on a single $2n$-cell, and $K = \hat{\mathbb{Z}}_p$. Hence $THH(W)\smash HK$ satisfies $$(S/p)_*(THH(W)\smash HK) \cong (H\mathbb{F}_p)_*(THH(W)) \cong P(x)\otimes_{\mathbb{F}_p} E(\sigma x),$$ by collapse of the Bökstedt spectral sequence for bidegree reasons. Hence $(S/p)_*(THH(H\hat{\mathbb{Z}}_p[x]))$ is isomorphic, as a graded $\mathbb{F}_p$-vector space (but not necessarily as an $\mathbb{F}_p$-algebra!), to $$E(\lambda_1,\sigma x)\otimes_{\mathbb{F}_p} P(\mu_1,x),$$ which has Poincaré series $\frac{(1 + (2p-1)t)(1 + (2n+1)t)}{(1 - 2nt)(1 - 2pt)}$.
Here are a few amusing consequences:
Let $n$ be a positive integer, $p$ a prime number, and let $E$ be an $E_{\infty}$-ring spectrum such that $\pi_*(E) \cong \hat{\mathbb{Z}}_p[x]$, with $x$ in grading degree $2n$.
- If $p$ does not divide $n$, then $THH_{2i}(E) \cong 0$ for all $i$ congruent to $-p$ modulo $n$ such that $i\leq pn-p-n$, and $THH_{2i}(E) \cong 0$ for all $i$ congruent to $-n$ modulo $p$ such that $i\leq pn-p-n$. In particular, $THH_{2(pn-p-n)}(E) \cong 0$.
- If $p$ divides $n$, then $THH_{i}(E)\cong 0$, unless $i$ is congruent to $-1,0,$ or $1$ modulo $2p$.
We split the proof into two cases: the case where $p \nmid n$ and the case where $p|n$.
- If $p$ does not divide $n$, then the largest integer $i$ such that the graded polynomial algebra $P(\mu_1,x)$ is trivial in grading degree $2i$ is $2(pn-p-n)$. (This is a standard exercise in elementary number theory. In schools in the United States it is often presented to students in a form like “What is the largest integer $N$ such that you cannot make exactly $5N$ cents using only dimes and quarters?”) Triviality of $P(\mu_1,x)$ in grading degree $2(pn-p-n)$ also implies triviality of $P(\mu_1,x)$ in grading degree $2(pn-p-n) - 2(p+n)$, hence the triviality of $E(\lambda_1,\sigma x)\otimes_{\mathbb{F}_p} P(\mu_1,x)$ in grading degree $2(pn-p-n)$, hence (multiplying by powers of $x$ or $\mu_1$) the triviality of $E(\lambda_1,\sigma x)\otimes_{\mathbb{F}_p} P(\mu_1,x)$ in all grading degrees $\leq 2(pn-p-n)$ which are congruent to $-2p$ modulo $2n$ or congruent to $-2n$ modulo $2p$.
So $(S/p)_{2i}(THH(E))$ vanishes if $i \leq pn-p-n$ and $i\equiv -p$ modulo $n$ or $i \equiv -n$ modulo $p$. The long exact sequence $$\dots \rightarrow (S/p)_{2i+1}(THH(E)) \rightarrow THH_{2i}(E) \stackrel{p}{\longrightarrow} THH_{2i}(E) \rightarrow (S/p)_{2i}(E) \rightarrow \dots$$ then implies that $THH_{2i}(E)$ is $p$-divisible. By Lemma \[finite generation of THH lemma\], $THH_{2i}(E)$ is finitely generated. Since $\pi_0(E) \cong \hat{\mathbb{Z}}_p$, $THH_{2i}(E)$ is a $\hat{\mathbb{Z}}_p$-module. The only finitely generated abelian group which is $p$-divisible and admits the structure of a $\hat{\mathbb{Z}}_p$-module is the trivial group.
- If $p$ divides $n$, then $E(\lambda_1,\sigma x)\otimes_{\mathbb{F}_p} P(\mu_1,x)$ is concentrated in grading degrees congruent to $-1,0$ and $1$ modulo $2p$. An argument exactly as in the previous part of this proof then shows that, if $i$ is not congruent to $-1,0,$ or $1$ modulo $2p$, then $THH_{i}(E)$ must be a $p$-divisible finitely generated abelian group which admits the structure of a $\hat{\mathbb{Z}}_p$-module, hence is trivial.
Appendix: construction of the spectral sequence with coefficients.
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In this appendix, we construct the spectral sequence of Theorem \[subprop on thh-may ss\] with coefficients in a symmetric bimodule. This has proven computationally useful in the paper [@K1localsphere] by G. Angelini-Knoll on topological Hochschild homology of $K(\mathbb{F}_q)$.
For clarity of exposition, we gave the construction of the topological Hochschild-May spectral sequence [*without*]{} coefficients (i.e., with coefficients in the commutative monoid object itself) in Section \[construction of ss section\]. The essential ideas in the construction of the spectral sequence are clearer in that case. Now we build the more general version of the spectral sequence in which we allow coefficients in a symmetric bimodule object. Since the necessary constructions are all quite similar to those of Section \[construction of ss section\], and the reader who understands Section \[construction of ss section\] will find no surprises here, we have relegated this material to an appendix.
We begin by extending Definition \[def of tensoring\] to include coefficients. For the following definitions, let $f\Sets_{+}$ be the category of pointed finite sets and basepoint preserving maps. Let $\mathcal{C}$ be a model category satisfying Running Assumptions \[ra:1\].
\[def of tensoring coeff\] For a cofibrant commutative monoid object $A$ in $\mathcal{C}$, we define a functor $$- \tilde{\otimes}(A; -): sf\Sets_{+} \times A\text{-mod} \rightarrow sA\text{-mod} ,$$ which we call the *pretensor product with coefficients*, as follows. If $Y_{\bullet}$ is a pointed simplicial finite set with basepoint $\{*_{Y_{\bullet}}\}$, $A$ is a commutative monoid in $\mathcal{C}$ and $M$ is a symmetric $A$-bimodule, then the simplicial object in $\mathcal{C}$ is given by:
- For all $n\in \mathbb{N}$, the $n$-simplex object $(Y_{\bullet}\tilde{\otimes}(A;M))_n$ is defined as $$(Y_{\bullet}\tilde{\otimes}(A;M))_n: = M \smash \bigsmash_{y\in Y_n-\{*_{Y_{n}}\}} A,$$ i.e, the smash product of $M$ and a copy of $A$ for each $n$-simplex $y\in Y_n-\{*_{Y_n}\}$.
- For all positive $n\in \mathbb{N}$ and all $0\le i\le n$, the $i$-th face map $$d_i : (Y_{\bullet} \tilde{\otimes} (A ;M) )_n \rightarrow (Y_{\bullet} \tilde{\otimes} (A ;M) )_{n-1}$$ is a smash product of two maps. The first map is defined as follows: for each $n$-simplex $y\in Y_n-\{y \in Y_n : \delta_i(y)\ne *_{Y_n-1}\}$, we associate a map $$A \rightarrow \bigsmash_{\left \{y\in Y_{n-1}-\{*_{Y_{n-1}}\} \right \} } A$$ which is inclusion into the coproduct in Comm($\mathcal{C}$) of the smash factor corresponding to the $n-1$-simplex $\delta_i(y)\in Y_{n-1}-\{*_{Y_{n-1}}\}$. The first map is then defined using the universal property of the coproduct in Comm($\mathcal{C}$) and then applying the forgetful functor to $\mathcal{C}$. The second map $$M \smash \bigsmash_{\left \{y\in Y_n-\{*_{Y_n}\}:\delta_i(y)=*_{Y_{n-1}}\right \}} A\rightarrow M$$ is given by composing the action map of $A$ on $M$ with itself in the evident way.
- For all positive $n\in \mathbb{N}$ and all $0\le i\le n$, the $i$-th degeneracy is a smash product of two maps. On the component corresponding to a $n$-simplex $$y\in Y_n-\{y\in Y_n: \sigma_i(y)\ne*_{Y_{n+1}}\}$$ we define the map $$A \rightarrow \bigsmash_{\left \{y\in Y_{n+1}-\{*_{Y_{n+1}}\}\right \}} A$$ as the inclusion of the smash factor corresponding to the $(n+1)$-simplex $\sigma_i(y)$ in the coproduct in Comm$(\mathcal{C})$. The first map is then defined using the universal property of the coproduct in Comm($\mathcal{C}$) and then applying the forgetful functor to $\mathcal{C}$. The second map, corresponding to the $i$-th degeneracy on $\{*_{Y_n}\}$, is the map $$M \smash \bigsmash_{\{y'\in Y_n : \sigma_i(y')=*_{Y_{n+1}}\}} A \rightarrow M$$ which is given by composing the action map of $A$ on $M$ with itself in the evident way.
The pretensor product is defined on morphisms in the evident way as in Definition \[def of tensoring\].
For $A$ a commutative monoid in $\mathcal{C}$, we define the tensor product with coefficients $$- \otimes (A;-): sf\Sets_{+} \times A\text{-mod}\rightarrow sA\text{-mod}$$ to be the geometric realization of the pretensor product: $$Y_{\bullet}\otimes (A;M) = | Y_{\bullet} \tilde{\otimes}(A;M)|.$$ One can check that when $M$ is a commutative symmetric $A$-bimodule algebra; i.e., the multiplication map is a map of $A$-bimodules and the unit $S\rightarrow M$ factors through the unit $S\rightarrow A$, then $Y_{\bullet}\otimes (A;M)$ is an object in $\Comm \mathcal{C}$.
As one would expect, if $Y_{\bullet}=(\Delta[1]/\delta\Delta[1])_{\bullet}$ where the basepoint is $\Delta[0]\subset \Delta[1]$, then $Y_{\bullet}\otimes (A;M)$ is identified with $THH(A;M)$; i.e. usual topological Hochschild homology with coefficients. We note that $THH(A;M)$ will be a module over $THH(A)$.
\[def of may filt coeff 0\] [**(Some important colimit diagrams with coefficients I.)**]{}
- - Let $S$ and $T$ be finite sets with distinguised basepoints $*_{S}$ and $*_T$ respectively, and suppose there is a basepoint preserving map $f:T\rightarrow S$. We can equip $\mathbb{N}^S$ with the strict direct product order as in Definition \[def of may filt 0 1\] and define a function of partially-ordered sets $\mathbb{N}_+^f:\mathbb{N}^T\rightarrow\mathbb{N}^S$ by $$(\mathbb{N}^{f}_{+}(x))(s)=\sum_{\{t\in T:f(t)=s\}}x(t)$$ as before. (The only way in which this differs from Definition \[def of may filt 0 1\] is that we are assured that $(\mathbb{N}^{f}_{+}(x))(*_S)$ has $x(*_T)$ as a summand.) This defines a functor $$\mathbb{N}_+^{(-)}: f\Sets_{+}\rightarrow \POSets.$$
- As in Definition \[def of may filt 0 1\], $\mathbb{N}_+^f$ will preserve the evident $L^1$ norm.
\[def of may filt coeff 1\] [**(Some important colimit diagrams with coefficients II.)**]{}
- - When $S$ is a pointed set, let $\mathcal{D}^S_n$ be the subposet of $\mathbb{N}^S$ consisting of $x\in \mathbb{N}^S$ such that $|x|\ge n$.
- A basepoint preserving function between finite pointed sets, $T\rightarrow S$, induces a map $\mathcal{D}^T_n \stackrel{{\mathcal{D}_n}^f_+}{\longrightarrow} \mathcal{D}^S_n$ of partially-ordered sets by restriction of $\mathbb{N}_+^f$.
- For each $n\in\mathbb{N}$, this defines a functor $${\mathcal{D}^{-}_n}_+: f\Sets_+ \rightarrow \POSets$$ from the category of finite pointed sets to the category of partially-ordered sets.
- - Let $S$ be a pointed finite set. For each $x\in \mathbb{N}^S$ and each $n\in \mathbb{N}$, let $\mathcal{D}^S_{n; x}$ denote the following sub-poset of $\mathbb{N}^S$: $$\mathcal{D}^S_{n; x} = \left\{ y \in \mathbb{N}^S: y \geq x, \mbox{\ and\ } \left| y\right| \geq n + \left| x\right| \right\}$$ as in Definition \[def of may filt 0 2\].
- If $T \stackrel{f}{\longrightarrow} S$ is a basepoint preserving function between finite pointed sets and $x\in \mathbb{N}^T$ and $n\in\mathbb{N}$, let $\mathcal{D}^T_{n;x} \stackrel{{\mathcal{D}^f_{n;x}}_+}{\longrightarrow} \mathcal{D}^S_{n;\mathcal{D}^f_n(x)}$ be the function of partially-ordered sets defined by restricting $\mathbb{N}_{+}^f$ to $\mathcal{D}^T_{n;x}$.
For each $n\in\mathbb{N}$ and each $x\in\mathbb{N}^{T}$, this defines a functor $${\mathcal{D}^{-}_{n;x}}_{+}: f\Sets_{+} \rightarrow \POSets$$ from the category of finite pointed sets to the category of partially-ordered sets.
\[def of may filt coeff 2\] [**(Some important colimit diagrams with coefficients III.)**]{}
- Let $S$ be a finite pointed set and let $n$ be a nonnegative integer. Write $\mathcal{E}^n_S$ for the set $$\mathcal{E}^n_S = \left\{ x\in \{ 0,1, \dots , n\}^{S} : \sum_{s\in S} x(s) \geq n \right\} .$$ We partially-order $\mathcal{E}^n_S$ by the strict direct product order, i.e., $x^{\prime}\leq x$ if and only if $x^{\prime}(s)\leq x(s)$ for all $s \in S$.
- The definition of $\mathcal{E}^n_S$ is natural in $S$ in the following sense: if $T\stackrel{f}{\longrightarrow} S$ is a basepoint preserving map of finite pointed sets and $x\in \mathbb{N}^T$, we have a map of partially-ordered sets $$\begin{aligned}
{\mathcal{E}^f_{n;x}}_{+}: \mathcal{E}_n^T & \rightarrow & \mathcal{E}_n^S \\
\left(\mathcal{E}^f_{n;x}(y)\right) (s) & = & \min \left\{ n, y(s) - \sum_{\{t\in T, f(t) = s\}} \left( x(t) + y(t) \right) \right\} \end{aligned}$$ Note that this functor depends on the choice of $x\in \mathbb{N}^T$. Functoriality follows in the same way as in Definition \[def of may filt 0 3\].
(The only way in which this differs from Definition \[def of may filt 0 3\] is that when we evaluate on the basepoint of $S$, $$\left({\mathcal{E}^f_{n;x}}_{+}(y)\right) (*_{S})= \min \left\{ n, y(*_S) - \sum_{\{t\in T_{+}, f(t) = *_S\}} \left( x(t) + y(t) \right) \right\}$$ the sum will be nonempty because it contains $x(*_{T})+y(*_{T})$.)
\[def of may filt coeff 3\] [**(Some important colimit diagrams with coefficients IV.)**]{}
- - Let $(I_{\bullet},M_{\bullet})$ be a pair with $I_{\bullet}$ a cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$ and $M_{\bullet}$ a cofibrant decreasingly filtered symmetric $I_{\bullet}$-module. Let S be a pointed set with basepoint $*_S$. In this case, let $$\mathcal{F}^{S}(I_{\bullet},M_{\bullet}): \left(\mathbb{N}^{S}\right)^{\op}\rightarrow \mathcal{C}$$ be the functor sending $x$ to $$M_{x(*_{S})} \smash \smash_{s\in S-\{*_S\}} I_{x(s)},$$ and let $$\mathcal{F}^{S}_n(I_{\bullet},M_{\bullet}): \left(\mathcal{D}^{S}_n\right)^{\op}\rightarrow \mathcal{C}$$ be the composite of $\mathcal{F}^{S}(I_{\bullet},M_{\bullet})$ with the inclusion of $\mathcal{D}^{S}_n$ into $\mathbb{N}^{S}$: $$\left(\mathcal{D}^{S}_n\right)^{\op} \hookrightarrow \left(\mathbb{N}^{S}\right)^{\op}\stackrel{\mathcal{F}^{S}(I_{\bullet},M_{\bullet})}{\longrightarrow}
\mathcal{C}.$$
- For $x\in\mathcal{D}^{S}_n$, we write $\mathcal{F}^{S}_{n; x}(I_{\bullet}, M_{\bullet})$ for the restriction of the diagram $\mathcal{F}^{S}(I_{\bullet},M_{\bullet})$ to $\mathcal{D}^{S}_{n; x}$, i.e. $\mathcal{F}^{S}_{n; x}(I_{\bullet},M_{\bullet})$ is the composite $$(\mathcal{D}^{S}_{n; x})^{\op} \hookrightarrow \left(\mathbb{N}^{S}\right)^{\op}\stackrel{\mathcal{F}^{S}(I_{\bullet},M_{\bullet})}{\longrightarrow}
\mathcal{C}.$$
- Finally, let $\mathcal{M}^{S}_n(I_{\bullet},M_{\bullet})$ denote the colimit $$\mathcal{M}^{S}_n(I_{\bullet},M_{\bullet}) = \colim \left(\mathcal{F}^{S}_n(I_{\bullet},M_{\bullet})\right)$$ in $\mathcal{C}$. We get a sequence in $C$ induced by the natural inclusion of $\mathcal{D}^{S}_n$ into $\mathcal{D}^{S}_{n-1}$, $$\label{may filt coeff diagram} \dots \rightarrow \mathcal{M}^{S}_2(I_{\bullet},M_{\bullet})
\rightarrow \mathcal{M}^{S}_1(I_{\bullet},M_{\bullet}) \rightarrow \mathcal{M}^{S}_0(I_{\bullet},M_{\bullet}) \cong M_0\smash \smash_{s\in S-\{*_S\}} I_0.$$ We refer to the functor $\mathbb{N}^{\op}\rightarrow \mathcal{C}$ given by sending $n$ to $\mathcal{M}^{S}_n(I_{\bullet},M_{\bullet})$ as the [*May filtration with coefficients*]{}.
- The May filtration with coefficients is functorial in $S$ in the following sense: if $T\stackrel{f}{\longrightarrow} S$ is a basepoint preserving function of finite pointed sets, we have a functor $$\begin{aligned}
{\mathcal{D}^f_n}_{+}: \mathcal{D}^{T}_n & \rightarrow & \mathcal{D}^{S}_n \\
\left({\mathcal{D}^f_n}_{+}(x)\right)(s) & \mapsto & \sum_{\left\{ t\in T: f(t) = s\right\}} x(t)\end{aligned}$$ and a map of diagrams from $\mathcal{F}^{T}_n(I_{\bullet},M_{\bullet})$ to $\mathcal{F}^{S}_n(I_{\bullet},M_{\bullet})$ given by sending the object $$\mathcal{F}^{T}_n(I_{\bullet},M_{\bullet})(x) = M_{x(*_T)}\smash \bigsmash_{t\in T-\{*_T\}}I_{x(t)}$$ by the map $$M_{x(*_T)} \smash \bigsmash_{t\in T-\{*_T\}}I_{x(t)} \rightarrow M_{\Sigma_{\{t\in T :f(t)=*_S\}}x(t)} \smash \bigsmash_{s\in S-\{*_S\}} I_{\Sigma_{\{ t\in T-\{*_T\}: f(t) = s\}} x(t)}$$ given as the smash product, across all $s\in S$, of the maps $$\smash_{\{ t\in T-\{*_T\}: f(t) = s\}} I_{x(t)} \rightarrow \smash I_{\Sigma_{\{ t\in T-\{*_T\}: f(t) = s\}} x(t)}$$ given by multiplication via the maps $\rho$ of Definition \[def of dec filt comm mon\] and the maps $$M_{x(*_T)} \smash \smash_{\{ t\in T-\{*_T\}: f(t) = *_S\}} I_{x(t)} \rightarrow M_{\Sigma_{\{t\in T :f(t)=*_S\}}x(t)}$$ given by module maps $\psi$ of Definition \[def of dec filt bimodule\].
To really make Definition \[def of may filt coeff 0\] precise, we should say in which order we multiply the factors using the maps $\rho$ and $\psi$; but the purpose of the associativity and commutativity axioms in Definition \[def of dec filt comm mon\] and Definition \[def of dec filt bimodule\] is that any two such choices commute, so any choice of order of multiplication will do.
\[def of may filt w coeffs\] [**(Definition of May filtration with coefficients.)**]{} Let $(I_{\bullet},M_{\bullet})$ be a pair with $I_{\bullet}$ a cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$ and $M_{\bullet}$ a filtered symmetric $I_{\bullet}$-bimodule in $\mathcal{C}$. Let $Y_{\bullet}$ be a pointed simplicial finite set. By the [*May filtration with coefficients on $Y_{\bullet}\tilde{\otimes} (I_0;M_{0})$*]{} we mean the functor $$\mathcal{M}^{Y_{\bullet}}(I_{\bullet},M_{\bullet}): \mathbb{N}^{\op} \rightarrow s\mathcal{C}$$ given by sending a natural number $n$ to the simplicial object of $\mathcal{C}$ $$\xymatrix{\mathcal{M}^{Y_0}_n(I_{\bullet},M_{\bullet}) \ar[r] &
\mathcal{M}^{Y_1}_n(I_{\bullet}, M_{\bullet}) \ar@<1ex>[l] \ar@<-1ex>[l] \ar@<1ex>[r] \ar@<-1ex>[r] &
\mathcal{M}^{Y_2}_n(I_{\bullet}, M_{\bullet}) \ar@<2ex>[l] \ar@<-2ex>[l] \ar[l] \ar@<2ex>[r] \ar@<-2ex>[r] \ar[r] &
\dots \ar@<-3ex>[l] \ar@<-1ex>[l] \ar@<1ex>[l] \ar@<3ex>[l] ,}$$ with $\mathcal{M}^{Y_i}_n(I_{\bullet},M_{\bullet})$ defined as in Definition \[def of may filt coeff 0\], and with face and degeneracy maps defined as follows:
- The face map $$d_i: \mathcal{M}^{Y_j}_n(I_{\bullet},M_{\bullet}) \rightarrow \mathcal{M}^{Y_{j-1}}_n(I_{\bullet},M_{\bullet})$$ is the colimit of the map of diagrams $$\mathcal{F}^{Y_j}_n(I_{\bullet},M_{\bullet}) \rightarrow \mathcal{F}^{Y_{j-1}}_n(I_{\bullet},M_{\bullet})$$ induced, as in Definition \[def of may filt coeff 0\], by $\delta_i: Y_j\rightarrow Y_{j-1}$.
- The degeneracy map $$s_i: \mathcal{M}^{Y_j}_n(I_{\bullet},M_{\bullet}) \rightarrow \mathcal{M}^{Y_{j+1}}_n(I_{\bullet},M_{\bullet})$$ is the colimit of the map of diagrams $$\mathcal{F}^{Y_j}_n(I_{\bullet},M_{\bullet}) \rightarrow \mathcal{F}^{Y_{j+1}}_n(I_{\bullet},M_{\bullet})$$ induced, as in Definition \[def of may filt coeff 0\], by $\sigma_i: Y_j\rightarrow Y_{j+1}$. The only way in which this differs from Definition \[def of may filt 0 4\] is that the maps $\delta_i$ and $\sigma_i$ are now the basepoint preserving structure maps of the pointed simplicial object $Y_{\bullet}$.
Note that the maps $\rho$ of Definition \[def of dec filt comm mon\] and $\psi$ of Definition \[def of dec filt bimodule\] yield, by taking smash products of the maps $\rho$ and $\psi$ associative and symmetric bimodule maps, $$\mathcal{F}_m^S(I_{\bullet})\smash \mathcal{F}_n^S(I_{\bullet},M_{\bullet}) \rightarrow \mathcal{F}_{n+m}^S(I_{\bullet},M_{\bullet})$$ hence after taking colimits, we produce maps $$\mathcal{M}_{m}^S(I_{\bullet})\smash \mathcal{M}_n^S(I_{\bullet},M_{\bullet}) \rightarrow \mathcal{M}_{n+m}^S(I_{\bullet},M_{\bullet});$$ i.e. the functor $$\mathbb{N}^{\op} \rightarrow \mathcal{C}$$ $$n \mapsto \mathcal{M}_n^S(I_{\bullet},M_{\bullet})$$ is a cofibrant decreasingly filtered symmetric $\mathcal{M}_n^S(I_{\bullet})$-bimodule in the sense of Definition \[def of dec filt bimodule\]. The same considerations as in Remark \[rem on structure\] give $|\mathcal{M}^{X_{\bullet}}(I_{\bullet},M_{\bullet})|$ the structure of a cofibrant decreasingly filtered symmetric $|\mathcal{M}^{X_{\bullet}}(I_{\bullet})|$-bimodule in the sense of Definition \[def of dec filt bimodule\].
We now need to adapt Lemmas \[cofiber computation lemma\], \[rather hard lemma\], and \[cofinality lemma\] to our situtation.
\[cofiber computation lemma 2\] Let $I_{\bullet}$ be a cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$ and let $M_{\bullet}$ be a cofibrant decreasingly filtered symmetric $I_{\bullet}$ bimodule. Let $S$ be a pointed set and $n\in \mathbb{N}$. There is a monomorphism $\iota : \mathcal{D}_{n+1}^S \rightarrow \mathcal{D}_{n}^S$. We write $$Kan: \mathcal{C}^{(\mathcal{D}_n^S)^{op}} \rightarrow \mathcal{C}^{(\mathcal{D}_{n+1}^S)^{op}}$$ for the left Kan extension of $\mathcal{F}_{n+1}^S(I_{\bullet},M_{\bullet})$ induced by $\iota^{op}$ and define $$\tilde{\mathcal{F}}_{n+1}^S(I_{\bullet},M_{\bullet}) :=Kan \left(\mathcal{F}_{n+1}^S(I_{\bullet},M_{\bullet})\right ).$$ The universal property of the Kan extension produces a canonical map $$c:\tilde{\mathcal{F}}_{n+1}^S(I_{\bullet},M_{\bullet}) \rightarrow \mathcal{F}_{n+1}^S(I_{\bullet},M_{\bullet} ).$$ With these definitions, the cofiber of the map $$\text{colim}(\tilde{\mathcal{F}}_{n+1}^S(I_{\bullet},M_{\bullet})) \stackrel{\text{colim }c}{\longrightarrow} \text{colim} \mathcal{F}_n^S(I_{\bullet},M_{\bullet}),$$ where colimits are computed in $\mathcal{C}$, is isomorphic to the coproduct in $\mathcal{C}$ $$\coprod_{\{x\in \mathbb{N}^S :|x|=n\}} \left ( \left ( M_{x(*_{S})}\smash \smash_{s\in S-\{*_S\}} I_{x(s)} \right ) / \left ( \text{colim} \mathcal{F}_{1;x}^S(I_{\bullet},M_{\bullet}) \right) \right ) .$$ This isomorphism is natural in the variable $S$.
We omit the proof because it follows from an evident generalization of Lemma \[cofiber computation lemma\].
\[rather hard lemma 2\] Let $S$ be a finite pointed set. Suppose the map $Y_{*_S,1}\rightarrow Y_{*_S,0}$ is a cofibration and for $s\in S-\{*_S\}$ the maps $Z_{s,1} \rightarrow Z_{s,0}$ are cofibrations. Suppose the objects $Y_{*_S,1}$ and $Y_{*_S,0}$ are cofibrant and that, for $s\in S-\{*_S\}$, $Z_{s,1}$ and $Z_{s,0}$ are cofibrant. Let $\mathcal{G}_S^{+}: (\mathcal{E}^S_1)^{op}\rightarrow \mathcal{C}$ be the functor given on objects by $$\mathcal{G}_{S}^{+}(x)= Y_{*_S,x(*_s)}\smash \smash_{s\in S} Z_{s,x(s)}$$ and given on morphisms in the obvious way. Then the smash product $$Y_{*_S,0}\smash \smash_{s\in S-\{*_S\}} Z_{s,0} \rightarrow Y_{*_S,0}/Y_{*_S,1} \smash \smash_{s\in S-\{*_S\}} Z_{s,0}/Z_{s,1}$$ of the cofiber projections $Z_{s,0} \rightarrow Z_{s,0}/Z_{s,1}$ and $Y_{*_S,0}\rightarrow Y_{*_S,0}/Y_{*_S,1}$ fits into a cofiber sequence: $$\text{colim} \mathcal{G}_S^{+} \rightarrow Y_{*_S,0} \smash \smash_{s\in S-\{*_S\}} Z_{s,0} \rightarrow Y_{*_S,0}/Y_{*_S,1} \smash \smash_{s\in S-\{*_S\}} Z_{s,0}/Z_{s,1} .$$
Letting $Z_{*_S,i}=Y_{*_S,i}$ for $i=0,1$, we can prove this lemma in almost the same way as Lemma \[rather hard lemma\]. The only difference is that in the proof we need to choose $s_0\in S-\{*_S\}$ so that $*_S$ is in $S'=S-\{s_0\}$.
\[cofinality lemma 2\] Suppose $S$ is a pointed finite set and $n\in \mathbb{N}$. Let $x\in \mathbb{N}^S$ and $\mathcal{E}^S_n$ and $\mathcal{D}_{n;x}^S$ be as in Definition \[def of may filt coeff 0\] Let $J_{n;x}^S$ be the functor defined by $$J_{n;x}^S : \mathcal{E}^S_n \rightarrow \mathcal{D}_{n;x}^S$$ $$\left ( J_{n;x}(y)\right ) (s) = x(s) +y(s).$$ Then $J_{n;x}$ has a right adjoint. Consequently $J_{n;x}$ is a cofinal functor; i.e. for any functor defined on $\mathcal{D}_{n;x}^S$ such that the limit $\text{lim} F$ exists, the limit $\text{lim}(F\circ J_{n;x}^S)$ also exists, and the canonical map $\text{lim}(F\circ J_{n;x}^S) \rightarrow \text{lim} F$ is an isomorphism.
The proof follows easily from the evident generalization of the proof of Lemma \[cofinality lemma\].
[**(Fundamental theorem of the May filtration with coefficients.)**]{} \[thm on fund thm 2\] Let $I_{\bullet}$ be a cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$, let $M_{\bullet}$ be a cofibrant decreasingly filtered symmetric $I_{\bullet}$-bimodule, and let $Y_{\bullet}$ be a simplicial pointed finite set. Then the associated graded commutative monoid $E_0^*\left| \mathcal{M}^{Y_{\bullet}}(I_{\bullet},M_{\bullet} )\right|$ of the geometric realization of the May filtration is weakly equivalent to the tensoring $Y_{\bullet}\otimes ( E_0^* I_{\bullet}; E_0^* M_{\bullet}) $ of $Y_{\bullet}$ with the associated graded commutative monoid of $I_{\bullet}$ with coefficients in the associated graded symmetric $E_0^*I_{\bullet}$-bimodule: $$\label{fund w e}E_0^*\left| \mathcal{M}^{Y_{\bullet}}(I_{\bullet},M_{\bullet})\right| \simeq Y_{\bullet}\otimes (E_0^* I_{\bullet};E_0^* M_{\bullet}).$$
Since geometric realization commutes with cofibers, there is an equivalence $$|\mathcal{M}^{Y_{\bullet}}_n(I_{\bullet},M_{\bullet}) |/|\mathcal{M}^{Y_{\bullet}}_{n+1}(I_{\bullet},M_{\bullet}) | \simeq |\mathcal{M}^{Y_{\bullet}}_{n}(I_{\bullet},M_{\bullet})/\mathcal{M}^{Y_{\bullet}}_{n+1}(I_{\bullet},M_{\bullet}) |$$ and we would like to identify this cofiber. Each $Y_i$ is some finite pointed set, so we will compute the cofiber of the map $$\mathcal{M}_{n+1}^{Y_{\bullet}}(I_{\bullet},M_{\bullet}) \rightarrow \mathcal{M}_{n}^{Y_{\bullet}}(I_{\bullet},M_{\bullet})$$ on each simplicial level as follows. We claim that for any finite pointed set $S$ $$\label{eq:cof}
\mathcal{M}_{n+1}^S(I_{\bullet},M_{\bullet}) \rightarrow \mathcal{M}_{n}^S(I_{\bullet},M_{\bullet}) \rightarrow \coprod_{x\in \mathbb{N}^S;|x|=n} M_{x(*_S)}/M_{x(*_S)+1} \smash \bigsmash_{s\in S-\{*_S\}} I_{x(s)} /I_{x(s)+1}$$ is a cofiber sequence. To prove this claim, first note that $\mathcal{M}_{n+1}^S$ is defined to be $\text{colim}\mathcal{F}^S_{n+1}(I_{\bullet},M_{\bullet})$. By Lemma \[cofiber computation lemma 2\], we can identify the cofiber of the left map in Equation \[eq:cof\] as the cofiber of the left map in the diagram: $$\text{colim} \tilde{\mathcal{F}}^S_{n+1} (I_{\bullet},M_{\bullet}) \longrightarrow \text{colim}\mathcal{F}^S_{n}(I_{\bullet},M_{\bullet}) \longrightarrow \coprod_{\{x\in \mathbb{N}^S;|x|=n\}} \left ( M_{*_S} \smash \bigsmash_{s\in S-\{*_S\}} I_{x(s)} \right ) / \left ( \text{colim } \mathcal{F}_{1;x}^S(I_{\bullet},M_{\bullet} ) \right ) .$$ Lemma \[cofiber computation lemma 2\] also demonstrates naturality in the variable $S$. By Lemma \[cofinality lemma 2\], the functor $J_{1;x}$ is cofinal and hence the map $$\text{colim }( \mathcal{F}_{1,x}^S(I_{\bullet},M_{\bullet}) \circ J_{1;x})\rightarrow \text{colim } (\mathcal{F}_{1;x}^S(I_{\bullet},M_{\bullet})$$ is an isomorphism. (We are applying the dual of the statement of Lemma \[cofinality lemma 2\], which also holds.) By Lemma \[rather hard lemma 2\], we identify $$\left ( M_{x(*_S)}\smash \smash_{s\in S-\{*_S\}} I_{x(s)}\right ) / \text{colim } ( \mathcal{F}_{1;x}(I_{\bullet};M_{\bullet})\circ J_{1;x} ) \simeq \left ( M_{x(*_S)}/M_{x(*_S)+1}\smash \smash_{s\in S-\{*_S\}} I_{x(s)} / I_{x(s)+1}\right )$$ as we needed to prove the cofiber sequence of Equation \[eq:cof\]. The same considerations as in the proof of Theorem \[thm on fund theorem\] apply, producing naturality in S.
We have a sequence of simplicial objects in $\mathcal{C}$ $$\xymatrix{
\vdots \ar[d] & \vdots \ar[d] & \vdots \ar[d] & \\
\mathcal{M}^{Y_0}_2(I_{\bullet},M_{\bullet} ) \ar[r]\ar[d] &
\mathcal{M}^{Y_1}_2(I_{\bullet},M_{\bullet} ) \ar@<1ex>[l] \ar@<-1ex>[l] \ar@<1ex>[r] \ar@<-1ex>[r]\ar[d] &
\mathcal{M}^{Y_2}_2(I_{\bullet},M_{\bullet} ) \ar@<2ex>[l] \ar@<-2ex>[l] \ar[l] \ar@<2ex>[r] \ar@<-2ex>[r] \ar[r]\ar[d] &
\dots \ar@<-3ex>[l] \ar@<-1ex>[l] \ar@<1ex>[l] \ar@<3ex>[l] \\
\mathcal{M}^{Y_0}_1(I_{\bullet},M_{\bullet} ) \ar[r]\ar[d] &
\mathcal{M}^{Y_1}_1(I_{\bullet},M_{\bullet} ) \ar@<1ex>[l] \ar@<-1ex>[l] \ar@<1ex>[r] \ar@<-1ex>[r]\ar[d] &
\mathcal{M}^{Y_2}_1(I_{\bullet},M_{\bullet} ) \ar@<2ex>[l] \ar@<-2ex>[l] \ar[l] \ar@<2ex>[r] \ar@<-2ex>[r] \ar[r]\ar[d] &
\dots \ar@<-3ex>[l] \ar@<-1ex>[l] \ar@<1ex>[l] \ar@<3ex>[l] \\
\mathcal{M}^{Y_0}_0(I_{\bullet},M_{\bullet} ) \ar[r] &
\mathcal{M}^{Y_1}_0(I_{\bullet},M_{\bullet} ) \ar@<1ex>[l] \ar@<-1ex>[l] \ar@<1ex>[r] \ar@<-1ex>[r] &
\mathcal{M}^{Y_2}_0(I_{\bullet},M_{\bullet} ) \ar@<2ex>[l] \ar@<-2ex>[l] \ar[l] \ar@<2ex>[r] \ar@<-2ex>[r] \ar[r] &
\dots \ar@<-3ex>[l] \ar@<-1ex>[l] \ar@<1ex>[l] \ar@<3ex>[l]
}$$
and geometric realization commuting with cofibers implies that the comparison map $$\label{fund iso} Y_{\bullet} \otimes ( E_0^*I_{\bullet};E_0^*M_{\bullet}) \rightarrow E_0^* | \mathcal{M}^{Y_{\bullet}} (I_{\bullet},M_{\bullet}) |$$ of objects in $\mathcal{C}$ is a weak equivalence. If $M_{\bullet}=I_{\bullet}$, then we recover Equation \[comparison map 1\].
One can show that the weak equivalence of Equation \[fund w e\] in Theorem \[thm on fund thm 2\] is an equivalence of symmetric $E_0^*|\mathcal{M}^{Y_{\bullet}}(I_{\bullet})|$-bimodules. Also, in the cases when both sides are objects in Comm($\mathcal{C}$) and $M_{\bullet}$ is a cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$ as well as a cofibrant decreasingly filtered symmetric $I_{\bullet}$-bimodule, one can show that the equivalence is an equivalence of commutative monoids in $\mathcal{C}$.
\[may thh ss\] Suppose $I_{\bullet}$ is a cofibrant decreasingly filtered commutative monoid object in $\mathcal{C}$, $M_{\bullet}$ is a cofibrant decreasingly filtered symmetric $I_{\bullet}$-bimodule, and $Y_{\bullet}$ is a simplicial pointed finite set. Let $H_*$ be a connective generalized homology theory on $\mathcal{C}$ as defined in Definition \[def of gen hom thy\], then by the topological Hochschild-May spectral sequence for $Y_{\bullet}\otimes (I_{\bullet};M_{\bullet})$, we mean the spectral sequence in $\mathcal{C}$ obtained by applying $H_*$ to the tower of cofiber sequences in $\mathcal{C}$ $$\xymatrix{
\vdots \ar[d] & \\
\left| \mathcal{M}_2^{Y_{\bullet}}(I_{\bullet},M_{\bullet} )\right| \ar[r]\ar[d] & \left| \mathcal{M}_2^{X^{\bullet}}(I_{\bullet},M_{\bullet} )\right|/\left| \mathcal{M}_3^{X^{\bullet}}(I_{\bullet},M_{\bullet} )\right| \\
\left| \mathcal{M}_1^{Y_{\bullet}}(I_{\bullet},M_{\bullet} )\right| \ar[r]\ar[d] & \left| \mathcal{M}_1^{X^{\bullet}}(I_{\bullet},M_{\bullet} )\right|/\left| \mathcal{M}_2^{X^{\bullet}}(I_{\bullet},M_{\bullet} )\right| \\
\left| \mathcal{M}_0^{Y_{\bullet}}(I_{\bullet},M_{\bullet} )\right| \ar[r] & \left| \mathcal{M}_0^{X^{\bullet}}(I_{\bullet},M_{\bullet} )\right|/\left| \mathcal{M}_1^{X^{\bullet}}(I_{\bullet},M_{\bullet} )\right| .}$$ The spectral sequence we refer to is the one associated to the exact couple $$\xymatrix{
D^1_{*,*} \cong \bigoplus_{i,j} H_i \left| \mathcal{M}_j^{Y_{\bullet}}(I_{\bullet},M_{\bullet} )\right| \ar[r] & \bigoplus_{i,j} H_i \left| \mathcal{M}_j^{Y_{\bullet}}(I_{\bullet},M_{\bullet} )\right| \cong D^1_{*,*} \ar[d] \\
& E^1_{*,*} \cong \bigoplus_{i,j} H_i \left| \mathcal{M}_j^{Y_{\bullet}}(I_{\bullet},M_{\bullet} )\right|/\left| \mathcal{M}_{j+1}^{Y_{\bullet}}(I_{\bullet},M_{\bullet} )\right| \ar[ul] & . }$$
\[thh-may coeff\] Given $I_{\bullet}$, $M_{\bullet}$, $Y_{\bullet}$ and $H_*$ as in Definition \[may thh ss\]. Suppose that $I_{\bullet}$ and $M_{\bullet}$ are Hausdorff as cofibrant decreasingly filtered objects in $\mathcal{C}$. Suppose $I_{\bullet}$ and $M_{\bullet}$ satisfy
Connectivity axiom
: $H_m(I_n)\cong 0$ for all $m<n$ and $H_m(M_\ell)\cong 0$ for all $m<\ell$.
Then the topological Hochschild-May spectral sequence is strongly convergent, and its input and output and differential are as follows: $$E^1_{s,t} \cong H_{s,t} (Y_{\bullet} \otimes (E_0^* I_{\bullet},E_0^* M_{\bullet} ) ) \Rightarrow H_s(Y_{\bullet} \otimes (I_{0},M_{0} ))$$ $$d^r : E_{s,t}^r \rightarrow E_{s-1,\text{ }t+r-1}^r$$
We need to check that $\left [\Sigma^{*}Z,\holim_i \left (H\smash \left |\mathcal{M}_i^{Y_{\bullet}}(I_{\bullet},M_{\bullet}) \right | \right ) \right ]$ is trivial, but by an evident generalization of Lemma \[conn lem 2\] $H_m \left (\left | \mathcal{M}_i^{Y_{\bullet}}(I_{\bullet},M_{\bullet}) \right | \right )\cong 0$ for all $m<i$, so by Lemma \[connectivity conditions lemma\], $\left[\Sigma^{*}Z,\holim_i \left(H\smash \left |\mathcal{M}_i^{Y_{\bullet}}(I_{\bullet},M_{\bullet}) \right| \right) \right]\cong 0$ as desired. Thus, the spectral sequence converges to $H_*\left ( \left | \mathcal{M}_0^{Y_{\bullet}}(I_{\bullet},M_{\bullet})\right | \right ) \cong H_*(X_{\bullet}\otimes I_0)$. A routine computation in the spectral sequence of a tower of cofibrations implies the bidegree of the differential.
The sequence $$\xymatrix{ \dots \ar[r] & \left | \mathcal{M}_2^{Y_{\bullet}}(I_{\bullet}, M_{\bullet}) \right | \ar[r] & \left | \mathcal{M}_1^{Y_{\bullet}}(I_{\bullet}, M_{\bullet}) \right | \ar[r] & \left | \mathcal{M}_0^{Y_{\bullet}}(I_{\bullet},M_{\bullet}) \right | }$$ is a cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$ whenever $M_{\bullet}$ is a cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$ compatible with the $I_{\bullet}$-action as observed in Remark \[rem on structure\] and therefore, in particular, there is a pairing of towers in the sense of [@mult1] under these conditions. Therefore by Proposition 5.1 of [@mult1] the differentials in the spectral sequence satisfy a graded Leibniz rule when $M_{\bullet}$ is a cofibrant decreasingly filtered commutative monoid in $\mathcal{C}$ compatible with the $I_{\bullet}$-action.
Convergence is standard and follows as stated in the proof of Theorem \[subprop on thh-may ss\].
We conclude with an example. Suppose $I_{\bullet}$ is a trivially filtered commutative monoid in $\mathcal{C}$; i.e., $I_n\simeq 0$ for $n\ge 1$. Suppose $M_{\bullet}$ is a cofibrant decreasingly filtered symmetric $I_{\bullet}$-bimodule object in $\mathcal{C}$ with $M_n\simeq 0$ for $n\ge 2$. Then the sequence of simplicial commutative monoids becomes $$\xymatrix{
\mathcal{M}^{Y_0}_1(I_{\bullet},M_{\bullet} ) \ar[r]\ar[d] &
\mathcal{M}^{Y_1}_1(I_{\bullet},M_{\bullet} ) \ar@<1ex>[l] \ar@<-1ex>[l] \ar@<1ex>[r] \ar@<-1ex>[r]\ar[d] &
\mathcal{M}^{Y_2}_1(I_{\bullet},M_{\bullet} ) \ar@<2ex>[l] \ar@<-2ex>[l] \ar[l] \ar@<2ex>[r] \ar@<-2ex>[r] \ar[r]\ar[d] &
\dots \ar@<-3ex>[l] \ar@<-1ex>[l] \ar@<1ex>[l] \ar@<3ex>[l] \\
\mathcal{M}^{Y_0}_0(I_{\bullet},M_{\bullet} ) \ar[r] &
\mathcal{M}^{Y_1}_0(I_{\bullet},M_{\bullet} ) \ar@<1ex>[l] \ar@<-1ex>[l] \ar@<1ex>[r] \ar@<-1ex>[r] &
\mathcal{M}^{Y_2}_0(I_{\bullet},M_{\bullet} ) \ar@<2ex>[l] \ar@<-2ex>[l] \ar[l] \ar@<2ex>[r] \ar@<-2ex>[r] \ar[r] &
\dots \ar@<-3ex>[l] \ar@<-1ex>[l] \ar@<1ex>[l] \ar@<3ex>[l]
}$$ where the realization of $\mathcal{M}^{Y_{\bullet}}_0(I_{\bullet},M_{\bullet})$ is $Y_{\bullet}\otimes (I_0 ;M_0)$, the realization of $\mathcal{M}^{Y_{\bullet}}_1(I_{\bullet},M_{\bullet})$ is $Y_{\bullet} \otimes (I_0; M_1)$ and the realization of the quotient $\mathcal{M}^{Y_{\bullet}}_0(I_{\bullet},M_{\bullet})/ \mathcal{M}^{Y_{\bullet}}_1(I_{\bullet},M_{\bullet})$ is $\mathcal{M}^{Y_{\bullet}}(I_0, M_0/M_1)$. The spectral sequence collapses to produce a long exact sequence coming from the cofiber sequence $$Y_{\bullet} \otimes (I_0; M_1) \rightarrow Y_{\bullet}\otimes (I_0 ;M_0) \rightarrow Y_{\bullet}\otimes (I_0, M_0/M_1) .$$ When $Y_{\bullet}=\Delta[1]/\delta \Delta[1]$ with the obvious basepoint, this specializes to a cofiber sequence, $$THH(I_0,M_1) \rightarrow THH(I_0;M_0) \rightarrow THH(I_0,M_0/M_1)$$ which recovers a folklore result; i.e., a cofiber sequence in coefficient bimodules induces a cofiber sequence in topological Hochschild homology. This seems to be well known, but we do not know where it appears explicitly in the literature.
|
---
author:
- 'Johanna Wald $^{1,}$ [^1]\'
- 'Helisa Dhamo $^{1,}$\'
- 'Nassir Navab $^{1}$\'
- 'Federico Tombari $^{1,2}$\'
- '$^{1}$ Technische Universität München'
- '$^{2}$ Google'
bibliography:
- 'egbib.bib'
title: Learning 3D Semantic Scene Graphs from 3D Indoor Reconstructions
---
[^1]: the authors contributed equally to this paper
|
---
abstract: 'We present infrared spectroscopy of the recurrent nova RS Ophiuchi, obtained 11.81, 20.75 and 55.71 days following its 2006 eruption. The spectra are dominated by hydrogen recombination lines, together with [[He]{}]{}, [[O]{}]{} and [[O]{}]{} lines; the electron temperature of $\sim10^4$K implied by the recombination spectrum suggests that we are seeing primarily the wind of the red giant, ionized by the ultraviolet flash when [RS Oph]{} erupted. However, strong coronal emission lines (i.e. emission from fine structure transitions in ions having high ionization potential) are present in the last spectrum. These imply a temperature of $930\,000$ K for the coronal gas; this is in line with x-ray observations of the 2006 eruption. The emission line widths decrease with time in a way that is consistent with the shock model for the x-ray emission.'
date: Revised bersion
title: 'Infrared observations of the 2006 outburst of the recurrent nova RS Ophiuchi: the early phase'
---
\[firstpage\]
stars: individual: RS Ophiuchi — infrared: stars — binaries: symbiotic — novae, cataclysmic variables
Introduction
============
RS Ophiuchi is a recurrent nova that has undergone nova eruptions in 1898, 1933, 1958, 1967, 1985, and possibly [@1907] 1907. As in the case of a classical nova, the eruption follows a thermonuclear runaway on the surface of the white dwarf [@tnr].
The key differences between classical and recurrent novae, in terms of both system properties and outburst behaviour, are reviewed by Anupama . The recurrents are a heterogeneous class of objects but the [RS Oph]{} type is characterized by a semi-detached binary consisting of a roche-lobe-filling M giant mass donor (M8III in the case of [RS Oph]{}; [@fekel]) and a massive (close to the Chandrasekhar limit) white dwarf ([*classical*]{} novae almost exclusively have cool [*dwarf*]{} mass donors).
--------------- ------- ------ ------ --------- ----- ------ ------ ------------- --
2006 UTC Date Day Comment
$IJ$ $HK$ $I$ $J$ $H$ $K$
Feb 24.64 11.81 80 80 350 350 1000 1000
Mar 5.58 20.75 80 80 350 350 1500 1500 Cloud
Apr 9.54 55.71 80 80 350 350 1500 1500 Thin cirrus
--------------- ------- ------ ------ --------- ----- ------ ------ ------------- --
\[obs\]
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(0.0,4.0)
\[data\]
The 1985 eruption of [RS Oph]{} was the first to have been observed over the entire electromagnetic spectrum, from the radio to the x-ray (see Bode 1987). What distinguishes the evolution of the eruption in the case of [RS Oph]{} is the fact that the ejected material runs into the dense red giant wind, which is shocked [@bk]. Observations of the 1985 eruption provided indirect evidence for the shocking of the wind and the ejecta: [RS Oph]{} was a strong and rapidly evolving x-ray [@mason] and radio source [@padin], and there was coronal emission over a wide range of wavelengths [@snijders; @evans88; @shore].
Infrared (IR) observations of the 1985 eruption are presented in Evans et al. , starting on day 23 of the outburst. These authors found that the 1–2.5 spectrum was dominated by hydrogen recombination lines, and [[He]{}]{}$\lambda$1.083. Coronal lines ([\[[Si]{}\]]{}$\lambda1.965$ and [\[[Si]{}\]]{}$\lambda2.481$) were present on day 143 of the eruption. They also tentatively noted the presence of first overtone CO emission, although this was based on low resolution circular variable filter data.
[RS Oph]{} was discovered in eruption [@hirosawa] on 2006 February 12.83, which we take to be day zero for this outburst. The discovery of the 2006 eruption triggered a multi-wavelength campaign of observations [@bode-iauc; @bode-b; @das; @evans06; @eyres06a; @eyres06b; @ness-a; @ness-b; @ness-c; @charo; @obrien-iauc; @obrien-nature] and yielded for the first time direct evidence, from VLBI imaging, for an expanding shock [@obrien-nature].
In this paper we present IR spectroscopy of [RS Oph]{}, obtained in the first 55 days of the 2006 eruption, covering a much earlier phase than that observed in 1985.
Observations
============
The observations were obtained with the UIST instrument on the United Kingdom Infrared Telescope (UKIRT), on 2006 February 24, March 5 and April 9. The data were obtained in the $I\!J\!H\!K$ bands, covering the wavelength range 0.87–2.51. First order sky subtraction was achieved by nodding along the slit; HR6493 was used to remove telluric features and for flux calibration. Wavelength calibration used an argon arc, and is accurate to $\pm0.0005$in the $I\!J$ bands, and to $\pm0.0003$ in the $HK$ bands.
We note that the precipitable water vapour on March 5 was high ($\sim5$ mm, cf. $\sim1.3$ mm for February 24 and $\sim1.8$ mm for April 9). As the March 5.58 data were taken through clouds, and the April 9.54 data were taken through thin cirrus, the telluric cancellation around 1.85 (and to a lesser extent the 1.4 region) is rather poor, especially for March 5.58. We estimate that the flux calibration for the February and April observations is accurate to $\pm20$%, that for the March observation is accurate only to $\pm60$%.
The observing log is given in Table \[obs\], in which the UT times are times of mid-observation. The data for February 24 and April 9 are shown in Fig. \[data\]; the data for March 5 are omitted in view of the greater uncertainty in flux calibration.
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\[vel\]
The spectra
===========
On all dates the spectra are dominated by hydrogen recombination lines, with [[He]{}]{}, [[O]{}]{} and [[O]{}]{} also present (see Table \[nebular\]). In particular the higher members of the hydrogen Pfund ($n\rightarrow5$) series are clearly resolved, and there is no evidence for the presence of first overtone CO in either emission or absorption. Thus the identification of CO in the 1985 eruption, reported by Evans et al. remains problematic. By April 9 (day 55) we also see strong emission in silicon ([\[[Si]{}\]]{}, [\[[Si]{}\]]{}, [\[[Si]{}\]]{}) and sulphur ([\[[S]{}\]]{}, [\[[S]{}\]]{}) lines (see Table \[coronal\]); the first two of these were also reported in the 1985 eruption [@evans88].
We have measured the full-width at half-maximum (FWHM) and full-width at zero intensity (FWZI) of several emission lines for each of the three dates (see Figs \[vel\],\[vel\_fwzi\]). While the FWHM of the emission lines indicates an expansion velocity $\sim500-600$, the emission line wings extend to ${\raisebox{-0.6ex}{$\,\stackrel
{\raisebox{-.2ex}{$\textstyle >$}}{\sim}\,$}}2500$. Swift observations [@bode-b] indicate that shock velocities $\sim3000$ are present, comparable with the IR line FWZI. After deconvolving the instrumental linewidth, we have converted the FWHM to an expansion velocity (cf. Tables \[nebular\],\[coronal\]) and, from a variety of lines, derived a mean value for each date.
We find that the mean expansion velocity declines with time, i.e. the emission lines tend to get narrower as the eruption progresses (Figs \[vel\], \[vel\_fwzi\]a); furthermore, the velocity implied by the broad wings also declines (see Fig. \[vel\_fwzi\]b). This effect, which arises as the ejected material decelerates as it ploughs into the giant wind, mirrors the behaviour reported by Shore et al. for optical emission lines and by Snijders for ultraviolet emission lines during the 1985 eruption. We note that the velocities determined from the line wings are comparable with, but somewhat greater than, the shock velocities deduced from the x-ray emission (see Fig. \[vel\_fwzi\]).
We also note that the FWHM of the coronal lines in 2006 April (day 55) is greater than that of the nebular lines. This is clearly seen in Fig. \[data\]b, which includes the [\[[Si]{}\]]{}$\lambda2.483$ line.
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\[vel\_fwzi\]
----------- --------- ------------ -------------------------- ------------------------- -------------------------------
Date Dereddened line flux FWHM
($10^{-14}$ W m$^{-2}$) ($\!{\mbox{\,km\,s$^{-1}$}}$)
Feb 24.64 1.0833 [[He]{}]{} $^3$S – $^3$P$^{\rm o}$ $1131\pm7$ 949
1.12895 [[O]{}]{} $^3$P – $^3$D$^{\rm o}$ $195\pm6.2$ 720
1.2084 [[O]{}]{} $^2$D$^{\rm o}$ – $^2$D $74.0\pm2.0$ 918
1.7002 [[He]{}]{} $^3$P$^{\rm o}$ – $^3$D $9.4\pm0.5$ 611
2.05869 [[He]{}]{} $^1$S – $^1$P$^{\rm o}$ $33.7\pm2.3$ 659
Mar 5.58 1.0833 [[He]{}]{} $^3$S – $^3$P$^{\rm o}$ $649\pm3.1$ 710
1.12895 [[O]{}]{} $^3$P – $^3$D$^{\rm o}$ $71.8\pm2.6$ 620
1.2084 [[O]{}]{} $^2$D$^{\rm o}$ – $^2$D $30.3\pm3.0$ 755
2.05869 [[He]{}]{} $^1$S – $^1$P$^{\rm o}$ $13.47\pm0.96$ 488
Apr 9.54 1.0833 [[He]{}]{} $^3$S – $^3$P$^{\rm o}$ $405.8\pm2.8$ 457
1.12895 [[O]{}]{} $^3$P – $^3$D$^{\rm o}$ $41.9\pm2.5$ 266
1.2084 [[O]{}]{} $^2$D$^{\rm o}$ – $^2$D $21.21\pm0.33$ 420
2.05869 [[He]{}]{} $^1$S – $^1$P$^{\rm o}$ $5.8\pm0.2$ 318
----------- --------- ------------ -------------------------- ------------------------- -------------------------------
\[nebular\]
------------------------- ---------------------- ----------------------------------- ------------------------- -------------------------------
$\lambda$ Dereddened line flux FWHM
($\!{\mbox{$\,\mu$m}}$) ($10^{-14}$ W m$^{-2}$) ($\!{\mbox{\,km\,s$^{-1}$}}$)
0.991 0.911 [\[[S]{}\]]{} $^2$P$^{\rm o}$ – $^2$P$^{\rm o}$ $14.21\pm0.56$ 531
1.252 1.252 [\[[S]{}\]]{} $^3$P – $^3$P $14.97\pm0.39$ 646
1.431 1.430 [\[[Si]{}\]]{} $^2$P$^{\rm o}$ - $^2$P$^{\rm o}$ $27.82\pm0.64$ 355
1.653 [\[[Si]{}\]]{} $^4$P - $^4$P $<0.15$ —
1.936 [\[[Si]{}\]]{} $^3$P$^{\rm o}$ - $^3$P$^{\rm o}$ $<0.08$ —
1.962 1.965 [\[[Si]{}\]]{} $^2$P$^{\rm o}$ - $^2$P$^{\rm o}$ $5.96\pm2.77$ 550
2.481 2.483 [\[[Si]{}\]]{} $^3$P - $^3$P $6.42\pm0.13$ 554
------------------------- ---------------------- ----------------------------------- ------------------------- -------------------------------
\[coronal\]
Discussion
==========
The spectra have been dereddened for $E(B-V)=0.73$ [@snijders] and the dereddened fluxes are reported in Table \[nebular\] for the He and O lines and in Table \[coronal\] for the coronal lines.
The hydrogen recombination lines
--------------------------------
Assuming that the continuum that is clearly visible in Fig. \[data\] is optically thin free-free and free-bound emission, we estimate the electron temperature to be $\sim10^4$ K for all three of our observations, but note that the flux calibration for the March observation is not reliable as the data were taken through cloud. This temperature is constrained primarily by the magnitude of the Brackett and Pfund discontinuities at 1.45 and 2.28 respectively (Fig. \[cont\]). The electron temperature derived from the optically thin emission is considerably less than that implied by the presence of IR coronal lines in the spectra (see below), or inferred from radio [@obrien-nature] and x-ray [@bode-b] observations. There remains an excess at wavelengths ${\raisebox{-0.6ex}{$\,\stackrel
{\raisebox{-.2ex}{$\textstyle <$}}{\sim}\,$}}1.5$, some (but not all) of which may be due to a contribution from the shocked gas.
Using flux ratios for the hydrogen recombination lines, and assuming Case B [@ferland], we find that the electron density for day 55.71 is $\sim10^7$ cm$^{-3}$. Assuming the mass-loss value given by O’Brien et al. , wind velocity 20 and shock velocity $\sim2000$ (cf. Fig. \[vel\_fwzi\]), the corresponding wind column, integrated from the base of the unshocked wind to infinity, is $\sim2.0\times10^{21}$ cm$^{-2}$, in good agreement with that obtained from the x-ray data (e.g. Fig. 3 of Bode et al. ).
The coronal lines
-----------------
We can use the dereddened fluxes of the silicon coronal lines for 2006 April 9 to estimate the temperature in the coronal region. The relative fluxes for lines in a coronal gas are discussed by Greenhouse et al. and we follow their analysis here, using collisional strengths from Osterbrock and Blaha , and ionization fractions as a function of temperature from Shull & van Steenberg . We find that the temperature of the coronal gas is $\simeq930\,000$ K ($\sim0.08$ keV). We note that Ness et al. deduced a temperature of a few $\times10^6$ K from the coronal x-ray lines in a Chandra observation on Jun 4.5; the temperature for the coronal gas obtained here is broadly consistent with the x-ray data.
Origin of the IR emission
-------------------------
The deduced electron temperature, $\sim10^4$ K, implies that the hydrogen IR emission on all three dates is primarily due to emission by the red giant wind, ionized by the ultraviolet flash when [RS Oph]{} erupted. Emission by the shocked wind must also contribute to the total emission; however IR observational evidence for this is apparent only on day 55.71 with the clear development of the S and Si coronal lines. As the shock propagates into, and eventually breaks out of, the wind (which is predicted to occur around $t\sim70$ days; see O’Brien, Bode & Kahn 1992) we expect that the contribution of the coronal gas will become dominant, and that of the cooler gas to decline and eventually disappear.
While there exist at least two regions with greatly differing temperatures in the environment of [RS Oph]{} the determination of abundances is problematic. However we anticipate that this will change when the shock breaks out of the giant wind. This next phase will be discussed in a forthcoming paper.
(5.0,4.5) (0.0,4.0)
Conclusions
===========
We have reported the early IR spectroscopy of the 2006 eruption of the recurrent nova [RS Oph]{}, covering the first 55 days. We find a spectrum dominated by hydrogen recombination lines arising from a gas at $\sim10^4$ K; silicon coronal lines prominent on day 55, implying a temperature for the coronal gas of 930000 K.
IR (and other) observations of this remarkable object are continuing and in subsequent papers we will present contemporaneous observations carried out with UKIRT and the Spitzer Space Observatory.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
We thank the UKIRT Director and the various UKIRT observers for supporting this project. The United Kingdom Infrared Telescope is operated by the Joint Astronomy Centre on behalf of the U.K. Particle Physics and Astronomy Research Council (PPARC). TRG is supported by the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., on behalf of the international Gemini partnership of Argentina, Australia, Brazil, Canada, Chile, the United Kingdom, and the United States of America. Use of the UKIRT by YM and YT is supported by National Astronomical Observatory of Japan. RDG and CEW are supported in part by the NSF (AST02-05814). The work of DKL, RJR and RWR is supported by The Aerospace Corporation’s Independent Research and Development Program. J.-U. N. gratefully acknowledges support provided by NASA through Chandra Postdoctoral Fellowship grant PF5-60039 awarded by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060. JPO and KLP acknowledge support from PPARC. SGS acknowledges partial support from NSF grants to Arizona State University. Data reduction was carried out using hardware and software provided by PPARC.
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abstract: 'We present and discuss our recent study of an eikonal two channel model, in which we reproduce the soft total, integrated elastic and diffractive cross sections, and the corresponding forward differential slopes in the ISR-Tevatron energy range. Our study is extended to provide predictions at the LHC and Cosmic Rays energies. These are utilized to assess the role of unitarity at ultra high energies, as well as predict the implied survival probability of exclusive diffractive central production of a light Higgs. Our approach is critically examined so as to estimate the margins of error of the calculated survival probability for diffractive Higgs production.'
author:
- 'E. Gotsman'
- 'E. Levin'
- 'U. Maor'
title: A Study of Soft Interactions at Ultra High Energies
---
[^1]
Introduction
============
The search for unambiguous s-channel unitarity signatures in ultra high energies soft hadronic scattering, is two folded: On the one hand, this is a fundamental issue on which we have only limited information from the ISR-Tevatron experiments. The only direct indication we have on the importance of unitarity considerations, derives from the observation that soft diffraction cross sections, essentially SD (single diffraction), have a much milder energy dependence than the seemingly similar, elastic cross sections. Enforcing unitarity constraints is a model dependent procedure. Thus, reliable modeling is essential for the execution of our study, leading to predictions of interest for LHC and AUGER experiments. On the other hand, unitarity considerations in soft scattering are instrumental for the assessment of inelastic hard diffraction rates, specifically, diffractive Higgs production at the LHC. Preliminary information on the importance and method of this calculation has been acquired in the study of hard diffractive di-jets at the Tevatron[@heralhc], leading to first generation estimates of the corresponding survival probabilities.
This presentation is based on our recent paper[@GLM07], which utilizes the GLM model[@heralhc; @1CH; @2CH; @SP2CH; @GKLMP; @GKLM] where we numerically solve the $s$-channel unitarity equation in an eikonal model. Our updated results, in the ISR-Tevatron range, were obtained from an improved two channel model calculations. The specific objectives of our study, based on the above, were: 1) To reproduce the total, integrated elastic and diffractive cross sections and corresponding forward differential slopes in the ISR-Tevatron energy range, and to obtain predictions for these observables at LHC and Cosmic Rays energies. 2) To calculate the survival probabilities of inelastic hard diffractive processes[@Bj; @GLM1]. This requires precise knowledge of the soft elastic and diffractive scattering amplitudes of the initial hadronic projectiles. As we noted, it is of particular importance for the assessment of the discovery potential for LHC Higgs production in an exclusive central diffractive process. 3) Some of the fundamental consequences of s-channel unitarity in the high energy limit are not clear, as yet. We examin the approach of the scattering amplitudes to the black disc bound. 4) We estimate the margin of error of our predicted survival probabilities, based on a critical analysis of our model.
The GLM Model
=============
The main assumption of the two channel GLM model is that hadrons are the correct degrees of freedom at high energies, diagonalizing the scattering matrix. In this Good-Walker type formalism, diffractively produced hadrons at a given vertex are considered as a single hadronic state described by the wave function $\Psi_D$, which is orthonormal to the wave function $\Psi_h$ of the incoming hadron, $<\Psi_h|\Psi_D>=0 $. We introduce two wave functions $\Psi_1$ and $\Psi_2$ which diagonalize the 2x2 interaction matrix ${\bf T}$ \[2CHM\] A\_[i,k]{}\^[i’,k’]{}=<\_[i]{}\_[k]{}||\_[i’]{}\_[k’]{}>= A\_[i,k]{}\_[i,i’]{}\_[k,k’]{}. In this representation the observed states are written \[2CHM31\] \_h=\_1+\_2, \[2CHM32\] \_D=-\_1+\_2, where, $\alpha^2+\beta^2=1$.
Using we can rewrite the unitarity equations \[UNIT\] ImA\_[i,k]{}(s,b)=|A\_[i,k]{}(s,b)|\^2 +G\^[in]{}\_[i,k]{}(s,b), where $G^{in}_{i,k}$ is the summed probability for all non diffractive inelastic processes induced by the initial $(i,k)$ states. The simple solution to has the form obtained in a single channel formalism[@1CH], \[2CHM1\] A\_[i,k]{}(s,b)=i [(]{}1 -[(]{}- [)]{}[)]{}, \[2CHM2\] G\^[in]{}\_[i,k]{}(s,b)=1-[(]{}- \_[i,k]{}(s,b)[)]{}. From we deduce the probability that the initial projectiles $(i,k)$ reach the final state interaction unchanged, regardless of the initial state re-scatterings, is given by $P^S_{i,k}=\exp {\left(}- \Omega_{i,k}(s,b) {\right)}$.
In general, we have to consider four possible $(i,k)$ re-scattering options. For initial $p$-$p$ (or $\bar p$-$p$) the two quasi-elastic amplitudes are equal $A_{1,2}=A_{2,1}$, and we have three re-scattering amplitudes. The corresponding elastic, SD and DD amplitudes are \[EL\] a\_[el]{}(s,b)= i{\^4A\_[1,1]{}+2\^2\^2A\_[1,2]{}+\^4[[A]{}]{}\_[2,2]{}}, \[SD\] a\_[sd]{}(s,b)= i{-\^2A\_[1,1]{}+(\^2-\^2)A\_[1,2]{}+\^2A\_[2,2]{}}, \[DD\] a\_[dd]{}= i\^2\^2{A\_[1,1]{}-2A\_[1,2]{}+A\_[2,2]{}}. Adjusted parameters are introduced to obtain explicit expressions for the opacities $\Omega_{i,k}(s,b)$.
In the following we shall consider Regge and non Regge options for the dynamics of interest. We use a simple general form for the input opacities, \[omega\] \_[i,k]{}[(]{}s,b [)]{}= \_[i,k]{}[(]{}s [)]{}[(]{}s,b [)]{}. \_[i,k]{}[(]{}s [)]{}= \^0\_[i,k]{}[(]{}[)]{}\^. The input b-profiles $\Gamma_{i,k} {\left(}s,b {\right)}$ are assumed to be Gaussians in b, corresponding to exponential differential cross sections in t-space, \_[i,k]{} [(]{}s,b [)]{}= [(]{}- [)]{}, \[radius\] R\^2\_[i,k]{}[(]{}s [)]{}= R\^2\_[0;i,k]{}+4C ln(s/s\_0). $R^2_{0;1,2}=\frac{1}{2}R^2_{0;1,1}$ and $R^2_{0;2,2}=0$. Our parametrization is compatible with, but not exclusive to, a Regge type input.
Fits and Predictions
====================
We have studied three models, with different parameterizations of $\Omega_{i,k}$, which were adjusted to the ISR-Tevatron experimental data base, specified above. Note that the fit has, in addition to the contribution in the form of , also a secondary Regge sector (see Ref.[@1CH; @2CH]). This is necessary, as the data base contains a relatively small number of experimental high energy measured values, which are independent of the Regge contribution. We do not quote the values of the Regge parameters, as the goal of this paper is to obtain predictions in the LHC and Cosmic Rays energy range. At W=1800$GeV$ the Regge sector contribution is less than 1$\%$. However, it is essential at the ISR energies.
Model A is a simplified two amplitude version of the two channel model, in which we assume that $\sigma_{dd}$ is small enough to be neglected. As such, this model breaks Regge factorization. The model was presented and discussed in Ref.[@2CH]. The parameters of Model A were obtained from a fit to a 55 experimental data points base and are listed in Table 1 with a corresponding $\chi^2/(d.o.f)$ of 1.50. Note that in Model A the (1,1) amplitude corresponds to $\Omega_{1,1}$, while the (1,2) amplitude corresponds to $\Delta \Omega=\Omega_{1,1}-\Omega_{1,2}$. See Ref.[@2CH].
Model A Model B(1) Model B(2)
------------------ ------------------ ------------------- -------------------
$\Delta$ 0.126 0.150 0.150
$\beta$ 0.464 0.526 0.776
$R^{2}_{0;1,1}$ 16.34 $GeV^{-2}$ 20.80 $GeV^{-2}$ 20.83 $GeV^{-2}$
$C$ 0.200 $GeV^{-2}$ 0.184 $GeV^{-2}$ 0.173 $GeV^{-2}$
$\sigma^0_{1,1}$ 12.99 $GeV^{-2}$ 4.84 $GeV^{-2}$ 9.22 $GeV^{-2}$
$\sigma^0_{2,2}$ N/A 4006.9 $GeV^{-2}$ 3503.5 $GeV^{-2}$
$\sigma^0_{1,2}$ 145.6$GeV^{-2}$ 139.3 $GeV^{-2}$ 6.5 $GeV^{-2}$
: Fitted parameters for Models A, B(1) and B(2).
Model B denotes our three amplitude model where the 5 published DD cross section points[@DDD] are contained in the fitted data base. The three opacities are taken to be Gaussians in $b$. If we assume the soft Pomeron to be a simple J pole, its coupling factorization implies $\sigma^0_{1,2} = \sqrt{\sigma^0_{1,1}\times \sigma^0_{2,2}}$. We denote this Model B(1). The fit obtained is not satisfactory, with a $\chi^2/(d.o.f.)$=2.30.
We have, also, studied Model B(2) in which coupling factorization is not assumed. Accordingly, $\sigma^0_{1,1}$, $\sigma^0_{1,2}$ and $\sigma^0_{2,2}$ are independent fitted parameters of the model. The model with a $\chi^2/(d.o.f.)$ = 1.25, provides a very good reproduction of our data base. In Model B(2) the leading t channel exchange is not a simple J pole. It is compatible with a model[@SAT] we have suggested a while ago in which the soft Pomeron dominated photo and low $Q^2$ DIS, is perceived as the saturated soft (low $Q^2$) limit of the hard Pomeron dominated (high $Q^2$) hard DIS. A major deficiency of Model B(2) is that it predicts dips in $\frac{d\sigma_{el}}{dt}$ at small $t$ values, which are not observed experimentally. This problem is common to all eikonal models which assume Gaussian b-profiles. Consequently, Model B(2) is valid only in the narrow forward $t$ cone, where it reproduces approximately 85$\%$ of the overall data very well. We shall discuss this problem in some detail in the Discussion Section.
--------------- ----------------- ------------------ ----------------- --------------- ------------ ---------- ------- --------------------------------------- --
$\sqrt{s}$ $\sigma_{tot} $ $ \sigma_{el} $ $ \sigma_{sd} $ $\sigma_{dd}$ $B_{el}$ $R_{el}$ $R_D$ $\frac {\sigma_{diff}} {\sigma_{el}}$
TeV mb mb mb mb $GeV^{-2}$
1.8 78.0 16.3 9.6 3.8 16.8 0.21 0.38 0.83
14 110.5 25.3 11.6 4.9 20.5 0.23 0.38 0.65
30 124.8 29.7 12.2 5.3 22.0 0.24 0.38 0.59
60 139.0 34.3 12.7 5.7 23.4 0.25 0.38 0.54
120 154.0 39.6 13.2 6.1 24.9 0.26 0.38 0.49
250 172.0 45.9 13.6 6.6 26.5 0.27 0.38 0.44
500 190.0 52.7 14.0 7.0 28.1 0.28 0.39 0.40
1000 209.0 60.2 14.3 7.4 29.8 0.29 0.39 0.10
$10^{11}$ 1070.0 451.2 21.6 19.5 109.9 0.42 0.46 0.09
1.22$10^{19}$ 1970.0 871.4 25.5 27.7 202.6 0.44 0.47 0.06
(Planck)
--------------- ----------------- ------------------ ----------------- --------------- ------------ ---------- ------- --------------------------------------- --
: Cross sections and elastic slope in Model B(2).
Model B(2) cross section and slope predictions at ultra high energies are summarized in Table 2. Note that $R_{el}=\sigma_{el}/\sigma_{tot}$ and $R_{D}=(\sigma_{el}+\sigma_{diff})/\sigma_{tot}$. At LHC (W=14 $TeV$) our predicted cross sections are: $\sigma_{tot}=110.5\,mb$, $\sigma_{el}=25.3\,mb$, $\sigma_{sd}=11.6 \,mb$ and $\sigma_{dd}=4.9\,mb$. These predictions are slightly higher than those obtained[@2CH] in Model A. The corresponding forward slopes are: $B_{el}=20.5 \,GeV^{-2}$, $B_{sd}=15.9\,GeV^{-2}$ and $B_{dd}=13.5\,GeV^{-2}$. We calculate, also, $\rho=0.125$. The calculations of $B_{sd}$, $B_{dd}$ and $\rho$ were executed with the fitted parameters of the model. For the record we have checked that we reproduce also the UA4, CDF and E710 $B_{sd}$ and $\rho$ data points.
Survival probabilities
======================
In the following we shall limit our discussion to the survival probability of Higgs production in an exclusive central diffractive process, calculated in our model. For a general review see Ref.[@heralhc].
In our model we assume an input Gaussian $b$-dependence also for the hard diffractive amplitude of interest. Its input, when convoluted with the soft (i,k) channel, is $$\label{3.6}
{\Omega_{i,k}^H}={\nu_{i,k}^H(s)} \Gamma_{i,k}^H(b),$$ $$\label{3.7}
\nu_{i,k}^H=\sigma_{i,k}^{H0}(\frac{s}{s})^{\Delta_H},$$ $$\label{3.8}
\Gamma_{i,k}^H(b)=\frac{1}{\pi {R_{i,k}^H}^2}\,e^{-\frac{\,b^2}{{R_{i,k}^H}^2}}.$$
![\[sp-dia\]Survival probability for exclusive central diffractive production of the Higgs boson](lrgdiagm.eps){width="65mm"}
The structure of the survival probability expression is shown in . The corresponding general formulae for the calculation of the survival probability for diffractive Higgs boson production have been discussed in Refs.[@SP2CH; @heralhc; @GKLMP]. Accordingly, \[SP\] [S \^2 ]{}=, &N(s) = d\^2b\_1d\^2b\_2 {A\_H(s,b\_1)A\_H(s,b\_2)\
& (1-A\_S (s,(\_1+\_2 )))}\^2, \[SP1\]\
&D(s) = d\^2b\_1d\^2b\_2 {A\_H(s,b\_1)A\_H(s,b\_2)}\^2. \[SP2\] $A_s$ denotes the soft strong interaction amplitude given by . Using -, the integrands of and are reduced by eliminating common $s$-dependent expressions. &&N(s) = d\^2 b\_1 d\^2 b\_2 {(1 - a\_[el]{}(s,b))A\^[pp]{}\_H(b\_1) A\^[pp]{}\_H(b\_2)\
&& - a\_[sd]{}(s,b)[(]{}A\^[pd]{}\_H(b\_1) A\^[pp]{}\_H(b\_2) + A\^[pp]{}\_H(b\_1) A\^[pd]{}\_H(b\_2) [)]{}\
&& - a\_[dd]{}(s,b) A\^[pd]{}\_H(b\_1) A\^[pd]{}\_H(b\_2)}\^2 \[SP3\], \[SP4\] D = d\^2 b\_1 d\^2 b\_2 {A\^[pp]{}\_H(b\_1) A\^[pp]{}\_H(b\_2)}\^2.
Following Refs.[@heralhc; @GLM07] we introduce two hard $b$-profiles A\^[pp]{}\_H(b) &=& [(]{}- [)]{}, \[2C10\]\
A\^[pd]{}\_H(b) &=& [(]{}-[)]{}. \[2C11\] The hard radii ${R_{i,k}^H}^2$ and cross section coefficients $V_{p \to p}$ and $V_{p \to d}$ are constants derived from HERA $J/\Psi$ elastic and inelastic photo and DIS production[@KOTE; @PSISL] (see, also, Ref.[@GKLMP]). $B_{el}^H=3.6 GeV^{-2}$, $B_{in}^H=1 GeV^{-2}$, $V_{p \to p}=\sqrt{3}$ and $V_{p \to d}=1$. have been taken from the experimental HERA data on $J/\Psi$ production in HERA[@KOTE; @PSISL].
Using - we calculate the survival probability $S^2_H$ for exclusive Higgs production in central diffraction. $S^2_H$ has been calculated[@heralhc] in the two amplitude Model A. The resulting $S^2_H\,=\,0.027$ is essentially the same as the predictions of KMR[@KKMR]. Our present results, obtained in the three amplitude B Models, indicate a reduction of the output value of $S^2_H$. Its LHC value in Model B(1) is 0.02, and in Model B(2) it is 0.007. We note that, our Model B(1) result is compatible with the result of Ref.[@KKMR]. We shall return to this issue in the Discussion Section.
Amplitude Analysis
==================
The basic amplitudes of the GLM two channel model are $A_{1,1}$, $A_{1,2}$ and $A_{2,2}$, whose $b$ structure is specified in ). These are the building blocks with which we construct $a_{el}$, $a_{sd}$ and $a_{dd}$ (-). The $A_{i,k}$ amplitudes are bounded by the black disc unitarity bound of unity. Checking Table 1, it is evident that in both Model B(1) and B(2) $\Omega_{2,2}$ is much larger than the other two fitted opacities. As a consequence, the amplitude $A_{2,2}(s,b)$ reaches the unitarity bound of unity at low energies. Similarly, the output amplitude $A_{1,2}(s,b)$ of Model A reaches unity at approximately LHC energy. The observation that one, or even two, of our $A_{i,k}(s,b)$=1 does not imply that the elastic scattering amplitude has reached the unitarity bound at these $(s,b)$ values. $a_{el}(s,b)$ reaches the black disc bound when, and only when, $A_{1,1}(s,b)$=$A_{1,2}(s,b)$=$A_{2,2}(s,b)$=1. In such a case we also obtain, that $a_{sd}(s,b)$=$a_{dd}(s,b)$=0. This result is independent of the fitted value of $\beta$.
Model B(2) predictions of $a_{el}$ over a wide range of energies are presented in . A fundamental feature of Models A, B(1) and B(2) is that $a_{el}$ approaches the black disc bound at $b=0$ very slowly, reaching the bound at energies higher than the GZK knee cutoff. If correct, this feature implies that $a_{el}$ does not reach the black disc bound over the entire accessible spectrum of Cosmic Rays energies, even though it gets monotonically darker.
The explanation of this behavior, in our presentation, is simple. Checking the values of $\beta$ and $\sigma_{i,k}^0$ corresponding to the 3 models (see Table 1), we note that $\Omega_{1,1}$ is smaller by 1-3 orders of magnitude relative to $\Omega_{2,2}$ ($\Omega_{1,2}$ in Model A). The consequent $a_{el}$ can reach the black disc bound only when $\Omega_{1,1}$ is large enough so that $A_{1,1}$ approaches unity. $\Omega_{1,1}$ grows slowly like $W^{0.3}$ (modulu $ln W$). Hence, the slow approach of $a_{el}$ toward the black disc bound. This result is incompatible with the output of Ref.[@KKMR] in which $a_{el}$ reaches the black disc bound approximately at the LHC. In our presentation it implies that unlike our models, in the KMR model there is relatively small variance in the weights of the 3 components of the proton wave function.
![b dependence of $a_{el}$ in Model B(2) at different energies[]{data-label="ampel"}](comaelg.eps){width="80mm"}
A consequence of the input $\Omega_{i,k}$ being large at small $b$, is that $P_{i,k}^S(s,b)$ is very small at $b$ = 0 and monotonically approaches its limiting value of 1, in the high $b$ limit. As a result, given a diffractive (non screened) input, its output (screened) amplitude is peripheral in $b$. This is a general feature, common to all eikonal models regardless of their b-profiles details. The same is, true, also, with regard to diffractive Good-Walker channels, which are contained in $\Omega_{i,k}$. This implies a non trivial $t$ dependence of $d\sigma_{diff}(M^2_{diff})/dt$ in the diffractive channels. These qualitative features are induced by Model A, B(1) and B(2), even though their detailed behavior are not identical. Given the deficiencies of our b-profiles, we refrain from giving any specific predictions beside the general observation stated above.
The general behavior indicated above becomes more extreme at ultra high energies, when $a_{el}$ continues to expand and gets darker. Consequently, the inelastic diffractive channels becomes more and more peripheral and relatively smaller when compared with the elastic channel. At the extreme, when $a_{el}(s,b)$ = 1, $a_{sd}\,=\,a_{dd}\,=\,0$. We demonstrate this feature and its consequence at the Planck mass in . As the black core of $a_{el}$ expands, the difference between Models A, B(1) and B(2), considered in this paper, diminishes, being confined to the narrow $b$ tail where $a_{el}(s,b)<1$. The above observations may be of interest in the analysis of Cosmic Ray experiments.
Discussion
==========
It is interesting to compare our model and its output with a different eikonal model recently proposed by KMR[@KMR] extending earlier versions[@KKMR]. The two models were constructed with very similar objectives but are fundamentally different in their conceptual theoretical input, data analysis and output results. 1) The input of KMR is a conventional Regge model in which high mass diffraction, initiated by Pomeron enhanced diagrams, is included. GLM is a phenomenological parametrization in which we assume diffraction to be strictly Good-Walker type, with no high mass diffraction distinction. We formulate our input in a general form consistent with Regge, but not exclusively so. Our statistically preferred non factorizeable Model B(2) is compatible with a partonic interpretation which considers the soft “Pomeron” to be a low $Q^2$ high density limit of the hard Pomeron[@SAT]. The GLM “Pomeron” is not a Regge simple J-pole, it does not include Pomeron enhanced diagrams, which are essential in the construction of KMR. 2) Since multi-Pomeron vertices are included in KMR, they had to fix $\alpha^{\prime}=0$. In order to maintain the experimentally observed forward $t$-cone shrinkage, they constructed a high absorption eikonal model in which the input is non conventional $\Delta=0.55$. With this input, KMR obtain an approximate DL behavior[@DL] in the ISR-Tevatron range. However, at higher energies their effective $\Delta$ becomes monotonically smaller (its value in the Tevatron-LHC range is reduced to 0.04) which results in a very slow rise of $\sigma_{tot}$ and $\sigma_{el}$. GLM is a weak screening eikonal model. Its fitted input is $\Delta=0.15$ and $C=\alpha^{\prime}=0.17$. With this input, GLM total cross sections are compatible with DL over the wide ISR-GZK range. 3) The goal of both GLM and KMR is to adjust the model parameters of their vacuum t exchange “Pomeron” input, so as to predict and calculate observables and factors of interest at the LHC and Cosmic Rays. Both models adjust more than 10 free parameters. Only CERN-UA4 and Tevatron energies are sufficiently high to justify neglecting the contribution of the secondary Regge sector. This limited data base is not sufficient to adjust the “Pomeron” free parameters. GLM chose, therefore, to construct a model containing also the secondary Regge sector and fit the extended data base spanning the ISR-Tevatron energy range. KMR constrain their parameter adjustment to the small data base of the highest energies. In our opinion the KMR procedure is not adequate. Indeed, their reconstruction of $\frac{d\sigma_{el}}{dt}$ at the 3 highest available energies is remarkably similar to a fit they made a few years ago with different parameters, notably a conventional $\Delta$ input. 4) GLM and KMR determine their input opacities in completely different procedures which define their (different) data bases. GLM approach is that a model which takes into account diffractive re-scatterings of the initial projectiles has to reconstruct properly the diffractive cross sections, which are, thus, included in its fitted data base. KMR goal is to reconstruct $a_{el}(s,b)$ for which the diffractive components are needed. To this end they fit $\frac{d\sigma_{el}}{dt}$ neglecting an explicit fit of the diffractive channels. Obviously, combining both GLM and KMR data bases is advisable. Regretfully, we were unable to obtain good simultaneous reproduction of such an extended data base. The question, is thus, which model provides a better approximation for the input opacities. 5) The b-distributions of $a_{el}(s,b)$ in GLM are significantly different from KMR. GLM obtain a relatively wide b distribution compared with a narrower one in KMR. $a_{el}(s,b=0)$ in KMR is consistently larger than in GLM, approaching the black disc bound much faster than in GLM. Regardless of these differences, the corresponding values of $\sigma_{tot}$ and $\sigma_{el}$ in both models in the UA4-Tevatron range are compatible. Such compatibility can exist only over a relatively narrow energy band and it cannot persist over a wide energy range. Indeed, the two models have different LHC and Cosmic Rays predictions, which hopefully will be tested soon. Our inability to reproduce $\frac{d\sigma_{el}}{dt}$ outside the narrow forward $t$ cone implies a deficiency in our $a_{el}$ at large $b$. We are not clear if this deficiency is reponsible for the small $S_H^2$ obtained in our Model B(2). Note, that even though our factorizable Model B(1) has the same feature of spurious dips outside the very forward at $t$ cone, its predicted $S_H^2$ is 0.02 which is compatible with KMR. 6) In our opinion, the data adjustment procedure adopted by KMR are not adequate. Our approach is to quantify our fit by minimizing its $\chi^2$. KMR reject any statistical approach to their data analysis. They tune many of their parameters by eye and refrain from a quantified assessment of their output. The difference between the procedures adopted by the two groups is cardinal, as one is unable to make a systemic evaluation of the KMR output. 7) The difference between the $S_H^2$ predictions of GLM and KMR are intriguing and reflect the sensitivity of $S_H^2$ to each model input. $S_H^2$ is calculated as a convolution of the hard amplitude for Higgs production and the soft probability $P^S_{i,k}(s,b)$. The hard amplitude features needed for this calculation in our model are the hard slopes $B^H_{el}$, $B^H_{in}$ and cross section coefficients $V^2_{p \to p}$ $V^2_{p \to d}$, determined from the HERA measured[@KOTE; @PSISL] in $J/\Psi$ photo and DIS elastic and inelastic production. Our sensitivity to these parameters is shown in . Note that when we change the value of $B_{in}^H$, we keep the ratio $V^2_{p \to d}/B^H_{in}$ unchanged. Doing so we do not change the cross section of the reaction $\gamma + p \to J/\Psi + X \mbox{(M $\leq$ 1.6 GeV)}$. KMR calculation is simpler in as much as they consider just the elastic hard slope. In our opinion there is a gap between the sophistication of KMR soft model and the simplicity of their hard approximation. Since $S_H^2$ is obtained from a convolution of the two terms it is not clear what is the contribution of KMR hard term to the margin of error in their calculation of $S_H^2$.
![ The dependence of $S^2$ at the LHC on $B^H_{el}$ and $ B^H_{in}$, the slopes for the hard cross sections.[]{data-label="BH"}](sphb.eps){width="70mm"}
KMR estimate their margin of uncertainty to be a factor of 2.5. Since our uncertainty derives from similar, though not identical, sources, our assessment is similar. As we saw, both GLM and KMR models are partially deficient. We noted that these are based on the different conceptual constructions and data analysis procedures of the two models. A discrimination between the two models depends on experimental results which are expected to become available within the next few years. In the following we list a few: 1) GLM predictions for $\sigma_{tot}$ and $\sigma_{el}$ at the LHC are 20$\%$ higher than the corresponding KMR values. This is a fundamental difference since the output energy dependence of GLM, which is a weak screening model, is compatible with an effective $\Delta=0.08$ all through the Tevatron-GZK energy range. In the KMR model the effective $\Delta$ is reduced rapidly due to the very strong screening which is inherent to this model. Hence, the KMR cross sections grow very moderately above the Tevatron energy. 2) The difference between the two models becomes more distinguished at Cosmic Rays energies. This may be checked by the Auger experiments where we expect soon some cross section results at energies spanning up to W = 100-150 $TeV$. 3) A basic feature particular to the KMR model is a contribution to diffraction which originates from the Pomeron induced diagrams which are not contained in GLM. As a result, both $\sigma_{sd}$ and $\sigma_{dd}$ predicted by KMR are larger than GLM. These differences are very significant for the DD channel where the KMR prediction at LHC is almost a factor of 3 larger than GLM. Note, that since diffraction in GLM is Good-Walker type, our predicted elastic and diffractive cross sections satisfy the Pumplin bound[@Pumplin], $\sigma_{el}(s,b)+\sigma_{diff}(s,b) \leq \frac{1}{2}\sigma_{tot}$. This bound does not aply to KMR, in which a significant part of its diffractive cross section originate from Pomeron enhanced contributions. 4) An estimate of $S^2_H$ value can be obtained, at an early stage of LHC operation, through a measurement of the rate of central hard LRG di-jets production (a GJJG configuration) coupled to a study of its expected rate in a non screened pQCD calculation. 0.5cm [**[Acknowledgments:]{}**]{} This research was supported in part by the Israel Science Foundation, founded by the Israeli Academy of Science and Humanities, by BSF grant $\#$ 20004019 and by a grant from Israel Ministry of Science, Culture and Sport and the Foundation for Basic Research of the Russian Federation.
[^1]: Talk given by U. Maor
|
---
abstract: 'Let $\ell$ be a commutative ring with unit. To every pair of $\ell$-algebras $A$ and $B$ one can associate a simplicial set ${\mathrm{Hom}}(A,B^\Delta)$ so that $\pi_0{\mathrm{Hom}}(A,B^\Delta)$ equals the set of polynomial homotopy classes of morphisms from $A$ to $B$. We prove that $\pi_n{\mathrm{Hom}}(A,B^\Delta)$ is the set of homotopy classes of morphisms from $A$ to $B^{{\mathfrak{S}}_n}_{\bullet}$, where $B^{{\mathfrak{S}}_n}_{\bullet}$ is the ind-algebra of polynomials on the $n$-dimensional cube with coefficients in $B$ vanishing at the boundary of the cube. This is a generalization to arbitrary dimensions of a theorem of Cortiñas-Thom, which addresses the cases $n\leq 1$. As an application we give a simplified proof of a theorem of Garkusha that computes the homotopy groups of his matrix-unstable algebraic $KK$-theory space in terms of polynomial homotopy classes of morphisms.'
address: |
Dep. Matemática-IMAS, FCEyN-UBA\
Ciudad Universitaria Pab 1\
1428 Buenos Aires\
Argentina
author:
- Emanuel Rodríguez Cirone
title: The homotopy groups of the simplicial mapping space between algebras
---
Introduction
============
Algebraic $kk$-theory was constructed by Cortiñas-Thom in [@cortho], as a completely algebraic analogue of Kasparov’s $KK$-theory. It is defined on the category ${{\mathrm{Alg}_\ell}}$ of associative, not necessarily unital algebras over a fixed unital commutative ring $\ell$. It consists of a triangulated category $kk$ endowed with a functor $j:{{\mathrm{Alg}_\ell}}\to kk$ that satisfies the following properties:
1. \[item1\] *Homotopy invariance.* The functor $j$ is polynomial homotopy invariant.
2. \[item2\] *Excision.* Every short exact sequence of $\ell$-algebras that splits as a sequence of $\ell$-modules gives rise to a distinguished triangle upon applying $j$.
3. \[item3\] *Matrix stability.* For any $\ell$-algebra $A$ we have $j({{M_\infty}}A)\cong j(A)$, where ${{M_\infty}}A$ denotes the algebra of finite matrices with coefficients in $A$ indexed by ${\mathbb{N}}\times{\mathbb{N}}$.
This functor $j$ is moreover universal with the above properties: any other functor from ${{\mathrm{Alg}_\ell}}$ into a triangulated category satisfying (H), (E) and (M) factors uniquely trough $j$. Another important property of $kk$-theory is that it recovers Weibel’s homotopy $K$-theory: $kk(\ell,A)\cong KH_0(A)$; see [@cortho]\*[Theorem 8.2.1]{}.
As a technical tool for defining algebraic $kk$-theory, Cortiñas-Thom introduced in [@cortho]\*[Section 3]{} a simplicial enrichment of ${{\mathrm{Alg}_\ell}}$. They associated a simplicial mapping space ${\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)$ to any pair of $\ell$-algebras $A$ and $B$, and they defined simplicial compositions $$\circ: {\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(B,C^\Delta)\times{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)\to{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,C^\Delta)$$ that make ${{\mathrm{Alg}_\ell}}$ into a simplicial category in the sense of [@quillen]\*[Section II.1]{}. The homotopy category of this simplicial category is Gersten’s homotopy category of algebras [@gersten]\*[Section 1]{}. This means that there is a natural bijection $$\pi_0{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)\cong [A,B],$$ where the right hand side denotes the set of polynomial homotopy classes of morphisms from $A$ to $B$. In the same vein, Cortiñas-Thom showed in [@cortho]\*[Theorem 3.3.2]{} that $$\pi_1{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)\cong [A,B^{{\mathfrak{S}}_1}],$$ where $B^{{\mathfrak{S}}_1}$ denotes the ind-algebra of polynomials on $S^1=\Delta^1/\partial\Delta^1$ with coefficients in $B$ that vanish at the basepoint. The main result of this paper is the following generalization of the latter to arbitrary dimensions.
\[thm:main\] For any pair of $\ell$-algebras $A$ and $B$ and any $n\geq 0$ there is a natural bijection $$\label{eq:main}\pi_n{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)\cong [A,B^{{\mathfrak{S}}_n}],$$ where $B^{{\mathfrak{S}}_n}$ is the ind-algebra of polynomials on the $n$-dimensional cube with coefficients in $B$ vanishing at the boundary of the cube.
To prove Theorem \[thm:main\] one has to compare two different notions of homotopy for ind-algebra homomorphisms $A\to B^{{\mathfrak{S}}_n}$: simplicial homotopy on the left hand side of and polynomial homotopy on the right. Simplicial homotopy implies polynomial homotopy by [@garku]\*[Hauptlemma (2)]{}. This key technical result of [@garku] —of which Garkusha provides a beautiful constructive proof— allows one to define a surjective function $$\label{eq:function}\pi_n{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)\to [A,B^{{\mathfrak{S}}_n}]$$ that turns out to be the desired bijection. We prove the injectivity of by showing that polynomial homotopy implies simplicial homotopy; this is done in Lemma \[lem:haupt\]. This lemma follows immediately from the existence of the multiplication morphisms defined in section \[sec:multi\].
With methods different from those of Cortiñas-Thom, Garkusha gave in [@garku] an alternative construction of $kk$-theory and moreover defined other universal bivariant homology theories of algebras. The latter are functors from ${{\mathrm{Alg}_\ell}}$ into some triangulated category that share properties (H) and (E) with algebraic $kk$-theory but satisfy different matrix-stability conditions. Garkusha showed in [@garku] that his bivariant homology theories are representable by spectra, and the simplicial mapping spaces between algebras are his main building blocks for these spectra. The idea of the isomorphism was already present in the proof of [@garku]\*[Comparison Theorem A]{}, where he computed the homotopy groups of the matrix-unstable algebraic $KK$-theory space in terms of polynomial homotopy classes of morphisms. However, Garkusha used [@garku]\*[Hauptlemma (3)]{} as a substitute for Lemma \[lem:haupt\] when proving injectivity. This makes his proof rely on nontrivial techniques from homotopy theory such as the construction of a motivic-like model category of simplicial functors on ${{\mathrm{Alg}_\ell}}$. As an application of Theorem \[thm:main\], we give a simplified proof of [@garku]\*[Comparison Theorem A]{} that uses no more homotopy theory than the definition of the homotopy groups of a simplicial set.
The rest of this paper is organized as follows. In section \[sec:conv\] we fix notation and we recall the definition of polynomial homotopy and the details of the simplicial enrichment of ${{\mathrm{Alg}_\ell}}$. In section \[sec:main\] we prove Lemma \[lem:haupt\] and Theorem \[thm:main\]. In section \[sec:spec\] we apply Theorem \[thm:main\] to give a simplified proof of [@garku]\*[Comparison Theorem A]{}.
Preliminaries {#sec:conv}
=============
Throughout this text, $\ell$ is a commutative ring with unit. We only consider associative, not necessarily unital $\ell$-algebras and we write ${{\mathrm{Alg}_\ell}}$ for the category of $\ell$-algebras and $\ell$-algebra homomorphisms. Simplicial $\ell$-algebras can be considered as simplicial sets using the forgetful functor ${{\mathrm{Alg}_\ell}}\to {\mathrm{Set}}$; this is usually done without further mention. The symbol $\otimes$ indicates tensor product over ${\mathbb{Z}}$.
Categories of directed diagrams {#sec:directed}
-------------------------------
Let ${\mathfrak{C}}$ be a category. A *directed diagram* in ${\mathfrak{C}}$ is a functor $X:I\to{\mathfrak{C}}$, where $I$ is a filtering partially ordered set. We often write $(X,I)$ or $X_{\bullet}$ for such a functor. We shall consider different categories whose objects are directed diagrams:
### Fixing the filtering poset
Let $I$ be a filtering poset. We will write ${\mathfrak{C}}^I$ for the category whose objects are the functors $X:I\to{\mathfrak{C}}$ and whose morphisms are the natural transformations.
### Varying the filtering poset
We will write ${\vec{{\mathfrak{C}}}}$ for the category whose objects are the directed diagrams in ${\mathfrak{C}}$ and whose morphisms are defined as follows: Let $(X,I)$ and $(Y,J)$ be two directed diagrams. A morphism from $(X,I)$ to $(Y,J)$ consists of a pair $(f,\theta)$ where $\theta:I\to J$ is a functor and $f:X\to Y\circ\theta$ is a natural transformation.
For a fixed filtering poset $I$, there is a faithful functor $a:{\mathfrak{C}}^I\to {\vec{{\mathfrak{C}}}}$ that acts as the identity on objects and that sends a natural transformation $f$ to the morphism $(f,{\mathrm{id}}_I)$.
### The category of ind-objects
The category ${\mathfrak{C}}^{\mathrm{ind}}$ of ind-objects of ${\mathfrak{C}}$ is defined as follows: The objects of ${\mathfrak{C}}^{\mathrm{ind}}$ are the directed diagrams in ${\mathfrak{C}}$. The hom-sets are defined by: $${\mathrm{Hom}}_{{\mathfrak{C}}^{\mathrm{ind}}}\left( (X,I),(Y,J)\right):=\lim_{i\in I}{\operatornamewithlimits{colim}}_{j\in J}{\mathrm{Hom}}_{\mathfrak{C}}(X_i,Y_j)$$
There is a functor ${\vec{{\mathfrak{C}}}}\to{\mathfrak{C}}^{\mathrm{ind}}$ that acts as the identity on objects and that sends a morphism $(f,\theta):(X,I)\to(Y,J)$ to the morphism: $$\left(f_i:X_i\to Y_{\theta(i)}\right)_{i\in I}\in \lim_{i\in I}{\operatornamewithlimits{colim}}_{j\in J}{\mathrm{Hom}}_{\mathfrak{C}}(X_i,Y_j)$$
Simplicial sets {#sec:simpli}
---------------
The category of simplicial sets is denoted by ${\mathbb{S}}$; see [@hovey]\*[Chapter 3]{}. Let ${\mathrm{Map}}(?,??)$ be the internal-hom in ${\mathbb{S}}$; we often write $Y^X$ instead of ${\mathrm{Map}}(X,Y)$.
### The iterated last vertex map {#subsec:lastvertex}
Let ${\mathrm{sd}}:{\mathbb{S}}\to{\mathbb{S}}$ be the subdivision functor. There is a natural transformation $\gamma:{\mathrm{sd}}\to {\mathrm{id}}_{\mathbb{S}}$ called the *last vertex map* [@goja]\*[Section III. 4]{}. For $X\in{\mathbb{S}}$, put $\gamma^1_X:=\gamma_X$ and define inductively $\gamma^n_X$ to be the following composite: $$\xymatrixcolsep{5em}\xymatrix{
{\mathrm{sd}}^nX={\mathrm{sd}}({\mathrm{sd}}^{n-1}X) \ar[r]^-{\gamma^1_{{\mathrm{sd}}^{n-1}X}} & {\mathrm{sd}}^{n-1}X \ar[r]^-{\gamma^{n-1}_X} & X}$$ It is immediate that $\gamma^n:{\mathrm{sd}}^n\to {\mathrm{id}}_{\mathbb{S}}$ is a natural transformation. Let ${\mathrm{sd}}^0:{\mathbb{S}}\to{\mathbb{S}}$ be the identity functor and let $\gamma^0:{\mathrm{sd}}^0\to{\mathrm{id}}_{\mathbb{S}}$ be the identity natural transformation.
For any $p,q\geq 0$ and any $X\in{\mathbb{S}}$ we have: $$\gamma^{p+q}_X=\gamma^p_X\circ {\mathrm{sd}}^p\left(\gamma^q_X\right)=\gamma^p_X\circ \gamma^q_{{\mathrm{sd}}^pX}$$
It follows from a straightforward induction on $n=p+q$.
### Simplicial cubes
Let $I:=\Delta^1$ and let $\partial I:=\{0,1\}\subset I$. For $n\geq 1$, let $I^n:=I\times\cdots\times I$ be the $n$–fold direct product and let $\partial I^n$ be the following simplicial subset of $I^n$: $$\partial I^n:=\left[(\partial I)\times I\times\cdots\times I\right]\cup\left[I\times (\partial I)\times\cdots\times I\right]\cup\cdots\cup\left[I\times\cdots\times I\times (\partial I)\right]$$ Let $I^0:=\Delta^0$ and let $\partial I^0:=\emptyset$. We identify $I^{m+n}=I^m\times I^n$ and $\partial (I^{m+n})=\left[(\partial I^m)\times I^n\right]\cup\left[I^m\times (\partial I^n)\right]$ using the associativity and unit isomorphisms of the direct product in ${\mathbb{S}}$.
### Iterated loop spaces {#subsec:loop}
Let $(X,*)$ be a pointed fibrant simplicial set. Recall from [@goja]\*[Section I.7]{} that the loopspace $\Omega X$ is defined as the fiber of a natural fibration $\pi_X:PX\to X$, where $PX$ has trivial homotopy groups. By the long exact sequence associated to this fibration, we have pointed bijections $\pi_{n+1}(X,*)\cong\pi_n(\Omega X,*)$ for $n\geq 0$ that are group isomorphisms for $n\geq 1$. Iterating the loopspace construction we get: $$\pi_0(\Omega^nX)\cong \pi_1(\Omega^{n-1}X,*)\cong \cdots \cong \pi_n(X,*)$$ Thus, $\pi_0\Omega^nX$ is a group for $n\geq 1$ and this group is abelian for $n\geq 2$. Moreover, a morphism $\varphi:X\to Y$ of pointed fibrant simplicial sets induces group homomorphisms $\varphi_*:\pi_0\Omega^n X\to \pi_0\Omega^n Y$ for $n\geq 1$. Let ${\mathrm{incl}}$ denote the inclusion $\partial I^n\to I^n$. It is easy to see that the iterated loop functor $\Omega^n$ on pointed fibrant simplicial sets can be alternatively defined by the following pullback of simplicial sets: $$\label{eq:Omega1}\begin{gathered}\xymatrix{\Omega^nX\ar[r]^-{\iota_{n,X}}\ar[d] & {\mathrm{Map}}(I^n,X)\ar[d]^-{{\mathrm{incl}}^*} \\
\Delta^0\ar[r]^-{*} & {\mathrm{Map}}(\partial I^n,X) \\}\end{gathered}$$ We will always use the latter description of $\Omega^n$. Occasionally we will need to compare $\Omega^n$ for different integers $n$; for this purpose we will explicitely describe how the diagram arises from successive applications of the functor $\Omega$. We start defining $\Omega X$ by the following pullback in ${\mathbb{S}}$: $$\xymatrix{\Omega X\ar[r]^-{\iota_{1,X}}\ar[d] & {\mathrm{Map}}(I,X)\ar[d]^-{{\mathrm{incl}}^*} \\
\Delta^0\ar[r]^-{*} & {\mathrm{Map}}(\partial I,X) \\}$$ For $n\geq 1$, define inductively $\iota_{n+1,X}:\Omega^{n+1}X\to{\mathrm{Map}}(I^{n+1},X)$ as the following composite: $$\xymatrix{\Omega\left(\Omega^nX\right)\ar[r]^-{\iota_{1,\Omega^nX}} & {\mathrm{Map}}\left(I,\Omega^nX\right)\ar[r]^-{(\iota_{n,X})_*} & {\mathrm{Map}}\left(I,{\mathrm{Map}}(I^n,X)\right)\cong{\mathrm{Map}}\left(I^n\times I,X\right)}$$ It is easily verified that is a pullback. Moreover, $\iota_{m+n,X}$ equals the following composite: $$\xymatrixcolsep{2em}\xymatrix{\Omega^n\left(\Omega^mX\right)\ar[r]^-{\iota_{n,\Omega^mX}} & {\mathrm{Map}}\left(I^n,\Omega^mX\right)\ar[r]^-{(\iota_{m,X})_*} & {\mathrm{Map}}\left(I^n,{\mathrm{Map}}(I^m,X)\right)\cong{\mathrm{Map}}\left(I^m\times I^n,X\right)}$$ Thus, under the identification of diagram , each time we apply $\Omega$ the new $I$-coordinate appears to the right.
Simplicial enrichment of algebras {#subsec:simpenrich}
---------------------------------
We proceed to recall some of the details of the simplicial enrichment of ${{\mathrm{Alg}_\ell}}$ introduced in [@cortho]\*[Section 3]{}. Let ${\mathbb{Z}}^\Delta$ be the simplicial ring defined by: $$[p]\mapsto{\mathbb{Z}}^{\Delta^p}:={\mathbb{Z}}[t_0,\dots ,t_p]/\langle 1-\textstyle\sum t_i\rangle$$ An order-preserving function $\varphi:[p]\to[q]$ induces a ring homomorphism ${\mathbb{Z}}^{\Delta^q}\to{\mathbb{Z}}^{\Delta^p}$ by the formula: $$t_i\mapsto \sum_{\varphi(j)=i}t_j$$ Now let $B\in{\mathrm{Alg}}_\ell$ and define a simplicial $\ell$-algebra $B^\Delta$ by: $$\label{eq:defideltap}[p]\mapsto B^{\Delta^p}:=B\otimes {\mathbb{Z}}^{\Delta^p}$$ If $A$ is another $\ell$-algebra, the simplicial set ${\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)$ is called the *simplicial mapping space* from $A$ to $B$. For $X\in {\mathbb{S}}$, put $B^X:={\mathrm{Hom}}_{\mathbb{S}}(X,B^\Delta)$; it is easily verified that $B^X$ is an $\ell$-algebra with the operations defined pointwise. When $X=\Delta^p$, this definition of $B^{\Delta^p}$ coincides with . We have a natural isomorphism as follows, where the limit is taken over the category of simplices of $X$: $$B^X\overset{\cong}\to \displaystyle\lim_{\Delta^p\downarrow X}B^{\Delta^p}$$ For $A,B\in{\mathrm{Alg}}_\ell$ and $X\in {\mathbb{S}}$ we have the following adjunction isomorphism: $${\mathrm{Hom}}_{\mathbb{S}}(X,{\mathrm{Hom}}_{{\mathrm{Alg}}_\ell}(A,B^\Delta))\cong{\mathrm{Hom}}_{{\mathrm{Alg}}_\ell}(A,B^X)$$
\[rem:simpliberreta\] Let $X$ and $Y$ be simplicial sets. In general $(B^X)^Y\not\cong B^{X\times Y}$ —this already fails when $X$ and $Y$ are standard simplices; see [@cortho]\*[Remark 3.1.4]{}.
\[rem:mult\] The simplicial ring ${\mathbb{Z}}^{\Delta}$ is commutative and hence the same holds for the rings ${\mathbb{Z}}^X={\mathrm{Hom}}_{\mathbb{S}}(X,{\mathbb{Z}}^\Delta)$, for any $X\in{\mathbb{S}}$. Thus, the multiplication in ${\mathbb{Z}}^X$ induces a ring homomorphism ${\mathbf{m}}_X:{\mathbb{Z}}^X\otimes{\mathbb{Z}}^X\to{\mathbb{Z}}^X$. Note that ${\mathbf{m}}_X$ is natural in $X$.
Polynomial homotopy {#sec:alghtp}
-------------------
Two morphisms $f_0,f_1:A\to B$ in ${{\mathrm{Alg}_\ell}}$ are *elementary homotopic* if there exists an $\ell$-algebra homomorphism $f:A\to B[t]$ such that ${\mathrm{ev}}_0\circ f=f_1$ and ${\mathrm{ev}}_1\circ f=f_0$. Here, ${\mathrm{ev}}_i$ stands for the evaluation $t\mapsto i$. Equivalently, $f_0$ and $f_1$ are elementary homotopic if there exists $f:A\to B^{\Delta^1}$ such that the following diagram commutes for $i=0,1$: $$\xymatrixcolsep{3em}\xymatrixrowsep{2em}\xymatrix{A\ar[d]_-{f_i}\ar[r]^-{f} & B^{\Delta^1}\ar[d]^-{(d^i)^*} \\
B\ar[r]^-{\cong} & B^{\Delta^0} \\}$$ Here the $d^i:\Delta^0\to\Delta^1$ are the coface maps. Elementary homotopy $\sim_e$ is a reflexive and symmetric relation, but it is not transitive —the concatenation of polynomial homotopies is usually not polynomial. We let $\sim$ be the transitive closure of $\sim_e$ and call $f_0$ and $f_1$ *(polynomially) homotopic* if $f_0\sim f_1$. It is easily shown that $f_0\sim f_1$ iff there exist $r\in{\mathbb{N}}$ and $f:A\to B^{{\mathrm{sd}}^r\Delta^1}$ such that the following diagrams commute: $$\xymatrixcolsep{3em}\xymatrixrowsep{2em}\xymatrix{A\ar[d]_-{f_i}\ar[r]^-{f} & B^{{\mathrm{sd}}^r\Delta^1}\ar[d]^-{(d^i)^*} \\
B\ar[r]^-{\cong} & B^{{\mathrm{sd}}^r\Delta^0} \\}$$ It turns out that $\sim$ is compatible with composition; see [@gersten]\*[Lemma 1.1]{} for details. Thus, we have a category $[{{\mathrm{Alg}_\ell}}]$ whose objects are $\ell$-algebras and whose hom-sets are given by $[A,B]:={\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A, B)/\hspace{-0.2em}\sim$. We also have an obvious functor ${{\mathrm{Alg}_\ell}}\to [{{\mathrm{Alg}_\ell}}]$.
Let $(A,I)$ and $(B,J)$ be two directed diagrams in ${{\mathrm{Alg}_\ell}}$ and let $f,g\in{\mathrm{Hom}}_{{{\mathrm{Alg}_\ell}}^{\mathrm{ind}}}\left((A,I),(B,J)\right)$. We call $f$ and $g$ *homotopic* if they correspond to the same morphism upon applying the functor ${{\mathrm{Alg}_\ell}}^{\mathrm{ind}}\to [{{\mathrm{Alg}_\ell}}]^{\mathrm{ind}}$. We also write: $$[A_{\bullet},B_{\bullet}]:={\mathrm{Hom}}_{[{{\mathrm{Alg}_\ell}}]^{\mathrm{ind}}}\left((A,I),(B,J)\right)=\lim_{i\in I}{\operatornamewithlimits{colim}}_{j\in J}[A_i,B_j]$$
Functions vanishing on a subset {#sec:vanish}
-------------------------------
A *simplicial pair* is a pair $(K,L)$ where $K$ is a simplicial set and $L\subseteq K$ is a simplicial subset. A *morphism of pairs* $f:(K',L')\to (K,L)$ is a morphism of simplicial sets $f:K'\to K$ such that $f(L')\subseteq L$. A simplicial pair $(K,L)$ is *finite* if $K$ is a finite simplicial set. We will only consider finite simplicial pairs, omitting the word “finite” from now on. Let $(K,L)$ be a simplicial pair, let $B\in{{\mathrm{Alg}_\ell}}$ and let $r\geq 0$. Put: $$B_r^{(K,L)}:=\ker\left(B^{{\mathrm{sd}}^rK}\to B^{{\mathrm{sd}}^rL}\right)$$ The last vertex map induces morphisms $B^{(K,L)}_r\to B^{(K,L)}_{r+1}$ and we usually consider $B^{(K,L)}_{\bullet}$ as a directed diagram in ${{\mathrm{Alg}_\ell}}$: $$B^{(K,L)}_{\bullet}:B^{(K,L)}_0\to B^{(K,L)}_1\to B^{(K,L)}_2\to\cdots$$ Notice that a morphism $f:(K',L')\to (K,L)$ induces a morphism $f^*:B^{(K,L)}_{\bullet}\to B^{(K',L')}_{\bullet}$ of ${{\mathbb{Z}_{\geq 0}}}$-diagrams.
\[lem:simppair\] Let $(K,L)$ be a simplicial pair and let $B\in{{\mathrm{Alg}_\ell}}$. Then ${\mathbb{Z}}^{(K,L)}_r$ is a free abelian group and there is a natural $\ell$-algebra isomorphism: $$\label{eq:natiso1}B\otimes {\mathbb{Z}}^{(K,L)}_r\overset{\cong}\to B^{(K,L)}_r$$
The following sequence is exact by definition of ${\mathbb{Z}}^{(K,L)}_r$ and [@cortho]\*[Lemma 3.1.2]{}: $$\label{eq:sec}\xymatrix{0\ar[r] & {\mathbb{Z}}^{(K,L)}_r\ar[r] & {\mathbb{Z}}^{{\mathrm{sd}}^rK}\ar[r] & {\mathbb{Z}}^{{\mathrm{sd}}^rL}\ar[r] & 0 \\}$$ The group ${\mathbb{Z}}^{{\mathrm{sd}}^rL}$ is free abelian by [@cortho]\*[Proposition 3.1.3]{} and thus the sequence splits. It follows that ${\mathbb{Z}}^{(K,L)}_r$ is free because it is a direct summand of the free abelian group ${\mathbb{Z}}^{{\mathrm{sd}}^rK}$. Moreover, the following sequence is exact: $$\xymatrix{0\ar[r] & B\otimes{\mathbb{Z}}^{(K,L)}_r\ar[r] & B\otimes{\mathbb{Z}}^{{\mathrm{sd}}^rK}\ar[r] & B\otimes{\mathbb{Z}}^{{\mathrm{sd}}^rL}\ar[r] & 0 \\}$$ To finish the proof we identify $B\otimes {\mathbb{Z}}^{{\mathrm{sd}}^rK}\overset{\cong}\to B^{{\mathrm{sd}}^rK}$ using the natural isomorphism of [@cortho]\*[Proposition 3.1.3]{}.
\[exa:fSn\] Following [@garku]\*[Section 7.2]{}, we will write $B^{{\mathfrak{S}}_n}_{\bullet}$ instead of $B^{(I^n,\partial I^n)}_{\bullet}$. Note that $B^{{\mathfrak{S}}_0}_{\bullet}$ is the constant ${{\mathbb{Z}_{\geq 0}}}$-diagram $B$.
Main theorem {#sec:main}
============
Multiplication morphisms {#sec:multi}
------------------------
Let $(K,L)$ and $(K',L')$ be simplicial pairs. It follows from Lemma \[lem:simppair\] that ${\mathbb{Z}}_r^{(K,L)}\otimes {\mathbb{Z}}_s^{(K',L')}$ identifies with a subring of ${\mathbb{Z}}^{{\mathrm{sd}}^rK}\otimes{\mathbb{Z}}^{{\mathrm{sd}}^sK'}$. Let $\mu^{K,K'}$ be the composite of the following ring homomorphisms: $$\xymatrixcolsep{7em}\xymatrix{{\mathbb{Z}}^{{\mathrm{sd}}^rK}\otimes{\mathbb{Z}}^{{\mathrm{sd}}^sK'}\ar@{..>}[d]_-{\mu^{K,K'}}\ar[r]^-{(\gamma^s)^*\otimes(\gamma^r)^*} & {\mathbb{Z}}^{{\mathrm{sd}}^{r+s}K}\otimes {\mathbb{Z}}^{{\mathrm{sd}}^{r+s}K'}\ar[d]^-{({\mathrm{pr}}_1)^*\otimes ({\mathrm{pr}}_2)^*} \\
{\mathbb{Z}}^{{\mathrm{sd}}^{r+s}(K\times K')} & {\mathbb{Z}}^{{\mathrm{sd}}^{r+s}(K\times K')}\otimes {\mathbb{Z}}^{{\mathrm{sd}}^{r+s}(K\times K')}\ar[l]_-{{\mathbf{m}}} \\ }$$ Here $\gamma^j$ is the iterated last vertex map defined in section \[subsec:lastvertex\], ${\mathrm{pr}}_i$ is the projection of the direct product into its $i$-th factor and ${\mathbf{m}}$ is the map described in Remark \[rem:mult\].
\[lem:multi\] The morphism $\mu^{K,K'}$ defined above induces a ring homomorphism $\mu^{(K,L),(K',L')}$ that fits into the following diagram: $$\xymatrixcolsep{5em}\xymatrix{{\mathbb{Z}}^{(K,L)}_r\otimes {\mathbb{Z}}^{(K',L')}_s\ar[r]^-{{\mathrm{incl}}}\ar[d]_-{\mu^{(K,L),(K',L')}} & {\mathbb{Z}}^K_r\otimes {\mathbb{Z}}^{K'}_s\ar[d]^-{\mu^{K,K'}} \\
{\mathbb{Z}}^{(K\times K',(K\times L')\cup (L\times K'))}_{r+s}\ar[r]^-{{\mathrm{incl}}} & {\mathbb{Z}}^{K\times K'}_{r+s}}$$ Moreover, $\mu^{(K,L),(K',L')}$ is natural in both variables with respect to morphisms of simplicial pairs. We call $\mu^{(K,L),(K',L')}$ a *multiplication morphism*.
Let $\varepsilon$ be the restriction of $\mu^{K,K'}$ to ${\mathbb{Z}}^{(K,L)}_r\otimes{\mathbb{Z}}^{(K',L')}_s$; we have to show that $\varepsilon$ is zero when composed with the morphism: $${\mathbb{Z}}^{{\mathrm{sd}}^{r+s}(K\times K')}\to {\mathbb{Z}}^{{\mathrm{sd}}^{r+s}((K\times L')\cup (L\times K'))}$$ Since the functor ${\mathbb{Z}}^{{\mathrm{sd}}^{r+s}(?)}:{\mathbb{S}}\to {\mathrm{Alg}}_{\mathbb{Z}}^{{\mathrm{op}}}$ commutes with colimits, it will be enough to show that $\varepsilon$ is zero when composed with the projections to ${\mathbb{Z}}^{{\mathrm{sd}}^{r+s}(K\times L')}$ and to ${\mathbb{Z}}^{{\mathrm{sd}}^{r+s}(L\times K')}$; this is a straightforward check. For example, the following commutative diagram shows that $\varepsilon$ is zero when composed with the projection to ${\mathbb{Z}}^{{\mathrm{sd}}^{r+s}(L\times K')}$; we write $i$ for the inclusion $L\subseteq K$. $$\xymatrixcolsep{3em}\begin{gathered}\xymatrix{{\mathbb{Z}}^{(K,L)}_r\otimes{\mathbb{Z}}^{(K',L')}_s \ar[d]\ar@/^/[dr]^-{0} & \\
{\mathbb{Z}}^{{\mathrm{sd}}^rK}\otimes {\mathbb{Z}}^{{\mathrm{sd}}^sK'}\ar[r]^-{i^*\otimes 1}\ar[d]_-{(\gamma^s)^*\otimes(\gamma^r)^*} & {\mathbb{Z}}^{{\mathrm{sd}}^rL}\otimes {\mathbb{Z}}^{{\mathrm{sd}}^sK'}\ar[d]^-{(\gamma^s)^*\otimes(\gamma^r)^*} \\
{\mathbb{Z}}^{{\mathrm{sd}}^{r+s}K}\otimes {\mathbb{Z}}^{{\mathrm{sd}}^{r+s}K'}\ar[r]^-{i^*\otimes 1}\ar[d]_-{({\mathrm{pr}}_1)^*\otimes ({\mathrm{pr}}_2)^*} & {\mathbb{Z}}^{{\mathrm{sd}}^{r+s}L}\otimes {\mathbb{Z}}^{{\mathrm{sd}}^{r+s}K'}\ar[d]^-{({\mathrm{pr}}_1)^*\otimes ({\mathrm{pr}}_2)^*} \\
{\mathbb{Z}}^{{\mathrm{sd}}^{r+s}(K\times K')}\otimes {\mathbb{Z}}^{{\mathrm{sd}}^{r+s}(K\times K')}\ar[r]^-{i^*\otimes i^*}\ar[d]_-{{\mathbf{m}}} & {\mathbb{Z}}^{{\mathrm{sd}}^{r+s}(L\times K')}\otimes {\mathbb{Z}}^{{\mathrm{sd}}^{r+s}(L\times K')}\ar[d]^-{{\mathbf{m}}} \\
{\mathbb{Z}}^{{\mathrm{sd}}^{r+s}(K\times K')}\ar[r]^-{i^*} & {\mathbb{Z}}^{{\mathrm{sd}}^{r+s}(L\times K')} \\}\end{gathered}$$ The assertion about naturality is clear.
\[rem:mular\] We can consider ${\mathbb{Z}}^{(K,L)}_{\bullet}\otimes{\mathbb{Z}}^{(K',L')}_{\bullet}$ as a directed diagram of rings indexed over ${{\mathbb{Z}_{\geq 0}}}\times {{\mathbb{Z}_{\geq 0}}}$. Let $\theta:{{\mathbb{Z}_{\geq 0}}}\times{{\mathbb{Z}_{\geq 0}}}\to{{\mathbb{Z}_{\geq 0}}}$ be defined by $\theta(r,s)=r+s$; it is clear that $\theta$ is a functor. Then the morphisms of Lemma \[lem:multi\] assemble into a morphism in ${\vec{{\mathrm{Alg}}_{\mathbb{Z}}}}$: $$\xymatrix{\left(\mu^{(K,L),(K',L')},\theta\right):{\mathbb{Z}}^{(K,L)}_{\bullet}\otimes{\mathbb{Z}}^{(K',L')}_{\bullet}\ar[r] & {\mathbb{Z}}^{(K\times K',(K\times L')\cup (L\times K'))}_{\bullet}}$$ We will often think of $\mu^{(K,L),(K',L')}$ in this way, omitting $\theta$ from the notation.
\[rem:mub\] Upon tensoring $\mu^{(K,L),(K',L')}$ with an $\ell$-algebra $B$ and using we obtain an $\ell$-algebra homomorphism: $$\xymatrix{\mu^{(K,L),(K',L')}_B:\left(B^{(K,L)}_r\right)^{(K',L')}_s\ar[r] & B^{(K\times K', (K\times L')\cup(L\times K'))}_{r+s}}$$ This morphism is obviously natural with respect to morphisms of simplicial pairs and with respecto to $\ell$-algebra homomorphisms. Again, we can think of it as a morphism in ${\vec{{{\mathrm{Alg}_\ell}}}}$. It is easily verified that this morphism is associative in the obvious way.
\[exa:muhtpyequiv\] For any $n\geq 0$ and any $B\in{{\mathrm{Alg}_\ell}}$ we have a morphism $\iota:B\to B^{\Delta^n}$ induced by $\Delta^n\to\ast$. It is well known that $\iota$ is a homotopy equivalence, as we proceed to explain. Let $v:B^{\Delta^n}\to B$ be the restriction to the $0$-simplex $0$. Explicitely, we have $v(t_i)=0$ for $i>0$ and $v(t_0)=1$. It is easily verified that $v\circ\iota={\mathrm{id}}_B$. Now let $H:B^{\Delta^p}\to B^{\Delta^n}[u]$ be the elementary homotopy defined by $H(t_i)=ut_i$ for $i>0$ and $H(t_0)=t_0+(1-u)(t_1+\cdots +t_n)$. We have ${\mathrm{ev}}_1\circ H={\mathrm{id}}_{B^{\Delta^n}}$ and ${\mathrm{ev}}_0\circ H=\iota\circ v$. This shows that $\iota\circ v={\mathrm{id}}_{B^{\Delta^n}}$ in $[{{\mathrm{Alg}_\ell}}]$.
The homotopy $H$ constructed above is natural with respect to the inclusion of faces of $\Delta^n$ that contain the $0$-simplex $0$. More precisely: if $f:[m]\to [n]$ is an injective order-preserving map such that $f(0)=0$, then the following diagram commutes: $$\xymatrix{B^{\Delta^n}\ar[d]_-{f^*}\ar[r]^-{H} & B^{\Delta^n}[u]\ar[d]^-{f^*[u]} \\
B^{\Delta^m}\ar[r]^-{H} & B^{\Delta^m}[u] }$$
Now let $p,q\geq 0$. Recall from the proof of [@hovey]\*[Lemma 3.1.8]{} that the simplices of $\Delta^p\times\Delta^q$ can be identified with the chains in $[p]\times[q]$ with the product order. The nondegenerate $(p+q)$-simplices of $\Delta^p\times\Delta^q$ are identified with the maximal chains in $[p]\times[q]$; there are exactly $\binom{p+q}{p}$ of these. Following [@hovey], let $c(i)$ for $1\leq i\leq \binom{p+q}{p}$ be the complete list of maximal chains of $[p]\times[q]$. Then $\Delta^p\times\Delta^q$ is the coequalizer in ${\mathbb{S}}$ of the two natural morphisms of simplicial sets $f$ and $g$ induced by the inclusions $c(i)\cap c(j)\subseteq c(i)$ and $c(i)\cap c(j)\subseteq c(j)$ respectively: $$\xymatrixcolsep{3em}\xymatrix{\displaystyle f,g:\coprod_{1\leq i<j\leq\binom{p+q}{p}}\Delta^{n_{c(i)\cap c(j)}}\ar@<-.5ex>[r] \ar@<.5ex>[r] & \displaystyle\coprod_{1\leq i\leq\binom{p+q}{p}}\Delta^{n_{c(i)}}}$$ Here $n_c$ is the number of edges in $c$; that is, the dimension of the nondegenerate simplex corresponding to $c$. Since $B^?:{\mathbb{S}}^{\mathrm{op}}\to{{\mathrm{Alg}_\ell}}$ preserves limits, it follows that $B^{\Delta^p\times\Delta^q}$ is the equalizer of the following diagram in ${{\mathrm{Alg}_\ell}}$: $$\label{eq:Hpq1}\xymatrixcolsep{3em}\xymatrix{\displaystyle f^*,g^*:\displaystyle\prod_{1\leq i\leq\binom{p+q}{p}}B^{\Delta^{n_{c(i)}}}\ar@<-.5ex>[r] \ar@<.5ex>[r] & \displaystyle\prod_{1\leq i<j\leq\binom{p+q}{p}}B^{\Delta^{n_{c(i)\cap c(j)}}}}$$ Moreover, since ${\mathbb{Z}}[u]$ is a flat ring, $?\otimes{\mathbb{Z}}[u]$ preserves finite limits and $B^{\Delta^p\times\Delta^q}[u]$ is the equalizer of the following diagram: $$\label{eq:Hpq2}\xymatrixcolsep{3em}\xymatrix{\displaystyle f^*[u],g^*[u]:\displaystyle\prod_{1\leq i\leq\binom{p+q}{p}}B^{\Delta^{n_{c(i)}}}[u]\ar@<-.5ex>[r] \ar@<.5ex>[r] & \displaystyle\prod_{1\leq i<j\leq\binom{p+q}{p}}B^{\Delta^{n_{c(i)\cap c(j)}}}[u]}$$ Notice that every maximal chain of $[p]\times[q]$ starts at $(0,0)$. This implies, by the discussion above on the naturality of $H$, that the following diagram commutes for every $i$ and $j$: $$\xymatrix{B^{\Delta^{n_{c(i)}}}\ar[d]\ar[r]^-{H} & B^{\Delta^{n_{c(i)}}}[u]\ar[d] \\
B^{\Delta^{n_{c(i)\cap c(j)}}}\ar[r]^-{H} & B^{\Delta^{n_{c(i)\cap c(j)}}}[u] }$$ Then the homotopy $H$ on the different $B^{\Delta^{n_{c(i)}}}$ gives a morphism of diagrams from to that induces $H:B^{\Delta^p\times\Delta^q}\to B^{\Delta^p\times\Delta^q}[u]$. Let $\iota:B\to B^{\Delta^p\times\Delta^q}$ be the morphism induced by $\Delta^p\times\Delta^q\to \ast$ and let $v:B^{\Delta^p\times\Delta^q}\to B$ be the restriction to the $0$-simplex $(0,0)$. It is easily verified that ${\mathrm{ev}}_1\circ H$ is the identity of $B^{\Delta^p\times\Delta^q}$ and that ${\mathrm{ev}}_0\circ H=\iota\circ v$; this shows that $\iota$ is a homotopy equivalence.
Finally, consider the following commutative diagram. Since each $\iota$ is a homotopy equivalence, it follows that $\mu^{\Delta^p,\Delta^q}:(B^{\Delta^p})^{\Delta^q}\to B^{\Delta^p\times\Delta^q}$ is a homotopy equivalence too. $$\xymatrix{B^{\Delta^p}\ar[r]^-{\iota} & (B^{\Delta^p})^{\Delta^q}\ar[d]^-{\mu} \\
B\ar[r]^-{\iota}\ar[u]^-{\iota} & B^{\Delta^p\times\Delta^q}}$$
The author does not know whether $\mu^{K,K'}:(B^K)^{K'}\to B^{K\times K'}$ is a homotopy equivalence for general $K$ and $K'$. It is true, though, that $\mu^{K,K'}$ induces an isomorphism in any homotopy invariant and excisive homology theory, as we explain below.
Let ${\mathscr{T}}$ be a triangulated category with desuspension $\Omega$ and let $j:{{\mathrm{Alg}_\ell}}\to{\mathscr{T}}$ be a functor satisfying properties (H) and (E). We will show that $j\left(\mu^{K,K'}:(B^K)^{K'}\to B^{K\times K'}\right)$ is an isomorphism, for any $K$, $K'$ and $B$.
Fix an object $U$ of ${\mathscr{T}}$ and consider a short exact sequence of $\ell$-algebras $${\mathscr{E}}:\xymatrix{A'\ar[r] & A\ar[r] & A''\\}$$ that splits as a sequence of $\ell$-modules. To alleviate notation, we will write ${\mathscr{T}}^U_n(A)$ instead of ${\mathrm{Hom}}_{\mathscr{T}}(U,\Omega^nj(A))$, for any $n\in{\mathbb{Z}}$. By property (E), we have a distinguished triangle $$\triangle_{\mathscr{E}}:\xymatrix{\Omega j(A'')\ar[r] & j(A')\ar[r] & j(A)\ar[r] & j(A'')\\}$$ that induces a long exact sequence of groups as follows: $$\xymatrix{\ar@{..>}[r]& {\mathscr{T}}_{n+1}^U(A'')\ar[r] & {\mathscr{T}}_n^U(A')\ar[r] & {\mathscr{T}}_n^U(A)\ar[r] & {\mathscr{T}}_n^U(A'')\ar@{..>}[r] & \\}$$ This sequence is moreover natural in ${\mathscr{E}}$. Now consider a cartesian square of $\ell$-algebras where the horizontal morphisms are split surjections of $\ell$-modules: $$\xymatrix{A\ar[r]\ar[d] & A'\ar[d] \\
A''\ar[r] & A'''}$$ Proceeding as in the proof of [@ralf]\*[Theorem 2.41]{}, we get an exact Mayer-Vietoris sequence: $$\xymatrix{\ar@{..>}[r]& {\mathscr{T}}_{n+1}^U(A''')\ar[r] & {\mathscr{T}}_n^U(A)\ar[r] & {\mathscr{T}}_n^U(A')\oplus {\mathscr{T}}_n^U(A'')\ar[r] & {\mathscr{T}}_n^U(A''')\ar@{..>}[r] & \\}$$ This sequence is natural with respect to morphisms of squares.
Let $q\geq 0$. We will show that $j(\mu^{K,\Delta^q})$ is an isomorphism by induction on the dimension of $K$. The case $\dim K=0$ follows from the facts that $j$ preserves finite products and that $\mu^{\Delta^0,\Delta^q}$ is an $\ell$-algebra isomorphism. Now let $n\geq 0$ and suppose $j(\mu^{K',\Delta^q})$ is an isomorphism for every finite $L$ with $\dim L\leq n$. If $\dim K=n+1$, we have a cocartesian square: $$\xymatrix{K & {\mathrm{sk}}^nK\ar[l] \\
\coprod_1^r\Delta^{n+1}\ar[u] & \coprod_1^r\partial\Delta^{n+1}\ar[u]\ar[l]}$$ Upon applying the functors $(B^{?})^{\Delta^q}$ and $B^{?\times\Delta^q}$ we get the following cartesian squares: $$\xymatrix{(B^K)^{\Delta^q}\ar[d]\ar[r] & (B^{{\mathrm{sk}}^nK})^{\Delta^q}\ar[d] \\
\prod_1^r(B^{\Delta^{n+1}})^{\Delta^q}\ar[r] & \prod_1^r(B^{\partial\Delta^{n+1}})^{\Delta^q}}
\hspace{1em}\xymatrix{B^{K\times\Delta^q}\ar[d]\ar[r] & B^{{\mathrm{sk}}^nK\times\Delta^q}\ar[d] \\
\prod_1^rB^{\Delta^{n+1}\times\Delta^q}\ar[r] & \prod_1^rB^{\partial\Delta^{n+1}\times\Delta^q}}$$ The horizontal morphisms in these diagrams are split surjections of $\ell$-modules. Indeed, the morphism ${\mathbb{Z}}^K\to {\mathbb{Z}}^{{\mathrm{sk}}^nK}$ is a split surjection of abelian groups since it is surjective by [@cortho]\*[Lemma 3.1.2]{} and ${\mathbb{Z}}^{{\mathrm{sk}}^nK}$ is a free abelian group by [@cortho]\*[Proposition 3.1.3]{}. Upon tensoring with $B$, we get that $B^K\to B^{{\mathrm{sk}}^nK}$ is a split surjection of $\ell$-modules. Similar arguments apply to the remaining horizontal morphisms. By naturality of $\mu$, we have a morphism of squares from the square on the left to the square on the right; this induces a morphism of long exact Mayer-Vietoris sequences upon applying ${\mathscr{T}}^U_*$. Note that ${\mathscr{T}}^U_*(\mu^{\Delta^{n+1},\Delta^q})$ is an isomorphism by Example \[exa:muhtpyequiv\]. It follows from the 5-lemma that $$(\mu^{K,\Delta^q})_*:{\mathrm{Hom}}_{\mathscr{T}}(U,j((B^K)^{\Delta^q}))\to{\mathrm{Hom}}_{\mathscr{T}}(U,j(B^{K\times\Delta^q}))$$ is an isomorphism. Since $U$ is arbitrary, this implies that $j(\mu^{K,\Delta^q})$ is an isomorphism. Now we can show that $j(\mu^{K,K'})$ is an isomorphism by induction on the dimension of $K'$.
Main theorem {#sec:mainthm}
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Following [@garku]\*[Section 7.2]{}, we put $\widetilde{B}^{{\mathfrak{S}}_n}_{\bullet}:=B^{(I^n\times I,\partial I^n\times I)}_{\bullet}$. The coface maps $d^i:\Delta^0\to I$ induce morphisms $(d^i)^*:\widetilde{B}^{{\mathfrak{S}}_n}_{\bullet}\to B^{{\mathfrak{S}}_n}_{\bullet}$.
\[lem:haupt2\] Let $f:A\to \widetilde{B}^{{\mathfrak{S}}_n}_r$ be an $\ell$-algebra homomorphism. Then the following composites are homotopic; i.e. they belong to the same class in $[A,B^{{\mathfrak{S}}_n}_r]$: $$A\overset{f}\longrightarrow \widetilde{B}^{{\mathfrak{S}}_n}_r\overset{(d^i)^*}\longrightarrow B^{{\mathfrak{S}}_n}_r\hspace{1em} (i=0,1)$$
This is [@garku]\*[Hauptlemma (2)]{}.
As noted by Garkusha in [@garku], Lemma \[lem:haupt2\] shows that if two morphisms are simplicially homotopic, then they are polynomially homotopic. The converse also holds, up to increasing the number of subdivisions:
\[lem:haupt\] Let $H:A\to (B^{{\mathfrak{S}}_n}_r)^{{\mathrm{sd}}^sI}$ be a homotopy between two $\ell$-algebra homomorphisms $A\to B^{{\mathfrak{S}}_n}_r$. Then there exists a morphism $\widetilde{H}:A\to \widetilde{B}^{{\mathfrak{S}}_n}_{r+s}$ in ${{\mathrm{Alg}_\ell}}$ such that the following diagram commutes for $i=0,1$: $$\xymatrix{A\ar[r]^-{H}\ar[d]_-{\widetilde{H}} & (B^{{\mathfrak{S}}_n}_r)^{{\mathrm{sd}}^sI}\ar[r]^-{(d^i)^*} & B^{{\mathfrak{S}}_n}_r\ar[d]^-{(\gamma^s)^*} \\
\widetilde{B}^{{\mathfrak{S}}_n}_{r+s}\ar[rr]^-{(d^i)^*} & & B^{{\mathfrak{S}}_n}_{r+s} \\}$$
Let $\widetilde{H}$ be the composite: $$\xymatrixcolsep{7em}\xymatrix{A\ar[r]^-{H} & (B^{(I^n,\partial I^n)}_r)^{(I,\emptyset)}_s\ar[r]^-{\mu^{(I^n,\partial I^n),(I,\emptyset)}} & B^{(I^n\times I, (\partial I^n)\times I)}_{r+s}\\}$$ It is immediate from the naturality of $\mu$ that $\widetilde{H}$ satisfies the required properties.
\[thm:bij\] For any pair of $\ell$-algebras $A$ and $B$ and any $n\geq 0$, there is a natural bijection: $$\label{eq:bij}\pi_n{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)\cong [A,B^{{\mathfrak{S}}_n}_{\bullet}]$$
We will show that $\pi_0\Omega^n{\mathit{Ex}}^\infty{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)\cong [A,B^{{\mathfrak{S}}_n}_{\bullet}]$. Consider ${\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)$ as a simplicial set pointed at the zero morphism. For every $p\geq 0$ we have a pullback of sets: $$\label{eq:pbzero}\begin{gathered}\xymatrix{\left(\Omega^n{\mathit{Ex}}^\infty{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)\right)_p\ar[d]\ar[r] & {\mathrm{Map}}\left(I^n,{\mathit{Ex}}^\infty{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)\right)_p\ar[d] \\
\ast\ar[r] & {\mathrm{Map}}\left(\partial I^n,{\mathit{Ex}}^\infty{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)\right)_p \\}\end{gathered}$$ For a finite simplicial set $K$ we have: $$\begin{aligned}
{\mathrm{Map}}\left(K,{\mathit{Ex}}^\infty{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)\right)_p&={\mathrm{Hom}}_{\mathbb{S}}\left(K\times \Delta^p,{\mathit{Ex}}^\infty{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)\right)\\
&\cong{\operatornamewithlimits{colim}}_r{\mathrm{Hom}}_{\mathbb{S}}\left(K\times\Delta^p,{\mathit{Ex}}^r{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)\right)\\
&\cong{\operatornamewithlimits{colim}}_r{\mathrm{Hom}}_{\mathbb{S}}\left({\mathrm{sd}}^r(K\times\Delta^p),{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)\right)\\
&\cong{\operatornamewithlimits{colim}}_r{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}\left(A,B^{{\mathrm{sd}}^r(K\times\Delta^p)}\right)\end{aligned}$$ It follows from these identifications, from and from the fact that filtered colimits of sets commute with finite limits, that we have the following bijections: $$\label{eq:bij0}\left(\Omega^n{\mathit{Ex}}^\infty{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)\right)_0\cong {\operatornamewithlimits{colim}}_r{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^{{\mathfrak{S}}_n}_r)$$ $$\label{eq:bij1}\left(\Omega^n{\mathit{Ex}}^\infty{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)\right)_1\cong {\operatornamewithlimits{colim}}_r{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,\widetilde{B}^{{\mathfrak{S}}_n}_r)$$ Using we get a surjection: $$\left(\Omega^n{\mathit{Ex}}^\infty{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)\right)_0\cong {\operatornamewithlimits{colim}}_r{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^{{\mathfrak{S}}_n}_r)\longrightarrow [A,B^{{\mathfrak{S}}_n}_{\bullet}]$$ We claim that this function induces the desired bijection. The fact that it factors through $\pi_0$ follows from the identification and Lemma \[lem:haupt2\]. The injectivity of the induced function from $\pi_0$ follows from Lemma \[lem:haupt\].
\[rem:groupstructure\] Let $A$ and $B$ be two $\ell$-algebras and let $n\geq 1$. Endow the set $[A,B^{{\mathfrak{S}}_n}_{\bullet}]$ with the group structure for which is a group isomorphism. This group structure is abelian if $n\geq 2$. Moreover, if $f:A\to A'$ and $g:B\to B'$ are morphisms in $[{{\mathrm{Alg}_\ell}}]$, then the following functions are group homomorphisms: $$f^*:[A', B^{{\mathfrak{S}}_n}_{\bullet}]\to [A, B^{{\mathfrak{S}}_n}_{\bullet}]$$ $$g_*:[A, B^{{\mathfrak{S}}_n}_{\bullet}]\to [A, (B')^{{\mathfrak{S}}_n}_{\bullet}]$$
\[exa:groupinverse\] Recall that $B^{\Delta^1}=B[t_0,t_1]/\langle 1-t_0-t_1\rangle$. Let $\omega$ be the automorphism of $B^{\Delta^1}$ defined by $\omega(t_0)=t_1$, $\omega(t_1)=t_0$; it is clear that $\omega$ induces an automorphism of $B^{{\mathfrak{S}}_1}_0=\ker(B^{\Delta^1}\to B^{\partial\Delta^1})$. Let $f:A\to B^{{\mathfrak{S}}_1}_0$ be an $\ell$-algebra homomorphism and let $[f]$ be its class in $[A,B^{{\mathfrak{S}}_1}_{\bullet}]$. We claim that $[\omega\circ f]=[f]^{-1}\in[A,B^{{\mathfrak{S}}_1}_{\bullet}]$. In order to prove this claim, we proceed to recall the definition of the group law $*$ in $\pi_1{\mathit{Ex}}^\infty{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)$. Consider $f$ and $\omega\circ f$ as $1$-simplices of ${\mathit{Ex}}^\infty{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)$ using the identification: $$\left({\mathit{Ex}}^\infty{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)\right)_1\cong{\operatornamewithlimits{colim}}_r{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^{{\mathrm{sd}}^r\Delta^1})$$ According to [@goja]\*[Section I.7]{}, if we find $\alpha\in\left({\mathit{Ex}}^\infty{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)\right)_2$ such that $$\label{eq:condi}\left\{\begin{array}{l}d_0\alpha=\omega\circ f\\
d_2\alpha=f\end{array}\right.$$ then we have $[f]*[\omega\circ f]=[d_1\alpha]$. Let $\varphi:B^{\Delta^1}\to B^{\Delta^2}$ be the $\ell$-algebra homomorphism defined by $\varphi(t_0)=t_0+t_2$, $\varphi(t_1)=t_1$. Let $\alpha$ be the $2$-simplex of ${\mathit{Ex}}^\infty{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(A,B^\Delta)$ induced by the composite: $$A\overset{f}\to B^{\Delta^1}\overset{\varphi}\to B^{\Delta^2}$$ It is easy to verify that the equations hold and that $d_1\alpha$ is the zero path.
\[exa:cmn\] Let $A$ and $B$ be two $\ell$-algebras and let $m,n\geq 1$. Let $c:I^m\times I^n\overset{\cong}\to I^n\times I^m$ be the commutativity isomorphism; $c$ induces an isomorphism $c^*:B^{{\mathfrak{S}}_{n+m}}_{\bullet}\to B^{{\mathfrak{S}}_{m+n}}_{\bullet}$. We claim that the following function is multiplication by $(-1)^{mn}$: $$c^*:[A,B^{{\mathfrak{S}}_{n+m}}_{\bullet}]\to [A,B^{{\mathfrak{S}}_{m+n}}_{\bullet}]$$ Indeed, this follows from Theorem \[thm:bij\] and the well known fact that permuting two coordinates in $\Omega^{m+n}$ induces multiplication by $(-1)$ upon taking $\pi_0$.
Garkusha’s Comparison Theorem A {#sec:spec}
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Matrix-unstable algebraic $KK$-theory consists of a triangulated category ${{D(\mathfrak{F}_{\mathrm{spl}})}}$ endowed with a functor $j:{{\mathrm{Alg}_\ell}}\to {{D(\mathfrak{F}_{\mathrm{spl}})}}$ that satisfies (H) and (E), and is moreover universal with respect to these two properties. It was constructed by Garkusha in [@garkuuni]\*[Section 2.4]{} by means of deriving a certain Brown category and then stabilizing the loop functor. Garkusha defined in [@garku] a space ${\mathscr{K}}(A,B)$ such that $\pi_0{\mathscr{K}}(A,B)\cong{\mathrm{Hom}}_{{D(\mathfrak{F}_{\mathrm{spl}})}}(j(A),j(B))$, for any pair of $\ell$-algebras $A$ and $B$. In this section we apply Theorem \[thm:main\] to give a simplified proof of [@garku]\*[Comparison Theorem A]{}, where $\pi_0{\mathscr{K}}(A,B)$ is computed in terms of polynomial homotopy classes of morphisms.
Extensions and classifying maps {#subsec:extensions}
-------------------------------
Let ${\mathrm{Mod}}_\ell$ be the category of $\ell$-modules and write $F:{{\mathrm{Alg}_\ell}}\to {\mathrm{Mod}}_\ell$ for the forgetful functor. An *extension* of $\ell$-algebras is a diagram $$\label{eq:extension}{\mathscr{E}}:\xymatrix{A\ar[r] & B\ar[r] & C\\}$$ in ${{\mathrm{Alg}_\ell}}$ that becomes a split short exact sequence upon applying $F$. A *morphism of extensions* is a morphism of diagrams in ${{\mathrm{Alg}_\ell}}$. We often consider specific splittings for the extensions we work with and we sometimes write $({\mathscr{E}},s)$ to emphasize that we are considering an extension ${\mathscr{E}}$ with splitting $s$. Let $({\mathscr{E}},s)$ and $({\mathscr{E}}',s')$ be two extensions with specified splittings; a *strong morphism of extensions* $({\mathscr{E}}',s')\to ({\mathscr{E}},s)$ is a morphism of extensions $(\alpha,\beta,\gamma):{\mathscr{E}}'\to {\mathscr{E}}$ that is compatible with the splittings; i.e. such that the folowing diagram commutes: $$\xymatrixrowsep{2em}\xymatrix{FB'\ar[d]_{F\beta} & FC'\ar[l]_-{s'}\ar[d]^-{F\gamma} \\
FB & FC\ar[l]_-{s} \\}$$
The functor $F:{{\mathrm{Alg}_\ell}}\to{\mathrm{Mod}}_\ell$ admits a right adjoint ${\widetilde{T}}:{\mathrm{Mod}}_\ell\to{{\mathrm{Alg}_\ell}}$; see [@garku]\*[Section 3]{} for details. Let $T$ be the composite functor ${\widetilde{T}}\circ F:{{\mathrm{Alg}_\ell}}\to{{\mathrm{Alg}_\ell}}$. Let $A\in{{\mathrm{Alg}_\ell}}$ and let $\eta_A:TA\to A$ be the counit of the adjunction. Notice that $F\eta_A$ is a retraction which has the unit map $\sigma_A:FA\to F{\widetilde{T}}(FA)=FTA$ as a section. Let $JA:=\ker \eta_A$. The *universal extension* of $A$ is the extension: $${\mathscr{U}}_A:\xymatrix{JA\ar[r] & TA\ar[r]^-{\eta_A} & A\\}$$ We will always consider $\sigma_A$ as a splitting for ${\mathscr{U}}_A$.
\[lem:classexists\] Let be an extension with splitting $s$ and let $f:D\to C$ be a morphism in ${{\mathrm{Alg}_\ell}}$. Then there exists a unique strong morphism of extensions ${\mathscr{U}}_{D}\to({\mathscr{E}},s)$ extending $f$: $$\label{eq:classexists}\begin{gathered}\xymatrix{{\mathscr{U}}_D\ar@{..>}[d]_-{\exists !} & JD\ar[r]\ar@{..>}[d]_-{\xi} & TD\ar@{..>}[d]\ar[r]^-{\eta_D} & D\ar[d]^-{f} \\
({\mathscr{E}},s) & A\ar[r] & B\ar[r] & C \\}\end{gathered}$$
It follows easily from the adjointness of ${\widetilde{T}}$ and $F$.
The morphism $\xi$ in is called the *classifying map of $f$ with respect to the extension $({\mathscr{E}},s)$*. When $D=C$ and $f={\mathrm{id}}_C$ we call $\xi$ the *classifying map of $({\mathscr{E}},s)$*.
\[lem:classhmtp\] In the hypothesis of Proposition \[lem:classexists\], the homotopy class of the classifying map $\xi$ does not depend upon the splitting $s$.
See, for example, [@garku]\*[Section 3]{}.
Because of Proposition \[lem:classhmtp\], it makes sense to speak of the classifying map of as a homotopy class $JC\to A$ without specifying a splitting for .
\[lem:Jhtp\] The functor $J:{{\mathrm{Alg}_\ell}}\to {{\mathrm{Alg}_\ell}}$ sends homotopic morphisms to homotopic morphisms. Thus, it defines a functor $J:[{{\mathrm{Alg}_\ell}}]\to[{{\mathrm{Alg}_\ell}}]$.
It is explained in [@cortho] in the discussion following [@cortho]\*[Corollary 4.4.4.]{}.
Path extensions {#subsec:pathext}
---------------
Let $B$ be an $\ell$-algebra and let $n,q\geq 0$. Put: $$P(n,B)^q_{\bullet}:=B^{(I^{n+1}\times\Delta^q,(I^n\times\{1\}\times\Delta^q)\cup(\partial I^n\times I\times\Delta^q))}_{\bullet}$$ On the one hand, the composite $I^n\times\Delta^q\cong I^n\times\{0\}\times\Delta^q\subseteq I^{n+1}\times\Delta^q$ induces morphisms $p_{n,B}^q:P(n,B)^q_r\to B^{(I^n\times\Delta^q,\partial I^n\times\Delta^q)}_r$. On the other hand, we have inclusions $B^{(I^{n+1}\times\Delta^q,\partial I^{n+1}\times\Delta^q)}_r\subseteq P(n,B)^q_r$. We claim that the following diagram is an extension: $${\mathscr{P}}_{n,B}^q:\xymatrix{B^{(I^{n+1}\times\Delta^q,\partial I^{n+1}\times\Delta^q)}_r\ar[r]^-{\mathrm{incl}} & P(n,B)^q_r\ar[r]^-{p_{n,B}^q} & B^{(I^n\times\Delta^q,\partial I^n\times\Delta^q)}_r}$$ Exactness at $P(n,B)_r^q$ holds because the functors $B^{{\mathrm{sd}}^r(-)}:{\mathbb{S}}\to {{\mathrm{Alg}_\ell}}^{\mathrm{op}}$ preserve pushouts and we have: $$\partial I^{n+1}\times \Delta^q=\left[ (I^n\times \{1\}\times \Delta^q)\cup(\partial I^n\times I\times \Delta^q)\right]\cup (I^n\times\{0\}\times \Delta^q)$$ Exactness at $B^{(I^{n+1}\times\Delta^q,\partial I^{n+1}\times\Delta^q)}_r$ follows from the fact that both this algebra and $P(n,B)_r^q$ are subalgebras of $B^{{\mathrm{sd}}^r(I^{n+1}\times \Delta^q)}$. A splitting of $p_{n,B}^q$ in the category of $\ell$-modules can be constructed as follows. Consider the element $t_0\in{\mathbb{Z}}^{\Delta^1}$; $t_0$ is actually in ${\mathbb{Z}}^{(I,\{1\})}_0$ since $d_0(t_0)=0$. Let $s_{n,B}^q$ be the composite: $$\xymatrixrowsep{2em}\xymatrix{B^{(I^n\times\Delta^q,\partial I^n\times\Delta^q)}_r\ar@/_5pc/@{..>}[dddr]_-{s_{n,B}^q}\ar[r]^-{?\otimes t_0} & B^{(I^n\times\Delta^q,\partial I^n\times\Delta^q)}_r\otimes {\mathbb{Z}}^{(I,\{1\})}_0\ar[d]^-{\cong} \\
& \left(B^{(I^n\times\Delta^q,\partial I^n\times\Delta^q)}_r\right)^{(I,\{1\})}_0\ar[d]^-{\mu} \\
& B^{(I^n\times\Delta^q\times I,(I^n\times\Delta^q\times\{1\})\cup (\partial I^n\times\Delta^q\times I))}_r\ar[d]^-{\cong} \\
& B^{(I^n\times I\times\Delta^q,(I^n\times\{1\}\times\Delta^q)\cup (\partial I^n\times I\times\Delta^q))}_r}$$ It is straightforward to check that $s_{n,B}^q$ is a section to $p_{n,B}^q$.
\[rem:functorialitiespathBnq\] It is clear that the extensions $({\mathscr{P}}^q_{n,B}, s_{n,B}^q)$ are:
1. natural in $B$ with respect to $\ell$-algebra homomorphisms;
2. natural in $r$ with respect to the last vertex map;
3. and natural in $q$ with respect to morphisms of ordinal numbers.
\[exa:Lambda\] Let $A$ and $B$ be two $\ell$-algebras, let $n\geq 0$ and let $f:A\to B^{{\mathfrak{S}}_n}_r$ be an $\ell$-algebra homomorphism. By Proposition \[lem:classexists\], there exists a unique strong morphism of extensions ${\mathscr{U}}_A\to{\mathscr{P}}_{n,B}^0$ that extends $f$: $$\xymatrixcolsep{3em}\xymatrix{{\mathscr{U}}_A\ar@{..>}[d]_-{\exists !} & JA\ar[r]\ar@{..>}[d]_-{\Lambda^n(f)} & TA\ar@{..>}[d]\ar[r] & A\ar[d]^-{f} \\
{\mathscr{P}}_{n,B}^0 & B^{{\mathfrak{S}}_{n+1}}_r\ar[r] & P(n,B)_r^0\ar[r] & B^{{\mathfrak{S}}_n}_r \\}$$ We will write $\Lambda^n(f)$ for the classifying map of $f$ with respect to ${\mathscr{P}}_{n,B}^0$. Notice that: $$\label{eq:LambdaJ}\Lambda^n(f)=\Lambda^n({\mathrm{id}}_{B^{{\mathfrak{S}}_n}})\circ J(f)$$ Indeed, this follows from the uniqueness statement in Proposition \[lem:classexists\] and the fact that the following diagram exhibits a strong morphism of extensions ${\mathscr{U}}_A\to {\mathscr{P}}_{n,B}^0$ that extends $f$: $$\xymatrixcolsep{3em}\xymatrix{{\mathscr{U}}_A\ar[d] & JA\ar[d]_-{J(f)}\ar[r] & TA\ar[r]\ar[d]^-{T(f)} & A\ar[d]^-{f} \\
{\mathscr{U}}_{B^{{\mathfrak{S}}_n}_r}\ar[d] & J(B^{{\mathfrak{S}}_n}_r)\ar[d]_-{\Lambda^n({\mathrm{id}}_{B^{{\mathfrak{S}}_n}})}\ar[r] & T(B^{{\mathfrak{S}}_n}_r)\ar[r]\ar[d] & B^{{\mathfrak{S}}_n}_r\ar[d]^-{{\mathrm{id}}} \\
{\mathscr{P}}_{n,B}^0 & B^{{\mathfrak{S}}_{n+1}}_r\ar[r] & P(n,B)_r^0\ar[r] & B^{{\mathfrak{S}}_n}_r \\}$$ By and Lemma \[lem:Jhtp\], we can consider $\Lambda^n$ as a function $\Lambda^n:[A,B^{{\mathfrak{S}}_n}_{\bullet}]\to[JA, B^{{\mathfrak{S}}_{n+1}}_{\bullet}]$.
Write ${\mathrm{Born}}$ for the category of bornological algebras; see [@ralf]\*[Definition 2.5]{}. Cuntz-Meyer-Rosenberg constructed in [@ralf]\*[Section 6.3]{} a triangulated category ${{\Sigma\mathrm{Ho}}}$ endowed with a functor ${\mathrm{Born}}\to{{\Sigma\mathrm{Ho}}}$ that is homotopy invariant and excisive in the bornological context, and is moreover universal with respect to these two properties; see [@ralf]\*[Section 6.7]{}. The matrix-unstable algebraic $KK$-theory category ${{D(\mathfrak{F}_{\mathrm{spl}})}}$ is the analogue of ${{\Sigma\mathrm{Ho}}}$ in the algebraic context. Garkusha proved in [@garku] that there is an isomorphism $$\label{eq:idea}
{\mathrm{Hom}}_{{D(\mathfrak{F}_{\mathrm{spl}})}}(j(A),j(B))\cong {\operatornamewithlimits{colim}}_n[J^nA, B^{{\mathfrak{S}}_n}_{\bullet}],$$ where the transition functions on the right hand side are the $\Lambda^n$ of Example \[exa:Lambda\]. These functions $\Lambda^n:[J^nA,B^{{\mathfrak{S}}_n}_{\bullet}]\to[J^{n+1}A, B^{{\mathfrak{S}}_{n+1}}_{\bullet}]$ are the algebraic analogues of the morphism $\Lambda$ of [@ralf]\*[Definition 6.23]{} that is used in [@ralf]\*[Section 6.3]{} to define the hom-sets in ${{\Sigma\mathrm{Ho}}}$.
Matrix-unstable algebraic $KK$-theory space {#sec:kkspace}
-------------------------------------------
Let $A$ and $B$ be two $\ell$-algebras and let $n\geq 0$. From the proof of Theorem \[thm:bij\], it follows that there is a natural bijection: $$\left(\Omega^n{\mathit{Ex}}^\infty{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(J^nA,B^\Delta)\right)_q\cong {\operatornamewithlimits{colim}}_r{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}\left(J^nA,B^{(I^n\times\Delta^q, \partial I^n\times\Delta^q)}_r\right)$$ Let $f\in{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(J^nA,B^{(I^n\times\Delta^q, \partial I^n\times\Delta^q)}_r)$ and define $\zeta^n(f)$ as the classifying map of $f$ with respect to the extension ${\mathscr{P}}_{n,B}^q$: $$\xymatrix{{\mathscr{U}}_{J^nA}\ar@{..>}[d]_-{\exists !} & J^{n+1}A\ar[r]\ar@{..>}[d]_-{\zeta^n(f)} & TJ^nA\ar@{..>}[d]\ar[r] & J^nA\ar[d]^-{f} \\
{\mathscr{P}}_{n,B}^q & B^{(I^{n+1}\times\Delta^q, \partial I^{n+1}\times\Delta^q)}_r\ar[r] & P(n,B)^q_r\ar[r] & B^{(I^n\times\Delta^q, \partial I^n\times\Delta^q)}_r \\}$$ It follows from Remark \[rem:functorialitiespathBnq\] that this defines a morphism of simplicial sets: $$\label{eq:zetaene}\zeta^n:\Omega^n{\mathit{Ex}}^\infty{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(J^nA,B^\Delta)\to \Omega^{n+1}{\mathit{Ex}}^\infty{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(J^{n+1}A,B^\Delta)$$
Let $A$ and $B$ be two $\ell$-algebras. The *matrix-unstable algebraic $KK$-theory space* of the pair $(A,B)$ is the simplicial set defined by $${\mathscr{K}}(A,B):={\operatornamewithlimits{colim}}_n\Omega^n{\mathit{Ex}}^\infty{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(J^nA,B^\Delta),$$ where the transition morphisms are the $\zeta^n$ defined in .
Note that ${\mathscr{K}}(A,B)$ is a fibrant simplicial set, since it is a filtering colimit of fibrant simplicial sets. This definition of ${\mathscr{K}}(A,B)$ is easily seen to be the same as the one given in [@garku]\*[Section 4]{}.
\[thm:comparisonthms1\] For any pair of $\ell$-algebras $A$ and $B$ and any $m\geq 0$, there is a natural isomorphism $$\pi_m{\mathscr{K}}(A,B)\cong {\operatornamewithlimits{colim}}_v[J^vA, B^{{\mathfrak{S}}_{m+v}}_{\bullet}]$$ where the transition functions on the right hand side are the $\Lambda^n$ of Example \[exa:Lambda\].
Since $\pi_m\cong\pi_0\Omega^m$ commutes with filtered colimits, we have: $$\begin{aligned}
\pi_m{\mathscr{K}}(A,B)&\cong {\operatornamewithlimits{colim}}_v\pi_0\Omega^m\Omega^v{\mathit{Ex}}^\infty{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(J^vA,B^\Delta) \\
&\cong {\operatornamewithlimits{colim}}_v\pi_0\Omega^{v+m}{\mathit{Ex}}^\infty{\mathrm{Hom}}_{{\mathrm{Alg}_\ell}}(J^vA,B^\Delta) \\
&\cong {\operatornamewithlimits{colim}}_v[J^vA,B^{{\mathfrak{S}}_{v+m}}_{\bullet}] && \text{(by Theorem \ref{thm:bij})}
\end{aligned}$$ Notice that $\Omega^m\Omega^v\cong\Omega^{v+m}$ because of our conventions on iterated loop spaces; see section \[subsec:loop\]. To finish the proof, we need to compare the function $\Lambda^{m+v}$ with: $$\pi_m\zeta^v:[J^vA,B^{{\mathfrak{S}}_{v+m}}_{\bullet}]\to [J^{v+1}A,B^{{\mathfrak{S}}_{v+1+m}}_{\bullet}]$$ Let $c_{v,m}:I^v\times I^m\overset{\cong}\to I^m\times I^v$ be the commutativity isomorphism; $c_{v,m}$ induces an isomorphism $(c_{v,m})^*:B^{{\mathfrak{S}}_{m+v}}_r\to B^{{\mathfrak{S}}_{v+m}}_r$. It is straightforward to verify that the following squares commute: $$\xymatrixcolsep{3em}\xymatrix{[J^vA, B^{{\mathfrak{S}}_{v+m}}_{\bullet}]\ar[r]^-{\pi_m\zeta^v} & [J^{v+1}A, B^{{\mathfrak{S}}_{v+1+m}}_{\bullet}] \\
[J^vA, B^{{\mathfrak{S}}_{m+v}}_{\bullet}]\ar[r]^-{\Lambda^{m+v}}\ar[u]_-{\cong}^-{(c_{v,m})^*} & [J^{v+1}A, B^{{\mathfrak{S}}_{m+v+1}}_{\bullet}]\ar[u]^-{\cong}_-{(c_{v+1,m})^*}}$$ These squares assemble into a morphism of diagrams that, upon taking colimit in $v$, induces the desired isomorphism $\pi_m{\mathscr{K}}(A,B)\cong {\operatornamewithlimits{colim}}_v[J^vA, B^{{\mathfrak{S}}_{m+v}}_{\bullet}]$.
It is posible to mimic the definition of ${{\Sigma\mathrm{Ho}}}$ in [@ralf]\*[Section 6.3]{} to give a new and more explicit construction of Garkusha’s matrix-unstable algebraic $KK$-theory category ${{D(\mathfrak{F}_{\mathrm{spl}})}}$. Indeed, we can take as a *definition* of the hom-sets in ${{D(\mathfrak{F}_{\mathrm{spl}})}}$. Theorem \[thm:bij\] provides $[J^nA, B^{{\mathfrak{S}}_n}_{\bullet}]$ with the group structure needed to make sense of the signs that appear when defining the composition rule; see [@ralf]\*[Lemmas 6.29 and 6.30]{}. The algebraic context is, however, a little more complicated than the bornological one. We will develop these ideas further in a future paper.
|
---
abstract: 'Considering a four dimensional parallelisable manifold, we develop a concept of Dirac-type tensor equations with wave functions that belong to left ideals of the set of nonhomogeneous complex valued differential forms.'
author:
- 'N.G.Marchuk [^1]'
title: 'A concept of Dirac-type tensor equations'
---
PACS: 04.20Cv, 04.62, 11.15, 12.10
Steklov Mathematical Institute, Gubkina st.8, Moscow 119991, Russia
nmarchuk@mi.ras.ru, www.orc.ru/\~nmarchuk
In the previous papers [@nona],[@nona1],[@nona2], developing ideas of [@Ivanenko]-[@Benn], we consider Dirac-type tensor equations with non-Abelian gauge symmetries on a four dimensional parallelisable manifold with wave functions that belong to the algebrae $\Lambda^\C,\Lambda,\Lambda^\C_\even,\Lambda_\even$ of real dimensions $32,16,16,8$ respectively, where $\Lambda$ is the set of nonhomogeneous differential forms, $\Lambda_\even$ is the set of even differential forms, and $\Lambda^\C,\Lambda^\C_\even$ are the corresponding sets of complex valued differential forms. Now we develop a concept of Dirac-type tensor equations with wave functions that belong to left ideals of $\Lambda^\C$. This concept gives us possibility to find an ${\rm SU}(3)$ invariant Dirac-type tensor equation with 12 complex valued components of wave function. In addition to unitary gauge symmetries Dirac-type tensor equations have a gauge symmetry with respect to the spinor group $\Spin(\W)$. In Section 9 we investigate in detail a connection between the (spinor) Dirac equation and the Dirac-type tensor equation. Of cause, Dirac spinors and tensors are different mathematical objects and, generally speaking, we can’t establish an equivalence between them. But it nevertheless, if we take Dirac spinors that are solutions of the Dirac equation and tensors (differential forms) that are solutions of the corresponding Dirac-type tensor equation, then, with the aid of above mentioned spinor group symmetry, we can establish a one-to-one correspondence between these spinors and tensors (Section 9).
Two pictures of the Dirac equation.
===================================
Let $\R^{1,3}$ be the Minkowski space with coordinates $x^\mu$ ($\mu=0,1,2,3$), with a metric tensor $\eta_{\mu\nu}=\eta^{\mu\nu}$ defined by the Minkowski matrix $$\eta=\|\eta_{\mu\nu}\|=\|\eta^{\mu\nu}\|=\diag(1,-1,-1,-1).$$
Consider the Dirac equation for an electron $$\gamma^\mu(\partial_\mu\psi+ia_\mu\psi)+im\psi=0,
\label{Dirac:eq}$$ where $\partial_\mu=\partial/\partial x^\mu$, $\psi=\psi(x)$ is a column of four complex valued smooth functions (an electron wave function), $a_\mu=a_\mu(x)$ is a real valued covector field (a potential of electromagnetic field), $m$ is a real constant (the electron mass), $i=\sqrt{-1}$, and $\gamma^\mu$ are (Dirac) matrices that satisfy the identities (${\bf 1}$ is the identity matrix) $$\gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu=2\eta^{\mu\nu}{\bf 1}.$$ In particular we may take $$\begin{aligned}
\gamma^0&=&\pmatrix{1 &0 &0 & 0\cr
0 &1 & 0&0 \cr
0 &0 &-1&0 \cr
0 &0 &0 &-1},\quad
\gamma^1=\pmatrix{0 &0 &0 & 1\cr
0 &0 & 1&0 \cr
0 &-1 &0 &0 \cr
-1 &0 &0 &0},
\label{gamma:matrices}\\
\gamma^2&=&\pmatrix{0 &0 &0 & -i\cr
0 &0 & i&0 \cr
0 & i&0 &0 \cr
-i &0 &0 &0},\quad
\gamma^3=\pmatrix{0 &0 & 1& 0\cr
0 &0 & 0&-1 \cr
-1 &0 &0 &0 \cr
0 &1&0 &0}.\nonumber\end{aligned}$$ Consider a change of coordinates $$x^\mu\to\acute x^\mu=p^\mu_\nu x^\nu
\label{change:coord}$$ from the proper orthochroneous Lorentz group $\SO^+(1,3)$, i.e., $$P^{\rm T}\eta P=\eta,\quad \det P=1,\quad p^0_0>0,
\label{proper}$$ where $P=\|p^\mu_\nu\|$. Then $$\partial_\mu\to\acute\partial_\mu=q^\nu_\mu\partial_\nu,\quad
a_\mu\to\acute a_\mu=q^\nu_\mu a_\nu,$$ where $q^\nu_\mu$ are elements of the inverse matrix $P^{-1}$, i.e., $$q^\nu_\mu p^\mu_\lambda=\delta^\nu_\lambda,\quad
q^\mu_\nu p^\lambda_\mu=\delta^\lambda_\nu$$ and $\delta^\mu_\nu$ is the Kronecker tensor. P. Dirac in [@Dirac] postulate that $\gamma^\mu$ is a vector with values in $\M^\C(4)$ ($\M^\C(4)$ is the algebra of complex valued $4\!\times\!4$-matrices). That means $$\gamma^\mu\to\acute\gamma^\mu=p^\mu_\nu\gamma^\nu$$ and eq. (\[Dirac:eq\]) in coordinates $(\acute x)$ has the form $$\acute\gamma^\mu(\acute\partial_\mu\psi+i\acute a_\mu\psi)+im\psi=0,
\label{Dirac:eq:prime}$$ where $\acute\partial_\mu=\partial/\partial\acute x^\mu$, $\psi=\psi(x(\acute x))$. It is proved in the theory of representations of Lorentz group that there exists a pair of nondegenerate matrices $\pm R\in\M^\C(4)$ such that $$R^{-1}\gamma^\mu R=p^\mu_\nu\gamma^\nu=\acute \gamma^\mu.
\label{gamma:prime}$$ Substituting $R^{-1}\gamma^\mu R$ for $\acute\gamma^\mu$ in (\[Dirac:eq:prime\]), we get $$R^{-1}\gamma^\mu R(\acute\partial_\mu\psi+i\acute a_\mu\psi)+im\psi=0,
\label{Dirac:eq:prime2}$$ or, equivalently, $$\gamma^\mu(\acute\partial_\mu(R\psi)+i\acute a_\mu(R\psi))+im(R\psi)=0,
\label{Dirac:eq:spinor}$$ We see that an invariance of the Dirac equation can be proved in two ways.
If we suppose the following transformation rules for $\gamma^\mu$ and $\psi$ under every change of coordinates (\[change:coord\],\[proper\]): $$\gamma^\mu\to R^{-1}\gamma^\mu R,\quad \psi\to\psi,$$ then we get [*a spinorless picture*]{} of the Dirac equation.
In the case of transformation rules $$\gamma^\mu\to\gamma^\mu,\quad \psi\to R\psi,$$ we get a conventional [*spinor picture*]{} of the Dirac equation. In sect.9, considering a connection between the Dirac equation and the corresponding Dirac-type tensor equation, we shall see that the two pictures of the Dirac equation are the consequence of a gauge symmetry of the Dirac-type tensor equation with respect to the spinor group.
A Differentiable manifold $X$. Tensors.
=======================================
Let $X$ be a four-dimensional orientable differentiable manifold with local coordinates $x^\mu$ ($\mu=0,1,2,3$). Summation convention over repeating indices is assumed. Let $\top^q_p$ be the set of smooth type $(p,q)$ real valued tensors (tensor fields) on $X$ $$\top^q_p=\{u^{\nu_1\ldots\nu_q}_{\mu_1\ldots\mu_p}=
u^{\nu_1\ldots\nu_q}_{\mu_1\ldots\mu_p}(x)\}.$$ Under a change of coordinates $(x)\to(\acute x)$ a tensor $u^{\nu_1\ldots\nu_q}_{\mu_1\ldots\mu_p}$ transforms as follows: $$u^{\nu_1\ldots\nu_q}_{\mu_1\ldots\mu_p}\to
\acute u^{\beta_1\ldots\beta_q}_{\alpha_1\ldots\alpha_p}=
u^{\nu_1\ldots\nu_q}_{\mu_1\ldots\mu_p}
\frac{\partial\acute x^{\beta_1}}{\partial x^{\nu_1}}\ldots
\frac{\partial\acute x^{\beta_q}}{\partial x^{\nu_q}}
\frac{\partial x^{\mu_1}}{\partial\acute x^{\alpha_1}}\ldots
\frac{\partial x^{\mu_p}}{\partial\acute x^{\alpha_p}}.$$ Operations of addition and tensor product are defined in the usual way $$(u+v)^{\nu_1\ldots\nu_q}_{\mu_1\ldots\mu_p}=
u^{\nu_1\ldots\nu_q}_{\mu_1\ldots\mu_p}+v^{\nu_1\ldots\nu_q}_{\mu_1\ldots\mu_p}
\in\top^q_p,\quad u,v\in\top^q_p;$$ $$(u\otimes v)^{\nu_1\ldots\nu_{q+s}}_{\mu_1\ldots\mu_{p+r}}=
u^{\nu_1\ldots\nu_q}_{\mu_1\ldots\mu_p}
v^{\nu_{q+1}\ldots\nu_{q+s}}_{\mu_{p+1}\ldots\mu_{p+r}}\in\top^{q+s}_{p+r},\quad
u\in\top^q_p,v\in\top^s_r.$$
A Parallelisable manifold with a tetrad.
========================================
An $n$-dimensional differentiable manifold is called [*parallelisable*]{} if there exist $n$ linear independent vector or covector fields on it. Let $X$ be a four dimensional parallelisable manifold with local coordinates $x=(x^\mu)$ and $$e_\mu{}^a=e_\mu{}^a(x),\quad a=0,1,2,3$$ be four covector fields on $X$. This set of four covectors is called [*a tetrad*]{}. The full set $\{X,e_\mu{}^a,\eta\}$, where $\eta$ is the Minkowski matrix, is denoted by $\W$. Here and in what follows we use Greek indices as tensorial indices and Latin indices as nontensorial (tetrad) indices.
We can define a metric tensor on $W$ $$g_{\mu\nu}=e_\mu{}^a e_\nu{}^b\eta_{ab}$$ such that $$g_{\mu\nu}=g_{\nu\mu},\quad g_{00}>0,\quad g=\det\|g_{\mu\nu}\|<0,$$ and the signature of the matrix $\|g_{\mu\nu}\|$ is equal to $-2$. Hence we may consider $\W$ as pseudo-Riemannian space with the metric tensor $g_{\mu\nu}$. We raise and lower Latin indices with the aid of the Minkowski matrix $\eta^{ab}=\eta_{ab}$ and Greek indices with the aid of the metric tensor $g_{\mu\nu}$ $$e^{\nu a}=g^{\mu\nu}e_\mu{}^a,\quad e_{\mu a}=\eta_{ab}e_\mu{}^b.$$ With the aid of the tetrad we can replace all or part of Greek indices in a tensor $u^{\nu_1\ldots\nu_q}_{\mu_1\ldots\mu_p}\in\top^q_p$ by Latin indices $$u^{b_1\ldots b_q}_{a_1\ldots a_p}=
u^{\nu_1\ldots\nu_q}_{\mu_1\ldots\mu_p} e^{\mu_1}{}_{a_1}\ldots
e^{\mu_p}{}_{a_p}e_{\nu_1}{}^{b_1}\ldots e_{\nu_q}{}^{b_q}.$$ As a result we get a set of invariants $u^{b_1\ldots b_q}_{a_1\ldots a_p}$ enumerated by Latin indices. On the contrary we can replace Latin indices by Greek indices $$u^{\nu_1\ldots\nu_q}_{\mu_1\ldots\mu_p}=
u^{b_1\ldots b_q}_{a_1\ldots a_p} e^{\nu_1}{}_{b_1}\ldots
e^{\nu_p}{}_{b_p}e_{\mu_1}{}^{a_1}\ldots e_{\mu_q}{}^{a_q}.$$ Note that $$e^\mu{}_a e_\mu{}^b=\delta^b_a,\quad
e^\mu{}_a e_\nu{}^a=\delta^\mu_\nu.$$ The metric tensor $g_{\mu\nu}$ defines the Levi-Civita connection, the curvature tensor, the Ricci tensor, and the scalar curvature $$\begin{aligned}
{\Gamma_{\mu\nu}}^\lambda&=&
\frac{1}{2}g^{\lambda\kappa}(\partial_\mu g_{\nu\kappa}+
\partial_{\nu}g_{\mu\kappa}-\partial_\kappa g_{\mu\nu}),
\label{Levi-Civita}\\
{R}_{\lambda\mu\nu}{}^\kappa&=&
\partial_\mu {\Gamma}_{\nu\lambda}{}^\kappa-\partial_\nu
{\Gamma}_{\mu\lambda}{}^\kappa+
{\Gamma}_{\mu\eta}{}^\kappa{\Gamma}_{\nu\lambda}{}^\eta-
{\Gamma}_{\nu\eta}{}^\kappa{\Gamma}_{\mu\lambda}{}^\eta,
\label{curv}\\
R_{\nu\rho}&=&{R^\mu}_{\nu\mu\rho},
\label{Ricci}\\
R&=&g^{\rho\nu}R_{\rho\nu}
\label{scalar-curv}\end{aligned}$$ with symmetries $${\Gamma_{\mu\nu}}^\lambda={\Gamma_{\nu\mu}}^\lambda,\quad
R_{\mu\nu\lambda\rho}=R_{\lambda\rho\mu\nu}=R_{[\mu\nu]\lambda\rho},\quad
R_{\mu[\nu\lambda\rho]}=0,\quad
R_{\nu\rho}=R_{\rho\nu}$$ Covariant derivatives $\nabla_\mu\,:\,\top^q_p\to\top^q_{p+1}$ are defined via the Levi-Civita connection by the usual rules
1\. If $u=u(x),\ x\in X$ is a scalar function, then $$\nabla_\mu u=\partial_\mu u.$$
2\. If $u^\nu\in\top^1$, then $$\nabla_\mu u^\nu\equiv u^\nu_{;\mu}=\partial_\mu u^\nu +
{\Gamma_{\lambda\mu}}^\nu u^\lambda.$$
3\. If $u_\nu\in\top_1$, then $$\nabla_\mu u_\nu\equiv u_{\nu;\mu}=\partial_\mu u_\nu -
{\Gamma_{\nu\mu}}^\lambda u_\lambda.$$
4\. If $u=(u^{\nu_1\ldots \nu_k}_{\lambda_1\ldots \lambda_l})\in\top^k_l$, $v=(v^{\mu_1\ldots \mu_r}_{\rho_1\ldots \rho_s})\in\top^r_s$, then $$\nabla_\mu(u\otimes v)=(\nabla_\mu u)\otimes v + u\otimes\nabla_\mu v.$$
With the aid of these rules it is easy to calculate covariant derivatives of arbitrary type tensors. Also we can check the formulas $$\begin{aligned}
&&\nabla_\mu g_{\alpha\beta}=0,\quad
\nabla_\mu g^{\alpha\beta}=0,\quad
\nabla_\mu \delta_\alpha^\beta=0,\\
&&\nabla_\alpha(R^{\alpha\beta}-\frac{1}{2}R g^{\alpha\beta})=0,\\
&&(\nabla_\mu \nabla_\nu-\nabla_\nu \nabla_\mu)a_\rho=
{R^\lambda}_{\rho\mu\nu}a_\lambda,\end{aligned}$$ for any $a_\rho\in\top_1$.
Differential forms.
===================
Let $\Lambda_k$ be the set of all exterior differential forms of rank $k=0,1,2,3,4$ ($k$-forms) on $\W$ and $$\Lambda=\Lambda^0\oplus\ldots\oplus\Lambda^4=\Lambda^\even\oplus\Lambda^\odd,$$ $$\Lambda^\even=\Lambda^0\oplus\Lambda^2\oplus\Lambda^4,\quad
\Lambda^\odd=\Lambda^1\oplus\Lambda^3.$$ The set of smooth real valued functions on $\W$ is identified with the set of 0-forms $\Lambda_0$. A $k$-form $U\in\Lambda_k$ can be written as $$U=\frac{1}{k!} u_{\nu_1\ldots \nu_k}dx^{\nu_1}\ww dx^{\nu_k}=
\sum_{\mu_1<\cdots<\mu_k} u_{\mu_1\ldots \mu_k}dx^{\mu_1}\ww dx^{\mu_k},
\label{k-form}$$ where $u_{\nu_1\ldots \nu_k}=u_{\nu_1\ldots \nu_k}(x)$ are real valued components of a covariant antisymmetric ($u_{\nu_1\ldots \nu_k}=u_{[\nu_1\ldots \nu_k]}$) tensor. Differential forms from $\Lambda$ can be written as linear combinations of the 16 basis elements $$1,dx^\mu,dx^{\mu_1}\wedge dx^{\mu_2},\ldots,dx^{0}\wedge\ldots\wedge
dx^3,
\quad
\mu_1<\mu_2<\ldots.
\label{basis}$$ The exterior product of differential forms is defined in the usual way. If $U\in\Lambda^r,V\in\Lambda^s$, then $$U\wedge V=(-1)^{rs}V\wedge U\in\la^{r+s}.$$ In the sequel we also use complex valued differential forms from $\Lambda^\C_k,\Lambda^\C,\Lambda^\C_\even,\Lambda^\C_\odd$.
The tetrad $e_\mu{}^a$ can be written with the aid of four 1-forms $$e^a=e_\mu{}^a dx^\mu\quad \hbox{or}\quad e_a=\eta_{ab}e^b.$$ The $k$-form $U$ from (\[k-form\]) can be written as $$U=\frac{1}{k!}u_{a_1\ldots a_k}e^{a_1}\ww e^{a_k},$$ where invariants $u_{a_1\ldots a_k}=u_{[a_1\ldots a_k]}$ connected with tensor components $u_{\mu_1\ldots \mu_k}$ by the formula $$u_{a_1\ldots a_k}=u_{\mu_1\ldots \mu_k}e^{\mu_1}{}_{a_1}\ldots e^{\mu_k}{}_{a_k}.$$ Differential forms from $\Lambda$ can be written as $$\sum_{k=0}^4 \frac{1}{k!}u_{a_1\ldots a_k}e^{a_1}\ww e^{a_k},\quad
u_{a_1\ldots a_k}=u_{[a_1\ldots a_k]}$$ and the 16 differential forms $$1,e^a,e^{a_1}\wedge e^{a_2},\ldots e^0\ww e^3,\quad
a_1<a_2<\ldots
\label{tetrad:basis}$$ can be considered as basis forms of $\Lambda$.
A central product of differential forms.
========================================
Let us define [*a central product*]{} of differential forms $U,V\to UV$ by the following rules:
1\. For $U,V,W\in\Lambda,\,\alpha\in\Lambda_0$ $$1U=U1=U,\quad
(\alpha U)V=U(\alpha V)=\alpha(UV),$$ $$U(VW)=(UV)W,\quad
(U+V)W=UW+VW.$$
2\. $e^a e^b=e^a\wedge e^b+\eta^{ab}$.
3\. $e^{a_1}\ldots e^{a_k}=e^{a_1}\ww e^{a_k}$ for $a_1<\ldots<a_k$.
Note that from the second rule we get the equality $$e^a e^b+e^b e^a=2\eta^{ab},
\label{Clifford:rel}$$ which appear in the Clifford algebra[^2]. Substituting into (\[Clifford:rel\]) $$e^a=e_\mu{}^a dx^\mu,\quad
e^b=e_\nu{}^b dx^\nu,\quad
\eta^{ab}=g^{\mu\nu}e_\mu{}^a e_\nu{}^b,$$ we get $$e_\mu{}^a e_\nu{}^b(dx^\mu dx^\nu+dx^\nu dx^\mu-2g^{\mu\nu})=0.$$ Thus, $$dx^\mu dx^\nu+dx^\nu dx^\mu=2g^{\mu\nu}.$$ Evidently, the operation of central product maps $\Lambda\times\Lambda\to\Lambda$. From this definition of the central product it is evident that for $u_{a_1\ldots a_k}=u_{[a_1\ldots a_k]}$ $$\frac{1}{k!}u_{a_1\ldots a_k}e^{a_1}\ww e^{a_k}=
\frac{1}{k!}u_{a_1\ldots a_k}e^{a_1}\ldots e^{a_k}.$$
In the sequel we use notation $$e^{a_1\ldots a_k}=e^{a_1}\ldots e^{a_k}=e^{a_1}\ww e^{a_k}\for
a_1<\cdots<a_k.$$
Let us define an operation of conjugation, which maps $\Lambda_k\to\Lambda_k$ or $\Lambda_k^\C\to\Lambda_k^\C$. For $U\in\Lambda_k^\C$ $$U^*=(-1)^{\frac{k(k-1)}{2}}\bar{U},$$ where $\bar{U}$ is the differential form with complex conjugated components (if $U\in\Lambda$, then $\bar{U}=U$). We see that $$(UV)^*=V^*U^*,\quad U^{**}=U,\quad
(e^{a_1}\ldots e^{a_k})^*=e^{a_k}\ldots e^{a_1}$$ and $$U^*=U\quad\hbox{for}\quad U\in\Lambda_0\oplus\Lambda_1\oplus\Lambda_4,$$ $$U^*=-U\quad\hbox{for}\quad U\in\Lambda_2\oplus\Lambda_3.$$
Let us define [*a trace*]{} of differential form as linear operation $\Tr:\Lambda\to\Lambda_0$ such that $$\Tr(1)=1,\quad \Tr(e^{a_1}\ww e^{a_k})=0\quad\hbox{for}\quad
k=1,2,3,4.$$ It is easy to prove that $$\Tr(UV-VU)=0\quad\hbox{for}\quad U,V\in\Lambda.$$
Let us take the set of forms $$\Spin(\W)=\{S\in\Lambda_\even\,:\,S^*S=1\},$$ which can be considered as a Lie group with respect to the central product. It can be shown that the real Lie algebra of the Lie group $\Spin(\W)$ is coincide with the set of 2-forms $\Lambda_2$ with the commutator $[U,V]=UV-VU$, which maps $\Lambda_2\times\Lambda_2\to\Lambda_2$. If $U\in\Lambda_2$, then $\pm\exp(U)\in\Spin(\W)$, where $$\exp(U)=1+\sum_{k=1}^\infty\frac{1}{k!}U^k.$$ In some cases it is suitable to use for calculations [*an exterior exponent*]{} [@Lounesto]. If $u_{a_1 a_2}=u_{[a_1 a_2]}$ and $U=\frac{1}{2} u_{a_1 a_2}e^{a_1}\wedge e^{a_2}\in\Lambda_2$ is such that $$\begin{aligned}
\lambda&=&1 - u_{01}{}^2 - u_{02}{}^2 - u_{03}{}^2 + u_{12}{}^2 -
u_{03}{}^2 u_{12}{}^2 + 2 u_{02} u_{03} u_{12} u_{13} \\
&& + u_{13}{}^2 - u_{02}{}^2 u_{13}{}^2 -
2 u_{01} u_{03} u_{12} u_{23} +
2 u_{01} u_{02} u_{13} u_{23} + u_{23}{}^2 -
u_{01}{}^2 u_{23}{}^2>0,\end{aligned}$$ then $$\pm\frac{1}{\sqrt{\lambda}}(1+U+\frac{1}{2}U\wedge U)\in\Spin(\W).$$ It is not hard to prove that if $U\in\Lambda_k$ and $S\in\Spin(\W)$, then $$S^{-1}US\in\Lambda_k,$$ i.e., $S^{-1}\Lambda_k S\subseteq\Lambda_k$. In particular, $$S^{-1}e^a S=p^a_b e^b,$$ where the matrix $P=\|p^a_b\|$ has the properties $$P^{\rm T}\eta P=\eta,\quad\det P=1,\quad p^0_0>0.$$ The transformation $$e^a\to \check e^a=S^{-1}e^a S,
\label{rot:tetrad}$$ where $S\in\Spin(\W)$, is called [*a Lorentz rotation of the tetrad*]{}. Obviously, $$\begin{aligned}
S^{-1}(\frac{1}{k!}u_{a_1\ldots a_k}e^{a_1}\ww e^{a_k})S&=&
S^{-1}(\frac{1}{k!}u_{a_1\ldots a_k}e^{a_1}\ldots e^{a_k})S\\
&=&\frac{1}{k!}u_{a_1\ldots a_k}(S^{-1}e^{a_1}S)\ldots(S^{-1} e^{a_k}S)\\
&=&\frac{1}{k!}u_{a_1\ldots a_k}\check e^{a_1}\ldots\check e^{a_k}
=\frac{1}{k!}u_{a_1\ldots a_k}\check e^{a_1}\ww\check e^{a_k}.\end{aligned}$$ From the identities $e^a e^b+e^b e^a=2\eta^{ab}$ it follows that $$\check e^a\check e^b+\check e^b\check e^a=2\eta^{ab}.$$
Tensors from $\Lambda_k\top_s^r$.
=================================
Let us take a tensor $$u^{\lambda_1\ldots\lambda_r}_{\nu_1\ldots\nu_s\mu_1\ldots\mu_k}=
u^{\lambda_1\ldots\lambda_r}_{\nu_1\ldots\nu_s[\mu_1\ldots\mu_k]}
\in\top^r_{s+k}$$ antisymmetric with respect to $k$ covariant indices. One may consider the following objects: $$U^{\lambda_1\ldots\lambda_r}_{\nu_1\ldots\nu_s}=
\frac{1}{k!}u^{\lambda_1\ldots\lambda_r}_{\nu_1\ldots\nu_s\mu_1\ldots\mu_k}
dx^{\mu_1}\ww dx^{\mu_k},
\label{lambda:tensor}$$ which are formally written as $k$-forms. Under a change of coordinates $(x)\to(\acute x)$ values (\[lambda:tensor\]) transform as components of tensor of type $(s,r)$, i.e., $$\acute U^{\alpha_1\ldots\alpha_r}_{\beta_1\ldots\beta_s}=
q^{\nu_1}_{\beta_1}\ldots q^{\nu_s}_{\beta_s}
p^{\alpha_1}_{\lambda_1}\ldots
p^{\alpha_r}_{\lambda_r}
U^{\lambda_1\ldots\lambda_r}_{\nu_1\ldots\nu_s},
\quad q^\nu_\beta=\frac{\partial x^\nu}{\partial\acute x^\beta},\quad
p^\alpha_\lambda=\frac{\partial\acute x^\alpha}{\partial x^\lambda}.$$ The objects (\[lambda:tensor\]) are called [*tensors of type $(s,r)$ with values in $\Lambda_k$*]{}. We write this as $$U^{\lambda_1\ldots\lambda_r}_{\nu_1\ldots\nu_s}\in\Lambda_k\top^r_s.$$ We take $$\Lambda\top^r_s=\Lambda_0\top^r_s\oplus\ldots\oplus \Lambda_4\top^r_s.$$
Note that $dx^\mu=\delta^\mu_\nu dx^\nu$, where $\delta^\mu_\nu$ is the Kronecker tensor ($\delta^\mu_\nu=0$ for $\mu\neq\nu$ and $\delta^\mu_\mu=1$). Hence, $dx^\mu\in\Lambda_1\top^1$.
Elements of $\Lambda_0\top^r_s$ are identified with tensors from $\top^r_s$. For the sequel it is suitable to write tensors (\[lambda:tensor\]) as $$U^{\lambda_1\ldots\lambda_r}_{\nu_1\ldots\nu_s}=
\frac{1}{k!}
u^{\lambda_1\ldots\lambda_r}_{\nu_1\ldots\nu_s a_1\ldots a_k}
e^{a_1}\ww e^{a_k}\in\Lambda_k\top^r_s.$$ Let us define a central product of elements $U^{\mu_1\ldots\mu_r}_{\nu_1\ldots\nu_s}\in\Lambda\top^r_s$ and $V^{\alpha_1\ldots\alpha_p}_{\beta_1\ldots\beta_q}\in\Lambda\top^p_q$ as a tensor from $\Lambda\top^{r+p}_{s+q}$ of the form $$W^{\mu_1\ldots\mu_r\alpha_1\ldots\alpha_p}_{\nu_1\ldots\nu_s\beta_1\ldots\beta_q}
=U^{\mu_1\ldots\mu_r}_{\nu_1\ldots\nu_s}
V^{\alpha_1\ldots\alpha_p}_{\beta_1\ldots\beta_q},$$ where on the right-hand side there is the central product of differential forms (the indices $\mu_1,\ldots,\mu_r,\alpha_1,\ldots,\alpha_p,\nu_1,\ldots,\nu_s,\beta_1,\ldots,\beta_q$ are fixed).
If $U^{\mu_1\ldots\mu_r}_{\nu_1\ldots\nu_s}\in\Lambda_0\top^r_s$ and $V^{\alpha_1\ldots\alpha_p}_{\beta_1\ldots\beta_q}\in\Lambda_0\top^p_q$, then the central product of these elements is identified with the tensor product.
Let us define operators (Upsilon) $\Upsilon_\mu$, which act on tensors from $\Lambda\top^r_s$ by the following rules:
a\) If $u=(u_{\nu_1\ldots \nu_s}^{\epsilon_1\ldots \epsilon_r})
\in\Lambda_0\top^r_s$, then $$\Upsilon_\mu u=\partial_\mu u.$$
b\) $\Upsilon_\mu dx^\nu = -{\Gamma^\nu}_{\mu\lambda}
dx^\lambda$.
c\) If $U\in\Lambda\top^r_s$, $V\in\Lambda\top^p_q$, then $$\Upsilon_\lambda(UV)=
(\Upsilon_\lambda U)V+U\Upsilon_\lambda V.$$
d\) If $U,V\in\Lambda\top^r_s$, then $$\Upsilon_\lambda(U+V)=\Upsilon_\lambda U+\Upsilon_\lambda V.$$
With the aid of these rules it is easy to calculate how operators $\Upsilon_\mu$ act on arbitrary tensor from $\Lambda\top_s^r$.
1\. If $U\in\Lambda_k$ has the form (\[k-form\]), then $$\Upsilon_\nu U=\frac{1}{k!}u_{\mu_1\ldots\mu_k;\nu}dx^{\mu_1}\ww dx^{\mu_k}\in
\Lambda_k\top_1.$$
The proof in straightforward.
From the rule b) we get $$(\Upsilon_\mu\Upsilon_\nu-\Upsilon_\nu\Upsilon_\mu)dx^\lambda=
-{R^\lambda}_{\rho\mu\nu}dx^\rho.
\label{R1}$$
2\. Under a change of coordinates $(x)\to(\acute x)$ operators $\Upsilon_\mu$ transform as components of a covector, i.e., $$\acute\Upsilon_\mu=q^\nu_\mu \Upsilon_\nu,\quad
q^\nu_\mu=\frac{\partial x^\nu}{\partial\acute x^\mu}.$$
. This fact follows from the transformation rule of Christoffel symbols $\Gamma_{\mu\nu}{}^\lambda$.
3\. $$e^a U e_a=\left\{\begin{array}{ll}
4U&\mbox{for $U\in\Lambda_0\top^p_q$}\\
-2U&\mbox{for $U\in\Lambda_1\top^p_q$}\\
0&\mbox{for $U\in\Lambda_2\top^p_q$}\\
2U&\mbox{for $U\in\Lambda_3\top^p_q$}\\
-4U&\mbox{for $U\in\Lambda_4\top^p_q$}
\end{array}
\right.$$
The proof is by direct calculations.
Let us take the tensor $$B_\mu=-\frac{1}{4}e^a\wedge\Upsilon_\mu e_a
=-\frac{1}{4}e^a\Upsilon_\mu e_a\in\Lambda_2\top_1.
\label{B-def}$$
4\. Under the Lorentz rotation of tetrad (\[rot:tetrad\]) the tensor $B_\mu$ transforms as a connection $$B_\mu\to\check B_\mu=S^{-1}B_\mu S-S^{-1}\Upsilon_\mu S.$$
. We have $$\begin{aligned}
&&-4\check B_\mu=\check e^a\Upsilon_\mu\check e_a=S^{-1}e^a S\Upsilon_\mu(S^{-1}e_a S)\\
&&=S^{-1}e^a S(\Upsilon_\mu S^{-1})e_a S+S^{-1}e^a \Upsilon_\mu(e_a) S+
S^{-1}e^a e_a\Upsilon_\mu S\\
&&=-4S^{-1}B_\mu S+4S^{-1}\Upsilon_\mu S+
S^{-1}e^a S(\Upsilon_\mu S^{-1})e_a S.\end{aligned}$$ Here we use the formula $e^a e_a=4$ from Theorem 3. It can be checked that $S\Upsilon_\mu S^{-1}\in\Lambda_2\top_1$. Consequently from Theorem 3 we have $$e^a S\Upsilon_\mu S^{-1} e_a S=0.$$ These completes the proof.
From the formula (\[B-def\]) it is easily shown that $$\Upsilon_\mu e^a=[B_\mu,e^a],\quad
\Upsilon_\mu e_a=[B_\mu,e_a].$$ Hence, if we take the operators $$\D_\mu=\Upsilon_\mu-[B_\mu,\,\cdot\,],$$ then $$\D_\mu e^a=0,\quad \D_\mu e_a=0.$$
5\. The operators $\D_\mu$ satisfy the Leibniz rule $$\D_\mu(UV)=(\D_\mu U)V+U\D_\mu V\quad\hbox{for}\quad
U\in\Lambda\top^r_s,\,V\in\Lambda\top^p_q$$ and $$\D_\mu\D_\nu-\D_\nu\D_\mu=0.$$
is by direct calculations.
Note that [*the volume form*]{} $$\ell=e^0\wedge e^1\wedge e^2\wedge e^3=\sqrt{-g}\,dx^0\ww dx^3\in\Lambda_4$$ is constant with respect to these operators, i.e., $$\D_\mu\ell=0.$$
We have the following consequences from the theorems 4,5. Under the Lorentz rotation of tetrad (\[rot:tetrad\]) $B_\mu,\D_\mu$ transform as follows $$\begin{aligned}
B_\mu&\to&\check B_\mu=B_\mu-S^{-1}\D_\mu S,\label{B:transform}\\
\D_\mu&\to&\check\D_\mu=\D_\mu+[S^{-1}\D_\mu S,\,\cdot\,].
\nonumber\end{aligned}$$
Let us denote 1-form $$H:=e^0.$$ We define the operator of [*Hermitian conjugation*]{} of tensors $\dagger\,:\,\Lambda\top^p_q\to\Lambda\top^p_q$ $$U^\dagger=H U^* H \quad\hbox{for}\quad U\in\Lambda\top^p_q.$$ Evidently, $$(UV)^\dagger=V^\dagger U^\dagger,\quad U^{\dagger\dagger}=U,
\quad i^\dagger=-i.$$ We shall see in Section 10 that this operator is connected to the operator of Hermitian conjugation of matrices.
We say that a tensor $U\in\Lambda\top^p_q$ is [*Hermitian*]{} if $U^\dagger=U$ and [*anti-Hermitian*]{} if $U^\dagger=-U$. Every tensor $U$ can be decomposed into Hermitian and anti-Hermitian parts $$U=\frac{1}{2}(U+U^\dagger)+\frac{1}{2}(U-U^\dagger).$$
Note that all discussed constructions, which were defined in this paper for tensors from $\Lambda\top^p_q$, are also valid for complex valued tensors from $\Lambda^\C\top^p_q$.
Now we may define the operation $$(\cdot,\cdot)\,:\,\Lambda^\C\times\Lambda^\C\to\Lambda^\C_0$$ by the formula $$(U,V)=\Tr(U^\dagger V).$$ This operation has all the properties of Hermitian scalar product at every point $x\in X$ $$\begin{aligned}
&&\alpha(U,V)=(\bar\alpha U,V)=(U,\alpha V),\\
&&(U,V)=\overline{(V,U)},\quad (U+W,V)=(U,V)+(W,V),\\
&&(U,U)>0\quad\hbox{for}\quad U\neq0,\end{aligned}$$ where $U,V,W\in\Lambda^\C$, $\alpha\in\Lambda^\C_0$, and a bar means complex conjugation. The operation $(\cdot,\cdot)$ converts $\Lambda^\C$ into the unitary space at every point $x\in X$.
Let us denote by $T_0,\ldots T_{15}$ the following differential forms: $$i,ie^0,e^1,e^2,e^3,ie^{01},ie^{02},ie^{03},e^{12},e^{13},e^{23},
e^{012},e^{013},e^{023},ie^{123},e^{0123}.
\label{basis:ah}$$ which form an orthonormal basis of $\Lambda^\C$ $$(T_\k,T^\n)=\delta_\k^\n,\quad {\textsc k},{\textsc n}=0,\ldots15$$ and $$T_\k=-T_\k^\dagger,\quad \D_\mu T_\k=0,$$ where $T^\n=T_\n$. This basis is said to be [*the anti-Hermitian basis of $\Lambda^\C$*]{}.
A differential form $t\in\Lambda^\C$ such that $$t^2=t,\quad \D_\mu t=0,\quad t^\dagger=t
\label{t}$$ is called [*an idempotent*]{}. We suppose that under a Lorentz rotation of tetrad $e^a\to\check e^a=S^{-1}e^a S$, $S\in\Spin(\W)$ an idempotent $t$ transforms as $$t\to\check t=S^{-1}tS.$$ In this case $$\begin{aligned}
&&\check t^2=\check t,\\
&&\check t^\dagger=\check H\check t^*\check H=\check t,\\
&&\check\D_\mu\check t=0.\end{aligned}$$
We may consider [*the left ideal*]{} generated by the idempotent $t$ $$\I(t)=\{Ut\,:\,U\in\Lambda^\C\}\subseteq\Lambda^\C.
\label{ideal}$$
Let us define the set of differential forms $$L(t)=\{U\in\I(t)\,:\,U^\dagger=-U,\,[U,t]=0\}.$$ This set is closed with respect to the commutator (if $U,V\in L(t)$, then $[U,V]\in L(t)$) and can be considered as a real Lie algebra. With the aid of the real Lie algebra $L(t)$ we define the corresponding Lie group $$G(t)=\{\exp(U)\,:\,U\in L(t)\}.$$ In Section 10 we consider $t,\I(t),L(t),G(t)$ in details.
Finally let us summarize properties of the operators $\D_\mu$ $$\begin{aligned}
&&\D_\mu(UV)=(\D_\mu U)V+U\D_\mu V,\\
&&\D_\mu\D_\nu-\D_\nu\D_\mu=0,\\
&&\D_\mu(U+V)=\D_\mu U+\D_\mu V,\\
&&\D_\mu e^a=0,\,\,\D_\mu e_a=0,\\
&&\D_\mu\ell=0,\\
&&\D_\mu(U^*)=(\D_\mu U)^*,\\
&&\D_\mu(U^\dagger)=(\D_\mu U)^\dagger,\\
&&\D_\mu(\Tr(U))=\partial_\mu(\Tr(U))=\Tr(\D_\mu U),\\
&&\D_\mu(U,V)=\partial_\mu(U,V)=(\D_\mu U,V)+(U,\D_\mu V).\end{aligned}$$ Under a change of coordinates $(x)\to(\acute x)$ operators $\D_\mu$ transform as components of a covector, i.e., $$\D_\mu\to\acute\D_\mu=\frac{\partial x^\nu}{\partial\acute x^\mu}\D_\nu.$$ Under a Lorentz rotation of tetrad $e^a\to\check e^a=S^{-1}e^a S$ ($S\in\Spin(\W)$) operators $\D_\mu$ transform as $\D_\mu\to\check\D_\mu=\D_\mu+[S^{-1}\D_\mu S,\,\cdot\,]$.
Dirac-type tensor equations. A general case.
============================================
We begin with the following equation in $\W$ (a tetrad $e^a$ is given and, consequently, the tensor $B_\mu\in\Lambda_2\top_1$ and the operators $\D_\mu$ are defined): $$dx^\mu(\D_\mu\Phi+\Phi A_\mu+B_\mu\Phi)+im\Phi=0,
\label{init:eq}$$ where $\Phi\in\Lambda^\C$, $i=\sqrt{-1}$, $m$ is a given real constant, and $A_\mu\in\Lambda^\C\top_1$ is such that $A_\mu{}^\dagger=-A_\mu$. We consider the differential form $\Phi$ as unknown (16 complex valued components) and $A_\mu$ as known. Writing eq. (\[init:eq\]) as a system of equations for components of $\Phi$, we see that the number of equations is equal to the number of unknown values.
An equation is said to be [*a tensor equation*]{} if all values in it are tensors and all operations in it take tensors to tensors.
In eq. (\[init:eq\]) we have $dx^\mu\D_\mu\Phi,dx^\mu\Phi A_\mu,dx^\mu B_\mu\Phi\in\Lambda^\C$. Hence eq. (\[init:eq\]) is a tensor equation.
Let $t\in\Lambda^\C$ be an idempotent and $A_\mu$ be a tensor from $L(t)\top_1$. Then we may consider the equation in $\W$ $$(dx^\mu(\D_\mu\Phi+\Phi A_\mu+B_\mu\Phi)+im\Phi)t=0,
\label{init2:eq}$$ From the identities $\D_\mu t=0$, $[A_\mu,t]=0$ it follows that eq. (\[init2:eq\]) can be written as the equation for $\Psi=\Phi t\in\I(t)$ $$dx^\mu(\D_\mu\Psi+\Psi A_\mu+B_\mu\Psi)+im\Psi=0,
\label{Dirac:type}$$ where the idempotent $t$, the real constant $m$, and the tensor $A_\mu\in L(t)\top_1$ are considered as known. The differential form $\Psi\in\I(t)$ is considered as unknown.
In Section 10 we shall see that there are four types of idempotents $t$. Consequently, there are four types of equations (\[Dirac:type\]). These equations are called [*Dirac-type tensor equations*]{}. A connection of eqs. (\[Dirac:type\]) with the Dirac equation will be discussed in Section 9.
Denoting $\alpha^\mu=H dx^\mu\in(\Lambda_0\oplus\Lambda_2)\top^1$, we see that $(\alpha^\mu)^\dagger=\alpha^\mu$.
6\. If $\Psi\in\I(t)$ satisfies eq. (\[Dirac:type\]), then the tensor $$J^\mu=i\Psi^\dagger\alpha^\mu\Psi$$ satisfies the equality $$\frac{1}{\sqrt{-g}}\D_\mu(\sqrt{-g}\,J^\mu)-[A_\mu,J^\mu]=0,
\label{conserv:law}$$ which is called [*a (non-Abelian) charge conservation law*]{}.
Note that $\Psi=\Psi t$ and $J^\mu=t J^\mu t$. Therefore, $$[J^\mu,t]=[t J^\mu t,t]=0,$$ i.e., $J^\mu\in L(t)\top^1$.
of Theorem 6. Let us multiply eq. (\[init2:eq\]) from the left by $H$ and denote the left-hand side of resulting equation by $$Q=\alpha^\mu(\D_\mu\Psi+\Psi A_\mu+B_\mu\Psi)+imH\Psi.
\label{Q}$$ Then $$Q^\dagger=(\D_\mu\Psi^\dagger-A_\mu\Psi^\dagger+\Psi^\dagger B_\mu^\dagger)\alpha^\mu-
im\Psi^\dagger H.$$ Consider the expression $$\begin{aligned}
i(\Psi^\dagger Q+Q^\dagger\Psi)&=&
i(\Psi^\dagger\alpha^\mu\D_\mu\Psi+
\D_\mu\Psi^\dagger\alpha^\mu\Psi+\Psi^\dagger(\D_\mu\alpha^\mu)\Psi)\nonumber
-[A_\mu,i\Psi^\dagger\alpha^\mu\Psi]\\
&&+i\Psi^\dagger(-D_\mu\alpha^\mu+\alpha^\mu B_\mu+B_\mu^\dagger\alpha^\mu)\Psi\label{tmp}\\
&=&\frac{1}{\sqrt{-g}}\D_\mu(\sqrt{-g}\,J^\mu)-[A_\mu,J^\mu].\nonumber\end{aligned}$$ Here we use the formulae $$\D_\mu\alpha^\mu=-\Gamma_{\mu\nu}{}^\mu \alpha^\nu+\alpha^\mu B_\mu+B_\mu^\dagger\alpha^\mu,$$ $$\D_\mu J^\mu+\Gamma_{\mu\nu}{}^\mu
J^\nu=\frac{1}{\sqrt{-g}}\D_\mu(\sqrt{-g}\,J^\mu).$$ which can be easily checked. In (\[tmp\]) equality $Q=0$ leads to the equality (\[conserv:law\]). These completes the proof.
Now we may write eq. (\[Dirac:type\]) together with [*Yang-Mills equations*]{} (\[YM1\],\[YM2\]) $$\begin{aligned}
&&dx^\mu(\D_\mu\Psi+\Psi A_\mu+B_\mu\Psi)+im\Psi=0, \label{Dirac:type1}\\
&&\D_\mu A_\nu-\D_\nu A_\mu-[A_\mu,A_\nu]=F_{\mu\nu},\label{YM1}\\
&&\frac{1}{\sqrt{-g}}\D_\mu(\sqrt{-g}\,F^{\mu\nu})-[A_\mu,F^{\mu\nu}]=J^\nu,\label{YM2}\\
&&J^\nu=i\Psi^\dagger\alpha^\nu\Psi,\label{J}\end{aligned}$$ where $\Psi\in\I(t)$, $A_\mu\in L(t)\top_1$, $F_{\mu\nu}\in L(t)\top_2$, $J^\nu\in L(t)\top^1$. In this system of equations we consider $\Psi,A_\mu,F_{\mu\nu},J^\nu$ as unknown values and $m,t$ as known values.
7\. Let us denote the left-hand side of eq. (\[YM2\]) by $R^\nu$ $$R^\nu:=\frac{1}{\sqrt{-g}}\D_\mu(\sqrt{-g}\,F^{\mu\nu})-[A_\mu,F^{\mu\nu}],$$ where $F_{\mu\nu}$ satisfy (\[YM1\]). Then $$\frac{1}{\sqrt{-g}}\D_\mu(\sqrt{-g}\,R^{\mu})-[A_\mu,R^{\mu}]=0.$$
The proof is by direct calculations.
This theorem means that eq. (\[YM2\]) is consistent with the charge conservation law (\[conserv:law\]).
Unitary and Spin gauge symmetries.
==================================
8\. Let $\Psi,A_\mu,F_{\mu\nu},J^\nu$ satisfy eqs. (\[Dirac:type1\]-\[J\]) with a given idempotent $t$ and constant $m$. And let $U\in G(t)$, where the Lie group $G(t)$ is defined in Section 6. Then the following values with tilde: $$\tilde\Psi=\Psi U,\quad\tilde A_\mu=U^{-1}A_\mu U-U^{-1}\D_\mu U,\quad
\tilde F_{\mu\nu}=U^{-1}F_{\mu\nu}U,$$ $$\tilde J^\nu=U^{-1}J^\nu U,\quad
\{\tilde t,\tilde B_\mu,\tilde\D_\mu\}=
\{t,B_\mu,\D_\mu\}$$ satisfy the same equations (\[Dirac:type1\]-\[J\]).
The proof is straightforward.
This theorem means that eqs. (\[Dirac:type1\]-\[J\]) are invariant under gauge transformations with the symmetry Lie group $G(t)$.
9\. Let $\Psi,A_\mu,F_{\mu\nu},J^\nu$ satisfy eqs. (\[Dirac:type1\]-\[J\]) with a given idempotent $t$ and constant $m$. And let $S\in\Spin(\W)$. Then the following values with check: $$\check\Psi=\Psi S,\quad\check A_\mu=S^{-1}A_\mu S,\quad
\check F_{\mu\nu}=S^{-1}F_{\mu\nu}S,\quad\check J^\nu=S^{-1}J^\nu S,
\label{theorem9}$$ $$\check t=S^{-1}tS,\quad \check B_\mu=B_\mu-S^{-1}\D_\mu S,
\quad \check\D_\mu=\D_\mu+[S^{-1}\D_\mu,\,\cdot\,]$$ satisfy the same equations (\[Dirac:type1\]-\[J\]).
The proof is by direct calculations.
Note that for values with check the operation of Hermitian conjugation is defined by $\check U^\dagger=\check H\check U^*\check H$, where $\check H=S^{-1}H S$.
This theorem means that eqs. (\[Dirac:type1\]-\[J\]) are invariant under gauge transformations with the symmetry Lie group $\Spin(\W)$.
Eqs. (\[Dirac:type1\]-\[J\]) can be derived from the following Lagrangian: $$\L=\frac{1}{4}i\sqrt{-g}\,\Tr(\Psi^\dagger H Q-Q^\dagger H\Psi)+
C\frac{1}{4}\sqrt{-g}\,Tr(\frac{1}{8}F_{\mu\nu}F^{\mu\nu}),
\label{lagr}$$ where $Q$ is from (\[Q\]), $F_{\mu\nu}=\D_\mu A_\nu-\D_\nu A_\mu-[A_\mu,A_\nu]
\in L(t)\top_2$, and $C$ is a real constant. If we have a basis $\{t_1,\ldots,t_D\}$ of $\I(t)$ and a basis $\{\tau_1,\ldots,\tau_d\}$ of $L(t)$ that satisfy (\[basis\]),(\[tau\]), then we may substitute $\Psi=\psi^\k t_\k$, $A_\mu=a_\mu^n \tau_n$ into the Lagrangian $\L$. Variating $\L$ with respect to $\psi^\k$ and $a_\mu^n$, we arrive at eqs. (\[Dirac:type1\]-\[J\]). We discuss bases of $\I(t)$ and $L(t)$ in Section 10.
In [@nona2] we discuss a gravitational term for the Lagrangian $\L$.
A connection between the Dirac-type tensor equation and the Dirac equation.
===========================================================================
Let $t\in\Lambda^\C$ be an idempotent with the properties (\[t\]) and $\I(t)$ be the left ideal (\[ideal\]) of complex dimension $D$. In the next section we shall see that $D$ may take one of four possible values $D=4,8,12,16$. We use an orthonormal basis $t_{\k}=t^{\k}$, ${\sc k}=1,\ldots,D$ of $\I(t)$ such that $$\D_\mu t_{\k}=0,\quad (t_{\n},t^{\k})=\delta^{\k}_{\n}.
\label{basis}$$ [Small Caps]{} font indices run from $1$ to $D$. Consider a linear operator $|\rangle $ that maps $\I(t)$ to $\C^D$. If $$\Omega=\omega^{\k}t_{\k}\in\I(t),$$ then $$|\Omega\rangle =(\omega^1\ldots\omega^D)^{\rm T}.$$ In particular, $$|t_{\k}\rangle =(0\,\ldots\,1\,\ldots\,0)^{\rm T}$$ with only $1$ on the $k$-th place of the column.
By $M^\C(D)\top_p^q$ denote the set of type $(p,q)$ tensors with values in $D\!\times\!D$ complex matrices. Let $\gamma\,:\,\Lambda^\C\top^q_p\to M^\C(D)\top^q_p$ be a map such that for $U=(U^{\nu_1\ldots\nu_q}_{\mu_1\ldots\mu_p})\in\Lambda^\C\top_p^q$ $$Ut_{\n}=\gamma(U)^{\k}_{\n}t_{\k}.
\label{gamma:map}$$ Hence, $$\gamma(U)^{\k}_{\n}=(t^{\k},Ut_{\n}),$$ where $\gamma(U)^{\k}_{\n}$ are elements of the matrix $\gamma(U)$ (an upper index enumerates rows and a lower index enumerates columns).
If $U\in\Lambda^\C$ and $\Omega\in\I(t)$, then $$U\Omega=U\omega^{\n}t_{\n}=\omega^{\n}\gamma(U)^{\k}_{\n}t_{\k}.$$ That means $$|U\Omega\rangle =\gamma(U)|\Omega\rangle .$$ If $U,V\in\Lambda^\C$, $\Omega\in\I(t)$, then $$|UV\Omega\rangle =\gamma(U)\gamma(V)|\Omega\rangle =\gamma(UV)|\Omega\rangle .$$ Consequently, $$\gamma(UV)=\gamma(U)\gamma(V),$$ i.e., $\gamma$ is a matrix representation of $\Lambda^\C$. For example, if we take $dx^\mu=\delta^\mu_\nu dx^\nu\in\Lambda_1\top^1$, then we get $$dx^\mu t_{\n}=\gamma(dx^\mu)^{\k}_{\n}t_{\k}.$$ Denoting $\gamma^\mu=\gamma(dx^\mu)$, we see that the equality $dx^\mu dx^\nu+dx^\nu dx^\mu=2g^{\mu\nu}$ leads to the equality $$\gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu=2g^{\mu\nu}{\bf 1},$$ where ${\bf 1}$ is the identity matrix of dimension $D$.
Let us take the set of differential forms $$\K(t)=\{V\in\Lambda^\C\,:\,[V,t]=0\},$$ which can be considered as an algebra (at any point $x\in X$). Now we define a map $$\theta\,:\,\K(t)\top^q_p\to M^\C(D)\top^q_p$$ such that for $V=(V^{\nu_1\ldots\nu_q}_{\mu_1\ldots\mu_p})\in\K(t)\top^q_p$ $$t_{\n}V=\theta(V)^{\k}_{\n}t_{\k}.$$ Therefore, $$\theta(V)^{\k}_{\n}=(t^{\k},t_{\n}V).$$ If $V\in\K(t)$ and $\Omega\in\I(t)$, then $$\Omega V=\omega^{\n}t_{\n}V=\omega^{\n}\theta^{\k}_{\n}t_{\k}.$$ Than means $$|\Omega V\rangle =\theta(V)|\Omega\rangle .$$ If $U,V\in\K(t)$, $\Omega\in\I(t)$, then $$|\Omega UV\rangle =\theta(V)\theta(U)|\Omega\rangle =\theta(UV)|\Omega\rangle .$$ Consequently, $$\theta(UV)=\theta(V)\theta(U).$$
If $U\in\Lambda^\C$, $V\in\K(t)$, $\Omega\in\I(t)$, then $U\Omega\in\I(t)$, $\Omega V\in\I(t)$ and $$\begin{aligned}
|U\Omega V\rangle &=&\gamma(U)|\Omega V\rangle =\gamma(U)\theta(V)|\Omega\rangle ,\\
|U\Omega V\rangle &=&\theta(V)|U\Omega\rangle =\theta(V)\gamma(U)|\Omega\rangle .\end{aligned}$$ Consequently, $$[\gamma(U),\theta(V)]=0.$$
Denoting the left-hand side of eq. (\[Dirac:type\]) by $$\Omega=dx^\mu(\D_\mu\Psi+\Psi A_\mu+B_\mu\Psi)+im\Psi
\label{Omega}$$ and using formulas $$\begin{aligned}
&&\psi:=|\Psi\rangle ,\\
&&|\D_\mu\Psi\rangle =|\D_\mu(t_{\k}\psi^{\k})\rangle =|t_{\k}\partial_\mu\psi^{\k}\rangle=
\partial_\mu\psi,\\
&&|dx^\mu\Psi A_\mu\rangle =\gamma^\mu|\Psi A_\mu\rangle =\gamma^\mu\theta(A_\mu)\psi,\\
&&|dx^\mu B_\mu\Psi\rangle =\gamma^\mu|B_\mu\Psi\rangle =\gamma^\mu\gamma(B_\mu)\psi,\end{aligned}$$ we see that $$|\Omega\rangle =\gamma^\mu(\partial_\mu+\theta(A_\mu)+\gamma(B_\mu))\psi+im\psi.
\label{Omega:column}$$ Note that $$B_\mu=\frac{1}{2}b_{\mu ab}e^a\wedge e^b=\frac{1}{4}b_{\mu ab}(e^a e^b-e^b e^a).$$ Therefore, $$\gamma(B_\mu)=\frac{1}{4}b_{\mu ab}[\gamma^a,\gamma^b],$$ where $\gamma^a=\gamma(e^a)$ and the equalities $e^a e^b+e^b e^a=2\eta^{ab}$ leads to the equalities $$\gamma^a\gamma^b+\gamma^b\gamma^a=2\eta^{ab}{\bf 1}.$$ If we take $V\in G(t)$, then $$\begin{aligned}
\Omega V&=&(dx^\mu(\D_\mu\Psi+\Psi A_\mu+B_\mu\Psi)+im\Psi)V\\
&=&dx^\mu(\D_\mu(\Psi V)+\Psi V(V^{-1}A_\mu V-V^{-1}\D_\mu V)+B_\mu(\Psi V))+im(\Psi V)\end{aligned}$$ and $$\begin{aligned}
|\Omega V\rangle &=&\theta(V)|\Omega\rangle =P(\gamma^\mu(\partial_\mu+\theta(A_\mu)+\gamma(B_\mu))\psi+im\psi)\\
&=&\gamma^\mu(\partial_\mu+(P\theta(A_\mu)P^{-1}-(\partial_\mu P)P^{-1})+\gamma(B_\mu))(P\psi)+imP\psi,\end{aligned}$$ where $P=\theta(V)$ and we use formulae $[P,\gamma^\mu]=0$, $[P,\gamma(B_\mu)]=0$. Thus the invariance of eq. (\[Dirac:type\]) under the gauge transformation ($V\in G(t)$) $$\Psi\to\Psi V,\quad A_\mu\to V^{-1}A_\mu V-V^{-1}\D_\mu V$$ leads to the invariance of the equation $$\gamma^\mu(\partial_\mu+\theta(A_\mu)+\gamma(B_\mu))\psi+im\psi=0
\label{psi}$$ under the gauge transformation ($P=\theta(V)$) $$\psi\to P\psi,\quad \theta(A_\mu)\to P\theta(A_\mu)P^{-1}-(\partial_\mu P)P^{-1}.$$ Let $S\in\Spin(\W)$. Consider the Lorentz rotation of the tetrad $e^a\to\check e^a=S^{-1}e^a S$, which leads to the transformation $$t\to\check t=S^{-1}tS,\quad \I(t)\to\I(\check t),\quad t_{\n}\to\check t_{\n}=S^{-1}t_{\n}S.$$ Let us define a map $|\check\rangle\,:\,\I(\check t)\to\C^D$. If $\Phi=\phi^{\k}\check t_{\k}\in\I(\check t)$, then $$|\Phi\check\rangle=(\phi^1\ldots\phi^D)^{\rm T}.$$ Evidently for $\Omega\in\I(t)$ $$|S^{-1}\Omega S\check\rangle=|\Omega\rangle.$$ Also we may define a map $\check\gamma\,:\,\Lambda^\C\top_p^q\to M^\C(D)\top^q_p$ such that $$U\check t_{\n}=\check\gamma(U)^{\k}_{\n}\check t_{\k}$$ and $$\check\gamma(UV)=\check\gamma(U)\check\gamma(V).$$ It is easily seen that $$\check\gamma(U)=\gamma(S)\gamma(U)\gamma(S^{-1}).$$ In particular, $$\check\gamma(S)=\gamma(S).$$ For $\Omega\in\I(t)$ we have $$|\Omega S\check\rangle=|S(S^{-1}\Omega S)\check\rangle=
\check\gamma(S)|S^{-1}\Omega S\check\rangle=\gamma(S)|\Omega\rangle.$$ If we apply these identities to $$\begin{aligned}
\Omega S&=&(dx^\mu(\D_\mu\Psi+\Psi A_\mu+B_\mu\Psi)+im\Psi)S\\
&=&dx^\mu(\check\D_\mu\check\Psi+\check\Psi\check A_\mu+\check
B_\mu\check\Psi)+im\check\Psi,\end{aligned}$$ then we get $$\begin{aligned}
|\Omega S\check\rangle&=&\gamma(S)|\Omega\rangle=
R(\gamma^\mu(\partial_\mu\psi+\theta(A_\mu)\psi+\gamma(B_\mu)\psi)+im\psi)\\
&=&R\gamma^\mu R^{-1}(\partial_\mu(R\psi)+\theta(A_\mu)R\psi+
(R\gamma(B_\mu)R^{-1}-(\partial_\mu R)R^{-1})R\psi) +imR\psi,\end{aligned}$$ where $R=\gamma(S)$. Note that $[R,\theta(A_\mu)]=0$.
This implies that the invariance of eq. (\[Dirac:type\]) under the gauge transformation (\[theorem9\]) leads to the invariance of eq. (\[psi\]) under the gauge transformation ($R=\gamma(S)$) $$\psi\to R\psi,\quad \theta(A_\mu)\to\theta(A_\mu)\quad
\gamma^\mu\to R\gamma^\mu R^{-1},$$ $$\gamma(B_\mu)\to R\gamma(B_\mu)R^{-1}-(\partial_\mu R)R^{-1}.$$
Now consider a transformation of eqs. (\[Dirac:type\]),(\[psi\]) under a change of coordinates $(x)\to(\acute x)$. Coordinates $(\acute x)$ we denote with the aid of primed indices $x^{\mu^\prime}$. In coordinates $(\acute x)$ eq. (\[Dirac:type\]) has the form $$\acute\Omega\equiv dx^{\mu^\prime}(\D_{\mu^\prime}\Psi+\Psi A_{\mu^\prime}+
B_{\mu^\prime}\Psi)+im\Psi=0,$$ where $$\D_{\mu^\prime}=\frac{\partial x^{\nu}}{\partial x^{\mu^\prime}}\D_\nu,\quad
dx^{\mu^\prime}=\frac{\partial x^{\mu^\prime}}{\partial x^\nu}dx^\nu,\quad
A_{\mu^\prime}=\frac{\partial x^{\nu}}{\partial x^{\mu^\prime}}A_\nu,\quad
B_{\mu^\prime}=\frac{\partial x^{\nu}}{\partial x^{\mu^\prime}}B_\nu,$$ and eq. (\[psi\]) has the form $$|\acute\Omega\rangle=
\gamma^{\mu^\prime}(\partial_{\mu^\prime}+\theta(A_{\mu^\prime})+
\gamma(B_{\mu^\prime}))\psi+im\psi=0,$$ where $$\partial_{\mu^\prime}=\frac{\partial}{\partial x^{\mu^\prime}},\quad
\gamma^{\mu^\prime}=
\frac{\partial x^{\mu^\prime}}{\partial x^\nu}\gamma^\nu.$$ Note that $$\gamma^\mu=\gamma(dx^\mu)=\gamma^a e^\mu{}_a,\quad
\gamma^a=\gamma(e^a)$$ and $$\gamma^{\mu^\prime}=\gamma^a e^{\mu^\prime}{}_a=
\gamma^a\frac{\partial x^{\mu^\prime}}{\partial x^\nu}e^\nu{}_a.$$
Finally, let us note that in the Lagrangian (\[lagr\]) $$\frac{1}{4}\sqrt{-g}\,\Tr(i(\Psi^\dagger H\Omega-\Omega^\dagger
H\Psi))=
\frac{1}{4}\sqrt{-g}\,i(\langle\Psi|\gamma^0|\Omega\rangle-
\langle\Omega|\gamma^0|\Psi\rangle),$$ where $\langle\Psi|=|\Psi\rangle^\dagger$.
Idempotents and bases of left ideals.
=====================================
Let us take the idempotent $$t_{(1)}=\frac{1}{4}(1+e^0)(1+i e^{12})=
\frac{1}{4}(1+e^0+i e^{12}+i e^{012}),$$ which satisfies conditions (\[t\]), and consider the left ideal $\I(t_{(1)})$. It can be shown that the complex dimension of $\I(t_{(1)})$ is equal to four (this $\I(t_{(1)})$ is [*a minimal left ideal*]{} of $\Lambda^\C$ and $t_{(1)}$ is [*a primitive idempotent*]{}) and the following differential forms $$\begin{aligned}
t^1&=&2 t_{(1)}=\frac{1}{2}(1+e^0+i e^{12}+i e^{012}),\\
t^2&=&-2e^{13} t_{(1)}=\frac{1}{2}(-e^{13}+ie^{23}-e^{013}+ie^{023}),\\
t^3&=&2e^{03} t_{(1)}=\frac{1}{2}(-e^3+e^{03}-ie^{123}+ie^{0123}),\\
t^4&=&2e^{01} t_{(1)}=\frac{1}{2}(-e^1+ie^2+e^{01}-ie^{02})\end{aligned}$$ can be taken as basis forms of $\I(t_{(1)})$, which satisfy (\[basis\]). This basis, according to the formula (\[gamma:map\]), defines the matrix representation of $\Lambda^\C$ ($\gamma_{(1)}$ is a one-to-one map) $$\gamma_{(1)}\,:\,\Lambda^\C\top^q_p\to M^\C\top^q_p.$$
In particular we get matrices $\gamma_{(1)}^\mu=\gamma_{(1)}(dx^\mu)$ identical to (\[gamma:matrices\]). Denote $$\underline U:=\gamma_{(1)}(U)\for U\in\Lambda^\C\top^q_p.$$ Let $Y^\n_\k\in\Lambda^\C$, ($\textsc k,\textsc n=1,2,3,4$) be differential forms such that $\underline Y^\n_\k$ are $4\!\times\!4$-matrices with only nonzero element that equal to $1$ on the intersection of ${\textsc n}$-th row and ${\textsc k}$-th column. We can calculate that $$\begin{aligned}
Y^1_1&=& (1+ e^{0}+ e^{012}i + e^{12}i)/4,\\
Y^1_2&=& (e^{013}+ e^{13}+ e^{023}i + e^{23}i)/4,\\
Y^1_3&=& (e^{03}+ e^{3}+ e^{0123}i + e^{123}i)/4,\\
Y^1_4 &=& (e^{01}+ e^{1}+ e^{02}i + e^{2}i)/4,\\
Y^2_1&=& (-e^{013}- e^{13}+ e^{023}i + e^{23}i)/4,\\
Y^2_2&=& (1+ e^{0}- e^{012}i - e^{12}i)/4,\\
Y^2_3&=& (e^{01}+ e^{1}- e^{02}i - e^{2}i)/4,\\
Y^2_4 &=& (-e^{03}- e^{3}+ e^{0123}i + e^{123}i)/4,\\
Y^3_1&=& (e^{03}- e^{3}+ e^{0123}i - e^{123}i)/4,\\
Y^3_2&=& (e^{01}- e^{1}+ e^{02}i - e^{2}i)/4,\\
Y^3_3&=& (1- e^{0}- e^{012}i + e^{12}i)/4,\\
Y^3_4 &=& (-e^{013}+ e^{13}- e^{023}i + e^{23}i)/4,\\
Y^4_1&=& (e^{01}- e^{1}- e^{02}i + e^{2}i)/4,\\
Y^4_2&=& (-e^{03}+ e^{3}+ e^{0123}i - e^{123}i)/4,\\
Y^4_3&=& (e^{013}- e^{13}- e^{023}i + e^{23}i)/4,\\
Y^4_4 &=& (1- e^{0}+ e^{012}i - e^{12}i)/4.\end{aligned}$$ We see that $$t_{(1)}=Y^1_1,\quad \underline t_{(1)}=\diag(1,0,0,0)$$ and $$t^1=2Y^1_1,\quad t^2=2Y^2_1,\quad t^3=2Y^3_1,\quad t^4=2Y^4_1.$$ Now we may define the idempotents $$\begin{aligned}
t_{(2)}&=&Y^1_1+Y^2_2=\frac{1}{2}(1+e^0),\\
t_{(3)}&=&Y^1_1+Y^2_2+Y^3_3=\frac{1}{4}(3+e^0+ie^{12}-ie^{012}),\\
t_{(4)}&=&Y^1_1+Y^2_2+Y^3_3+Y^4_4=1\end{aligned}$$ such that $$\underline t_{(2)}=\diag(1,1,0,0),
\quad \underline t_{(3)}=\diag(1,1,1,0),
\quad \underline t_{(4)}=\diag(1,1,1,1).$$ Also we can take the following differential forms $t^1,\ldots, t^{16}$: $$t^{4(\n-1)+\k}=2Y^\k_\n,\quad \textsc k,\textsc n=1,2,3,4,$$ which satisfy conditions $\D_\mu t^\k=0$, $(t_\k,t^\n)=\delta^\n_\k$. Evidently, $\{t_1,\ldots,t_8\}$ is a basis of $\I(t_{(2)})$, $\{t_1,\ldots,t_{12}\}$ is a basis of $\I(t_{(3)})$, and $\{t_1,\ldots,t_{16}\}$ is a basis of $\I(t_{(4)})=\Lambda^\C$. In accordance with the formula (\[gamma:map\]), these bases define the maps $$\gamma_{(k)}\,:\,\Lambda^\C\top^q_p\to M^\C(4k)\top^q_p,\quad
k=1,2,3,4$$ such that $$\gamma_{(k)}(UV)=\gamma_{(k)}(U)\gamma_{(k)}(V)\for
U\in\Lambda^\C\top^q_p,\,V\in\Lambda^\C\top_r^s.$$ Also the maps $\gamma_{(k)}$ have the important property $$\gamma_{(k)}(U^\dagger)=(\gamma_{(k)}(U))^\dagger\for
U\in\Lambda^\C\top^q_p,$$ where $U^\dagger=H U^* H$ is the Hermitian-conjugated differential form and $(\gamma_{(k)(U)})^\dagger$ is the Hermitian conjugated matrix (transposed matrix with complex conjugated elements). Consider the set of differential forms $$\K_0(t)=\{U\in\I(t)\,:\,[U,t]=0\}=\K(t)\cap\I(t)$$ and the corresponding set of $4\!\times\!4$-matrices $$\underline\K_0(t)=\{\underline U\,:\,U\in\K_0(t)\}.$$ Evidently $\underline\K_0(t_{(k)})$, ($k=1,2,3,4$) are sets of matrices with all zero elements except elements in the left upper $k\!\times\!k$-block.
Considering the sets of differential forms $$L(t_{(k)})=\{U\in\K_0(t_{(k)})\,:\,U^\dagger=-U\}$$ and the corresponding sets of matrices $\underline L(t_{(k)})$ as real Lie algebrae, we see that $$L(t_{(k)})\simeq\underline L(t_{(k)})\simeq {\rm u}(k)\simeq
{\rm u}(1)\oplus{\rm su}(k),$$ where ${\rm u}(k)$, ($k=1,2,3,4$) are Lie algebrae of anti-Hermitian $k\!\times\!k$-matrices, ${\rm su}(k)$ are the Lie algebrae of traceless anti-Hermitian matrices, and the sign $\simeq$ denote isomorphism. We have the Lie groups $$G(t_{(k)})\simeq\underline G(t_{(k)})\simeq
{\rm U}(1)\oplus{\rm SU}(k),$$ where ${\rm SU}(k)$ are the Lie groups of unitary $k\!\times\!k$-matrices with determinants equal to $1$.
For elements of $L(t_{(k)})$ we define the normalized scalar product $$(u,v)_{(k)} := \frac{4}{k} (u,v)$$ such that $$(it_{(k)},it_{(k)})_{(k)}=1,\quad k=1,2,3,4.$$ Now we show that for every $k=1,2,3,4$ we may take generators $\tau$ of $L(t_{(k)})$ such that $$\D_\mu\tau_n=0,\quad(\tau_n,\tau^m)_{(k)}=\delta^m_n,\quad
\tau^\dagger_n=-\tau_n,\quad [\tau_n,\tau_l]=c_{nl}^m\tau_m,
\label{tau}$$ where $\tau_n=\tau^n$ and $c_{nl}^m$ are real structure constants of the Lie algebra $L(t_{(k)})$.
We use differential forms $$\begin{aligned}
\lambda_1 &=& Y^1_2 + Y^2_1\\
\lambda_2 &=& -i Y^1_2 + i Y^2_1\\
\lambda_3 &=& Y^1_1 - Y^2_2\\
\lambda_4 &=& Y^1_3 + Y^3_1\\
\lambda_5 &=& -i Y^1_3 + i Y^3_1\\
\lambda_6 &=& Y^2_3 + Y^3_2\\
\lambda_7 &=& -i Y^2_3 + i Y^3_2\\
\lambda_8 &=& \frac{1}{\sqrt3} (Y^1_1 + Y^2_2 - 2 Y^3_3)\end{aligned}$$ such that $\{\underline\lambda_1,\ldots,\underline\lambda_8\}$ is equivalent to the Gell-Mann basis of the real Lie algebra ${\rm su}(3)$. We take the following generators of $L(t_{(k)})$:
1\. For $L(t_{(1)}) \simeq {\rm u}(1)$ $$\tau_0=it_{(1)}.$$ 2. For $L(t_{(2)})\simeq {\rm u}(1)\oplus{\rm su}(2)$ $$\tau_0=it_{(2)},\quad \tau_n=i\lambda_n,\quad
n=1,2,3.$$ 3. For $L(t_{(3)})\simeq {\rm u}(1)\oplus{\rm su}(3)$ $$\tau_0=it_{(3)},\quad \tau_n=i\sqrt{\frac{3}{2}}\lambda_n,\quad
n=1,\ldots,8.$$ 4. For $L(t_{(4)})\simeq {\rm u}(1)\oplus{\rm su}(4)$ we take as a basis $\tau_0,\ldots,\tau_{15}$ the anti-Hermitian basis (\[basis:ah\])
Therefore, $$L(t_{(k)})=\{f_n\tau^n\},$$ where $f_n=f_n(x)$ are real valued scalar functions.
Let us define the set of differential forms $$\underbrace{L(t)}=\{U\in L(t)\,:\,\Tr U=0\}.$$ We see that $$\underbrace{L(t_{(k)})}\simeq{\rm su}(k),\quad k=2,3,4.$$ If we replace $L(t)$ by $\underbrace{L(t)}$ in above considerations, then we get that all results are valid (we must take $J^\mu=i\Psi^\dagger\alpha^\mu\Psi-\Tr(i\Psi^\dagger\alpha^\mu\Psi)$ in Theorem 6 and in eq. (\[J\])).
Special cases
=============
Let us denote $$I=-ie^{12}.$$ Then $$t_{(1)}=\frac{1}{4}(1+H)(1-iI)$$ and $$it_{(1)}=It_{(1)}.$$ 10. For a given $\Phi\in\I(t_{(1)})$ the equation $$\Psi t_{(1)}=\Phi,
\label{one-to-one}$$ has a unique solution $\Psi\in\Lambda_{\even}$.
. We have the orthonormal basis of $\I(t_{(1)})$ $$t_\k=F_\k t_{(1)},\quad {\textsc k}=1,2,3,4,$$ where $F_\k\in\Lambda_\even$. Decomposing $\Phi\in\I(t_{(1)})$ with respect to the basis $t_\k$ $$\Phi=(\alpha^\k+i\beta^\k)t_\k,\quad \alpha^\k,\beta^\k\in\Lambda_0,
\label{Phi}$$ and using the relation $it_{(1)}=It_{(1)}$, we see that the differential form $$\Psi=F_\k(\alpha^\k+I\beta^\k)\in\Lambda_\even$$ is a solution of eq. (\[one-to-one\]).
We claim that if $$U=u+u_{01}e^{01}+u_{02}e^{02}+u_{03}e^{03}+u_{12}e^{12}+u_{13}e^{13}+
u_{23}e^{23}+u_{0123}e^{0123}\in\Lambda_\even$$ is a solution of the homogeneous equation $$Ut_{(1)}=0,$$ then $U=0$. Indeed, the differential form $Ut_{(1)}\in\I(t_{(1)})$ can be decomposed into the basis $t_\k$ $$Ut_{(1)}=\frac{1}{2}(u-iu_{12})t_1+\frac{1}{2}(-u_{13}-iu_{23})t_2+\\
\frac{1}{2}(u_{03}-iu_{0123})t_3+\frac{1}{2}(u_{01}+iu_{02})t_4.$$ Thus the identity $Ut_{(1)}=0$ implies $U=0$. So the solution (\[Phi\]) of eq. (\[one-to-one\]) is unique. These completes the proof.
11\. For a given $\Phi\in\I(t_{(2)})$ the equation $$\Psi t_{(2)}=\Phi,
\label{one-to-one2}$$ has a unique solution $\Psi\in\Lambda^\C_{\even}$.
. We have the orthonormal basis of $\I(t_{(2)})$ $$t_\k=F_\k t_{(2)},\quad {\textsc k}=1,\ldots,8,$$ where $F_\k\in\Lambda^\C_\even$. Decomposing $\Phi\in\I(t_{(2)})$ with respect to the basis $t_\k$ $$\Psi=(\alpha^\k+i\beta^\k)t_\k,\quad \alpha^\k,\beta^\k\in\Lambda_0,
\label{Phi2}$$ we see that the differential form $$\Psi=F_\k(\alpha^\k+i\beta^\k)\in\Lambda^\C_\even$$ is a solution of eq. (\[one-to-one2\]).
We claim that if $$U=(v+iw)+\sum_{0\leq
a<b\leq3}(v_{ab}+iw_{ab})e^{ab}+(v_{0123}+iw_{0123})e^{0123}$$ is a solution of the homogeneous equation $$Ut_{(2)}=0,$$ then $U=0$. Indeed, the differential form $Ut_{(2)}\in\I(t_{(2)})$ can be decomposed into the basis $t_\k$ $$Ut_{(2)}=\phi_\k t^\k,\quad \phi_\k=(t_\k,Ut_{(2)}).$$ We get $$\begin{aligned}
\phi_1&=&(v_{}-iv_{12}+iw_{}+w_{12})/2\\
\phi_2&=&(-v_{13}-iv_{23}-iw_{13}+w_{23})/2\\
\phi_3&=&(v_{03}-iv_{0123}+iw_{03}+w_{0123})/2\\
\phi_4&=&(v_{01}+iv_{02}+iw_{01}-w_{02})/2\\
\phi_5&=&(v_{13}-iv_{23}+iw_{13}+w_{23})/2\\
\phi_6&=&(v_{}+iv_{12}+iw_{}-w_{12})/2\\
\phi_7&=&(v_{01}-iv_{02}+iw_{01}+w_{02})/2\\
\phi_8&=&(-v_{03}-iv_{0123}-iw_{03}+w_{0123})/2.\end{aligned}$$ Evidently the identity $Ut_{(2)}=0$ implies $U=0$. So the solution (\[Phi2\]) of eq. (\[one-to-one2\]) is unique. These completes the proof.
Denote $$\stackrel1 L(t)=\{U\in\Lambda_\even\,:\,U^\dagger=-U, [U,t_{(1)}]=0\}.$$ The Lie algebra $\stackrel1 L(t)$ is isomorphic to the Lie algebra ${\rm
u}(1)$ and as a generator of $\stackrel1 L(t)$ we may take $\tau_0=I$.
12\. Differential forms $\stackrel1\Psi\in\Lambda_\even$, $\stackrel1
A_\mu\in\stackrel1 L(t)\top_1$ satisfy the equation $$dx^\mu(\D_\mu\stackrel1\Psi+\stackrel1\Psi\stackrel1 A_\mu+B_\mu\stackrel1\Psi)H+m\stackrel1\Psi I=0
\label{even:eq}$$ iff differential forms $\Psi=\stackrel1\Psi t_{(1)}\in\I(t_{(1)})$, $A_\mu=\stackrel1 A_\mu t_{(1)}\in L(t)\top_1$ satisfy eq. (\[Dirac:type\]).
. Multiplying (\[even:eq\]) from the right by $t_{(1)}$ and using relations $H t_{(1)}=t_{(1)}$, $I t_{(1)}=it_{(1)}$, we obtain that $\Psi=\stackrel1\Psi t_{(1)}$, $A_\mu=\stackrel1 A_\mu t_{(1)}$ satisfy (\[Dirac:type\]).
Conversely, let $\Psi\in\I(t)$, $A_\mu\in L(t)\top_1$ satisfy (\[Dirac:type\]). By Theorem 10 there exists a unique solution $\stackrel1\Psi\in\Lambda_\even$, $\stackrel1 A_\mu\in\Lambda_\even\top_1$ of the system of equations $\stackrel1\Psi t_{(1)}=\Psi$, $\stackrel1 A_\mu t_{(1)}=A_\mu$. It can be shown that $\stackrel1 A_\mu\in\stackrel1 L(t)\top_1$. Substituting $\stackrel1\Psi t_{(1)}, \stackrel1 A_\mu t_{(1)}$ for $\Psi,A_\mu$ in (\[Dirac:type\]), we arrive at the equality $$(dx^\mu(\D_\mu\stackrel1\Psi+\stackrel1\Psi\stackrel1 A_\mu+B_\mu\stackrel1\Psi)H+m\stackrel1\Psi
I)t_{(1)}=0,$$ which we rewrite as $$\stackrel1\Omega t_{(1)}=0.$$ We see that $\stackrel1\Omega\in\Lambda_\even$ and, according to Theorem 10, we get $$\stackrel1\Omega=0.$$ These completes the proof.
Denote $$\stackrel2 L(t)=\{U\in\Lambda^\C_\even\,:\,U^\dagger=-U, [U,t_{(2)}]=0\}.$$ The Lie algebra $\stackrel2 L(t)$ is isomorphic to the Lie algebra ${\rm
u}(1)\oplus{\rm su}(2)$ and we may take the following generators of $\stackrel2 L(t)$: $$\tau_0=i,\quad
\tau_1=e^{23},\quad
\tau_2=-e^{13},\quad
\tau_3=e^{12}.$$
13\. Differential forms $\stackrel2\Psi\in\Lambda^\C_\even$, $\stackrel2
A_\mu\in\stackrel2 L(t)\top_1$ satisfy the equation $$dx^\mu(\D_\mu\stackrel2\Psi+\stackrel2\Psi\stackrel2 A_\mu+B_\mu\stackrel2\Psi)H+im\stackrel2\Psi=0
\label{even:C}$$ iff differential forms $\Psi=\stackrel2\Psi t_{(2)}\in\I(t_{(2)})$, $A_\mu=\stackrel2 A_\mu t_{(2)}\in L(t)\top_1$ satisfy eq. (\[Dirac:type\]).
is word for word identical to the proof of the preceding theorem.
Equations (\[even:eq\]),(\[even:C\]) together with correspondent Yang-Mills equations were considered in [@nona]-[@nona2].
[99]{} Dirac P.A.M., Proc. Roy. Soc. Lond. A117 (1928) 610.
Marchuk N.G., Nuovo Cimento, 117B, 01, (2002) 95.
Marchuk N.G., Nuovo Cimento, 117B, 05, (2002) 613.
Marchuk N.G., Dirac-type tensor equations on a parallelisable manifold, to appear in Nuovo Cimento B.
Marchuk N.G., Nuovo Cimento, 115B, N.11, (2000) 1267.
Ivanenko D., Landau L., Z. Phys., 48 (1928)340.
Kähler E., Randiconti di Mat. (Roma) ser. 5, 21, (1962) 425.
Riesz M., pp.123-148 in C.R. 10 Congres Math. Scandinaves, Copenhagen, 1946. Jul. Gjellerups Forlag, Copenhagen, 1947. Reprinted in L.G[å]{}rding, L.Hërmander (eds.): [*Marcel Reisz, Collected Papers*]{}, Springer, Berlin, 1988, pp.814-832.
Gürsey F., Nuovo Cimento, 3, (1956) 988.
Hestenes D., [*Space-Time Algebra*]{}, Gordon and Breach, New York, 1966.
Hestenes D., J. Math. Phys., 8, (1967) 798-808.
Benn I.M., Tucker R.W., [*An introduction to spinors and geometry with applications to physics*]{}, Bristol, 1987.
Lounesto P., [*Clifford Algebras and Spinors*]{}, Cambridge Univ. Press (1997, 2001)
Grassmann H., Math. Commun. 12, 375 (1877). Doran C., Hestenes D., Sommen F., Van Acker N., J.Math.Phys. 34(8), (1993) 3642.
[^1]: Research supported by the Russian Foundation for Basic Research grants 00-01-00224, 00-15-96073.
[^2]: In a special case the central product was invented by H. Grassmann [@Grassmann] in 1877 as an attempt to unify the exterior calculus (the Grassmann algebra) with the quaternion calculus. A discussion on that matter see in [@Doran]. In some papers the central product is called a Clifford product.
|
---
abstract: 'In the Aharonov-Albert-Vaidman (AAV) weak measurement, it is assumed that the measuring device or the pointer is in a quantum mechanical pure state. In reality, however, it is often not the case. In this paper, we generalize the AAV weak measurement scheme to include more generalized situations in which the measuring device is in a mixed state. We also report an optical implementation of the weak value measurement in which the incoherent pointer is realized with the pseudo-thermal light. The theoretical and experimental results show that the measuring device under the influence of partial decoherence could still be used for amplified detection of minute physical changes and are applicable for implementing the weak value measurement for massive particles.'
author:
- 'Young-Wook Cho'
- 'Hyang-Tag Lim'
- 'Young-Sik Ra'
- 'Yoon-Ho Kim'
title: Weak value measurement with an incoherent measuring device
---
Introduction
============
The projection postulate of the quantum theory states that the outcome of a measurement on a quantum system must be one of the eigenvalues of the measurement operator. The weak value introduced by Aharonov, Albert, and Vaidman, however, is quite peculiar in that the measurement outcomes of the weak value may lie well outside the normal range of the eigenvalues of the measurement operator [@AAV]. The weak value measurement, nevertheless, does not violate standard quantum theory and the effect is understood to be due to quantum interference of complex amplitudes [@Duck].
The Aharonov-Albert-Vaidman (AAV) weak value measurement is accomplished in two steps: the weak measurement followed by postselection. The postselection step is the standard projection measurement (i.e., strong measurement) but, for the weak measurement, the measuring device or the pointer is assumed to be in a quantum mechanical pure state [@AAV; @Duck]. In the case of quantum mechanical particles with mass whose center-of-mass coordinates are considered as the pointer for the measuring device, it becomes extremely difficult to achieve the measuring device in a pure state because the coupling to the environment causes decoherence of the pointer state as demonstrated, for example, in decoherence of matter waves [@Kleckner; @Arndt; @decoherence] and degradation of an atom laser beam [@atomlaser]. It is thus not surprising that the AAV weak value measurement to date has been implemented only with light whose spatial or temporal coherence can be used to represent the pointer in a pure state [@Ritchie; @Resch; @Solli; @Wang; @Hosten; @lundeen; @Dixon].
In this paper, we generalize the AAV weak value measurement to include more generalized situations in which the measuring device (or the pointer state) is in a mixed state. We also report an optical implementation of the weak value measurement in which the incoherent pointer is realized with the pseudo-thermal light. The theoretical and experimental results suggest that the measuring device under the influence of partial decoherence, i.e., the pointer state with partial coherence (or a density matrix with non-zero off-diagonal elements), could still be used for amplified detection of weak effects.
Theory
======
Weak value measurement with a coherent measuring device
-------------------------------------------------------
We start with a brief description of the AAV weak value measurement [@AAV; @Duck]. The impulse interaction Hamiltonian between the pointer and the system, whose observable $\hat A$ is to be measured, is given in general as $$\hat H = \delta ( t - t_0 )\hat p\hat A, \label{hamiltonian}$$ where $\hat p$ represents the momentum operator for the measuring device (with the conjugate position operator $\hat q$) and $t_0$ is the time of measurement (i.e., interaction). In the AAV weak value measurement [@AAV; @Duck], the system is prepared in a pure state $|\psi_{in}\rangle$. The initial state of the pointer (i.e., the measuring device) is also assumed to be in a pure state (q-representation) as $$\left| {\phi _{{{in}}} } \right\rangle = \left(\frac{2}{\pi w_0^2}\right)^{1/4} \int {dq} \, \exp (-{q^2}/{w_0 ^2} ) | q \rangle, \label{md}$$ where $w_0$ quantifies the pointer spread [@notep].
After the interaction in Eq. (\[hamiltonian\]), the quantum state of both the system and the pointer is evolved to $$\exp{(-i \hat{p} \hat{A} / \hbar)} |\psi_{in}\rangle |\phi_{in}\rangle.$$ If we now make a projection measurement on the system in the $|\psi_f\rangle$ basis (i.e., postselection of the system having the quantum state $|\psi_f\rangle$), the pointer state is found to be [@notep] $$\begin{aligned}
\lefteqn{\langle {\psi _f }| \exp{(-i \hat{p} \hat{A} / \hbar)} |\psi_{in}\rangle |\phi_{in}\rangle} \nonumber \\
&\simeq& \left( {\langle \psi _f | \psi _{in} \rangle - i \hat p \langle {\psi _f } |\hat A| {\psi _{in} } \rangle/\hbar + \ldots } \right)| {\phi _{in} } \rangle \nonumber\\
&\simeq& N \langle {{\psi _f }} | {{\psi _{in} }} \rangle \int{dp} \, \exp \left( - \frac{{w_0 ^2 p^2 + 4 i A_w p}}{{4 \hbar^2}} \right) | {p } \rangle,\label{evolve}\end{aligned}$$ where $N \equiv ({w_0^2}/{2\pi\hbar^2})^{1/4}$ and $A_w$ is the weak value defined as [@AAV; @Duck] $$A_w \equiv \frac{\langle {\psi _f } | \hat A | {\psi _{in} } \rangle } {\langle {{\psi _f }} | {{\psi _{in} }} \rangle }. \label{wv}$$ Note that Eq. (\[evolve\]) has been derived with the assumption $$\max_{n=2,3..} \left| \frac{{\langle \psi_f | \hat{A}^n | \psi_{in} \rangle (p/\hbar)^n}}{{\langle \psi_f |\psi_{in} \rangle}} \right| \ll |p A_w /\hbar| \ll 1.\label{wvlimit}$$
Using Eq. (\[md\]), we can re-write Eq. (\[evolve\]) as $$\left(\frac{2}{\pi w_0^2}\right)^{1/4} \langle {{\psi _f }} | {{\psi _{in} }} \rangle \int {dq} \, \exp\left[ - \frac{(q - A_w )^2}{ w_0 ^2} \right] | q \rangle.\label{wvm}$$ The essence of the weak value measurement is illustrated in Eq. (\[wvm\]): the pointer displays, as an outcome of the measurement, the weak value $A_w$ which may be much larger than any eigenvalues of $\hat A$ if $| {\psi _{in} } \rangle$ and $| {\psi _{f} } \rangle$ are nearly orthogonal to each other.
Although approximations were used to derive Eq. (\[wvm\]), it is in fact possible to calculate the effect of the AAV weak value measurement without any approximation. By expanding $|\psi_{in}\rangle$ and $|\psi_f\rangle$ in the eigenbasis of $\hat A$ as $| {\psi _{in} } \rangle = \sum\nolimits_k {\alpha _k | {a_k } \rangle}$ and $| {\psi _f } \rangle = \sum\nolimits_{l} {\beta_{l} | {a_{l} } \rangle}$, the probability distribution $P_{\psi}(q)$ of the pointer $q$ is explicitly calculated to be $$\begin{aligned}
P_{\psi}( q ) &=& \left| {\langle q |\langle {\psi _f } | \hat U | {\psi _{{in}} } \rangle | {\phi _{{in}} } \rangle } \right|^2 \nonumber \\
&=&\left| \left(\frac{2}{\pi w_0^2}\right)^{1/4} \sum\limits_{k} {\alpha_k} {\beta_k^{*}} \exp \left[ - \frac{(q-a_k )^2}{w_0^2} \right] \right| ^{2} \nonumber\\
&=& \sqrt{\frac{2}{\pi w_0^2}} \sum\limits_{k,j} {\alpha _k } \beta_k^{*} \alpha_j^* \beta_j \nonumber \\
& & \times \exp \left[ - \frac{(q - a_k )^2 +(q - a_j )^2 }{ w_0 ^2 }\right], \label{purecase}\end{aligned}$$ where $\hat U=\exp{(-i \hat{p} \hat{A} / \hbar)}$, $a_j$ and $a_k$ are the eigenvalues of $\hat A$, and we have used the orthonormality condition ${\langle {a _l} | {{a _k }} \rangle }=\delta _{l,k}$. Note that, in the weak value measurement limit shown in Eq. (\[wvlimit\]), $P_\psi(q)$ in Eq. (\[purecase\]) approximates to a single Gaussian peaked at the weak value $A_w$.
Weak value measurement with an incoherent measuring device
----------------------------------------------------------
So far, we have considered the case in which the measuring device is in a pure state, i.e., the pointer spread is completely coherent as shown in Eq. (\[md\]). Let us now generalize the problem by considering that the measuring device (having the same pointer spread $w_0$) is no longer in a pure state, rather in a mixed state with some partial coherence quantified with $w_c$. The pointer state is then expressed as a density matrix $$\begin{aligned}
\rho _\phi &=& \frac{\sqrt{2}}{\pi w_0 w_c}\int dq_0 dq'dq'' \exp \left[ - \frac{{{q_0 ^2 } }} {{w_0 ^2 }} \right] \nonumber \\
& \times & \exp \left[ - \frac{{{(q' - q_0 )^2 } }} {{w_c ^2 }} \right] \exp \left[ - \frac{{{(q'' - q_0 )^2 } }} {{w_c ^2 }} \right] | {q'} \rangle \langle {q''} | ,\end{aligned}$$ and the initial system-pointer quantum state is described as $$| {\psi _{in} } \rangle \rho _\phi \langle {\psi _{in} } |.$$ After the weak measurement, the initial system-pointer density matrix $| {\psi _{in} } \rangle \rho _\phi \langle {\psi _{in} } | $ is evolved due to the interaction Hamiltonian in Eq. (\[hamiltonian\]) into $$\hat U | {\psi _{in} } \rangle \rho _\phi \langle {\psi _{in} } | \hat U^\dag.$$ Making a projection measurement on the system in the $ | {\psi _f } \rangle $ basis (i.e., postselecting the system having the state $ | {\psi _f } \rangle $), the pointer state is found to be $$\begin{aligned}
\rho_f &=& \langle {\psi _f } |\hat U| {\psi _{in} }
\rangle \rho _\phi \langle {\psi _{in} } |\hat U^\dag | {\psi _f } \rangle \nonumber \\
&=&\sum\limits_{k,j,l,m} \beta_l^{*} \alpha _k \alpha _{j}^{*} }{\beta_{m} \langle {a _l } |\hat U| {a _{k} }
\rangle \rho _\phi \langle {a _{j} } |\hat U^\dag | {a _m } \rangle.\end{aligned}$$ The probability distribution for the pointer $P_\rho(q)$ is then calculated as $$\begin{aligned}
P_\rho(q) &=& \langle q | \rho_f | q\rangle \nonumber\\
&=& \frac{\sqrt{2}}{\pi w_0 w_c } \sum\limits_{k,j} \alpha _k \beta_k^{*} \alpha_j^{*} \beta_{j} \int dq_0 \nonumber \\
& & \times \exp\left[-\frac{q_0^2}{w_0^2} -\frac{(q-a_k-q_0)^2}{w_c^2} -\frac{(q-a_j-q_0)^2}{w_c^2}\right] \nonumber \\
&=& \sqrt{\frac{2}{\pi (2w_0^2+w_c^2)}} \sum\limits_{k,j} \alpha _k \beta_k^{*} \alpha_j^{*} \beta_{j} \nonumber \\
& & \times \exp \left[ w_0 ^{-2} \left( - \frac{( q - a_k )^2 } {\gamma^2 } \right.\right. - \frac{( q - a_j )^2 } {\gamma^2}\nonumber \\
& & \left.\left. + \frac{(2q - a_k - a_j)^2 } {\gamma^4 + 2\gamma^2} \right)\right], \label{mixedcase}\end{aligned}$$
where we have used the orthonormality conditions ${\langle {a _l} | {{a _k }} \rangle }=\delta _{l,k}$ and ${\langle {a _j} | {{a _m }} \rangle }=\delta _{j,m}$. Note that the degree of partial coherence is defined as $\gamma \equiv w_c/w_0$.
Equation (\[mixedcase\]) shows that the weak value effect should still be observable even though the measuring device (i.e., the pointer) is in a mixed state whose degree of partial coherence is quantified with $\gamma$. Note that the pointer is effectively in a pure state in the limit $\gamma \gg 1$ and, in this limit, Eq. (\[mixedcase\]) approximates to Eq. (\[purecase\]).
Experiment
==========
The weak value measurement setup which incorporates the pointer in a mixed state is schematically shown in Fig. \[setup\]. The system state is the polarization state of the photon (analogous to a spin 1/2 particle) and is assumed to be in a pure state. The transverse position of the photon corresponds to the pointer (i.e., the measuring device) for measuring the system state [@Duck; @Ritchie].
The incoherent pointer state is realized with the pseudo-thermal light source based on scattering of a focused laser beam (a He-Ne laser operating at 632.8 nm) at a rotating ground disk (RD) [@Martienssen]. The focusing (L1) and collimating (L2) lenses have 30 mm and 75 mm focal lengths, respectively. By moving L1 longitudinally, thereby changing the beam size on the rotating ground disk, the degree of transverse spatial coherence of the collimated beam can be varied. Therefore, we can easily adjust the degree of partial coherence $\gamma$ of the pointer state in Eq. (\[mixedcase\]) by simply moving the focusing lens L1. The collimated beam is then split into two by a beam splitter (BS1); one beam is for the weak value measurement and the other is for characterizing the pointer state.
For the weak value measurement, the photon is prepared in a definite polarization state with polarizer P1. An iris placed after P1 defines the $e^{-2}$ beam waist radius $w'_0=0.697$ mm. The lens L3 ($f= 100$ mm) then focuses the beam so that the beam waist is $w_0=\lambda f/(\pi w'_0)=28.9$ $\mu$m at the focus. The weak measurement on the system (i.e., the polarization state of the photon) is then implemented with a 0.5 mm thick quartz plate Q with its optic axis oriented vertically. The quartz plate Q causes a small polarization dependent displacement between the two orthogonal polarization components due to the birefringence. The expected displacement is much smaller (in our case $a=1.316$ $\mu$m) than the beam width $w_0$ and, therefore, the quartz plate Q acts as the weak measurement operator on the system [@Duck; @Ritchie]. In order to ensure that the incoming polarization of the photon is not changed by Q, i.e, the net phase difference of $2\pi$ between the vertical and horizontal polarization of the photon, Q was tilted (about the optic axis) to $43.5^\circ$.
After the weak measurement by quartz plate Q, the postselection measurement (on polarization) is implemented by the polarizer P2 placed at the focus of L3. Finally, an imaging lens L4 ($f=50$ mm) and a CCD camera are used to measure the transverse spatial profile of the photon at the P2 location.
![Schematic of the experiment. The incoherent pointer state is realized with a pseudo-thermal light source whose transverse spatial coherence can be varied. A He-Ne laser beam is focused on a rotating ground disk (RD) with a movable lens L1 and the scattered light is collimated with another lens L2. BS1 and BS2 are 50/50 beam splitters. The transverse spatial coherence of the beam is measured with detectors D1 and D2 and is used for determining $\gamma$, the degree of partial coherence of the pointer state. P1 and P2 are polarizers for state preparation and postselection, respectively. The weak measurement occurs at the tilted quartz plate Q.[]{data-label="setup"}](fig1.eps){width="3.2in"}
Results and analysis
====================
Characterizing the incoherent measuring device (pointer state)
--------------------------------------------------------------
As mentioned earlier, we can vary the transverse spatial coherence of the collimated beam by changing the beam size on the rotating ground disk and the degree of transverse spatial coherence is directly related to the degree of partial coherence $\gamma$ in Eq. (\[mixedcase\]). Therefore, the first step in experimentally demonstrating the weak value measurement with an incoherent measuring device is to properly and accurately characterize the transverse spatial coherence of the collimated beam.
In the experiment, the light scattered at the rotating ground disk RD is collimated with L2. Beam splitter BS2 then splits the collimated beam: the transmitted beam is used for the weak value measurement and the reflected beam is used for measuring the transverse spatial coherence of the pointer, see Fig. \[setup\]. The degree of transverse spatial coherence of the beam is measured with a Hanbury-Brown$-$Twiss type interferometer, consisting of a 50/50 beam splitter (BS2) and two detectors D1 and D2. Detectors D1 and D2 are multi-mode fiber coupled so that the effective diameter of the detectors is 62.5 $\mu$m, the core diameter of the fiber. The fiber connected to D1 can be scanned and the photocurrents from the detectors D1 and D2 are digitized and stored on a computer. The cross-correlation between the two split modes (of BS2) are then calculated using the AC components of the digitized photocurrents, $\Delta I (t)$, using the relation $$C(x)=\frac{\langle \Delta I_1(x,t) \Delta I_2(t)\rangle_t }{\sqrt{\langle(\Delta I_1(x,t))^2\rangle_t} \sqrt{\langle(\Delta I_2(t))^2\rangle_t}},$$ where the subscripts 1 and 2 refer to detectors D1 and D2, respectively, and $\langle \ldots \rangle_t$ represents time averaging.
![The normalized cross-correlation, $C(x)$, as a function of D1 position, $x$, shows the degree of transverse spatial coherence of the collimated beam. The measured $e^{-2}$ widths $w'_c$ are 1.42 mm, 0.93 mm, 0.60 mm, and 0.28 mm. The $w_c$ values for the weak value measurement can be calculated from $w'_c$ and they are 59.0 $\mu$m, 38.5 $\mu$m, 25.0 $\mu$m, and 11.7 $\mu$m, respectively. The degree of partial coherence $\gamma$ can then be calculated using the relation $\gamma = w_c/w_0$. See text for details.[]{data-label="corr"}](fig2.eps){width="3.3in"}
The experimental results are shown in Fig. \[corr\]. The cross-correlation measurements show that defocusing of L1 causes reduction of the transverse spatial coherence of the collimated beam. The measured $e^{-2}$ widths $w'_c$ are 1.42 mm, 0.93 mm, 0.60 mm, and 0.28 mm, depending on the L1 position. Since the weak value measurement setup uses lens L3 ($f=100$ mm) to focus the beam, see Fig. \[setup\], the measured value $w'_c$ should be converted to the value relevant in the weak value measurement setup using the relation $w_c = w'_c w_0 /w'_0$. As discussed in the previous section, $w'_0 = 0.698$ mm and $w_0 = \lambda f/(\pi w'_0)$ = 28.9 $\mu$m. The degree of partial coherence $\gamma$ in Eq. (\[mixedcase\]) is then calculated using the relation $\gamma = w_c/w_0$. The $\gamma$ values are 2.04 (focusing), 1.33 (1 mm defocusing), 0.865 (2 mm defocusing), and 0.404 (5 mm defocusing).
Weak value measurement with an incoherent measuring device
----------------------------------------------------------
Since the degree of partial coherence $\gamma$ is determined for the measuring device (i.e., the pointer state), we now can proceed to test weak value measurement with an incoherent measuring device. We start by re-writing the general result in Eq. (\[mixedcase\]) using experimentally relevant parameters.
In the experiment, the initial and final polarization states of the photon are assumed linear (P1 and P2 are linear polarizers) so that $| {\psi _{in} } \rangle = \cos \alpha | H \rangle + \sin \alpha | V \rangle$ and $ | {\psi _f } \rangle = \cos \beta | H \rangle + \sin \beta | V \rangle$. Since $|\psi_{in}\rangle$ and $|\psi_f\rangle$ should be almost orthogonal to observe the weak value effect, P1 and P2 angles are set at $\alpha=\pi/4$ and $\beta=-\pi/4+\epsilon$, respectively. Also, the eigenvalues of the observable, corresponding to the expected beam shift for each polarization state, are $a_H = - a$ and $a_V = 0$. Under these conditions, Eq. (\[mixedcase\]) can be re-written as $$\begin{aligned}
P_\rho(q) &\propto& \cos^2 \beta \exp\left[ w_0 ^{-2} \left( - \frac{2(a + q)^2} {\gamma^2} + \frac{4(a + q)^2}{\gamma^4 + 2\gamma^2} \right) \right] \nonumber\\
& &+ \sin 2\beta \exp \left[ w_0 ^{-2} \left( - \frac{a^2 + 2aq + 2q^2 } {\gamma^2} \right.\right. \nonumber \\
&&+ \left.\left.\frac{(a + 2q)^2 } {\gamma^4 + 2\gamma^2 } \right) \right] \nonumber \\
& & + \sin ^2 \beta \exp\left[ w_0 ^{-2} \left( { - \frac{{2q^2 }} {{\gamma^2 }} + \frac{{4q^2 }} {{\gamma^4 + 2\gamma^2 }}} \right) \right]. \label{theory}\end{aligned}$$
![Weak value measurement with an incoherent measuring device. (a) and (c) are experimental data and (b) and (d) are corresponding theoretical results plotted using Eq. (\[theory\]). For (a) and (b), $\epsilon= 1.00 \times 10^{ - 3}$ rad. For (c) and (d), $\epsilon= 2.79\times10^{ - 2}$ rad. All plots are normalized to unity and vertically shifted for clarity. []{data-label="weakvalue"}](fig3.eps){width="3.30in"}
We have performed the weak value measurement with the incoherent pointer for several values of $\gamma$, which characterizes the degree of partial coherence of the pointer (i.e., the measuring device), and for several values of $\epsilon$, which determines the $\langle \psi_f | \psi_{in}\rangle$ value. Note that, since the weak value $A_w$ does not make sense if $\langle \psi_f | \psi_{in}\rangle = 0$, $\epsilon$ should not be zero. (If $\langle \psi_f | \psi_{in}\rangle = 0$, the weak value is not defined by the definition Eq. (\[wv\]), and the assumption Eq. (\[wvlimit\]) cannot be not satisfied.) To result a large weak value $A_w$, however, $\epsilon$ should be close to zero.
The experimental results and corresponding theoretical results are shown in Fig. \[weakvalue\]. In experiment, the peak position of the measured transverse spatial profile of the beam on the CCD represents the weak value $A_w$. The experimental results show that the weak value $A_w$, which is larger than the eigenvalue of the operator $a_H=-a=-1.316$ $\mu$m, is observable even with the pointer (measuring device) in a mixed state if the pointer has some degree of partial coherence, i.e., nonzero off-diagonal elements of the density matrix representing the pointer state. Note also that, the larger the degree of partial coherence $\gamma$, the larger the resulting weak value $A_w$.
The experimental data also show that, if $\epsilon$ is too close to zero ($\epsilon = 1.00 \times 10^{-3}$ rad), the weak value is not well defined, see Fig. \[weakvalue\](a). The spatial profile shows two peaks when the $\gamma$ is large enough, but it reduces to a single Gaussian peak centered nearly at zero when $\gamma$ is much smaller than 1. This clearly is due to the lack of quantum interference.
The weak value effect is more clearly visible for a slightly larger value of $\epsilon$, $\epsilon = 2.7 \times 10^{ - 2}$ rad. As shown in Fig. \[weakvalue\](c), the weak value effect is reduced gradually as $\gamma$ gets smaller. Even for a rather small value $\gamma=0.404$ (an incoherent pointer with small partial coherence), a rather large weak value $A_w=-4.58$ $\mu$m is observed. The experimental results, Figs. \[weakvalue\](a) and \[weakvalue\](c), are in good agreement with the theoretical plots, Figs. \[weakvalue\](b) and \[weakvalue\](d), calculated using Eq. (\[theory\]).
Finally, the weak value effect as a function of $\gamma$ is summarized in Fig. \[amp\], which shows that the amplification (defined as the ratio between the peak position of the spatial profile and the expectation value of the observable $\hat A$, i.e., $\langle \psi_{in} |\hat A|\psi_{in} \rangle = -a/2=-0.658$ $\mu$m) depends heavily on $\gamma$ and $\epsilon$. For $\gamma \gg 1$, the pointer state becomes effectively pure so that the amplification factor is bounded to a specific value. In the intermediate range of $\gamma$, the amplification factor increases as $\gamma$ gets larger. We would like to point out that the weak value amplification can still be observed even for $\gamma < 1$. Of course, if the pointer is completely incoherent (i.e., $\gamma=0$), no weak value effect can occur (i.e. no amplification).
 for $\epsilon = 2.79 \times 10^{-2}$ rad. For each data point, the error bars are smaller than the solid circle. The solid lines are due to the theoretical result in Eq. (\[theory\]). The upper and lower solid lines are for the weak value amplification calculated for $\epsilon = 1.92 \times 10^{-2}$ rad and $\epsilon = 3.67 \times 10^{-2}$ rad, respectively. These $\epsilon$ values correspond to relative angle setting errors of $\pm 0.5^\circ$ between polarizers P1 and P2 [@last]. []{data-label="amp"}](fig4.eps){width="3.3in"}
Conclusion
==========
In conclusion, we have generalized the AAV weak value effect to include the situations in which the measuring device (the pointer) is in a mixed state and have demonstrated the generalized weak value effect in an optical experiment in which the pointer in a mixed state is realized with the pseudo-thermal light source of a varying degree of partial spatial coherence. We have also introduced an experimentally measurable quantity which effectively quantifies the partial coherence of the pointer. Our results show that the pointer state no longer in a pure state but in a mixed state (with some partial coherence) can still exhibit the weak value effect and thus may be used for amplified detection of a very small physical changes. The result reported in this paper should be directly applicable to weak value measurement schemes involving a beam of massive particles whose pointer states (the transverse profile of the beam) cannot be expected to be in a pure state due to the decoherence causing interactions, such as, inter-particle collisions, strong coupling with the environment, etc.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the National Research Foundation of Korea (2009-0070668) and POSTECH BSRI Research Fund. YWC, HTL, and YSR acknowledge partial support from the Korea Research Foundation (KRF-2008-314-C00075).
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The angle setting error of $\pm 0.5^\circ$ comes from the fact that our rotation mounts were graded in $2^\circ$ increment.
|
---
abstract: 'We revisit a two right-handed neutrino model with two texture zeros, namely an indirect model based on $A_4$ with the recently proposed new type of constrained sequential dominance (CSD2), involving vacuum alignments along the $(0,1,-1)^T$ and $(1,0,2)^T$ directions in flavour space, which are proportional to the neutrino Dirac mass matrix columns. In this paper we construct a renormalizable and unified indirect $A_4 \times SU(5)$ model along these lines and show that, with spontaneous CP violation and a suitable vacuum alignment of the phases, the charged lepton corrections lead to a reactor angle in good agreement with results from Daya Bay and RENO. The model predicts a right-angled unitarity triangle in the quark sector and a Dirac CP violating oscillation phase in the lepton sector of $\delta \approx 130^\circ$, while providing a good fit to all quark and lepton masses and mixing angles.'
---
MPP-2013-13\
SISSA 04/2013/FISI
[ **Spontaneous CP violation in $\boldsymbol{A_4 \times SU(5)}$ with Constrained Sequential Dominance 2** ]{}\
Stefan Antusch$^{\star}$ [^1], Stephen F. King$^{\dagger}$ [^2], Martin Spinrath$^{\ddag}$ [^3],\
$^{\star}$ *Department of Physics, University of Basel,*
*Klingelbergstr. 82, CH-4056 Basel, Switzerland*
$^{\star}$ *Max-Planck-Institut für Physik (Werner-Heisenberg-Institut),*
*Föhringer Ring 6, D-80805 München, Germany*
$^{\dagger}$ *School of Physics and Astronomy, University of Southampton,*
*SO17 1BJ Southampton, United Kingdom*
$^{\ddag}$ *SISSA/ISAS and INFN,*
*Via Bonomea 265, I-34136 Trieste, Italy*
Introduction
============
The lepton mixing angles have the distinctive feature that the atmospheric angle $\theta_{23}$ and the solar angle $\theta_{12}$, are both rather large [@pdg]. Direct evidence for the reactor angle $\theta_{13}$ was first provided by T2K, MINOS and Double Chooz [@Abe:2011sj; @Adamson:2011qu; @Abe:2011fz]. Subsequently Daya Bay [@DayaBay], RENO [@RENO], and Double Chooz [@DCt13] Collaborations have measured $\sin^2(2\theta_{13})$: $$\begin{aligned}
\label{t13}
\begin{array}{cc}
\text{Daya Bay: } & \sin^2(2\theta_{13})=0.089\pm0.011 \text{(stat.)}\pm0.005
\text{(syst.)}\ ,\\
\text{RENO: } & \sin^2(2\theta_{13})= 0.113\pm0.013\text{(stat.)}\pm0.019
\text{(syst.)\ ,}\\
\text{Double Chooz: }& \sin^2(2\theta_{13})=0.109\pm0.030\text{(stat.)}\pm0.025
\text{(syst.)}\ .\\
\end{array}\end{aligned}$$
This rules out the hypothesis of exact tri-bimaximal (TB) mixing [@Harrison:2002er], and many alternative proposals have recently been put forward [@flurry], although there are relatively few examples which also include unification [@Antusch:2011qg; @Marzocca:2011dh; @GUTs; @Meroni:2012ty]. For example, an attractive scheme based on trimaximal (TM) mixing remains viable [@Haba:2006dz], sometimes referred to as TM$_2$ mixing since it maintains the second column of the TB mixing matrix and hence preserves the solar mixing angle prediction $\sin\theta_{12} \approx 1/\sqrt{3}$. However there is another variation of TM mixing which also preserves this good solar mixing angle prediction by maintaining the first column of the TB matrix, namely TM$_1$ mixing [@Lam:2006wm].
Although there were models of TM$_2$ mixing which can account for the smallness of the reactor angle [@King:2011zj], the first model in the literature for TM$_1$ mixing, which also fixed the value of the reactor angle, was proposed in [@Antusch:2011ic]. The model discussed in [@Antusch:2011ic] was actually representative of a general strategy for obtaining TM$_1$ mixing using sequential dominance (SD) [@King:1998jw] and vacuum alignment. The strategy of combining SD with vacuum alignment is familiar from the constrained sequential dominance (CSD) approach to TB mixing [@King:2005bj] where a neutrino mass hierarchy is assumed and the dominant and subdominant flavons responsible for the atmospheric and solar neutrino masses are aligned in the directions of the third and second columns of the TB mixing matrix, namely ${\langle}\phi_1^{\nu}{\rangle}\propto (0,1,-1)^T$ and ${\langle}\phi_2^{\nu}{\rangle}\propto (1,1,1)^T$. The new idea was to maintain the usual vacuum alignment for the dominant flavon, ${\langle}\phi_1^{\nu}{\rangle}\propto (0,1,-1)^T$ as in CSD, but to replace the effect of the subdominant flavon vacuum alignment by a different one, namely either ${\langle}\phi_{120}{\rangle}\propto (1,2,0)^T$ or ${\langle}\phi_{102}{\rangle}\propto (1,0,2)^T$, where such alignments may be naturally achieved from the standard ones using orthogonality arguments.
We referred to this new approach as CSD2 [^4] and showed that it leads to TM$_1$ mixing and a reactor angle which, at leading order, is predicted to be proportional to the ratio of the solar to the atmospheric neutrino masses, $\theta_{13} = \frac{\sqrt{2}}{3} \,
\frac{m^\nu_2}{m^\nu_3}$. The model was proposed before the results from Daya Bay and RENO, and the prediction turned out to be rather too small compared to the results in Eq. . More generally it has been shown that any type I seesaw model with two right-handed neutrinos and two texture zeros in the neutrino Yukawa matrix (as in Occam’s razor) is not compatible with the experimental data for the case of a normal neutrino mass hierarchy [@Harigaya:2012bw]. However this conclusion ignores the effect of charged lepton corrections, and so an “Occam’s razor” model which includes such corrections may become viable.
In the present paper we construct a fully renormalisable unified $A_4 \times SU(5)$ model in which the neutrino sector satisfies the CSD2 conditions, and show that, with spontaneous CP violation and a suitable vacuum alignment of the phases, the charged lepton corrections can correct the reactor angle, bringing it into agreement with results from Daya Bay and RENO. We shall use here similar techniques as in [@Antusch:2011sx], where spontaneous CP violation with flavon phases determined by the vacuum alignment was discussed for the first time, in order to ensure that the charged lepton mixing angle correction (typically about $\sim 3^\circ$) adds constructively to the $\theta_{13}^{\nu}$ angle from the neutrino sector (typically about $\sim 5^\circ - 6^\circ$) leading to $\theta_{13} \sim 8^\circ - 9^\circ$, within the range of the measured value from Daya Bay and RENO. In fact the present model is more ambitious, since it describes all quark and lepton masses and mixing angles, including predictions for all the CP violating phases.
We demonstrate the viability of the model by performing a global fit to the charged lepton masses and the quark masses and mixing parameters. For the neutrino mixing angles we make a parameter scan and find very good agreement with the experimental data. We emphasise that the present $A_4 \times SU(5)$ model represents one of the first unified “indirect” family symmetry models in the literature that has been constructed to date that is consistent with all experimental data on quark and lepton mass and mixing parameters where “indirect” simply means that the family symmetry is completely broken by the vacuum alignment.[^5] For a review see [@King:2013eh].
We emphasise that the idea of spontaneous CP violation has a long history [@Branco:2011zb]. However, in explicit flavour models using this idea only the positions of the phases in the mass matrices was predicted, but not the phases of the flavon fields themselves (see e.g. [@Ross:2004qn]). Spontaneous CP violation with calculable flavon phases from vacuum alignment was first discussed in [@Antusch:2011sx] and demonstrated in example models based on $A_4$ and $S_4$. In this paper we shall use a similar approach where the $A_4$ model is formulated in the real $SO(3)$ basis (see e.g. [@King:2006np]) and where we only consider the real representations $\mathbf{1}$ and $\mathbf{3}$. In such a framework, one can either use a “simple” CP symmetry under which the components of the scalar fields transform trivially as $\phi_i \rightarrow \phi_i^{*}$, or a “generalised” CP symmetry which intertwines CP with $A_4$ (see e.g. [@Feruglio:2012cw] and references therein). In the latter case, in our basis, the triplet fields would transform as $\phi_i \rightarrow U_3 \phi_i^{*}$, where $U_3$ interchanges the second and third component. When complex $\mathbf{1}'$ and $\mathbf{1}''$ representations are used in a model, the $U_3$ transformation then takes care of the fact that under CP the two complex singlets are interchanged with each other. However, as already mentioned above, in our model this will make no difference. CP symmetry leads to real coupling constants in a suitable field basis (after “unphysical” phases have been absorbed by field redefinitions). CP is subsequently spontaneously broken by the flavon vacuum alignment, which is controlled by additional Abelian symmetries $\mathbb{Z}_3$ and $\mathbb{Z}_4$, resulting in calculable complex flavon phases as in [@Antusch:2011sx].
The layout of the rest of the paper is as follows: in the next section we discuss the general strategy we will adopt in our model. After a brief review of CSD2 we discuss charged lepton sector corrections to TM$_1$ mixing before we describe the method which we use to fix the flavon vevs. In section \[Sec:Model\] we describe our model, the field content and symmetries and the resulting Yukawa and mass matrices. The justification for the chosen vacuum alignment including phases is given in section \[Sec:Flavon\]. In the subsequent sections we comment on the Higgs mass and then we give the numerical results from our global fit and scans. In section \[Sec:Summary\] we summarize and conclude and in the appendix we define our notations and conventions and give the messenger sector of our model.
The strategy {#Sec:Strategy}
============
Let us now describe our general idea in somewhat more detail, before we present an explicit GUT model example in the next section. As outlined in the introduction, we are combining three ingredients which finally result in a highly predictive unified flavour model. These ingredients are:
- CSD2 for the neutrino mixing angles $\theta_{ij}^\nu$,
- charged lepton mixing contributions as they are typical in GUTs,
- spontaneous CP violation with aligned phases.
We now briefly describe these three concepts and the resulting new class of models.
### CSD2 in the neutrino sector {#csd2-in-the-neutrino-sector .unnumbered}
In models with CSD2 [@Antusch:2011ic], the neutrino mass matrix is dominated by two right-handed neutrinos with mass matrix $M_R={\rm diag}(M_A, M_B)$ and couplings to the lepton doublets $A = (0,a,-a)^T$ and $B = (b,0,2b)^T$ [^6] such that the neutrino Yukawa matrix takes the form $Y_{\nu} = (A, B)$, in left-right convention. A summary of the used conventions is given in the Appendix.
After the seesaw mechanism is implemented, CSD2 leads to the following light effective neutrino Majorana mass matrix: $$M_\nu = m_a \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & -1 & 1 \end{pmatrix} + m_b \begin{pmatrix} 1 & 0 & 2 \\ 0 & 0 & 0 \\ 2 & 0 & 4 \end{pmatrix} = m_a \begin{pmatrix} \epsilon \, \text{e}^{\operatorname{\text{i}}\alpha} & 0 & 2 \epsilon \, \text{e}^{\operatorname{\text{i}}\alpha} \\ 0 & 1 & -1 \\ 2 \epsilon \, \text{e}^{\operatorname{\text{i}}\alpha} & -1 & 1 + 4 \epsilon \, \text{e}^{\operatorname{\text{i}}\alpha} \end{pmatrix},
\label{eq:numassmatrix}$$ where $m_a = \frac{v_u^2 a^2}{M_A}$, $m_b = \frac{v_u^2 b^2}{M_B}$, and where $\alpha$ is the relative phase difference between $m_a$ and $m_b$. We define $\epsilon = |m_b|/|m_a|$, and assume $ \epsilon \ll 1$ leading to a normal mass hierarchy in accordance with SD. As discussed in Appendix \[App:Conventions\] we use here different conventions than in the original CSD2 paper [@Antusch:2011ic] which are more convenient in the context of $SU(5)$ GUTs.
Only three parameters, e.g. $m_a$, $\epsilon$ and $\alpha$, govern the neutrino masses and mixing parameters. For the mixing parameters, the predicted values are, to leading order in $\epsilon$ (from [@Antusch:2011ic] with adapted conventions[^7]): $$\begin{aligned}
s_{23}^\nu &\approx \frac{1}{\sqrt{2}} - \frac{\epsilon}{\sqrt{2}} \cos \alpha \;, & \delta_{13}^\nu &\approx \pi - \alpha + \epsilon \frac{5}{2} \sin \alpha \;, \\
s_{13}^\nu &\approx \frac{\epsilon}{\sqrt{2}} \;, & \alpha_2 &\approx - \alpha + 2 \epsilon \sin \alpha \;, \\
s_{12}^\nu &\approx \frac{1}{\sqrt{3}} \;.\end{aligned}$$ The mixing scheme resulting from CSD2 can be identified as trimaximal mixing of type 1 (i.e. TM$_1$ [@Lam:2006wm]) but with a predicted value of the neutrino 1-3 mixing, $\theta^\nu_{13} = \frac{\sqrt{2}}{3}\frac{m^\nu_2}{m^\nu_3} \sim 5^\circ - 6^\circ$. With neutrino mass $m^\nu_1=0$, only one Majorana CP phase is physical. Without charged lepton corrections, $\delta_{13}^\nu$ would be identical with the leptonic Dirac CP phase $\delta$. Let us also note that CSD2 predicts a deviation of $\theta_{23}^\nu$ from $45^\circ$, depending on the phase $\alpha$.
### Charged lepton mixing contribution in GUTs {#charged-lepton-mixing-contribution-in-guts .unnumbered}
In GUT models the charged lepton Yukawa matrix is generically non-diagonal in the flavour basis, due to the close link between the charged lepton and the down-type quark Yukawa matrices, which typically provides the main origin of the flavour mixing in the quark sector. With the Cabibbo angle $\theta_C$ being the largest mixing in the quark sector, the mixing in $Y_e$ is often dominated by a 1-2 mixing $\theta^e_{12}$ as well, such that the relevant part of (the hierarchical matrix) $Y_e$ can be written as $$Y_e \approx \begin{pmatrix}
0 & c \, \text{e}^{\operatorname{\text{i}}\beta} & 0 \\
* & d & 0 \\
0 & * & * \end{pmatrix} \;,$$ where $c$, $d$, and $\beta$ are real and where the entries marked by a ’$*$’ are not relevant for our discussion here. With $c \ll d$ one can read off to leading order the values for the complex 1-2 mixing angle (for more details see also Appendix \[App:Conventions\]) that are $$\theta_{12}^e \approx \left| \frac{c}{d} \right| \text{ and } \delta^e_{12} = \begin{cases} -\beta & \text{for } c/d >0 \\ - \beta + \pi & \text{for } c/d < 0 \end{cases} \;.$$ Since we will have $c/d < 0$ in our example GUT model in the next section, let us consider this case also in the following discussion.
In explicit GUT models, $\theta^e_{12}$ is typically related to the Cabibbo angle by group theoretical Clebsch factors from GUT breaking, as has been discussed recently, e.g. in [@Antusch:2011qg; @Marzocca:2011dh]. In many GUT models, in particular in those where the muon and the strange quark mass at the GUT scale is predicted by such a Clebsch factor as $m_\mu/m_s = 3$ [@Georgi:1979df], but also if the Yukawa matrices $Y_e$ (and $Y_d$) are (nearly) symmetric with a zero in the (0,0)-element [@Antusch:2011qg], $\theta_{12}^e$ is predicted as $$\theta_{12}^e \approx \frac{ \theta_C}{3} \;.$$ In the example GUT model in the next section we will see explicitly how such a prediction arises in an $SU(5)$ GUT.
The leptonic mixing parameters, defined via $U_{\text{PMNS}} = U_{e} U_{\nu}^\dagger$, are a combination of the mixing from the neutrino and the charged lepton sectors. Making use of the fact that, to leading order, $\theta_{23}^e = \theta_{13}^e = \delta_{12}^\nu = \delta_{23}^\nu = 0$, and using the CSD2 expressions from above for the neutrino sector, and general formulae for the charged lepton mixing contributions of [@Antusch:2005kw; @King:2005bj; @Antusch:2008yc] $$\begin{aligned}
s_{23} \text{e}^{- \operatorname{\text{i}}\delta_{23}} &= s_{23}^\nu \text{e}^{- \operatorname{\text{i}}\delta_{23}^\nu} - \theta_{23}^e c_{23}^\nu \text{e}^{- \operatorname{\text{i}}\delta_{23}^e} \;, \\
s_{13} \text{e}^{- \operatorname{\text{i}}\delta_{13}} &= \theta_{13}^\nu \text{e}^{- \operatorname{\text{i}}\delta_{13}^\nu} - \theta_{12}^e s_{23}^\nu \text{e}^{- \operatorname{\text{i}}(\delta_{23}^\nu + \delta_{12}^e)} \;, \\
s_{12} \text{e}^{- \operatorname{\text{i}}\delta_{12}} &= s_{12}^\nu \text{e}^{- \operatorname{\text{i}}\delta_{12}^\nu} - \theta_{12}^e c_{23}^\nu c_{12}^\nu \text{e}^{- \operatorname{\text{i}}\delta_{12}^e} \;,\end{aligned}$$ we obtain (up to $\mathcal{O}(\epsilon)$) $$\begin{aligned}
\theta_{23} &\approx 45^\circ - \epsilon \cos \alpha \;, \\
\theta_{13} &\approx \frac{\epsilon}{\sqrt{2}} - \cos(\beta - \alpha) \,\frac{\theta_{12}^e}{\sqrt{2}} \;, \\
\theta_{12} &\approx 35.3^\circ + \cos \beta\, \frac{\theta_{12}^e}{\sqrt{2} } \;,\end{aligned}$$ where $\epsilon \approx \frac{2}{3}\frac{m^\nu_2}{m^\nu_3} \approx 8.4^\circ$. When the phases $\alpha$ and $\beta$ are fixed by the vacuum alignment, and when also $\theta_{12}^e$ is predicted from the GUT structure, as both will be the case in our model, all three mixing angles and also the CP phases $\delta$ and $\alpha_2$, are predicted. Thus, the resulting models of this type can be highly predictive.
We would like to note here already that in the explicit GUT model in the next section, we will construct a vacuum alignment such that $\alpha = \pi / 3$, leading to[^8] $$\theta_{23} \approx 45^\circ - \frac{\epsilon}{2} \approx 41^\circ \:,$$ close to the best fit value for the normal hierarchy case from global fits to the neutrino data [@Fogli:2012ua]. The alignment of $\beta$ will satisfy $\beta = \alpha + \pi$, such that the neutrino and charged lepton contributions to $\theta_{13}$ simply add up, leading to (with $\theta_{12}^e = \theta_C/3$) $$\theta_{13} \approx \frac{\epsilon}{\sqrt{2}} + \frac{ \theta_C}{3 \sqrt{2}} \approx 8^\circ - 9^\circ\;,$$ in agreement with the recent measurements. With these values of $\alpha$ and $\beta$, it also turns out that $\theta_{12}$ is predicted somewhat smaller than $35^\circ$, namely $$\theta_{12} \sim 33^\circ \;.$$ This value of $\theta_{12}$ could be distinguished from the tribimaximal value by a future reactor experiment with $\sim 60$ km baseline [@Minakata:2004jt].
### Spontaneous CP violation with aligned phases {#spontaneous-cp-violation-with-aligned-phases .unnumbered}
Finally, the third ingredient is spontaneous CP violation with aligned phases of the flavon vevs, using the method proposed in [@Antusch:2011sx]. To give a brief summary of this method, let us note that phase alignment can very simply be achieved using discrete symmetries when the flavon vevs effectively depend on one parameter, i.e. when the direction of the vevs is given by the form of the potential. This remains true even in the presence of “generalised” CP transformations as long as these CP transformations fix the phases of the involved coupling constants. Working example models with $A_4$ and $S_4$ family symmetry can be found in [@Antusch:2011sx]. Note that $S_4$ is in agreement only with “simple” CP, while the “generalised” CP transformation for $A_4$ interchanges the complex singlet representations [@Feruglio:2012cw]. In both cases all the coupling constants are forced to be real in a suitable field basis.
To illustrate the phase alignment, let us consider a case with a flavon field $\xi$ which is a singlet under the family symmetry and singly charged under a $\mathbb{Z}_n$ shaping symmetry (with $n\ge2$). Then typical terms in the flavon superpotential, which “drive” the flavon vev non-zero, have the form $$\label{eq:flavonpotentialZn}
P \left( \frac{\xi^n}{\Lambda^{n-2}} \mp M^2 \right).$$ The field $P$ is the so-called “driving superfield”, meaning that the $F$-term $|F_P|^2$ generates the potential for $\xi$ which enforces a non-zero vev. $\Lambda$ is the (real and positive) suppression scale of the effective operator, and $M$ here is simply a (real) mass scale. From the potential for $\xi$, $$|F_P|^2 = \left| \frac{\xi^n}{\Lambda^{n-2}} \mp M^2 \right|^2 ,$$ the vev of $\xi$ has to satisfy $$\label{eq:flavonvevsdiscrete}
\xi^n = \pm \,\Lambda^{n-2} M^2\;.$$ Since the right side of the equation is real, we obtain that $$\label{eq:phaseswithZn}
\arg(\langle \xi \rangle) = \left\{ \begin{array}{ll}
\frac{2 \pi}{n}q \;,\quad q = 1, \dots , n & \mbox{\vphantom{$\frac{f}{f}$} for ``$-$'' in Eq.~(\ref{eq:flavonpotentialZn}),}\\
\frac{2 \pi}{n} q +\frac{\pi}{n} \;,\quad q = 1, \dots , n & \mbox{\vphantom{$\frac{f}{f}$} for ``$+$'' in Eq.~(\ref{eq:flavonpotentialZn}).}
\end{array}
\right.$$ For example, with a $\mathbb{Z}_3$ shaping symmetry and a “$+$” in Eq. (\[eq:flavonpotentialZn\]), only multiples of $\pi/3$ are allowed for $\arg( \langle \xi \rangle )$. We will use this method for the relevant flavons to constrain their phases. In the ground state, one of the vacua (with a fixed phase) is selected, which finally determines also the two phases $\alpha$ and $\beta$ relevant for the predictions in the lepton sector.
Furthermore, we note that we will also use the phase alignment to generate the CP violation in the quark sector, predicting a right-angled unitarity triangle, which is in excellent agreement with the present data (making use of the quark phase sum rule from [@Antusch:2009hq]).
We now turn to an explicit GUT model, where the above described strategy is applied.
The model {#Sec:Model}
=========
In the following we will construct an $A_4 \times SU(5)$ model with CSD2 [@Antusch:2011ic] in the neutrino sector. The model follows the strategy described in the previous section, such that the charged lepton mixing contribution to $\theta_{13}$ adds up constructively with the 1-3 mixing in the neutrino sector to $\theta_{13} \sim 8^\circ - 9^\circ$, with the phases fixed by the “discrete vacuum alignment” mechanism [@Antusch:2011sx].
$T_1$ $T_2$ $T_3$ $F$ $N_1$ $N_2$ $H_5$ $\bar H_5$ $H_{45}$ $\bar H_{45}$ $H_{24}$ $S$
---------------- --------------- --------------- --------------- ------------------- -------------- -------------- -------------- ------------------- --------------- -------------------------- --------------- --------------
$SU(5)$ $\mathbf{10}$ $\mathbf{10}$ $\mathbf{10}$ $\mathbf{\bar 5}$ $\mathbf{1}$ $\mathbf{1}$ $\mathbf{5}$ $\mathbf{\bar 5}$ $\mathbf{45}$ $\mathbf{\overline{45}}$ $\mathbf{24}$ $\mathbf{1}$
$A_4$ $\mathbf{1}$ $\mathbf{1}$ $\mathbf{1}$ $\mathbf{3}$ $\mathbf{1}$ $\mathbf{1}$ $\mathbf{1}$ $\mathbf{1}$ $\mathbf{1}$ $\mathbf{1}$ $\mathbf{1}$ $\mathbf{1}$
$U(1)_R$ 1 1 1 1 1 1 0 0 0 0 0 2
$\mathbb{Z}_4$ 3 3 3 0 0 2 2 0 0 2 1 2
$\mathbb{Z}_4$ 3 3 3 0 2 2 2 0 2 0 1 2
$\mathbb{Z}_3$ 1 2 0 0 1 2 0 0 0 0 2 0
$\mathbb{Z}_3$ 1 1 0 0 2 0 0 0 1 2 2 0
$\mathbb{Z}_3$ 0 2 2 1 0 2 2 0 1 1 2 1
$\mathbb{Z}_3$ 0 0 0 2 0 0 0 0 1 2 1 0
: \[Tab:Matter+HiggsFields\] The matter and Higgs fields in our model and their quantum numbers.
$SU(5)$ $A_4$ $U(1)_R$ $\mathbb{Z}_4$ $\mathbb{Z}_4$ $\mathbb{Z}_3$ $\mathbb{Z}_3$ $\mathbb{Z}_3$ $\mathbb{Z}_3$
---------------------- -------------- -------------- ---------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- -- -- -- -- --
$\phi_{102} $ $\mathbf{1}$ $\mathbf{3}$ 0 0 0 1 0 1 1
$\phi_{23} $ $\mathbf{1}$ $\mathbf{3}$ 0 2 0 2 1 0 1
$\phi_{1} $ $\mathbf{1}$ $\mathbf{3}$ 0 1 3 1 0 0 1
$\phi_{2} $ $\mathbf{1}$ $\mathbf{3}$ 0 0 3 0 0 0 0
$\phi_{3} $ $\mathbf{1}$ $\mathbf{3}$ 0 1 1 0 0 0 1
$\phi_{111} $ $\mathbf{1}$ $\mathbf{3}$ 0 3 3 0 0 0 0
$\phi_{211} $ $\mathbf{1}$ $\mathbf{3}$ 0 0 0 2 1 1 0
$\xi_{u} $ $\mathbf{1}$ $\mathbf{1}$ 0 0 0 0 2 1 0
$\xi_{1} $ $\mathbf{1}$ $\mathbf{1}$ 0 0 0 1 2 0 0
$\xi_{2} $ $\mathbf{1}$ $\mathbf{1}$ 0 0 0 2 0 2 0
$\theta_{2} $ $\mathbf{1}$ $\mathbf{1}$ 0 0 1 0 0 0 0
$\theta_{102} $ $\mathbf{1}$ $\mathbf{1}$ 0 0 0 1 0 0 2
$\rho_{111} $ $\mathbf{1}$ $\mathbf{1}$ 0 3 3 0 0 0 0
$\tilde \rho_{111} $ $\mathbf{1}$ $\mathbf{1}$ 0 3 3 0 0 0 0
$\rho_{23} $ $\mathbf{1}$ $\mathbf{1}$ 0 0 0 2 1 0 1
$\rho_{102} $ $\mathbf{1}$ $\mathbf{1}$ 0 0 0 1 0 1 1
: \[Tab:FlavonFields\] The flavon field content of our model.
The matter and the Higgs sector of the model is summarised in Table \[Tab:Matter+HiggsFields\] while the required flavons are shown in Table \[Tab:FlavonFields\]. The superpotential after integrating out the heavy messenger fields, see Appendix \[App:Messenger\], and suppressing order one coefficients reads $$\begin{aligned}
\mathcal{W}_N &= \xi_1 N_1^2 + \xi_2 N_2^2 \;,\\
\mathcal{W}_\nu &= \frac{1}{\Lambda} (H_5 F) (\phi_{23} N_1) + \frac{1}{\Lambda} (H_5 F) (\phi_{102} N_2) \;,\\
\mathcal{W}_d &= \frac{1}{\Lambda^3} \theta_2 \bar H_5 F (T_1 \phi_2) H_{24} + \frac{1}{\Lambda^3} \theta_{102} \bar H_5 F (T_2 \phi_{102}) H_{24}
+ \frac{1}{\Lambda^2} F (T_2 \phi_{23}) \bar H_{45} H_{24} + \frac{1}{\Lambda} \bar H_5 F (T_3 \phi_3) \;,\\
\mathcal{W}_u &= \frac{1}{\Lambda^2} T_1^2 H_5 \xi_u \xi_1 + \frac{1}{\Lambda^2} T_1 T_2 H_5 \xi_u^2 + \frac{1}{\Lambda^2} T_2^2 H_5 \xi_1^2 + \frac{1}{\Lambda} T_2 T_3 H_5 \xi_1 + T_3^2 H_5 \;,\end{aligned}$$ where $\Lambda$ denotes the messenger scale. The flavon potential, which gives rise to the vevs of the fields $\phi_i$, $\xi_i$ and $\theta_i$ will be discussed separately in the next section. Note that the flavons of type $\phi$ which enter the Yukawa couplings will be aligned with real vevs while the flavons of type $\theta$ and $\xi$ will generally acquire complex vevs with precisely determined phases. The above superpotential gives rise to the flavour structures in the neutrino sector, in the down-type quark and charged lepton sectors, and in the up-type quark sector.
[**Neutrino sector:**]{} From the flavon potential, to be discussed in the next section, the two triplet flavons entering the neutrino Yukawa sector are aligned along the directions $$\begin{aligned}
\label{eq:FlavonDirectionsNu}
{\langle}\phi_{23} {\rangle}\sim \begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}, \ \ \ \
{\langle}\phi_{102} {\rangle}\sim \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} \;, \quad
\end{aligned}$$ where both alignments are real. Inserting the above vacuum alignments, the real vev ${\langle}\xi_1 {\rangle}$ and the vev ${\langle}\xi_2 {\rangle}$ with a phase of $-\pi / 3$ into the superpotential leads to a Dirac Yukawa matrix and a right-handed heavy Majorana mass matrix of the form: $$Y_\nu = \begin{pmatrix}
0 & b \\
a & 0 \\
-a & 2 b
\end{pmatrix} \text{ and } \;\;
M_R = \begin{pmatrix}
M_A & 0 \\
0 & M_B
\end{pmatrix} .$$ where $M_A$, $a$ and $b$ are real and $M_B$ has a complex phase of $- \pi/3$. The (type-I) seesaw formula leads to a simple effective light neutrino mass matrix of the form given in eq. where the relative phase difference $\alpha$ between $m_a$ and $m_b$ is now fixed to be $\pi/3$. This form of $M_\nu$ gives $\theta^\nu_{13} \sim 5^\circ - 6^\circ$ for the 1-3 mixing in the neutrino sector, which will finally add up with the charged lepton mixing contribution.
[**Down-type quark and charged lepton sector:**]{} Turning to the down quark and charged lepton sector, two further triplet flavons enter: $$\begin{aligned}
\label{eq:FlavonDirectionsDown}
{\langle}\phi_{2} {\rangle}&\sim \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \;, \quad
{\langle}\phi_{3} {\rangle}\sim \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \;,
\end{aligned}$$ where ${\langle}\phi_{2} {\rangle}$ is aligned to be real. The phase of ${\langle}\phi_{3} {\rangle}$ will turn out to be unphysical. Furthermore the singlet $\theta_2$ with a phase of $\pi/2$ and the singlet $\theta_{102}$ with a phase of $4 \pi/3$ enters. Plugging in the vevs of the flavon fields leads to the following structure of the Yukawa matrices (in left-right convention) for the down-type quarks and charged leptons: $$\begin{aligned}
\label{eq:YdYe}
Y_d = \begin{pmatrix}
0 & \operatorname{\text{i}}\epsilon_2 & 0 \\
\bar\omega \epsilon_{102} & \epsilon_{23} & 2 \bar\omega \epsilon_{102} - \epsilon_{23} \\
0 & 0 & \epsilon_3
\end{pmatrix} \text{ and }
Y_e = \begin{pmatrix}
0 & -3/2 \bar\omega \epsilon_{102} & 0 \\
- 3/2 \operatorname{\text{i}}\epsilon_2 & 9/2 \, \epsilon_{23} & 0 \\
0 & \left(-3 \bar\omega \epsilon_{102} - 9/2 \, \epsilon_{23} \right) & \epsilon_3
\end{pmatrix} \;,\end{aligned}$$ where $\bar\omega = \text{e}^{4 \pi \operatorname{\text{i}}/3}$, cf. section \[Sec:Strategy\]. The $\epsilon_i$ are proportional to the order one couplings which we have not written down explicitly and possible Higgs mixing angles. For the sake of simplicity we only show here the proportionality to the dimensionful quantities $$\begin{aligned}
\epsilon_2 \sim \frac{v_{24}}{\Lambda^3} |{\langle}\theta_2 {\rangle}{\langle}\phi_2 {\rangle}| \text{, }
\epsilon_{102} \sim \frac{v_{24}}{\Lambda^3} |{\langle}\theta_{102} {\rangle}{\langle}\phi_{102} {\rangle}| \text{, }
\epsilon_{23} \sim \frac{v_{24}}{\Lambda^2} |{\langle}\phi_{23} {\rangle}| \text{, }
\epsilon_{3} \sim \frac{1}{\Lambda} |{\langle}\phi_{3} {\rangle}| \text{,}\end{aligned}$$ where $v_{24}$ is the vev of $H_{24}$. We also note that we do not use the common Georgi-Jarlskog relation $m_\mu / m_s = 3$ [@Georgi:1979df] at the GUT scale but rather $m_\mu / m_s = 9/2$ [@Antusch:2009gu; @Antusch:2008tf]. The reason for this is that recent lattice results, see, e.g.[@Juttner:2011jg] suggest a much smaller error for the strange quark mass than the PDG quotes. And since we are in the small $\tan \beta$ regime and no large SUSY threshold corrections can correct the second generation GUT scale Yukawa coupling ratios we have to use the more realistic relation mentioned above. Explicitly, from the vevs of $H_{24}$ and $\bar H_5$ we get a relative factor of $-3/2$ for $\epsilon_2$ and $\epsilon_{102}$ and the $9/2$ from $H_{24}$ and $\bar H_{45}$. For the third generation we use $b-\tau$ Yukawa unification which is possible for small $\tan \beta$ due to the large RGE effects induced by the top mass.
For the 1-2 mixing in the charged lepton sector, we nevertheless obain $\theta_{12}^e \approx \theta_C/3$, where $\theta_C
\approx 0.23$ is the Cabibbo angle. The corresponding phase $\delta_{12}^e$ is chosen (see section \[Sec:Strategy\] and appendix \[App:Conventions\] for conventions), such that the charged lepton mixing angle correction $\theta_{12}^e$ is in phase with the neutrino reactor angle $\theta_{13}^{\nu}$ and the two angles add together constructively to yield the physical reactor angle $\theta_{13}$.
[**Up-type quark sector:**]{} Finally the up-type quark sector only involves singlet flavons with real vevs and gives a real symmetric Yukawa matrix of the form, $$Y_u = \begin{pmatrix}
a_u & b_u & 0 \\
b_u & c_u & d_u \\
0 & d_u & e_u
\end{pmatrix} \;,$$ where the dependence on $\Lambda$ and the flavon vevs reads $$a_u \sim \frac{|{\langle}\xi_u {\rangle}{\langle}\xi_1 {\rangle}|}{\Lambda^2} \text{, }
b_u \sim \frac{|{\langle}\xi_u {\rangle}|^2}{\Lambda^2} \text{, }
c_u \sim \frac{|{\langle}\xi_1 {\rangle}|^2}{\Lambda^2} \text{, }
d_u \sim \frac{|{\langle}\xi_u {\rangle}|}{\Lambda} \text{.}$$ Note that $e_u$ is coming from a renormalisable coupling and we have not explicitly written down all coefficients. For instance, $\Lambda$ is only a simplified notation for the various messenger masses as given in Appendix \[App:Messenger\], and hence $a_u^2 \ll |b_u c_u|$ as in our numerical fit in Section \[sec:fit\] is possible. The zero texture in the quark sector means that we can successfully apply the quark phase sum rule of [@Antusch:2009hq] due to our choice of phases.
The vacuum alignment {#Sec:Flavon}
====================
We have in total seven flavon fields which transform as triplets under $A_4$, see Table \[Tab:FlavonFields\], pointing in the following directions in flavour space, $$\begin{aligned}
\label{eq:FlavonDirections}
{\langle}\phi_{1} {\rangle}\sim \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \;, \quad
{\langle}\phi_{2} {\rangle}&\sim \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \;, \quad
{\langle}\phi_{3} {\rangle}\sim \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \;, \quad \\
{\langle}\phi_{211} {\rangle}\sim \begin{pmatrix} -2 \\ 1 \\ 1 \end{pmatrix} \;, \quad
{\langle}\phi_{111} {\rangle}& \sim \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \;, \quad
{\langle}\phi_{23} {\rangle}\sim \begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix} \;.
\end{aligned}$$ Apart from ${\langle}\phi_{1} {\rangle}$ and ${\langle}\phi_{3} {\rangle}$, the vevs of the above listed flavons will be aligned real using the phase alignment mechanism proposed in [@Antusch:2011sx]. The phases of ${\langle}\phi_{1} {\rangle}$ and ${\langle}\phi_{3} {\rangle}$ have no physical implications and hence will be set real for definiteness. The first three vevs form a basis in flavour space, while the second three alignments are proportional to the (real) columns of the tri-bimaximal mixing matrix. In our model, instead of $\phi_{111}$ (which is used in the CSD [@King:1998jw; @King:2005bj] models), we require the following (real) alignment, $$\begin{aligned}
{\langle}\phi_{102} {\rangle}\sim \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} \;,
\end{aligned}$$ in the neutrino sector, similar to a recently proposed flavon alignment [@Antusch:2011ic] but with the phase fixed as explicitly shown and discussed below.
The principal assumption of our model is that CP is conserved above the flavour breaking scale, and is spontaneously broken by the CP violating phases of flavon fields. With this assumption we can not only reproduce the correct mixing angles but can also make definite testable predictions for the CP violating phases in the lepton sector. In order to do this we will fix the phases of the following flavon vevs to $$\begin{aligned}
\label{Eq:Phases}
\alpha_{111} = 0 \;, \quad
\alpha_{211} = 0 \;, \quad
\alpha_{23} = 0 \;, \quad
\alpha_{2} = 0 \;, \quad
\alpha_{102} = 0 \;, \end{aligned}$$ where $\alpha_i$ stands for the phase of ${\langle}\phi_i {\rangle}$. Furthermore we have some singlet flavons with non-vanishing vevs of which some will have non-trivial phases. In this choice we have also ignored possible signs which means that the phases are fixed up to $\pm \pi$. We can fix the phases by using appropriate $\mathbb{Z}_n$ shaping symmetries as described in our previous paper [@Antusch:2011sx], see also section \[Sec:Strategy\].
$SU(5)$ $A_4$ $U(1)_R$ $\mathbb{Z}_4$ $\mathbb{Z}_4$ $\mathbb{Z}_3$ $\mathbb{Z}_3$ $\mathbb{Z}_3$ $\mathbb{Z}_3$
---------------- -------------- -------------- ---------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- -- -- -- -- --
$O_{1;2} $ $\mathbf{1}$ $\mathbf{1}$ 2 3 2 2 0 0 2
$O_{1;3} $ $\mathbf{1}$ $\mathbf{1}$ 2 2 0 2 0 0 1
$O_{2;3} $ $\mathbf{1}$ $\mathbf{1}$ 2 3 0 0 0 0 2
$O_{111;211} $ $\mathbf{1}$ $\mathbf{1}$ 2 1 1 1 2 2 0
$O_{111;23} $ $\mathbf{1}$ $\mathbf{1}$ 2 3 1 1 2 0 2
$O_{23;211} $ $\mathbf{1}$ $\mathbf{1}$ 2 2 0 2 1 2 2
$O_{2;102} $ $\mathbf{1}$ $\mathbf{1}$ 2 0 1 2 0 2 2
$O_{211;102} $ $\mathbf{1}$ $\mathbf{1}$ 2 0 0 0 2 1 2
$O_{1;23} $ $\mathbf{1}$ $\mathbf{1}$ 2 1 1 0 2 0 1
$A_{1} $ $\mathbf{1}$ $\mathbf{3}$ 2 2 2 1 0 0 1
$A_{2} $ $\mathbf{1}$ $\mathbf{3}$ 2 0 2 0 0 0 0
$A_{3} $ $\mathbf{1}$ $\mathbf{3}$ 2 2 2 0 0 0 1
$A_{111} $ $\mathbf{1}$ $\mathbf{3}$ 2 2 2 0 0 0 0
$P$ $\mathbf{1}$ $\mathbf{1}$ 2 0 0 0 0 0 0
: \[Tab:DrivingFields\] The driving field content of our model. Note that we only show here one $P$ field. Indeed one has to introduce as many $P$ fields as operators to fix the phases of the flavon fields. Since they will have all the same quantum numbers they will mix and we can go to a basis where the terms to fix the phase for each flavon is separated from the others. This was discussed in the appendix of [@Antusch:2011sx].
The method can be understood easily for the $A_4$ singlet flavon vevs. Their superpotential reads $$\begin{aligned}
\mathcal{W} &= \frac{P}{\Lambda} (\xi_1^3 - M^3) + \frac{P}{\Lambda} (\xi_2^3 + M^3) + \frac{P}{\Lambda} (\xi_u^3 - M^3) \nonumber\\
&+ \frac{P}{\Lambda^2} (\theta_2^4 - M^4) + \frac{P}{\Lambda} (\theta_{102}^3 - M^3) + \frac{P}{\Lambda} (\rho_{102}^3 - M^3) + \frac{P}{\Lambda} (\rho_{23}^3 - M^3) \;,\end{aligned}$$ where $M$ is a generic mass scale which we assume to be positive. The list of the driving fields is given in Table \[Tab:DrivingFields\]. The $F$ terms for $P$ will then fix the flavon vevs of the singlets up to a discrete choice. Note that for the sake of simplicity we have only introduced one $P$ field. Indeed, we need one $P$ field for every singlet. Since they all have the same quantum numbers they will mix and we can go to a basis where all terms are disentangled as in the equation above, see the appendix of [@Antusch:2011sx]. For the singlet flavons here we choose ${\langle}\xi_{1,u} {\rangle}$ and ${\langle}\rho_{102,23} {\rangle}$ to be real, ${\langle}\theta_{2} {\rangle}$ to be imaginary, ${\langle}\theta_{102} {\rangle}$ to have a phase of $4 \pi/3$ and ${\langle}\xi_2 {\rangle}$ to have a phase of $- \pi/3$.
We come now back to the phases of the triplet flavon vevs which can be fixed in the same way after the direction in flavour space is fixed. Note that the phases $\alpha_1$ and $\alpha_3$ are not fixed in our model. This is also not necessary. The flavon $\phi_1$ does not couple to the matter sector and hence its phase does not appear in the mass matrices. It will only be used in orthogonality relations where the phase of the vev does not matter. The flavon $\phi_3$ couples nevertheless to the matter sector. But as we have seen before it determines the 3-3 element of the down-type quark and charged lepton Yukawa matrix and its phase can be absorbed in the right-handed fields such that this phase renders unphysical. In this section we will use an explicit notation for the contraction of the $A_4$ indices. We use the standard “$SO(3)$ basis” for which the singlet of ${\bf 3 \otimes 3}$ is given by the $SO(3)$-type inner product ’$\cdot$’. The two triplets of ${\bf 3 \otimes 3}$ are constructed from the usual (antisymmetric) cross product ’$\times$’ and the symmetric star product ’$\star$’ (see, for example, [@King:2006np]).
We start with the alignment of the triplet flavons $\phi_i$, $i=1,2,3$, which can be aligned via $$\mathcal{W} = A_i \cdot (\phi_i \star \phi_i) + O_{i;j} (\phi_i \cdot \phi_j) + \frac{P}{\Lambda^2} \left( (\phi_2 \cdot \phi_2)^2 - M^4 \right) \;.$$ Solving the $F$-term conditions of $A_i$ aligns the flavons in one of the three standard directions and the $F$-term conditions of $O_{i;j}$ makes them orthogonal to each other. By convention we let them point in the directions as given in eq. . For $\alpha_2$ we choose the value $0$ ($\alpha_1$ and $\alpha_3$ remain undetermined). In Appendix \[App:Messenger\] we will discuss the messenger sector of our model. After integrating out heavy messenger fields we end up only with the effective operators written here and in the following.
We now turn to the flavons $\phi_{23}$, $\phi_{111}$ and $\phi_{211}$: For $\phi_{111}$ we use a slight modification of the alignment in the recent $SU(5) \times T^{\prime}$ model [@Meroni:2012ty] without auxiliary flavons, $$\begin{aligned}
\mathcal{W} &=
A_{111} \cdot \left( \phi_{111} \star \phi_{111} + \phi_{111} \rho_{111}+ \phi_{111} \tilde\rho_{111}\right)
+ \frac{P}{\Lambda^2} \left( (\phi_{111} \cdot \phi_{111})^2 - M^4 \right) \nonumber\\
& + \frac{P}{\Lambda^2} \left(\rho_{111}^4 + \rho_{111}^2 \tilde \rho_{111}^2 + \tilde \rho_{111}^4 - M^4 \right) \;.\end{aligned}$$ It gives the desired alignment and ${\langle}\phi_{111} {\rangle}$ can be chosen to be real.
Starting from this the other two alignments can be realised by $$\begin{split}
\mathcal{W} &= O_{1;23} (\phi_1 \cdot \phi_{23}) + O_{111;23} (\phi_{111} \cdot \phi_{23})
+ O_{111;211} (\phi_{111} \cdot \phi_{211} ) + O_{23;211} (\phi_{23} \cdot \phi_{211} ) \\
&\quad + \frac{P}{\Lambda} \left( ( \phi_{211} \star \phi_{211}) \cdot \phi_{211} - M^3 \right) + \frac{P}{\Lambda} \left( (\phi_{23} \cdot \phi_{23} ) \rho_{23} - M^3 \right) \;.
\end{split}$$ The orthogonality gives the desired directions and ${\langle}\phi_{211} {\rangle}$ can be chosen to be real. The phase of ${\langle}\phi_{23} {\rangle}$ is a bit peculiar. Above we have fixed ${\langle}\rho_{23} {\rangle}$ to be real and hence also ${\langle}\phi_{23} {\rangle}$ can be chosen to be real. In the first operator the vev of $\phi_1$ enters again and independent of the phases a $(0,1,-1)$ alignment is always orthogonal to a $(1,0,0)$ alignment.
Now we have everything together for the last missing non-trivial alignment $$\mathcal{W} = O_{211;102} (\phi_{102} \cdot \phi_{211}) + O_{2;102} (\phi_{102} \cdot \phi_2) + \frac{P}{\Lambda} \left( (\phi_{102} \cdot \phi_{102}) \rho_{102} - M^3 \right) \;.$$ The direction is again fixed by orthogonality conditions. The vev of $\phi_{102}$ can be chosen to be real (remember that also ${\langle}\rho_{102} {\rangle}$ is real).
The Higgs mass
==============
In our model we assume $b-\tau$ Yukawa coupling unification at the GUT scale. This happens in the MSSM only for large $\tan \beta$ via SUSY threshold corrections or small $\tan
\beta$ due to large RGE corrections by the top mass. We have decided for the second solution such that we can also neglect SUSY threshold corrections in our fit later on.
Nevertheless, the MSSM with small $\tan \beta$ prefers very light Higgs masses which is in conflict with the recent discovery of a Higgs-like particle with a mass of about 126 GeV [@Higgs].
A possible solution to this problem is given by the NMSSM, for a review see [@Ellwanger:2009dp] where the Higgs can have the right mass even for small $\tan \beta$. In fact our symmetries forbid a $\mu$-term because the combinations $H_5 \bar H_5$ and $H_{45} \bar H_{45}$ are charged under the shaping symmetries. But we have checked that we can add a singlet field $S$ which couples simultaneously to this two combinations. For convenience we have listed the field $S$ in Table \[Tab:Matter+HiggsFields\].
An explicit $S^3$ term in the superpotential is forbidden in the limit of unbroken $U(1)_R$ symmetry (i.e. before SUSY breaking) and by the shaping symmetries but is needed to stabilize the Higgs potential in the scale invariant NMSSM. But we note that there are still various possibilities to stabilize the potential for $S$. This could be done, for instance, by introducing an additional $U(1)'$ gauge group where the potential is stabilized by the $U(1)'$ $D$-terms. For a description of this and references, see the review article [@Ellwanger:2009dp]. We only note that it is straightforward to introduce such a $U(1)'$ in our model by charging the Higgs and matter fields appropriately which does not alter the flavour sector. Alternatively, the $S^3$ term could be generated non-perturbatively, breaking the shaping symmetries in an $F$-theory framework, see, for instance, [@Callaghan:2012rv]. We will not go here into more detail on this model building aspect and only like to note that our flavour model is compatible with some NMSSM variants and hence we can have a realistic Higgs mass.
The fit and numerical results {#sec:fit}
=============================
Here we will present the results of a numerical $\chi^2$-fit of the high energy parameters of the Yukawa matrices to the low energy charged lepton and quark masses and quark mixing parameters. Afterwards we will present the predictions for neutrino masses and mixing.
![Pictorial representation of the deviation of our predictions from low energy experimental data for the charged lepton Yukawa couplings and quark Yukawa couplings and mixing parameters. The deviations of the charged lepton masses are given in 1% while all other deviations are given in units of standard deviations $\sigma$. \[Fig:FitResultsPlot\] ](FitResults)
Parameter Value
------------------ -----------------------
$a_u$ $-3.01 \cdot 10^{-5}$
$b_u$ $-2.66 \cdot 10^{-4}$
$c_u$ $-2.57 \cdot 10^{-3}$
$d_u$ $ 3.09 \cdot 10^{-2}$
$e_u$ $2.05$
$\epsilon_2$ $-3.57 \cdot 10^{-5}$
$\epsilon_{102}$ $3.17 \cdot 10^{-5}$
$\epsilon_{23}$ $1.62 \cdot 10^{-4}$
$\epsilon_3$ $1.24 \cdot 10^{-2}$
$\tan \beta$ $1.49$
: Values of the effective parameters of the quark and charged lepton Yukawa matrices and $\tan \beta$ for $M_{\text{SUSY}} = 750$ GeV. The numerical values are determined from a $\chi^2$-fit to experimental data with a $\chi^2$ per degree of freedom of 2.05/3. \[Tab:Parameters\]
Quantity (at $m_t(m_t)$) Experiment Model Deviation
-------------------------------------- ------------------------------ ---------- -----------
$y_\tau$ in $10^{-2}$ 1.00 $1.00$ $-0.277$
$y_\mu$ in $10^{-4}$ 5.89 $5.89$ $ 0.097$
$y_e$ in $10^{-6}$ 2.79 $2.79$ $-0.016$
$y_b$ in $10^{-2}$ $1.58 \pm 0.05$ $1.64$ $1.088$
$y_s$ in $10^{-4}$ $2.99 \pm 0.86$ $2.95$ $-0.226$
$y_d$ in $10^{-6}$ $15.9^{+6.8}_{-6.6}$ $11.7$ $-0.639$
$y_t$ $0.936 \pm 0.016$ $0.939$ $ 0.159$
$y_c$ in $10^{-3}$ $3.39 \pm 0.46$ $3.40$ $ 0.223$
$y_u$ in $10^{-6}$ $7.01^{+2.76}_{-2.30}$ $7.59$ $0.209$
$\theta_{12}^{\text{CKM}}$ $0.2257^{+0.0009}_{-0.0010}$ $0.2257$ $ 0.026$
\[0.3pc\] $\theta_{23}^{\text{CKM}}$ $0.0415^{+0.0011}_{-0.0012}$ $0.0409$ $-0.488$
\[0.1pc\] $\theta_{13}^{\text{CKM}}$ $0.0036 \pm 0.0002$ $0.0036$ $-0.002$
\[0.1pc\] $\delta_{\text{CKM}}$ $1.2023^{+0.0786}_{-0.0431}$ $1.1975$ $-0.113$
: Fit results for the quark Yukawa couplings and mixing and the charged lepton Yukawa couplings at low energy compared to experimental data. The values for the Yukawa couplings are extracted from [@Xing:2007fb] and the CKM parameters from [@PDG10]. Note that the experimental uncertainty on the charged lepton Yukawa couplings are negligible small and we have assumed a relative uncertainty of 1 % for them. The $\chi^2$ per degree of freedom is 2.05/3. A pictorial representation of the agreement between our predictions and experiment can be found as well in Fig. \[Fig:FitResultsPlot\]. \[Tab:FitResults\]
For the RGE running of the Yukawa matrices we have used the [REAP]{} package [@Antusch:2005gp] and calculated with it the masses and mixing angles at low energies. Note that we have used the RGEs of the MSSM. Possible RGE effects due to including a variant of the NMSSM are neglected. On the one hand we can expect this effect to be flavour blind leading only to a rescaling of the GUT scale parameters and on the other hand, in the scale-invariant NMSSM for example, the RGE effects come from the coupling $\lambda$ which can be small [@Kowalska:2012gs] although $\tan \beta$ given there is preferred to be larger than 10. For small $\tan \beta$ the coupling $\lambda$ has to be rather large to be in agreement with recent Higgs data, see, e.g. [@Cheng:2013fma]. Furthermore, SUSY threshold corrections are negligibly small due to the small $\tan \beta$ and hence are not included in the fit.
For the charged lepton and quark masses and their errors at the top scale $m_t(m_t)$ we have taken the values from [@Xing:2007fb] and for the CKM parameters the PDG values [@PDG10]. Note that the experimental errors for the charged lepton masses are tiny and we have estimated the theoretical uncertainty from higher order effects to 1 %, and we will assume this as their errors instead.
The Yukawa matrices depend on nine real parameters (five from the up-type quarks and four from the down-type quarks and charged leptons). Furthermore we have included $\tan \beta$ as a free parameter in the fit. The unification of the $b$ and the $\tau$ Yukawa coupling at the GUT scale depends strongly on this parameter. On the contrary, the masses and mixing angles depend only very weakly on the SUSY scale which we have therefore fixed to $M_{\text{SUSY}} = 750$ GeV.
The fit results are summarised in Figure \[Fig:FitResultsPlot\] and Tables \[Tab:Parameters\] and \[Tab:FitResults\]. We have fitted ten parameters to thirteen observables with a $\chi^2$ of 2.05 and hence we can say that our model describes the data very well.[^9] Note that we followed here the strategy of our previous paper [@Antusch:2009hq] where we have found that for Yukawa matrices with negligibly small 1-3 mixings we find the correct value for the CKM phase and the Cabibbo angle $\theta_C$ with $\theta_{12}^d \approx \epsilon_2/\epsilon_{23} \lesssim \theta_C$ and $\theta_{12}^u \approx b_u/c_u \approx \theta_C/2$ if these two angles have a relative phase difference of 90$^\circ$.
![The correlations between $\theta_{13}$ and the other two mixing angles and the two physical phases in PCSD2. The regions compatible with the 1$\sigma$ (3$\sigma$) ranges of the mass squared differences and the mixing angles, taken from [@Fogli:2012ua], are depicted by the red (blue) points and delimited by dashed lines in corresponding colours. The 1$\sigma$ region for the Dirac CP phase is shown in black. \[Fig:Correlations1\]](th23th13 "fig:") ![The correlations between $\theta_{13}$ and the other two mixing angles and the two physical phases in PCSD2. The regions compatible with the 1$\sigma$ (3$\sigma$) ranges of the mass squared differences and the mixing angles, taken from [@Fogli:2012ua], are depicted by the red (blue) points and delimited by dashed lines in corresponding colours. The 1$\sigma$ region for the Dirac CP phase is shown in black. \[Fig:Correlations1\]](th12th13 "fig:") ![The correlations between $\theta_{13}$ and the other two mixing angles and the two physical phases in PCSD2. The regions compatible with the 1$\sigma$ (3$\sigma$) ranges of the mass squared differences and the mixing angles, taken from [@Fogli:2012ua], are depicted by the red (blue) points and delimited by dashed lines in corresponding colours. The 1$\sigma$ region for the Dirac CP phase is shown in black. \[Fig:Correlations1\]](dth13 "fig:") ![The correlations between $\theta_{13}$ and the other two mixing angles and the two physical phases in PCSD2. The regions compatible with the 1$\sigma$ (3$\sigma$) ranges of the mass squared differences and the mixing angles, taken from [@Fogli:2012ua], are depicted by the red (blue) points and delimited by dashed lines in corresponding colours. The 1$\sigma$ region for the Dirac CP phase is shown in black. \[Fig:Correlations1\]](p2th13 "fig:") ![The correlations between $\theta_{13}$ and the other two mixing angles and the two physical phases in PCSD2. The regions compatible with the 1$\sigma$ (3$\sigma$) ranges of the mass squared differences and the mixing angles, taken from [@Fogli:2012ua], are depicted by the red (blue) points and delimited by dashed lines in corresponding colours. The 1$\sigma$ region for the Dirac CP phase is shown in black. \[Fig:Correlations1\]](dmsq21th13 "fig:") ![The correlations between $\theta_{13}$ and the other two mixing angles and the two physical phases in PCSD2. The regions compatible with the 1$\sigma$ (3$\sigma$) ranges of the mass squared differences and the mixing angles, taken from [@Fogli:2012ua], are depicted by the red (blue) points and delimited by dashed lines in corresponding colours. The 1$\sigma$ region for the Dirac CP phase is shown in black. \[Fig:Correlations1\]](dmsq321th13 "fig:")
We turn our discussion now to the neutrino sector. Here we did not fit the parameters to the observables because here we are more interested in the allowed ranges and correlations between different observables which help in distinguishing this model from other models.
The effective neutrino mass matrix from eq. depends on three parameters. The neutrino mass scale $m_a$ the perturbation parameter $\epsilon$ and the relative phase $\alpha$. The phase $\alpha$ in our model is $\pi/3$ as discussed in section \[Sec:Strategy\]. Hence, only two real parameters $m_a$ and $\epsilon$ completely determine all observables in the neutrino sector.
We have varied these two parameters randomly and the results are shown in Figure \[Fig:Correlations1\] where we have used as constraint the fit results of the Bari group [@Fogli:2012ua]. The blue dots agree with all experimental data within 3$\sigma$ while the red dots agree even within 1$\sigma$. The dashed lines in the plots label the corresponding allowed ranges of the observables on the axes. The 1$\sigma$ range of the leptonic Dirac phase $\delta$ is shown in black because it is not measured directly and the fit results should be taken with a grain of salt. In the scan we also did not include it as a constraint.
We are everywhere in good agreement with the experimental data and we find clear correlations. Especially, noteworthy is the value for $\theta_{23}$ which lies around $38.5^\circ$. We also make precise predictions for the CP violating phases. One of the Majorana phases is unphysical because one neutrino remains massless. The Dirac CP phase has a value of $\delta \approx 130^\circ$ and the physical Majorana phase is $\alpha_2 \approx
315^\circ$. The Jarlskog determinant $J_\text{CP}$ is around 0.025 and the effective neutrino mass for neutrinoless double beta decay $m_{ee}$ is of the order of $3 \times 10^{-3}$ eV, beyond the reach of current experiments.
Summary {#Sec:Summary}
=======
We have constructed a unified $A_4 \times SU(5)$ model featuring the new type of constrained sequential dominance CSD2 proposed recently in [@Antusch:2011ic]. The $A_4 \times SU(5)$ model, with the CSD2 vacuum alignments $(0,1,1)^T$ and $(1,0,2)^T$, provides an excellent fit to the present data on quark and lepton masses and mixings, including the measured value of the leptonic mixing angle $\theta_{13}$ from Daya Bay and RENO, with testable predictions for the yet unknown parameters of the leptonic mixing matrix.
The main idea of the present model is that, with a strong normal hierarchical spectrum (with $m^{\nu}_1 = 0$ by construction since there are only two right-handed neutrinos) the 1-3 angle in the neutrino sector, $\theta_{13}^\nu$, is related to a ratio of neutrino masses by $\theta_{13}^\nu = \frac{\sqrt{2}}{3}\frac{m^\nu_2}{m^\nu_3}$, leading to $\theta^\nu_{13} \sim 5^\circ - 6^\circ$. In addition, the reactor angle receives another contribution from mixing in the charged lepton sector. The charged lepton mixing induces a correction to $\theta_{13}$ of $\sim 3^\circ$ which adds up constructively with $\theta^\nu_{13}$ to give $$\theta_{13} \sim 8^\circ - 9^\circ \;,$$ within the range of the measured value from Daya Bay and RENO. The constructive addition of the neutrino and charged lepton mixing angles is achieved by assuming high energy CP invariance which is spontaneously broken by flavon fields whose phases are controlled using Abelian $\mathbb{Z}_3$ and $\mathbb{Z}_4$ symmetries as proposed in [@Antusch:2011sx]. We emphasise that in our approach one can either use a “simple” CP symmetry, under which the components of the scalar fields transform trivially as $\phi_i \rightarrow \phi_i^{*}$, or a “generalised” CP symmetry (see e.g. [@Feruglio:2012cw] and references therein) where, in our basis, the triplet fields would transform as $\phi_i \rightarrow U_3 \phi_i^{*}$, with $U_3$ interchanging the second and third component of a triplet representation.
The resulting unified flavour model is highly predictive, as described in section \[sec:fit\], since only two parameters determine the neutrino mass matrix, while the charged lepton corrections are fixed by the GUT framework: In particular, for the Dirac CP phase $\delta$, for the one physical Majorana CP phase $\alpha_2$ and for the atmospheric angle $\theta_{23}$ we obtain the predictions $$\delta \approx 130^\circ \; , \quad \alpha_2 \approx 315^\circ \quad \mbox{and} \quad \theta_{23} \approx 38.5^\circ \;.$$ The predictions for $\delta$ and $\theta_{23}$ will be tested by the ongoing and future neutrino oscillation experiments. In addition, for $\theta_{12}$, we predict a value of $$\theta_{12} \sim 33^\circ \;,$$ which is slightly smaller than the tribimaximal mixing value but may be tested by a future reactor experiment with $\sim 60$ km baseline, which could measure $\theta_{12}$ with much improved precision [@Minakata:2004jt]. Furthermore, in the quark sector, we obtain a right-angled unitarity triangle (with $\alpha \approx 90^\circ$) from the same vacuum alignment techniques for the phases [@Antusch:2011sx], realizing the phase sum rule of [@Antusch:2009hq].
In summary, we have presented a highly predictive new unified model for fermion masses and mixing, which, in fact, represents the first unified indirect family symmetry model in the literature that has been constructed to date that is consistent with all experimental data on quark and lepton mass and mixing angles, and makes definite predictions for CP phases in both the quark and lepton sectors.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Michael A. Schmidt and Martin Holthausen for useful discussions about $A_4$ and generalised CP transformations and Christoph Luhn for useful discussions during the early stages of the project. S.A. acknowledges support by the Swiss National Science Foundation, S.F.K. from the STFC Consolidated ST/J000396/1 and M.S. by the ERC Advanced Grant no. 267985 “DaMESyFla”. S.F.K. and M.S. also acknowledge partial support from the EU Marie Curie ITN “UNILHC” (PITN-GA-2009-237920) and all authors were partially supported by the European Union under FP7 ITN INVISIBLES (Marie Curie Actions, PITN-GA-2011-289442).
Conventions and Notations {#App:Conventions}
=========================
In this section we want to summarize briefly our conventions and define some notation used throughout the main text. We will follow mainly the notation of [@Antusch:2008yc]. The only difference is a sign in the Majorana phases.
The Yukawa couplings follow the left-right convention $$\mathcal{L}_{\text{Yuk}} = - Y_{ij} \overline{\psi_L^i} \psi_R^j H + H.c. \,,$$ and for the effective light neutrino mass matrix we use the convention $$\mathcal{L}_\nu = - \frac{1}{2} \bar L_i (M_\nu)_{ij} L^c_j + H.c. \,,$$ where $L$ is the lepton doublet.
In the quark sector we define the CKM matrix by $$U_{\text{CKM}} = U_{u} U_{d}^\dagger = R_{23} U_{13} R_{12} \;,$$ where $U_{u}$ ($U_{d}$) is a unitary matrix diagonalising $Y_u Y_u^\dagger$ ($Y_d Y_d^\dagger$) and $$U_{12} = \begin{pmatrix}
c_{12} & s_{12} \text{e}^{-\operatorname{\text{i}}\delta_{12}} & 0\\
-s_{12} \text{e}^{\operatorname{\text{i}}\delta_{12}}&c_{12} & 0\\
0&0&1 \end{pmatrix} \;,$$ and similar for $U_{23}$ and $U_{13}$. We use $c_{12}$ and $s_{12}$ as abbreviations for $\cos \theta_{12}$ and $\sin \theta_{12}$. The matrices $R_{23}$ and $R_{12}$ are $U_{23}$ and $U_{12}$ with the complex phases set to zero. In this case $\delta_{13}$ coincides with the CKM phase $\delta_{\text{CKM}}$.
For the PMNS matrix we use $$U_{\text{PMNS}} = U_{e} U_{\nu}^\dagger = R_{23} U_{13} R_{12} \text{ diag}(\text{e}^{-\operatorname{\text{i}}\alpha_1/2},\text{e}^{-\operatorname{\text{i}}\alpha_2/2},1) \;,$$ where the neutrino mass matrix is diagonalized via $$U_\nu M_\nu M_\nu^\dagger U_\nu^\dagger = \text{diag}(m_1^2,m_2^2,m_3^2) \;.$$ and $U_\nu^\dagger = U^{\nu}_{23} U^{\nu}_{13} U^{\nu}_{12}$ (note the Hermitian conjugation). This conventions imply a complex conjugation of the neutrino mass matrix $M_\nu$ compared to our previous CSD2 paper [@Antusch:2011ic] and also the sign of the Majorana phases here is different.
The Renormalizable Superpotential {#App:Messenger}
=================================
$SU(5)$ $A_4$ $U(1)_R$ $\mathbb{Z}_4$ $\mathbb{Z}_4$ $\mathbb{Z}_3$ $\mathbb{Z}_3$ $\mathbb{Z}_3$ $\mathbb{Z}_3$
---------------------------------- ----------------------------------------- ---------------------------- ---------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- --
$\Sigma_1$, $\bar{\Sigma}_1$ $\mathbf{5}$, $\bar{\mathbf{5}}$ $\mathbf{1}$, $\mathbf{1}$ 1, 1 2, 2 0, 0 1, 2 2, 1 2, 1 0, 0
$\Sigma_2$, $\bar{\Sigma}_2$ $\mathbf{5}$, $\bar{\mathbf{5}}$ $\mathbf{1}$, $\mathbf{1}$ 1, 1 0, 0 0, 0 2, 1 0, 0 1, 2 0, 0
$\Sigma_3$, $\bar{\Sigma}_3$ $\mathbf{5}$, $\bar{\mathbf{5}}$ $\mathbf{1}$, $\mathbf{1}$ 1, 1 3, 1 3, 1 0, 0 0, 0 2, 1 0, 0
$\Upsilon_1$, $\bar{\Upsilon}_1$ $\mathbf{10}$, $\overline{\mathbf{10}}$ $\mathbf{3}$, $\mathbf{3}$ 1, 1 1,3 3, 1 1, 2 2, 1 2, 1 1, 2
$\Upsilon_2$, $\bar{\Upsilon}_2$ $\mathbf{10}$, $\overline{\mathbf{10}}$ $\mathbf{3}$, $\mathbf{3}$ 1, 1 3, 1 3, 1 0, 0 1, 2 0, 0 1, 2
$\Upsilon_3$, $\bar{\Upsilon}_3$ $\mathbf{10}$, $\overline{\mathbf{10}}$ $\mathbf{3}$, $\mathbf{3}$ 1, 1 3, 1 2, 2 1, 2 1, 2 0, 0 0, 0
$\Xi_1$, $\bar{\Xi}_1$ $\mathbf{5}$, $\bar{\mathbf{5}}$ $\mathbf{3}$, $\mathbf{3}$ 1, 1 3, 1 3, 1 1, 2 1, 2 0, 0 0, 0
$\Xi_2$, $\bar{\Xi}_2$ $\mathbf{5}$, $\bar{\mathbf{5}}$ $\mathbf{3}$, $\mathbf{3}$ 1, 1 3, 1 3, 1 0, 0 1, 2 0, 0 1, 2
$\Xi_3$, $\bar{\Xi}_3$ $\mathbf{5}$, $\bar{\mathbf{5}}$ $\mathbf{3}$, $\mathbf{3}$ 1, 1 3, 1 2, 2 1, 2 1, 2 0, 0 0, 0
$\Omega_1$, $\bar{\Omega}_1$ $\mathbf{10}$, $\overline{\mathbf{10}}$ $\mathbf{1}$, $\mathbf{1}$ 1, 1 3, 1 3, 1 1, 2 2, 1 2, 1 0, 0
$\Omega_2$, $\bar{\Omega}_2$ $\mathbf{10}$, $\overline{\mathbf{10}}$ $\mathbf{1}$, $\mathbf{1}$ 1, 1 3, 1 3, 1 1, 2 0, 0 1, 2 0, 0
$\Omega_3$, $\bar{\Omega}_3$ $\mathbf{10}$, $\overline{\mathbf{10}}$ $\mathbf{1}$, $\mathbf{1}$ 1, 1 3, 1 3, 1 2, 1 2, 1 1, 2 0, 0
$\Gamma_1$, $\bar{\Gamma}_1$ $\mathbf{1}$, $\mathbf{1}$ $\mathbf{1}$, $\mathbf{1}$ 0, 2 0, 0 0, 0 2, 1 0, 0 0, 0 1, 2
$\Gamma_2$, $\bar{\Gamma}_2$ $\mathbf{1}$, $\mathbf{1}$ $\mathbf{1}$, $\mathbf{1}$ 0, 2 0, 0 0, 0 2, 1 0, 0 2, 1 2, 1
$\Gamma_3$, $\bar{\Gamma}_3$ $\mathbf{1}$, $\mathbf{1}$ $\mathbf{1}$, $\mathbf{1}$ 0, 2 0, 0 0, 0 1, 2 2, 1 0, 0 2, 1
$\Gamma_4$, $\bar{\Gamma}_4$ $\mathbf{1}$, $\mathbf{1}$ $\mathbf{1}$, $\mathbf{1}$ 0, 2 0, 0 0, 0 2, 1 1, 2 0, 0 0, 0
$\Gamma_5$, $\bar{\Gamma}_5$ $\mathbf{1}$, $\mathbf{1}$ $\mathbf{1}$, $\mathbf{1}$ 0, 2 0, 0 0, 0 1, 2 0, 0 1, 2 0, 0
$\Gamma_6$, $\bar{\Gamma}_6$ $\mathbf{1}$, $\mathbf{1}$ $\mathbf{3}$, $\mathbf{3}$ 0, 2 0, 0 0, 0 1, 2 2, 1 2, 1 0, 0
$\Gamma_7$, $\bar{\Gamma}_7$ $\mathbf{1}$, $\mathbf{1}$ $\mathbf{1}$, $\mathbf{1}$ 0, 2 2, 2 2, 2 0, 0 0, 0 0, 0 0, 0
$\Gamma_8$, $\bar{\Gamma}_8$ $\mathbf{1}$, $\mathbf{1}$ $\mathbf{1}$, $\mathbf{1}$ 0, 2 0, 0 2, 2 0, 0 0, 0 0, 0 0, 0
$\Gamma_9$, $\bar{\Gamma}_9$ $\mathbf{1}$, $\mathbf{1}$ $\mathbf{1}$, $\mathbf{1}$ 0, 2 0, 0 0, 0 0, 0 1, 2 2, 1 0, 0
: \[Tab:MessengerFields\] The messenger field content of our model. Every line represents a messenger pair which receives a mass larger than the GUT scale and no cross terms are allowed. In the main text we labelled the messenger mass scale generically with $\Lambda$.
In this appendix we discuss the full renormalizable superpotential including the messenger fields which after being integrated out give the effective operators as discussed before.
We start with the superpotential bilinear in the fields which is in our case only the mass terms for the messengers $$\mathcal{W}^{\text{ren}}_\Lambda = M_{\Sigma_i} \Sigma_i \bar \Sigma_i + M_{\Upsilon_i} \Upsilon_i \bar \Upsilon_i + M_{\Xi_i} \Xi_i \bar \Xi_i + M_{\Omega_i} \Omega_i \bar \Omega_i + M_{\Gamma_i} \Gamma_i \bar \Gamma_i \;.$$ The full list of messenger fields is given in Table \[Tab:MessengerFields\] where every line is a messenger pair which receives a mass larger than the GUT scale so that they can be integrated out to give the desired effective operators. To simplify the notation before we have introduced the messenger scale $\Lambda$ as shorthand which is related to the individual messenger masses with order one coefficients.
Note that in the superpotential bilinear in the fields no $\mu$-term for the Higgs fields appears. This term is forbidden by symmetries and in combination with a NMSSM like mechanism helps to increase the Higgs mass to the experimentally determined value. A possible singlet field $S$ with couplings $S (H_5 \bar H_5 + H_{45} \bar H_{45})$ would not appear anywhere else in the superpotential with the symmetries and field content as specified in Tables \[Tab:Matter+HiggsFields\], \[Tab:FlavonFields\], \[Tab:DrivingFields\] and \[Tab:MessengerFields\].
![ The supergraphs before integrating out the messengers for the flavon sector (only diagrams are shown which give non-renormalizable contributions). \[Fig:FlavonMessenger\] ](Flavons){width="\textwidth"}
The next step in our discussion of the renormalizable superpotential is the flavon sector. The full potential for this sector reads (dropping for the sake of simplicity order one coefficients) $$\begin{aligned}
\mathcal{W}^{\text{ren}}_{\text{flavon}} &= O_{1;2} \phi_1 \phi_2 + O_{1;3} \phi_1 \phi_3 + O_{2;3} \phi_2 \phi_3 + O_{111;211} \phi_{111} \phi_{211} + O_{111;23} \phi_{111} \phi_{23} \nonumber\\
& + O_{23;211} \phi_{23} \phi_{211} + O_{2;102} \phi_2 \phi_{102} + O_{211;102} \phi_{211} \phi_{102} + O_{1;23} \phi_1 \phi_{23} \nonumber\\
& + A_1 \phi_1 \phi_1 + A_2 \phi_2 \phi_2 + A_3 \phi_3 \phi_3 + A_{111} \left( \phi_{111}^2 + \phi_{111} \rho_{111}+ \tilde \phi_{111} \rho_{111}\right) \nonumber\\
& + P \Gamma_9 \xi_u + \bar \Gamma_9 \xi_u^2 + P \Gamma_8^2 + \bar \Gamma_8 \phi_2^2 + \bar \Gamma_8 \theta_2^2 + P \Gamma_7^2 + \bar \Gamma_7 ( \phi_{111}^2 + \rho_{111}^2 + \tilde \rho_{111}^2 ) \nonumber\\
& + P \phi_{211} \Gamma_6 + \phi_{211}^2 \bar \Gamma_6 + P \xi_2 \Gamma_5 + \xi_2^2 \bar \Gamma_5 + P \xi_1 \Gamma_4 + \xi_1^2 \bar \Gamma_4 + P \rho_{23} \Gamma_3 + (\phi_{23}^2 + \rho_{23}^2) \bar \Gamma_3 \nonumber\\
& + P \rho_{102} \Gamma_2 + ( \phi_{102}^2 + \rho_{102}^2) \bar \Gamma_2+ P \theta_{102} \Gamma_1 + \theta_{102}^2 \bar \Gamma_1 \;.\end{aligned}$$ The first three lines of this superpotential have already been discussed in the flavon alignment section \[Sec:Flavon\] while the last four lines are needed to fix the phases of the various flavon vevs. For instance, the messenger pair $\Gamma_1$ and $\bar \Gamma_1$ gives after integrating out the effective operator $1/\Lambda P \theta_{102}^3$ where in this case $\Lambda$ stands for $M_{\Gamma_1}$ multiplied by real order one couplings. This operator fixes the phase of $\langle \theta_{102} \rangle$ up to a discrete choice as discussed before.
We will not list here all of the effective operators because they have already appeared in our superpotential for the flavon alignment and they can also be read off from the diagrams in Figure \[Fig:FlavonMessenger\] after contracting the messenger propagators to points.
![ The supergraphs before integrating out the messengers for the down-type quark and charged lepton sector. \[Fig:DownMessenger\] ](Down)
![ The supergraphs before integrating out the messengers for the up-type quark sector. \[Fig:UpMessenger\] ](Up)
![ The supergraphs before integrating out the messengers for the neutrino sector. \[Fig:NeutrinoMessenger\] ](Neutrinos)
For the renormalizable couplings including the matter and Higgs fields we find the renormalizable superpotential (again dropping order one coefficients) $$\begin{aligned}
\mathcal{W}^{\text{ren}}_d &= T_3 \bar H_5 \bar \Sigma_3 + F \phi_3 \Sigma_3 + T_2 \phi_{23} \bar \Upsilon_1 + \bar H_{45} \Upsilon_1 \bar \Xi_1 + F H_{24} \Xi_1 + T_2 \phi_{102} \bar \Upsilon_2 + \bar H_5 \Upsilon_2 \bar \Xi_2 \nonumber\\
& + \theta_{102} \Xi_2 \bar \Xi_1 + T_1 \phi_2 \bar \Upsilon_3 + \bar H_5 \Upsilon_3 \bar \Xi_3 + \theta_2 \Xi_3 \bar \Xi_1 \;,\\
\mathcal{W}^{\text{ren}}_u &= T_1 H_5 \Omega_3 + \xi_1 \Omega_2 \bar \Omega_3 + T_1 \xi_u \bar \Omega_2 + \Omega_2 \xi_u \bar \Omega_1 + T_2 \Gamma_4 \bar \Omega_1 \nonumber\\
& + \bar \Gamma_4 \xi_1^2 + T_2 H_5 \Omega_1 + T_3 \xi_1 \bar \Omega_1 + T_3^2 H_5 \;, \\
\mathcal{W}^{\text{ren}}_\nu &= \xi_1 N_1^2 + \xi_2 N_2^2 + F \phi_{23} \Sigma_1 + N_1 H_5 \bar \Sigma_1 + F \phi_{102} \Sigma_2 + N_2 H_5 \bar \Sigma_2 \;. \end{aligned}$$ After integrating out the heavy messenger fields we end up with the non-renormalizable operators as discussed in section \[Sec:Model\], cf. also Figures \[Fig:DownMessenger\]-\[Fig:NeutrinoMessenger\].
In addition to the renormalizable operators discussed so far there are six more operators allowed by the symmetries which are $$\mathcal{W}^{\text{ren}}_{\text{neg}} =
T_1 \Gamma_9 \bar \Omega_1 + T_2 \Gamma_9 \bar \Omega_3 + \Xi_1 \bar \Xi_3 \phi_2 + \Gamma_9 \Omega_1 \bar \Omega_2 + \Gamma_4 \bar \Omega_2 \Omega_3 + \Gamma_1 \Xi_1 \bar \Xi_2 \;.$$ The first two operators contribute effectively to the $T_1 T_2 H_5 \xi_u^2$ operator already present and for the sake of simplicity we have not shown them in Figure \[Fig:UpMessenger\]. The third operator generates the dimension six operator $F T_2 \bar H_{45}
H_{24} \phi_2 \phi_{23}$ which gives a contribution to the 2-2 element of the down-type quark and charged lepton Yukawa matrix. In fact the correction has the same phase and the same $SU(5)$ Clebsch–Gordan coefficient as the leading order coefficient so that we can safely neglect it. The last three operators finally give, after integrating out the heavy messengers, dimension seven and eight operators which give only small corrections (in our model we have discussed operators up to dimension six). The dimension seven operators, for instance, are induced by $\Gamma_9
\Omega_1 \bar \Omega_2$ which gives corrections to the $1-3$ and $2-3$ elements of the up-type quark Yukawa matrix which are very small compared to all other elements which are generated at maximum by a dimension five operator.
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[^1]: E-mail: `stefan.antusch@unibas.ch`
[^2]: E-mail: `king@soton.ac.uk`
[^3]: E-mail: `spinrath@sissa.it`
[^4]: It is interesting to compare the predictions of CSD2 to another alternative to CSD that has been proposed to account for a reactor angle called partially constrained sequential dominance (PCSD) [@King:2009qt]. PCSD involves a vacuum misalignment of the dominant flavon alignment to $(\varepsilon ,1,-1)^T$, with a subdominant flavon alignment $(1,1,1)^T$, leading to tri-bimaximal-reactor (TBR) mixing [@King:2009qt] in which only the reactor angle is switched on, while the atmospheric and solar angles retain their TB values.
[^5]: In fact the only other example of a unified indirect model with a realistic reactor angle that we are aware of is the last paper in [@GUTs] based on Pati-Salam unification, however that model predicts an atmospheric angle in second octant.
[^6]: $B = (b,2b,0)^T$ was also considered in [@Antusch:2011ic], but here we shall not consider it further.
[^7]: Compared to the notation of [@Antusch:2011ic], we have changed, for instance, $\alpha \rightarrow - \alpha$.
[^8]: We note that the choice $\alpha = \pi/3$ is motivated by the current data which favours $\theta_{23}$ in the first octant. On the other hand, one can in principle also construct other models with different values of $\alpha$, and there are also other options for $\beta$ and $\theta_{12}^e$, which may lead to interesting alternative models. In this sense, the strategy described here leads to a whole new class of possible models.
[^9]: We note that while we get an excellent fit for the quark masses themselves, as given in the PDG review, there is some tension with QCD results which favour $y_s/y_d \approx 19$ [@Leutwyler:2000hx], while our fit yields $y_s/y_d = 25.3$. We remark that this tension is the same that one also gets with the more conventional GJ relation instead of the Clebsch factors $9/2$ and $3/2$ used here, so it is not particular for our model. In our fit, we have not included $y_s/y_d$ as constraints, but we would like to note that future even more precise results on the quark masses, including lattice results, can provide powerful additional constraints on unified flavour models.
|
---
abstract: 'We extend the notions of CR GJMS operators and $Q$-curvature to the case of partially integrable CR structures. The total integral of the CR $Q$-curvature turns out to be a global invariant of compact nondegenerate partially integrable CR manifolds equipped with an orientation of the bundle of contact forms, which is nontrivial in dimension at least five. It is shown that its variation is given by the curvature-type quantity called the CR obstruction tensor, which is introduced in the author’s previous work. Moreover, we consider the linearized CR obstruction operator. Based on a scattering-theoretic characterization, we discuss its relation to the CR deformation complex of integrable CR manifolds. The same characterization is also used to determine the Heisenberg principal symbol of the linearized CR obstruction operator.'
address: 'Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro, Tokyo 152-8551, Japan'
author:
- Yoshihiko Matsumoto
bibliography:
- 'myrefs.bib'
title: 'GJMS operators, $Q$-curvature, and obstruction tensor of partially integrable CR manifolds'
---
Introduction {#introduction .unnumbered}
============
Recent development of general theory of parabolic geometries predicts the possibility of understanding geometric objects defined for any particular type of geometries in a broader context. With this general idea in mind, in the current article we discuss the CR versions of the GJMS operators, Branson’s $Q$-curvature, and the Fefferman–Graham obstruction tensor, which are originally studied in conformal geometry [@Graham_Jenne_Mason_Sparling_92; @Branson_95; @Fefferman_Graham_85].
The CR versions of the first two have been known for quite a long time. The CR $Q$-curvature was defined by Fefferman and Hirachi [@Fefferman_Hirachi_03] as the push-forward of the conformal $Q$-curvature associated to the Fefferman metric, which is a certain Lorentzian metric on the total space of a trivial circle bundle over a given manifold [@Fefferman_76; @Lee_86]. Gover and Graham [@Gover_Graham_05] constructed two independent families of CR-invariant powers of the sub-Laplacian, one by the tractor calculus approach and the other by the Fefferman metric approach, the latter of which we call the CR GJMS operators. Hislop, Perry, and Tang [@Hislop_Perry_Tang_08] discussed them from the viewpoint of scattering theory following the argument of Graham and Zworski [@Graham_Zworski_03] in the conformal case.
However, the studies described above were all restricted to the case of *integrable* CR structures. They are actually only a part of CR geometries, if we mean by this term parabolic geometries modeled on the standard CR sphere $S^{2n+1}\subset\mathbb{C}^{n+1}$. More precisely, let $G={\mathit{PSU}}(n+1,1)$ be the group of CR automorphisms of $S^{2n+1}$ and $P$ the stabilizer of any fixed point on $S^{2n+1}$. Then Čap and Schichl [@Cap_Schichl_00] showed (see also [@Cap_Slovak_09]\*[Subsection 4.2.4]{}) that the category of normal regular parabolic geometries of type $(G,P)$ is equivalent to that of strictly pseudoconvex partially integrable CR manifolds, where an almost CR manifold $(M,T^{1,0}M)$ is said to be *partially integrable* if the following is satisfied: $$\label{eq:partial_integrability}
[\Gamma(T^{1,0}M),\Gamma(T^{1,0}M)]\subset\Gamma(T^{1,0}M\oplus{\overline{T^{1,0}M}}).$$ Therefore, from the viewpoint of parabolic geometries, it is natural to consider this extended class of CR structures.
In this article, we shall work on nondegenerate partially integrable CR manifolds with the bundle of contact forms oriented, which we simply call *partially integrable CR manifolds*. When we take a contact form $\theta$, we always implicitly assume that it is positive with respect to the given orientation.
The purpose of this paper is twofold: on one hand we strengthen the parallelism between the conformal and CR cases, and on the other hand we discuss a rather subtle nature of the CR case. Firstly, we construct CR GJMS operators $P_{2k}$ and $Q$-curvature $Q_\theta$ for partially integrable CR structures by generalizing the scattering-theoretic approach of Hislop, Perry, and Tang to asymptotically complex hyperbolic Einstein metrics[^1]. Then the total integral of the $Q$-curvature becomes a global CR invariant, and we further discuss its variational properties. Secondly, we consider the linearized obstruction operator $\mathcal{O}^\bullet$, and explore its properties in connection with the CR deformation complex of Rumin [@Rumin_94] and Akahori, Garfield, Lee [@Akahori_Garfield_Lee_02]. The important ingredient is a scattering-theoretic characterization of $\mathcal{O}^\bullet$ based on the idea of [@Matsumoto_13].
The concept of asymptotically complex hyperbolic metrics (hereafter ACH metrics) is due to Epstein, Melrose, and Mendoza [@Epstein_Melrose_Mendoza_91]. They are generalizations of Bergman-type complete Kähler metrics on bounded strictly pseudoconvex domains $\Omega\subset\mathbb{C}^{n+1}$, i.e., Kähler metrics given by a potential function of the form $\log(1/r)$ up to constant multiple, where $r$ is a boundary defining function of $\Omega$. The definition goes as follows: we first introduce $\Theta$-structures on manifolds-with-boundary, by which the notion of $\Theta$-metrics makes sense, and then define ACH metrics as $\Theta$-metrics satisfying some extra conditions. The idea of $\Theta$-structures comes from the fact that, in the case of strictly pseudoconvex domains, $\frac{i}{2}(\partial r-{\overline{\partial}}r)|_{{\partial}\Omega}$ gives a conformal class of sections of $T^*{\overline{\Omega}}$ over ${\partial}\Omega$ that is independent of $r$. If $X$ is a $\Theta$-manifold, i.e., a $C^\infty$-smooth manifold-with-boundary equipped with a $\Theta$-structure, then it has two important features:
- the boundary $M=\partial X$ is equipped with a contact distribution $H$, with an orientation of the bundle $H^\perp\subset T^*M$ of contact forms;
- the usual tangent bundle $TX$ is blown up along the boundary to define the *$\Theta$-tangent bundle* ${{\fourIdx{\Theta}{}{}{}{TX}}}$ in a way that, in the case of strictly pseudoconvex domains, Bergman-type metrics continuously extend up to the boundary as $\Theta$-metrics, i.e., metrics of ${{\fourIdx{\Theta}{}{}{}{TX}}}$.
Then ACH metrics form a special class of $\Theta$-metrics. They are defined in such a way that each ACH metric determines a partially integrable CR structure $T^{1,0}M$ on $M$, which we call the *CR structure at infinity*. It is *compatible* to the $\Theta$-structure in the sense that $\operatorname{Re}T^{1,0}M=H$. In order to allow the Levi form to be of indefinite signature, we do not assume that ACH metrics are definite.
Conversely, given a $\Theta$-manifold $X$, one can think of a compatible partially integrable CR structure on $M=\partial X$ as a Dirichlet data for ACH metrics on $X$. Then one naturally tries to solve the Einstein equation under this boundary condition. While a perturbation result is obtained by Biquard [@Biquard_00], the author has constructed in [@Matsumoto_14], for any given partially integrable CR structure on $\partial X$, a best possible approximate solution which is $C^\infty$-smooth up to the boundary, and determined its arbitrariness. Let $\rho\in C^\infty(X)$ be any boundary defining function of $X$. Then our approximate solution satisfies
\[eq:approx\_Einstein\] $$\label{eq:approx_Einstein_ricci}
\operatorname{Ric}=-\frac{n+2}{2}g+O(\rho^{2n+2})$$ and $$\label{eq:approx_Einstein_trace}
\operatorname{Scal}=-(n+1)(n+2)+O(\rho^{2n+3}),$$
where $\operatorname{Ric}$ and $\operatorname{Scal}$ are the Ricci tensor and the scalar curvature of $g$, respectively, in the sense that $E:={\operatorname{Ric}}+\tfrac{n+2}{2}g\in \rho^{2n+2}C^\infty(X;\operatorname{Sym}^2{{\fourIdx{\Theta}{}{}{}{T^*X}}})$ and ${\operatorname{Scal}}+(n+1)(n+2)\in\rho^{2n+3}C^\infty(X)$. Moreover, associated to each contact form $\theta$ is a preferred local frame $\set{\bm{Z}_\infty,\bm{Z}_0,\bm{Z}_\alpha,\bm{Z}_{{\overline{\alpha}}}}$ of $\mathbb{C}{{\fourIdx{\Theta}{}{}{}{TX}}}$, and the proof of the existence of such a $g$ simultaneously shows that, if $\rho$ is the model boundary defining function associated to $\theta$ (see Subsection \[subsec:ACH\_metrics\]) and $E=\rho^{2n+2}\tilde{E}$, then the boundary values of the components $\tensor{\tilde{E}}{_\alpha_\beta}=\tilde{E}(\bm{Z}_\alpha,\bm{Z}_\beta)$ invariantly define a density-weighted tensor $\tensor{\mathcal{O}}{_\alpha_\beta}\in\tensor{\mathcal{E}}{_\alpha_\beta}(-n,-n)$. This is what we call the *CR obstruction tensor*, and by the construction, the trivialization of $\tensor{\mathcal{O}}{_\alpha_\beta}$ with respect to $\theta$ has a universal expression in terms of the Tanaka–Webster connection of $\theta$, in the sense that it is written as a universal polynomial of covariant derivatives of the Nijenhuis tensor, those of the Tanaka–Webster torsion and curvature, the Levi form, and its dual. It is known [@Matsumoto_14]\*[Theorem 0.2]{} that $\tensor{\mathcal{O}}{_\alpha_\beta}$ vanishes for integrable CR manifolds; hence it is of interest when the dimension is at least $5$, for any 3-dimensional almost CR manifold is integrable. The content of this paragraph is recalled in Section \[sec:ACHE\] in greater detail[^2].
The CR GJMS operators $P_{2k}$ are characterized as Dirichlet-to-Neumann-type operators associated to the generalized eigenfunction problem for the Laplacian of our approximate ACH-Einstein metric $g$. Since the construction of such operators can be executed without assuming that $g$ is (approximately) Einstein, we consider the following loose setting in the first place: $g$ is an ACH metric on a $(2n+2)$-dimensional $\Theta$-manifold $X$ that is $C^\infty$-smooth up to the boundary, and $T^{1,0}M$ is its CR structure at infinity. We take any contact form $\theta$ and the model boundary defining function associated to $\theta$ is denoted by $\rho$. Under this setting, we consider the following equation for any positive integer $k$: $$\label{eq:eigenfunction_equation}
\left(\Delta-\frac{(n+1)^2-k^2}{4}\right)u=0.$$ Our convention of the Laplacian is such that it has negative principal symbol when $g$ is a positive definite metric. Since the indicial roots of the operator $\Delta-((n+1)^2-k^2)/4$ are $n+1\pm k$, any sufficiently regular solution of must behave asymptotically like $\rho^{n+1-k}$ or $\rho^{n+1+k}$. If we try power series expansions starting with $\rho^{n+1-k}f$ to solve , where $f\in C^\infty(M)$, then a possible obstruction appears at the power $\rho^{n+1+k}$, which defines a differential operator $P^g_{2k}$ on the boundary. In other words, $P^g_{2k}f$ is the coefficient of the first logarithmic term of the solution, in which way Theorem \[thm:dn\_operator\] is stated. In particular, $P^g_{2n+2}$ is the obstruction to the $C^\infty$-smooth harmonic extension of a given function on the boundary. This is the “compatibility operator” which Graham [@Graham_83] studied when $(\mathring{X},g)$ is the complex hyperbolic space. Note also that, while $P^g_{2k}$ depends on $g$ in a complicated way, its dependency on $\theta$ is rather trivial. This is because $\theta$ is involved only in the choice of $\rho$ and has nothing to do with the equation itself. As a consequence, $P^g_{2k}$ invariantly defines an operator $\mathcal{E}(-(n+1-k)/2,-(n+1-k)/2)\longrightarrow\mathcal{E}(-(n+1+k)/2,-(n+1+k)/2)$ between densities.
The associated quantity $Q^g_\theta$ is defined in Theorem \[thm:log\_expansion\_definition\_of\_Q\] as the first logarithmic term coefficient of $U_0$ solving $\Delta(\log\rho+U_0)=(n+1)/2$ formally. This characterization imitates the work of Fefferman and Graham [@Fefferman_Graham_02] in the conformal case, and behind this is Branson’s original “analytic continuation in dimension” argument: $Q^g_\theta$ is the “derivative at $N=n$” of the function $P^g_{2n+2}1$ in dimension $2N+2$, where $N$ is considered as a continuous variable. Suppose $u_N$ is the formal solution of for $k=n+1$ in dimension $2N+2$ starting with $\rho^{N-n}$ (we take $f$ to be $1$). Then, by “differentiating $u_N$ at $N=n$,” we obtain a function $U$ satisfying $\Delta U=(n+1)/2$ with leading term $\log\rho$ and the next logarithmic term $Q^g_\theta\rho^{2n+2}\log\rho$.
We apply Theorems \[thm:dn\_operator\] and \[thm:log\_expansion\_definition\_of\_Q\] to the special case where $g$ satisfies to get $P_{2k}$ and $Q_\theta$, which are *CR GJMS operators* and the *CR $Q$-curvature*, as justified by the following result proved in Subsection \[subsec:GJMS\]. We say that a differential operator $D\colon C^\infty(M;E)\longrightarrow C^\infty(M;F)$ has *Heisenberg order $\le m$* if it is locally expressed as a matrix with entries of the form $$\sum_{2\alpha_0+\alpha_1+\dots+\alpha_{2n}\le m}
a_{(\alpha_0,\alpha_1,\dots,\alpha_{2n})}Y_0^{\alpha_0}Y_1^{\alpha_1}\dotsb Y_{2n}^{\alpha_{2n}},$$ where $\set{Y_0, Y_1, \dots, Y_{2n}}$ is a local frame of $TM$ such that $\set{Y_1, \dots, Y_{2n}}$ spans the contact distribution $H=\operatorname{Re}T^{1,0}M$.
\[thm:dn\_operator\_of\_ACHE\] Let $g$ be a $C^\infty$-smooth ACH metric on a $(2n+2)$-dimensional $\Theta$-manifold satisfying the approximate Einstein condition . Then $P_{2k}=P^g_{2k}$ for $k\le n+1$ and $Q_\theta=Q^g_\theta$ are independent of the ambiguity in $g$ and determined only by the CR structure at infinity and a contact form $\theta$. Each $P_{2k}$ has the form $$P_{2k}=\prod_{j=0}^{k-1}(\Delta_b+i(k-1-2j)T)
+(\text{a differential operator with Heisenberg order $\le 2k-1$}),$$ where $\Delta_b$ is the sub-Laplacian and $T$ is the Reeb vector field of $\theta$. If $\Hat{\theta}=e^\Upsilon\theta$, where $\Upsilon\in C^\infty(M)$, then $Q_\theta$ transforms as $$\label{eq:Q_transformation}
Q_{\Hat{\theta}}=e^{-(n+1)\Upsilon}(Q_\theta+P_{2n+2}\Upsilon).$$ Moreover, $P_{2k}$ is formally self-adjoint with respect to $\theta\wedge(d\theta)^n$ for all $k\le n+1$, and the *critical CR GJMS operator* $P_{2n+2}$ annihilates constant functions.
By the construction of $g$ and the proofs of Theorems \[thm:dn\_operator\] and \[thm:log\_expansion\_definition\_of\_Q\], $P_{2k}$ and $Q_\theta$ have universal expressions in terms of the Tanaka–Webster connection (for $Q_\theta$, this is the same sense as for $\tensor{\mathcal{O}}{_\alpha_\beta}$; the expressions for $P_{2k}$ involve the Tanaka–Webster covariant differentiations, of course).
Now we assume that $M=\partial X$ is compact. Then by the transformation law , the self-adjointness of $P_{2n+2}$, and the fact that $P_{2n+2}1=0$, the *total CR $Q$-curvature* $$\overline{Q}=\int_M Q_\theta\theta\wedge(d\theta)^n$$ is a global CR invariant. This is also characterized as the first log-term coefficient of the volume expansion of $g$, and this viewpoint leads to the first variational formula of $\overline{Q}$ with respect to deformations of partially integrable CR structures via the technique of Graham and Hirachi [@Graham_Hirachi_05]. Since it is known that $\overline{Q}=0$ for any 3-dimensional compact CR manifold [@Fefferman_Hirachi_03], we assume $2n+1\ge 5$. Recall from [@Matsumoto_14]\*[Proposition 6.1]{} that an infinitesimal deformation of partially integrable CR structure is given by a density-weighted tensor $\tensor{\psi}{_\alpha_\beta}\in\tensor{\mathcal{E}}{_(_\alpha_\beta_)}(1,1)$. In Subsection \[subsec:variational\_formula\], we prove the following.
\[thm:first\_variational\_formula\] Let $(M,T^{1,0}M)$ be a compact partially integrable CR manifold of dimension $2n+1\ge 5$. Let $\psi=\tensor{\psi}{_\alpha_\beta}$ be an infinitesimal deformation of partially integrable CR structure and $T^{1,0}_t$ a smooth 1-parameter family of partially integrable CR structures with fixed underlying contact structure that is tangent to $\psi$ at $t=0$. Let $\smash{\overline{Q}}^t$ be the total CR $Q$-curvature of $(M,T^{1,0}_t)$. Then, $$\label{eq:Variation_TotalQ}
\left.\left(\frac{d}{dt}\smash{\overline{Q}}^t\right)\right|_{t=0}=
\frac{4\cdot(-1)^n\cdot n!(n+1)!}{n+2}\int_M
\operatorname{Re}(\tensor{\mathcal{O}}{^\alpha^\beta}\tensor{\psi}{_\alpha_\beta}).$$ Here, $\tensor{\mathcal{O}}{_\alpha_\beta}\in\tensor{\mathcal{E}}{_(_\alpha_\beta_)}(-n,-n)$ is the CR obstruction tensor of $(M,T^{1,0}M)$, and the indices are raised by the weighted Levi form.
This result in particular shows the nontriviality of $\overline{Q}$ in the partially integrable category, while it seems a reasonable conjecture that $\overline{Q}=0$ for any integrable CR manifold (see [@Fefferman_Hirachi_03]). We furthermore derive a formula of the Heisenberg principal symbol of $\mathcal{O}^\bullet$, the linearization of the operator that gives the CR obstruction tensor, in Theorem \[thm:linearized\_obstruction\]. It assures that $\mathcal{O}^\bullet$ is nowhere vanishing when $2n+1\ge 5$, and consequently, that the locus $\set{\overline{Q}=0}$ in the space of partially integrable CR structures on a given contact manifold is the complement of an open dense subset with respect to the natural Fréchet topology, for that $\mathcal{O}^\bullet$ is nowhere zero implies that the second variation of $\overline{Q}$ cannot vanish at the critical points of $\overline{Q}$.
Theorem \[thm:linearized\_obstruction\] follows from an interpretation of $\mathcal{O}^\bullet$ as a Dirichlet-to-Neumann-type operator, which is established in Proposition \[prop:lichnerowicz\_equation\]. From the technical point of view, we remark that the proof of Proposition \[prop:lichnerowicz\_equation\] involves a careful analysis of Laplacians and divergence operators using the asymptotic Kählerity of ACH metrics (Lemma \[prop:asymptotic\_kahlerity\]) and the behavior of the curvature tensor at the boundary (Proposition \[prop:asymptotic\_complex\_hyperbolicity\]). One might be able to see the latter as the specialization of Stenzel’s work [@Stenzel_97] to ACH metrics.
If we restrict ourselves to the case of integrable CR manifolds, then the characterization of $\mathcal{O}^\bullet$ given in Proposition \[prop:lichnerowicz\_equation\] can be studied more deeply with the language of Kähler differential geometry, and we finally get the following properties of $\mathcal{O}^\bullet$ in Subsection \[subsec:further\_properties\].
\[thm:obstruction\_operator\_observation\] Let $(M,T^{1,0}M)$ be an integrable CR manifold of dimension $2n+1\ge 5$.
\(1) The linearized CR obstruction operator $\mathcal{O}^\bullet\colon\tensor{\mathcal{E}}{_(_\alpha_\beta_)}(1,1)\longrightarrow
\tensor{\mathcal{E}}{_(_\alpha_\beta_)}(-n,-n)$ is a complex-linear, formally self-adjoint operator.
\(2) Let $D\colon\mathcal{E}(1,1)\longrightarrow\tensor{\mathcal{E}}{_(_\alpha_\beta_)}(1,1)$ be defined by $\tensor{(Df)}{_\alpha_\beta}
=\tensor*{\nabla}{^{\mathrm{TW}}_\alpha}\tensor*{\nabla}{^{\mathrm{TW}}_\beta}f+i\tensor{A}{_\alpha_\beta}f$ and $D^*$ its adjoint, where the density bundles are trivialized by some $\theta$ and $\nabla^\mathrm{TW}$ and $A$ denote the associated Tanaka–Webster connection and torsion, respectively. Then the following holds: $$\label{eq:double_divergence_free}
\mathcal{O}^\bullet D=0\qquad\text{and}\qquad
D^*\mathcal{O}^\bullet=0.$$
\(3) Let $\mathcal{E}_\mathrm{N}(1,1)\subset\tensor{\mathcal{E}}{_\alpha_\beta_\gamma}(1,1)$ be the subspace of tensors with Nijenhuis-type symmetry, and $N^\bullet\colon\tensor{\mathcal{E}}{_(_\alpha_\beta_)}(1,1)\longrightarrow\mathcal{E}_\mathrm{N}(1,1)$ the linearized Nijenhuis operator: $\tensor{(N^\bullet\psi)}{_\alpha_\beta_\gamma}
=2\tensor*{\nabla}{^{\mathrm{TW}}_[_\alpha}\tensor{\psi}{_\beta_]_\gamma}$. Then $\mathcal{O}^\bullet$ can be decomposed as follows, where $B_\theta$ is a certain differential operator given by a universal formula in terms of the Tanaka–Webster connection: $$\label{eq:linearized_obstruction_decomposition}
\mathcal{O}^\bullet=B_\theta N^\bullet.$$ In particular, $\mathcal{O}^\bullet$ vanishes on the space $\ker N^\bullet$ of integrable infinitesimal deformations.
The operators involved here are schematically described as follows, where the weight $(w,w)$ is abbreviated as $(w)$: $$\begin{tikzcd}[column sep=scriptsize]
\mathcal{E}(1) \arrow{r}{D} &
\tensor{\mathcal{E}}{_(_\alpha_\beta_)}(1) \arrow{r}{N^\bullet} \arrow[bend left]{rrrrr}{\mathcal{O}^\bullet} &
\mathcal{E}_\mathrm{N}(1) \arrow[swap,bend right]{rrrr}{B_\theta} & & &
\mathcal{E}_\mathrm{N}(-n+1) \arrow{r}{(N^\bullet)^*} &
\tensor{\mathcal{E}}{_(_\alpha_\beta_)}(-n) \arrow{r}{D^*} &
\mathcal{E}(-n-2).
\end{tikzcd}$$ The two arrows $\mathcal{E}(1)\overset{D}{\longrightarrow}\tensor{\mathcal{E}}{_(_\alpha_\beta_)}(1)
\overset{N^\bullet}{\longrightarrow}\mathcal{E}_\mathrm{N}(1)$ on the left are the first two operators in the CR deformation complex [@Rumin_94; @Akahori_Garfield_Lee_02]; for the operator $D$, which maps the infinitesimal “Kuranishi wiggle” to the resulting infinitesimal change of the CR structure, see also [@Hirachi_Marugame_Matsumoto_inprep]. The two arrows on the right are their formal adjoints. Recall that, in the case of the flat CR structure, there exists a CR-invariant “long” operator from each of the three spaces on the left to the corresponding dual space on the right, which is unique by the composition series of the associated generalized Verma modules (see [@Collingwood_85]\*[Section 8.2]{} and [@Collingwood_88]\*[6.2]{}): $$\begin{tikzcd}[column sep=scriptsize]
\mathcal{E}(1) \arrow{r}{D} \arrow[bend left]{rrrrrrr}{L_1} &
\tensor{\mathcal{E}}{_(_\alpha_\beta_)}(1) \arrow{r}{N^\bullet} \arrow[bend left]{rrrrr}{L_2} &
\mathcal{E}_\mathrm{N}(1) \arrow[bend left]{rrr}{L_3} & & &
\mathcal{E}_\mathrm{N}(-n+1) \arrow{r}{(N^\bullet)^*} &
\tensor{\mathcal{E}}{_(_\alpha_\beta_)}(-n) \arrow{r}{D^*} &
\mathcal{E}(-n-2).
\end{tikzcd}$$ It is known that the composition of any two operators in the latter diagram vanishes if $n\ge 3$, and even when $n=2$ this is still true except for the compositions at $\mathcal{E}_\mathrm{N}(1)$ and $\mathcal{E}_\mathrm{N}(-n+1)$. Part (2) of the theorem above implies that the sequence $\mathcal{E}(1)\longrightarrow\tensor{\mathcal{E}}{_(_\alpha_\beta_)}(1)
\longrightarrow\tensor{\mathcal{E}}{_(_\alpha_\beta_)}(-n)\longrightarrow\mathcal{E}(-n-2)$ remains to be a complex for arbitrary integrable CR manifolds; this should be compared with the work of Branson and Gover [@Branson_Gover_07] in conformal geometry. Moreover, in the flat case, one can check (by our Proposition \[prop:linearized\_obstruction\_heisenberg\]) that the operator $L_2$ locally factors into the composition of $N^\bullet$, some non-CR-invariant differential operator $\mathcal{E}_\mathrm{N}(1)\longrightarrow\mathcal{E}_\mathrm{N}(-n+1)$, and $(N^\bullet)^*$. Part (3) of the theorem can be considered as a partial generalization of this to the curved case.
In Theorem \[thm:obstruction\_operator\_observation\], the nontrivial statements are part (1), in particular the complex-linearity of $\mathcal{O}^\bullet$, and part (3). The other things are more or less easy to see. The first equality of is most obvious among them: since $(\mathcal{O}^\bullet Df)(p)$ depends only on finite jets of $f$ and the CR structure $T^{1,0}M$ at $p\in M$, we can formally embed $M$ to $\mathbb{C}^{n+1}$ (see Kuranishi’s article [@Kuranishi_99] for example), and in this case the assertion is clear because $Df$ always integrates to a genuine deformation of integrable CR structure (see [@Akahori_Garfield_Lee_02; @Hirachi_Marugame_Matsumoto_inprep]). If part (1) is taken as a given, then the second equality of is also clear. Therefore what we should really discuss are parts (1) and (3), and this is done in Subsection \[subsec:further\_properties\]. Also in this subsection, we describe a direct proof of $D^*\mathcal{O}^\bullet=0$ via the Dirichlet-to-Neumann-type characterization of $\mathcal{O}^\bullet$ for the interest of comparing with the proof of [@Matsumoto_14]\*[Theorem 0.2 (3)]{}, from which we knew that the *imaginary part of $D^*\mathcal{O}^\bullet$* vanishes. The reason why we have a stronger conclusion for $\mathcal{O}^\bullet$ here are, firstly we have a Kähler structure for the bulk ACH-Einstein metric in this case, and secondly we are able to characterize $\mathcal{O}^\bullet$ in terms of a PDE associated to this metric and do not have to concern anymore about the bulk metrics for perturbed partially integrable CR structures.
I thank Kengo Hirachi for various discussions and suggestions on the exposition of the paper. I am also grateful to Robin Graham for pointing out that Lemma \[lem:Masakis\_lemma\] in this article is discussed in [@Graham_83]. Regarding this lemma, I acknowledge that I was benefited as well from discussions with Hideaki Hosaka, Masaki Mori, and Masaki Watanabe.
Partially integrable CR manifolds {#sec:partially_integrable}
=================================
Basic definitions
-----------------
Let $(M,T^{1,0}M)$ be a partially integrable CR manifold (that is not necessarily nondegenerate). A measurement of the failure of $(M,T^{1,0}M)$ not being integrable is given by the *Nijenhuis tensor* $N$, which is the real $(2,1)$-tensor over $H=\operatorname{Re}T^{1,0}M$ whose complexification is given by $$N(X,Y)=[X_{1,0},Y_{1,0}]_{0,1}+[X_{0,1},Y_{0,1}]_{1,0},\qquad
X,\ Y\in C^\infty(M,\mathbb{C}H),$$ where the subscripts “$1,0$” and “$0,1$” denote the projections from $\mathbb{C}H=T^{1,0}M\oplus{\overline{T^{1,0}M}}$ onto each summand. If $\set{Z_\alpha}$ is a local frame of $T^{1,0}M$, putting $Z_{{\overline{\alpha}}}={\overline{Z_\alpha}}$ we introduce the index notation with respect to $\set{Z_\alpha,Z_{{\overline{\alpha}}}}$ and its dual coframe $\set{\theta^\alpha,\theta^{{\overline{\alpha}}}}$. Since the Nijenhuis tensor $N$ is a $(1,2)$-tensor, $N$ is represented by a collection of functions indexed with two lower indices and a single upper index. In this case the only components that can be nonzero are $\tensor{N}{_\alpha_\beta^{{\overline{\gamma}}}}$ and their complex conjugates $\tensor{N}{_{{\overline{\alpha}}}_{{\overline{\beta}}}^\gamma}$. Moreover, it is clear from the definition that $\tensor{N}{_\alpha_\beta^{{\overline{\gamma}}}}$ is skew-symmetric in $\alpha$ and $\beta$. Expressing symmetrization (resp. skew-symmetrization) by round (resp. square) brackets, we can write as $$\label{eq:nijenhuis_skew_symmetry}
\tensor{N}{_(_\alpha_\beta_)^{{\overline{\gamma}}}}=0,\qquad\text{or equivalently,}\qquad
\tensor{N}{_[_\alpha_\beta_]^{{\overline{\gamma}}}}=\tensor{N}{_\alpha_\beta^{{\overline{\gamma}}}}.$$ In Penrose’s *abstract index notation* [@Penrose_Rindler_84], which is used throughout this paper, the symbol $\tensor{N}{_\alpha_\beta^{{\overline{\gamma}}}}$ is regarded as denoting the $(T^{1,0}M)^*\otimes(T^{1,0}M)^*\otimes{\overline{T^{1,0}M}}$ part of the tensor $N$ itself, not just its components. Equations is considered as an abstract expression of the skew-symmetry of $N$. Furthermore, in abstract index notation, the vector bundle $(T^{1,0}M)^*\otimes(T^{1,0}M)^*\otimes{\overline{T^{1,0}M}}$ in which $\tensor{N}{_\alpha_\beta^{{\overline{\gamma}}}}$ takes values is denoted by $\tensor{E}{_\alpha_\beta^{{\overline{\gamma}}}}$, and $\tensor{\mathcal{E}}{_\alpha_\beta^{{\overline{\gamma}}}}$ is the space of its sections. By $\tensor{E}{_[_\alpha_\beta_]^{{\overline{\gamma}}}}$ we mean the bundle of tensors with the symmetry as in , and hence $\tensor{N}{_\alpha_\beta^{{\overline{\gamma}}}}\in\tensor{\mathcal{E}}{_[_\alpha_\beta_]^{{\overline{\gamma}}}}$ has the same meaning as .
Suppose that $\theta$ is any (possibly locally-defined) nowhere-vanishing 1-form on $M$ that annihilates $H$. Then the *Levi form* $h$ is defined as follows: $$\label{eq:levi_form}
h(Z,{\overline{W}}):=-i\,d\theta(Z,{\overline{W}})=i\,\theta([Z,{\overline{W}}]),
\qquad\text{$Z$, $W\in C^\infty(M,T^{1,0}M)$}.$$ The Levi form itself depends on $\theta$, but $\Hat\theta=e^\Upsilon\theta$ implies $\Hat{h}=e^\Upsilon h$. Invariantly, we can define the $\mathbb{C}$-linear map $$\label{eq:weighted_levi_form}
T^{1,0}M\otimes{\overline{T^{1,0}M}}\longrightarrow\mathbb{C}(TM/H),
\qquad Z\otimes{\overline{W}}\longmapsto(i[Z,{\overline{W}}]\text{ mod $\mathbb{C}H$}).$$ It is natural to call this $\mathbb{C}(TM/H)$-valued hermitian form the *weighted Levi form* by the reason explained in Subsection \[subsec:density\_bundles\].
A partially integrable CR structure $T^{1,0}M$ is *nondegenerate* if the Levi form is nondegenerate at each point on $M$, which is equivalent to saying that $H$ is a contact distribution. In this case, any choice of $\theta$ is called a *contact form* or a *pseudohermitian structure.* The global existence of a contact form is equivalent to the triviality of $H^\perp\subset T^*M$. A particular example of such situation is when the Levi form has definite signature, in which case $T^{1,0}M$ is *strictly pseudoconvex*. As declared in Introduction, in the sequel we always assume that $T^{1,0}M$ is nondegenerate and $H^\perp$ is oriented.
If a contact form $\theta$ is specified, then by the nondegeneracy of the Levi form, one can lower and raise indices of various tensors. Note that implies $$\label{eq:levi_form_differential}
d\theta=i\,\tensor{h}{_\alpha_{{\overline{\beta}}}}\,\theta^\alpha\wedge\theta^{{{\overline{\beta}}}}\mod\theta.$$ For example, we define $\tensor{N}{_\alpha_\beta_\gamma}:=\tensor{h}{_\gamma_{{\overline{\sigma}}}}\tensor{N}{_\alpha_\beta^{{\overline{\sigma}}}}$. Then by differentiating one can show that $\tensor{N}{_\alpha_\beta_\gamma}+\tensor{N}{_\beta_\gamma_\alpha}+\tensor{N}{_\gamma_\alpha_\beta}=0$. Choosing a contact form $\theta$ also enables us to pick a canonical vector field $T$, called the *Reeb vector field*, that is characterized by $\theta(T)=1$ and $T\contraction d\theta=0$. Note that $T$ is transverse to $H$. If $\set{Z_\alpha}$ is a local frame of $T^{1,0}M$, then the associated *admissible coframe* $\set{\theta^\alpha}$ is the collection of $1$-forms vanishing on $\mathbb{C}T\oplus{\overline{T^{1,0}M}}$ such that $\set{\theta^\alpha|_{T^{1,0}M}}$ is the dual coframe for $\set{Z_\alpha}$. This makes $\set{\theta,\theta^\alpha,\theta^{{\overline{\alpha}}}}$ into the dual coframe for $\set{T,Z_\alpha,Z_{{\overline{\alpha}}}}$. In our index notation of tensors, the index $0$ is used for components corresponding with $T$ or $\theta$.
We next introduce the Tanaka–Webster connection $\nabla$ on a nondegenerate partially integrable CR manifold. Just as in the integrable case, $\nabla$ is a connection of $TM$ characterized by the fact that $H$, $T$, $J$, $h$ are all parallel with respect to $\nabla$ and the torsion tensor $\operatorname{Tor}(X,Y):=\nabla_XY-\nabla_YX-[X,Y]$ satisfies
\[eq:torsion\_tanaka\_webster\] $$\begin{aligned}
{2}
\label{eq:torsion_tanaka_webster_1}
&\operatorname{Tor}(X, JY)-\operatorname{Tor}(JX,Y)=2\,h(X,Y)T, &\qquad &X, Y\in \Gamma(H),\\
\label{eq:torsion_tanaka_webster_2}
&\operatorname{Tor}(T, JX)=-J\operatorname{Tor}(T, X), &\qquad &X\in \Gamma(H).
\end{aligned}$$
This definition leads to the following first structure equation, where $\set{\theta^\alpha}$ is an admissible coframe and $\tensor{\omega}{_\alpha^\beta}$ are the connection forms: $$\label{eq:FirstStructureEquation2}
d\theta^{\gamma}
=\theta^{\alpha}\wedge\tensor{\omega}{_\alpha^\gamma}
-\tensor{A}{_{{\overline{\alpha}}}^\gamma}\theta^{{\overline{\alpha}}}\wedge\theta
-\tfrac{1}{2}\tensor{N}{_{{\overline{\alpha}}}_{{\overline{\beta}}}^\gamma}
\theta^{{\overline{\alpha}}}\wedge\theta^{{\overline{\beta}}}.$$ The tensor $A$ is the *Tanaka–Webster torsion tensor*. Moreover, if $\Pi$ is the curvature of $\nabla$, then the component $\tensor{\Pi}{_\alpha^\beta_\sigma_{{\overline{\tau}}}}$ is called the *Tanaka–Webster curvature tensor* and denoted by $\tensor{R}{_\alpha^\beta_\sigma_{{\overline{\tau}}}}$. The other components of $\Pi$ can be written in terms of $N$, $A$, $R$, and their covariant derivatives.
Density bundles {#subsec:density_bundles}
---------------
Suppose first that we can take an $(n+2)$-nd root of the CR canonical bundle $K=\smash{\bigwedge}^{n+1}({\overline{T^{1,0}M}})^\perp$. We fix such a line bundle $E(-1,0)$ and write its dual $E(1,0)$. We set $$E(w,w'):=E(1,0)^{\otimes w}\otimes \smash{{\overline{E(1,0)}}}^{\otimes w'},\qquad\text{$w$, $w'\in\mathbb{Z}$},$$ and call it the *density bundle of biweight $(w,w')$*. The space of sections of $E(w,w')$ is denoted by $\mathcal{E}(w,w')$, and its elements are called *densities*. Since there is a canonical isomorphism $E(-n-2,0)\cong K$, we can uniquely define a compatible connection $\nabla$ on $E(1,0)$. The bundles and the spaces of density-weighted tensors are indicated by the usual symbols followed by the weight: for example, $\tensor{E}{_\alpha_{{\overline{\beta}}}}(w,w'):=\tensor{E}{_\alpha_{{\overline{\beta}}}}\otimes E(w,w')$ and the space of its sections is $\tensor{\mathcal{E}}{_\alpha_{{\overline{\beta}}}}(w,w')$.
Farris [@Farris_86] observed that, if $\zeta$ is a locally-defined nonvanishing section of $K$, then there is a unique contact form $\theta$ satisfying $$\label{eq:farris_volume_normalization}
\theta\wedge(d\theta)^{n}
=i^{n^{2}}n!(-1)^{q}\theta\wedge(T\contraction\zeta)\wedge(T\contraction{\overline{\zeta}}),$$ where $q$ is the number of the negative eigenvalues of the Levi form. We say that this $\theta$ is *volume-normalized* by $\zeta$. If we replace $\zeta$ with $\lambda\zeta$, where $\lambda\in C^\infty(M,\mathbb{C}^{\times})$, then $\theta$ changes to $\lvert\lambda\rvert^{2/(n+2)}\theta$. We set $${\lvert\zeta\rvert}^{2/(n+2)}=\zeta^{1/(n+2)}\otimes\smash{{\overline{\zeta}}}^{1/(n+2)}\in\mathcal{E}(-1,-1),$$ which is independent of the choice of the $(n+2)$-nd root of $\zeta$. Let $\lvert\zeta\rvert^{-2/(n+2)}\in\mathcal{E}(1,1)$ be its inverse. Then we obtain a CR-invariant section $\bm{\theta}$ of $T^*M\otimes E(1,1)$: $$\bm{\theta}:=\theta\otimes\lvert\zeta\rvert^{-2/(n+2)}.$$ Since $\bm{\theta}$ determines a trivialization of $\mathbb{C}H^\perp\otimes E(1,1)$, there is a canonical identification $$\label{eq:contact_form_density_identification}
\mathbb{C}H^\perp\cong E(-1,-1).$$ This is compatible with any Tanaka–Webster connection $\nabla$ because it is easily observed that $\nabla\bm{\theta}=0$ (see [@Gover_Graham_05]\*[Proposition 2.1]{}). Dually, there is an identification $$\mathbb{C}(TM/H)\cong E(1,1),\qquad (v\text{ mod $\mathbb{C}H$})\longmapsto\bm{\theta}(v).$$ We may use these isomorphisms to *define* $E(w,w)$ even if we cannot take an $(n+2)$-nd root of $K$. Since the Levi form $\tensor{h}{_\alpha_{{\overline{\beta}}}}$ and $\theta$ have the same scaling factor, $$\tensor{\bm{h}}{_\alpha_{{\overline{\beta}}}}:=
\tensor{h}{_\alpha_{{\overline{\beta}}}}\otimes\theta^{-1}\in\tensor{\mathcal{E}}{_\alpha_{{\overline{\beta}}}}(1,1)$$ is a parallel CR-invariant tensor, where $\theta$ is considered as a density in $\mathcal{E}(-1,-1)$ via . This is exactly the weighted Levi form given by . Moreover, $\theta\wedge(d\theta)^n$ multiplies by $e^{(n+1)\Upsilon}$ when $\theta$ is replaced by $e^\Upsilon\theta$, and thus $E(-n-1,-n-1)$ is identified with the bundle of volume densities.
We say that any weighted symmetric tensor $\tensor{\psi}{_\alpha_\beta}\in\tensor{\mathcal{E}}{_(_\alpha_\beta_)}(1,1)$ determines an *infinitesimal deformation of partially integrable CR structure*. This is because, if $T^{1,0}M$ is modified to a new almost CR structure $\Hat{T}^{1,0}=\operatorname{span}\set{\Hat{Z}_\alpha=Z_\alpha+\tensor{\varphi}{_\alpha^{{\overline{\beta}}}}Z_{{\overline{\beta}}}}$ by $\tensor{\varphi}{_\alpha^{{\overline{\beta}}}}\in\tensor{\mathcal{E}}{_\alpha^{{\overline{\beta}}}}$, then $\Hat{T}^{1,0}$ is partially integrable if and only if $\tensor{\varphi}{_\alpha_\beta}=\tensor{\bm{h}}{_\alpha_{{\overline{\gamma}}}}\tensor{\varphi}{_\beta^{{\overline{\gamma}}}}$ is symmetric. In the case where we are given a 1-parameter family $T^{1,0}_t$ of partially integrable CR structures such that $T^{1,0}_0=T^{1,0}M$ and each $T^{1,0}_t$ is described as above by $\tensor*{\varphi}{^t_\alpha_\beta}$ that is smooth in $t$, then we say that the family $T^{1,0}_t$ is *tangent* to the infinitesimal deformation $\tensor{\psi}{_\alpha_\beta}=\tensor*{\varphi}{^\bullet_\alpha_\beta}$. If moreover the original partially integrable CR structure $T^{1,0}M$ is integrable, then the differential $N^\bullet$ of the Nijenhuis tensor is given by, for any choice of a contact form, $$\tensor{(N^\bullet\psi)}{_\alpha_\beta_\gamma}
=2\tensor*{\nabla}{^{\mathrm{TW}}_[_\alpha}\tensor{\psi}{_\beta_]_\gamma},$$ where the last index of $\tensor{(N^\bullet\psi)}{_\alpha_\beta^{{\overline{\gamma}}}}\in\tensor{\mathcal{E}}{_\alpha_\beta^{{\overline{\gamma}}}}$ is lowered by the weighted Levi form.
Summary on asymptotically complex hyperbolic Einstein metrics {#sec:ACHE}
=============================================================
$\Theta$-structures
-------------------
Let $X$ be a $C^\infty$-smooth manifold-with-boundary of dimension $2n+2$. Suppose we are given a section $\Theta\in C^\infty({\partial}X,T^*X|_{{\partial}X})$. We assume the following conditions are satisfied, where $\iota\colon{\partial}X\hookrightarrow X$ is the inclusion map:
1. $\iota^*\Theta$ is a nowhere vanishing 1-form on ${\partial}X$; \[item:nowhere\_vanishing\_condition\]
2. The kernel $H$ of $\theta=\iota^*\Theta$ is a contact distribution on $\partial X$. \[item:contact\_condition\]
A *$\Theta$-structure* on $X$ is a conformal class $[\Theta]$ of elements of $C^\infty({\partial}X,T^*X|_{{\partial}X})$ satisfying (\[item:nowhere\_vanishing\_condition\]) and (\[item:contact\_condition\]) above, and a pair $(X,[\Theta])$ is called a *$\Theta$-manifold*. Any contact form on ${\partial}X$ that belongs to the class $\iota^*[\Theta]$ is called a *compatible contact form*. Note that picking up the conformal class $\iota^*[\Theta]$ amounts to fixing an orientation of $H^\perp\subset T^*{\partial}X$.
If we are given a $\Theta$-manifold $X$, there is a canonical smooth vector bundle ${{\fourIdx{\Theta}{}{}{}{TX}}}$, which we call the *$\Theta$-tangent bundle* of $X$. Over the interior $\mathring{X}$, $({{\fourIdx{\Theta}{}{}{}{TX}}})|_{\mathring{X}}$ is canonically isomorphic to the usual tangent bundle $T\mathring{X}$, while its structure near the boundary is described as follows. Let $p\in{\partial}X$, and $\set{N,T,Y_i}=\set{N,T,Y_1,\dots,Y_{2n}}$ a local frame of $TX$ in a neighborhood $U$ of $p$ such that
- $N|_{{\partial}X}$ is annihilated by $[\Theta]$,
- $T$, $Y_1$, $\dots$, $Y_{2n}$ are tangent to ${\partial}X\cap U$, and
- $\set{Y_1|_{{\partial}X},\dots,Y_{2n}|_{{\partial}X}}$ span the contact distribution $H$.
Then ${{\fourIdx{\Theta}{}{}{}{TX}}}|_U$ is spanned by $\set{\rho N,\rho^2T,\rho Y_i}$, where $\rho\in C^\infty(X)$ is any boundary defining function. The dual vector bundle of ${{\fourIdx{\Theta}{}{}{}{TX}}}$ is denoted by ${{\fourIdx{\Theta}{}{}{}{T^*X}}}$, and sections of tensor products of ${{\fourIdx{\Theta}{}{}{}{TX}}}$’s and ${{\fourIdx{\Theta}{}{}{}{T^*X}}}$’s are called *$\Theta$-tensors* in general. A fiber metric of ${{\fourIdx{\Theta}{}{}{}{TX}}}$ is called a *$\Theta$-metric*.
A virtue of the concept of the $\Theta$-tangent bundle is that the space $\mathcal{V}_\Theta$ of its sections is closed under the Lie bracket. Due to this fact, the Levi-Civita connection associated to any $\Theta$-metric can be naturally considered as a $\Theta$-connection on ${{\fourIdx{\Theta}{}{}{}{TX}}}$. Here we say that $\nabla$ is a *$\Theta$-connection* on a vector bundle $E$ if it is an $\mathbb{R}$-linear mapping $$\nabla\colon C^\infty(X;E)\longrightarrow C^\infty(X;{{\fourIdx{\Theta}{}{}{}{T^*X}}}\otimes E),\qquad
s\longmapsto(v\longmapsto\nabla_vs),$$ satisfying the usual Leibniz rule. As a consequence, the Riemann curvature tensor of a $\Theta$-metric is regarded as a $\Theta$-tensor, and so is the Ricci tensor.
A differential operator on functions is called a *$\Theta$-differential operator* if it is locally expressed as a polynomial in elements of $\mathcal{V}_\Theta$, and the set of such operators is denoted by ${{\fourIdx{\Theta}{}{}{}{\mathrm{Diff}}}}(X)$. If $E$ and $F$ are vector bundles over $X$, then the bundle version ${{\fourIdx{\Theta}{}{}{}{\mathrm{Diff}}}}(X;E,F)$ is similarly defined. As is easily observed, $\Theta$-connections are typical examples of elements of ${{\fourIdx{\Theta}{}{}{}{\mathrm{Diff}}}}(X;E,{{\fourIdx{\Theta}{}{}{}{T^*X}}}\otimes E)$.
ACH metrics {#subsec:ACH_metrics}
-----------
We explain what we call ACH metrics only in a rather practical form; a more intrinsic definition can be found in [@Epstein_Melrose_Mendoza_91; @Guillarmou_SaBarreto_08; @Matsumoto_14]. We start with a partially integrable CR manifold $(M,T^{1,0}M)$. The *standard $\Theta$-structure* on the product manifold $M\times[0,\infty)$ is the class $[\Theta]$ that annihilates the vector field $\partial_\rho=\partial/\partial\rho$, where $\rho$ is the second coordinate of $M\times[0,\infty)$, and pulls back to the given conformal class of contact forms on the boundary $M=M\times\set{0}$. The manifold $X=M\times[0,\infty)$ equipped with the standard $\Theta$-structure is called the *product $\Theta$-manifold*. Suppose we take a contact form $\theta$ on $M$ and a local frame $\set{Z_\alpha}$ of $T^{1,0}M$. Then we write $$\bm{Z}_\infty=\rho\partial_\rho,\qquad
\bm{Z}_0=\rho^2T,\qquad
\bm{Z}_\alpha=\rho Z_\alpha,\qquad\text{and}\qquad
\bm{Z}_{{\overline{\alpha}}}=\rho Z_{{\overline{\alpha}}},$$ where $T$ is the Reeb vector field. The set $\set{\bm{Z}_I}=\set{\bm{Z}_\infty,\bm{Z}_0,\bm{Z}_\alpha,\bm{Z}_{{\overline{\alpha}}}}$ spans the complexified $\Theta$-tangent bundle $\mathbb{C}{{\fourIdx{\Theta}{}{}{}{TX}}}$.
If $U\subset M\times[0,\infty)$ is an open neighborhood of $M$, we say that a $\Theta$-metric $g$ defined on $U$ is an *ACH metric* if, for some choice of $\theta$, the boundary values of its components with respect to $\set{\bm{Z}_I}$ are as follows: $$\begin{aligned}
{5}
\tensor{g}{_\infty_\infty}&=4,&\qquad
\tensor{g}{_\infty_0}&=0,&\qquad
\tensor{g}{_\infty_\alpha}&=0,\\
\tensor{g}{_0_0}&=1,&\qquad
\tensor{g}{_0_\alpha}&=0,&\qquad
\tensor{g}{_\alpha_{{\overline{\beta}}}}&=\tensor{h}{_\alpha_{{\overline{\beta}}}},&\qquad
\tensor{g}{_\alpha_\beta}&=0&\qquad
&\text{on $M$}.\end{aligned}$$ Here $\tensor{h}{_\alpha_{{\overline{\beta}}}}$ is the components of the Levi form with respect to $\set{Z_\alpha}$.
When $X$ is an arbitrary $\Theta$-manifold, by a *compatible partially integrable CR structure* on $M={\partial}X$ we mean a partially integrable CR structure whose underlying contact distribution is the one induced by the $\Theta$-structure. Then we define ACH metrics on $X$ as follows, where an $\Theta$-diffeomorphism between $\Theta$-manifolds means a diffeomorphism that preserves the $\Theta$-structures.
Let $X$ be a $\Theta$-manifold of dimension $2n+2$. Then a $\Theta$-metric $g$ on $X$ is called an *ACH metric* if there exist following:
1. A compatible partially integrable CR structure $T^{1,0}M$ on $M={\partial}X$;
2. An open neighborhood $U$ of $M$ in the product $\Theta$-manifold $M\times[0,\infty)$, an open neighborhood $V$ of $M$ in $X$, and a $\Theta$-diffeomorphism $\Phi\colon U\longrightarrow V$ such that $\Phi|_M=\operatorname{id}_M$ and $\Phi^*g$ is an ACH metric on $U$.
While $\Phi$ is not unique, it is known that $T^{1,0}M$ is determined by $g$. This is called the *CR structure at infinity*.
An ACH metric $g$ on a neighborhood $U$ of $M\subset M\times[0,\infty)$ is called *normalized* if $$\tensor{g}{_\infty_\infty}=4,\qquad
\tensor{g}{_\infty_0}=0,\qquad
\tensor{g}{_\infty_\alpha}=0\qquad
\text{everywhere in $U$}.$$ The following proposition, whose proof is given in [@Guillarmou_SaBarreto_08], makes this terminology reasonable.
\[prop:ACH\_normalization\] Let $X$ be a $\Theta$-manifold and $g$ a $C^\infty$-smooth ACH metric. If $\theta$ is any compatible contact form on $M={\partial}X$, then one can take a $\Theta$-diffeomorphism $\Phi\colon U\longrightarrow V$ so that $\Phi^*g$ is normalized. For any given $\theta$, the germ of $\Phi$ along $M$ is uniquely determined.
This asserts in particular that there is a distinguished (germ of) boundary defining function(s) for each $\theta$. This is called the *model boundary defining function* in [@Guillarmou_SaBarreto_08].
\[ex:complex\_hyperbolic\] The model of ACH metrics is, of course, the complex hyperbolic metric. In the Siegel upper-half space model, it is the Kähler metric $g$ on $\Omega=\set{(z',w)\in\mathbb{C}^n\times\mathbb{C}|\operatorname{Im}w>{\lvertz'\rvert}^2}$ with a global potential $4\log(1/r)$, where $r=\operatorname{Im}w-{\lvertz'\rvert}^2$: $$g=4\frac{\partial^2}{\partial z^i\partial{\overline{z}}^j}\left(\log\frac{1}{r}\right)dz^id{\overline{z}}^j
=4\left(\frac{r_ir_{{\overline{j}}}}{r^2}-\frac{r_{i{\overline{j}}}}{r}\right)dz^id{\overline{z}}^j.$$ The boundary of $\Omega$ is the Heisenberg group $\mathcal{H}$, which is identified with $\mathbb{C}^n\times\mathbb{R}$ by $(z',w)\longmapsto(z',\operatorname{Re}w)$. Extending this identification, we consider the following diffeomorphism between $\overline{\Omega}$ and $\mathcal{H}\times[0,\infty)=\mathbb{C}^n\times\mathbb{R}\times[0,\infty)$: $$\overline{\Omega}\longrightarrow\mathbb{C}^n\times\mathbb{R}\times[0,\infty),\qquad
(z',w)\longmapsto(z',\operatorname{Re}w,r=\operatorname{Im}w-{\lvertz'\rvert}^2).$$ Let $\Phi$ be the inverse of this mapping: $\Phi(z',t,r)=(z',t+i(r+{\lvertz'\rvert}^2))$. Then, $$\Phi^*(r_idz^i)=\Phi^*\left(-\sum_{\alpha=1}^n{\overline{z}}^\alpha dz^\alpha+\frac{1}{2i}dw\right)
=\frac{1}{2}dr-\frac{i}{2}dt
-\frac{1}{2}\sum_{\alpha=1}^n({\overline{z}}^\alpha dz^\alpha-z^\alpha d{\overline{z}}^\alpha)
=\frac{1}{2}dr-i\theta,$$ where $\theta$ is the standard contact form. Therefore, by setting $\rho=\sqrt{r/2}$ we obtain $$\Phi^*g=\frac{dr^1}{r^2}+4\frac{\theta^2}{r^2}
+\frac{4}{r}\sum_{\alpha=1}^ndz^\alpha d{\overline{z}}^\alpha
=4\frac{d\rho^2}{\rho^2}+\frac{\theta^2}{\rho^4}
+\frac{2}{\rho^2}\sum_{\alpha=1}^ndz^\alpha d{\overline{z}}^\alpha.$$ This shows that $g$ is an ACH metric on $\overline{\Omega}$ with $C^\infty$-structure replaced by the one that $\set{(z',t,\rho)}$ defines, and the CR structure at infinity is the standard one.
\[ex:Bergman\_type\] We had to take the square root of $r$ in the example above. This generalizes to the square root construction of Epstein, Melrose, and Mendoza [@Epstein_Melrose_Mendoza_91]. If $\Omega$ is a domain in a complex manifold $\mathcal{N}$ with $C^\infty$-smooth Levi nondegenerate boundary, then we can canonically define a $\Theta$-manifold $X$ called the *square root* of $\overline{\Omega}$, which is
- ${\overline{\Omega}}$ with $C^\infty$-structure replaced by adjoining the square roots of $C^\infty$-smooth boundary defining functions; and
- equipped with the $\Theta$-structure given by the pullback of $\frac{i}{2}(\partial r-{\overline{\partial}}r)|_{T{\partial}\Omega}$ by the identity map $\iota\colon X\longrightarrow{\overline{\Omega}}$.
The map $\iota$ is $C^\infty$-smooth but not vice versa; nevertheless it gives diffeomorphisms $\mathring{X}\cong\Omega$ and ${\partial}X\cong{\partial}\Omega$. Let $g$ be any Bergman-type metric on $\Omega$: $$\label{eq:Bergman_type_metric}
g=4\frac{\partial^2}{\partial z^i\partial{\overline{z}}^j}\left(\log\frac{1}{r}\right)dz^id{\overline{z}}^j,$$ where $r$ is a boundary defining function with respect to the original $C^\infty$-structure of $\overline{\Omega}$ that is at least $C^2$. Then $g$ can be naturally interpreted as an ACH metric on $X$. The CR structure at infinity ${\partial}X=M$ is exactly the integrable CR structure induced by the complex structure of $\mathcal{N}$.
Approximate solutions to the Einstein equation {#subsec:approximate_ACHE}
----------------------------------------------
Now we describe the existence and uniqueness result of approximate solutions of the Einstein equation under the assumption of $C^\infty$ boundary regularity.
\[thm:existence\] Let $X$ be a $\Theta$-manifold of dimension $2n+2$ and $T^{1,0}M$ a compatible partially integrable CR structure on $M={\partial}X$. Then there exists a $C^\infty$-smooth ACH metric $g$ on $X$ with CR structure at infinity $T^{1,0}M$ for which is satisfied. Such a metric $g$ is, up to $\Theta$-diffeomorphism actions that restrict to the identity on $M$, unique modulo $O(\rho^{2n+2})$ symmetric 2-$\Theta$-tensors with $O(\rho^{2n+3})$ traces.
This is essentially [@Matsumoto_14]\*[Theorem 1.1]{} (see Theorem 6.2 in the same article for the uniqueness), but the approximate Einstein condition is slightly modified. We only sketch the proof of this version here as the complete one can be found in [@Matsumoto_13_Thesis]. It is sufficient by Proposition \[prop:ACH\_normalization\] to consider the case where $g$ is defined near the boundary of $M\times[0,\infty)$ and is normalized. One proves under this assumption the existence of a metric $g$ satisfying and the uniqueness up to $O(\rho^{2n+2})$ symmetric 2-$\Theta$-tensors with $O(\rho^{2n+3})$ traces. This is done inductively: we perturb $g$ by an $O(\rho^m)$ 2-$\Theta$-tensor $\psi$, and compute the change $\Psi$ of $E:=\operatorname{Ric}+\frac{n+2}{2}g$ modulo $O(\rho^{m+1})$. The conclusion is that, if $1\le m\le 2n+1$, the correspondence $\psi\longmapsto\Psi$ is a one-to-one $\mathbb{R}$-linear mapping from $\mathcal{S}_m/\mathcal{S}_{m+1}$ onto itself, where $\mathcal{S}_m$ denotes the space of $C^\infty$-smooth symmetric 2-$\Theta$-tensors that are $O(\rho^m)$. As a result we obtain an ACH metric $g$ for which holds. Then one fixes the trace of $g$ modulo $O(\rho^{2n+3})$ by requiring , which needs a bit subtler observation of the map $\psi\longmapsto\Psi$.
Let $g$ be a $C^\infty$-smooth ACH metric satisfying . If $\theta$ is a compatible contact form, then for the normalization of $g$ with respect to $\theta$, we set $E=\rho^{2n+2}\tilde{E}$. Then $\tensor{\tilde{E}}{_\alpha_\beta}|_M=\tilde{E}(\bm{Z}_\alpha,\bm{Z}_\beta)|_M$ is uniquely determined by the CR structure at infinity and $\theta$, and it invariantly defines the *CR obstruction tensor* $\tensor{\mathcal{O}}{_\alpha_\beta}\in\tensor{\mathcal{E}}{_(_\alpha_\beta_)}(-n,-n)$.
The approximate solution $g$ is necessarily even (up to the ambiguous terms) in the sense of Guillarmou and Sà Barreto [@Guillarmou_SaBarreto_08]\*[Section 3.2]{}. This is because remains to be satisfied even if we formally replace $\rho$ with $-\rho$ in the expansion of $g$. However, we do not need the evenness in our subsequent discussion.
In particular, consider the case in which $M$ is the boundary of a domain $\Omega$ in $\mathbb{C}^{n+1}$. Then the Bergman-type metric given by Fefferman’s approximate solution $r$ to the complex Monge–Ampère equation [@Fefferman_76] satisfies, if considered as a $\Theta$-metric on the square root $X$ of ${\overline{\Omega}}$, $$\label{eq:approx_Einstein_Kahler}
\operatorname{Ric}=-\frac{n+2}{2}g+O(\rho^{2n+4}).$$ Hence we have $\tensor{\mathcal{O}}{_\alpha_\beta}=0$ in this case. Moreover, since $\tensor{\mathcal{O}}{_\alpha_\beta}$ admits an expression in terms of the Tanaka–Webster connection, we conclude by formal embedding that $\tensor{\mathcal{O}}{_\alpha_\beta}=0$ holds for an arbitrary integrable CR manifold. For details, see [@Matsumoto_14]\*[Proposition 5.5]{}.
Dirichlet problems and volume expansion {#sec:dirichlet_prob}
=======================================
Laplacian on functions
----------------------
We consider an arbitrary $C^\infty$-smooth normalized ACH metric $g$ defined near the boundary of a $(2n+2)$-dimensional product $\Theta$-manifold $X=M\times[0,\infty)$. The purpose of this subsection is to prove the following formula of the Laplacian of $g$.
\[prop:ACH\_Laplacian\] The Laplacian of a $C^\infty$-smooth ACH metric $g$ normalized with respect to $\theta$ is a $\Theta$-differential operator of the form $$\label{eq:ACHLaplacianWithSublaplacian}
\Delta=-\frac{1}{4}(\rho\partial_\rho)^2+\frac{n+1}{2}\rho\partial_\rho
+\rho^2\Delta_b-\rho^4T^2+\rho\Psi,
\qquad \Psi\in{{\fourIdx{\Theta}{}{}{}{\mathrm{Diff}}}}(X),$$ where $\Delta_b$ is the sub-Laplacian and $T$ is the Reeb vector field associated to $\theta$.
We start with the following expression of $g$, where $k$ is a 2-tensor over the subbundle whose complexification is spanned by $\set{\bm{Z}_0,\bm{Z}_\alpha,\bm{Z}_{{\overline{\alpha}}}}$. Here, $\set{\bm{\theta},\bm{\theta}^\alpha,\bm{\theta}^{{\overline{\alpha}}}}$ denotes the dual coframe: $$g=4\frac{d\rho^2}{\rho^2}+k,\qquad
k=\bm{\theta}^2+2\tensor{h}{_\alpha_{{\overline{\beta}}}}\bm{\theta}^\alpha\bm{\theta}^{{\overline{\beta}}}+O(\rho).$$ If we identify $({{\fourIdx{\Theta}{}{}{}{TX}}})|_{\mathring{X}}$ with $T\mathring{X}$, then on each hypersurface $M_\rho=M\times\set{\rho}$, $k$ gives a Riemannian metric of $M_\rho$. By the standard identification $M_\rho\cong M$, we can regard $k$ as a 1-parameter family $k_\rho$ of Riemannian metrics on $M$. In this sense, by abusing the notation we write $$\label{eq:normalized_ACH}
g=4\frac{d\rho^2}{\rho^2}+k_\rho.$$ Let $\nabla^{k_\rho}$ be the Levi-Civita connection of $k_\rho$ and $\Delta^{k_\rho}\colon C^\infty(M)\longrightarrow C^\infty(M)$ the associated Laplacian on functions. Then, as stated in [@Guillarmou_SaBarreto_08]\*[Equation (5.1)]{}, $\Delta$ is given by $$\label{eq:ACHLaplacian}
\Delta=-\frac{1}{4}(\rho\partial_\rho)^2+\frac{n+1}{2}\rho\partial_\rho+\Delta^{k_\rho}
-\frac{1}{8}\rho\partial_\rho(\log{\lvert\det k_\rho\rvert})\rho\partial_\rho,$$ where $\det k_\rho$ is the determinant of the matrix representing $k_\rho$ with respect to $\set{\rho^2T,\rho Y_i}$. Therefore, to show Proposition \[prop:ACH\_Laplacian\], it suffices to verify the following lemma.
\[lem:parametrized\_laplacian\] In the situation above, $\Delta^{k_\rho}$ is a $\Theta$-differential operator and is expressed as $$\label{eq:Laplacian_of_metric_on_level_sets}
\Delta^{k_\rho}=\rho^2\Delta_b-\rho^4T^2+\rho\Psi,
\qquad\Psi\in{{\fourIdx{\Theta}{}{}{}{\mathrm{Diff}}}}(X).$$
We compute using a local frame $\set{\bm{Z}_i}=\set{\bm{Z}_0,\bm{Z}_\alpha,\bm{Z}_{{\overline{\alpha}}}}$ of $TM\cong TM_\rho$. Let us introduce the tensor $K$ defined by $$\nabla^{k_\rho}_{\bm{Z}_i}\bm{Z}_j=\nabla^\mathrm{TW}_{\bm{Z}_i}\bm{Z}_j+\tensor{K}{^k_i_j}\bm{Z}_k,$$ which measures the difference of $\nabla^{k_\rho}$ and the Tanaka–Webster connection $\nabla^\mathrm{TW}$. Then we obtain $$\Delta^{k_\rho}f
=-\tensor{(k_\rho^{-1})}{^i^j}\tensor*{\nabla}{^{\mathrm{TW}}_i}\tensor*{\nabla}{^{\mathrm{TW}}_j}f
-\tensor{(k_\rho^{-1})}{^i^j}\tensor{(k_\rho^{-1})}{^k^l}\tensor{K}{_k_i_j}
\tensor*{\nabla}{^{\mathrm{TW}}_l}f,
\qquad f\in C^\infty(M),$$ where the upper index of $K$ is lowered by $k_\rho$. The first term of the right-hand side expands as $\rho^2\Delta_bf-\rho^4T^2f+\rho\Psi'f$, where $\Psi'\in{{\fourIdx{\Theta}{}{}{}{\mathrm{Diff}}}}(X)$. On the other hand, one can show that $$\tensor{K}{_k_i_j}=
\frac{1}{2}(\tensor*{\nabla}{^{\mathrm{TW}}_i}\tensor{(k_\rho)}{_j_k}
+\tensor*{\nabla}{^{\mathrm{TW}}_j}\tensor{(k_\rho)}{_i_k}
-\tensor*{\nabla}{^{\mathrm{TW}}_k}\tensor{(k_\rho)}{_i_j}
-\tensor{\operatorname{Tor}}{_k_i_j}+\tensor{\operatorname{Tor}}{_i_j_k}+\tensor{\operatorname{Tor}}{_j_i_k}),$$ where $\tensor{\operatorname{Tor}}{^k_i_j}$ is the torsion of the Tanaka–Webster connection and again the first index is lowered by $k_\rho$. Since $\tensor{(k_\rho^{-1})}{^i^j}\tensor{(k_\rho^{-1})}{^k^l}
(\tensor{\operatorname{Tor}}{_k_i_j}-\tensor{\operatorname{Tor}}{_i_j_k}-\tensor{\operatorname{Tor}}{_j_i_k})$ equals $-2\tensor{(k_\rho^{-1})}{^k^l}\tensor{\operatorname{Tor}}{^j_j_k}$ and $\tensor{\operatorname{Tor}}{^j_j_k}$ is actually zero, we have $$\label{eq:diff_LC_TW_contracted}
\tensor{(k_\rho^{-1})}{^i^j}\tensor{(k_\rho^{-1})}{^k^l}\tensor{K}{_k_i_j}=
\frac{1}{2}\tensor{(k_\rho^{-1})}{^i^j}\tensor{(k_\rho^{-1})}{^k^l}
(\tensor*{\nabla}{^{\mathrm{TW}}_i}\tensor{(k_\rho)}{_j_k}
+\tensor*{\nabla}{^{\mathrm{TW}}_j}\tensor{(k_\rho)}{_i_k}
-\tensor*{\nabla}{^{\mathrm{TW}}_k}\tensor{(k_\rho)}{_i_j}).$$ Since $\nabla^\mathrm{TW}$ annihilates $\bm{\theta}^2+2\tensor{h}{_\alpha_{{\overline{\beta}}}}\bm{\theta}{^\alpha}\bm{\theta}{^{{\overline{\beta}}}}$, the right-hand side vanishes at $\rho=0$, from which we conclude that holds.
Construction of $P^g_{2k}$ and $Q^g_\theta$
-------------------------------------------
Proposition \[prop:ACH\_Laplacian\] provides sufficient knowledge of $\Delta$ to analyze our Dirichlet-type problems.
\[thm:dn\_operator\] Let $g$ be a $C^\infty$-smooth ACH metric on a $(2n+2)$-dimensional $\Theta$-manifold $X$, and $T^{1,0}M$ its CR structure at infinity. We take a contact form $\theta$ on $M$ and $\rho$ denotes the associated model boundary defining function. Let $k$ be a positive integer. Then, for any real-valued function $f\in C^\infty(M)$, there exists $u\in C^\infty(\mathring{X})$ of the form $$\label{eq:form_of_generalized_eigenfunction}
u=\rho^{n+1-k}F+\rho^{n+1+k}\log\rho\cdot G,\qquad F,G\in C^\infty(X),\qquad F|_{\partial X}=f$$ that solves $$\label{eq:formal_eigenfunction_equation}
\left(\Delta-\frac{(n+1)^2-k^2}{4}\right)u=O(\rho^\infty).$$ The function $F$ is unique modulo $O(\rho^{2k})$, and $G$ is unique modulo $O(\rho^\infty)$. Moreover, there is a differential operator $P^g_{2k}\colon C^\infty(M)\longrightarrow C^\infty(M)$ determined by $g$ and $\theta$ such that $$\label{eq:definition_of_P}
G|_M=c_kP^g_{2k}f,\qquad c_k=\frac{2\cdot(-1)^{k+1}}{k!(k-1)!}$$ with the form $$\label{eq:principal_part_of_GJMS}
P^g_{2k}=\prod_{j=0}^{k-1}(\Delta_b+i(k-1-2j)T)+(\text{an operator with Heisenberg order $\le 2k-1$}),$$ where $\Delta_b$ is the sub-Laplacian and $T$ is the Reeb vector field. The operators $P^g_{2k}$ are formally self-adjoint, and $P^g_{2n+2}$ annihilates constant functions.
By Proposition \[prop:ACH\_normalization\], we may assume that $g$ is normalized. For any $s\in\mathbb{R}$, let $$\Delta_s:=\Delta-s(n+1-s).$$ We consider $F\in C^\infty(M\times[0,\infty))$ with expansion $$F\sim\sum_{j=0}^\infty\rho^jf^{(j)},\qquad f^{(j)}\in C^\infty(M),$$ and try to solve $\Delta_s(\rho^{2(n+1-s)}F)=O(\rho^\infty)$ by determining $f^{(j)}$’s. Proposition \[prop:ACH\_Laplacian\] implies $$\Delta_s(\rho^{2(n+1-s)+j}f^{(j)})
\sim\rho^{2(n+1-s)+j}\cdot\left(-\frac{1}{4}j(j-4s+2n+2)f^{(j)}+\rho D^j_sf^{(j)}\right),$$ where $D^j_s$ is a formal power series in $\rho$ with coefficients in the space of linear differential operators on $M$. This formula is used to determine the expansion of $F$. First we set $f^{(0)}$ to be the given function $f$. If we write $F_0=f^{(0)}=f$, then $$\Delta_s(\rho^{2(n+1-s)}F_0)=\rho^{2(n+1-s)}(0+\rho D_{0,s}f)=O(\rho^{2(n+1-s)+1}).$$ We inductively define $f^{(j)}$, as far as $j-4s+2n+2\not=0$, by $$\frac{1}{4}j(j-4s+2n+2)f^{(j)}=
(\text{the $\rho^{2(n+1-s)+j}$-coefficient of $\Delta_s(\rho^{2(n+1-s)}F_{j-1})$}),$$ and set $F_j:=F_{j-1}+\rho^jf^{(j)}$ so that $\Delta_sF_j=O(\rho^{2(n+1-s)+j+1})$. Then $F_j$ is written as follows using some linear differential operators $p_{l,s}$ on $M$: $$\label{eq:Expansion_F_with_p}
F_j=f+\rho p_{1,s}f+\rho^2p_{2,s}f+\dots+\rho^{j}p_{j,s}f.$$ If we furthermore set $p_{0,s}:=1$, then $p_{j,s}$ is recursively given by $$\label{eq:recursive_formula_p}
p_{j,s}
=\frac{4}{j(j-4s+2n+2)}\sum_{l=0}^{j-1}(\text{the $\rho^{j-1-l}$-coefficient of $D^l_s$})p_{l,s}.$$
In our situation, $s$ is taken to be $(n+1+k)/2$, which implies that $4s-2n-2=2k$ is a positive integer. Hence the procedure above only works while $j<2k$. As a result, $F_{2k-1}$ is determined so that $\Delta_s(\rho^{n+1-k}F_{2k-1})=O(\rho^{n+1+k})$. In the next step, we have seen that in general we cannot solve $\Delta_s(\rho^{n+1-k}F_{2k})=O(\rho^{n+1+k+1})$ by polynomials in $\rho$, and so here we need the first logarithmic term. For $g^{(j)}\in C^\infty(M)$, we have $$\begin{gathered}
\Delta_{(n+1+k)/2}(\rho^{n+1+k+j}\log\rho\cdot g^{(j)})
=-\frac{1}{4}j(j+2k)\rho^{n+1+k+j}\log\rho\cdot g^{(j)}
-\frac{1}{2}(j+k)\rho^{n+1+k+j}g^{(j)}\\
\mod \rho^{n+1+k+j+2}(C^\infty(M\times[0,\infty))+\log\rho\cdot C^\infty(M\times[0,\infty))).
\end{gathered}$$ So we can uniquely take $g^{(0)}$ so that $\Delta_s(\rho^{n+1-k}F_{2k-1}+\rho^{n+1+k}\log\rho\cdot g^{(0)})=O(\rho^{n+1+k})$ holds: $$\label{eq:LowestTerm_G}
g^{(0)}:=\frac{2}{k}\tilde{p}_{2k,(n+1+k)/2}f,\qquad\text{where}\quad
\tilde{p}_{2k,s}=\sum_{l=0}^{2k-1}(\text{the $\rho^{2k-1-l}$-coefficient of $D^l_s$})p_{l,s}.$$ We set $G_0:=g^{(0)}$ and $F_{2k}:=F_{2k-1}+\rho^{2k}f^{(2k)}$ by choosing $f^{(2k)}\in C^\infty(M)$ arbitrarily. Then $\Delta_s(\rho^{n+1-k}F_{2k}+\rho^{n+1+k}\log\rho\cdot G_0)=O(\rho^{n+1+k})$, or more precisely, $$\Delta_s(\rho^{n+1-k}F_{2k}+\rho^{n+1+k}\log\rho\cdot G_0)=\rho^{n+1+k}(R_0+\log\rho\cdot S_0),$$ where $R_0$, $S_0\in C^\infty(M\times[0,\infty))$ vanish on the boundary. Now we can continue the induction to determine $F_{2k+j}$ and $G_j$, which are polynomials in $\rho$ of degrees $2k+j$ and $j$, respectively, by adding higher-order terms to $F_{2k}$ and $G_0$ so that $$\Delta_s(\rho^{n+1-k}F_{2k+j}+\rho^{n+1+k}\log\rho\cdot G_j)=\rho^{n+1+k}(R_j+\log\rho\cdot S_j)$$ with some $R_j$, $S_j\in C^\infty(M\times[0,\infty))$ that are $O(\rho^{j+1})$, which are again uniquely achieved. By Borel’s Lemma, we obtain $F$ and $G\in C^\infty(M\times[0,\infty))$ such that $$\Delta_s(\rho^{n+1-k}F+\rho^{n+1+k}\log\rho\cdot G)=O(\rho^\infty).$$ Thus we obtain a solution of the form , and the ambiguity lives only in $f_{2k}$.
Self-adjointness of $P^g_{2k}$ can be shown by, as [@Fefferman_Graham_02]\*[Proposition 3.3]{}, looking at the logarithmic term in the expansion of the integral of $\braket{du_1,du_2}-\frac{1}{4}((n+1)^2-k^2)u_1u_2$, where $u_1$ and $u_2$ solve . We omit the details. The fact that $P^g_{2n+2}1=0$ is immediate from the definition because $u=1$ is of course harmonic.
Formula is shown as follows, as in [@Graham_83]. If we write $\rho D^j_s=\rho^2\Delta_b-\rho^4T^2+\tilde{D}^j_s$, then by Proposition \[prop:ACH\_Laplacian\], the Heisenberg order of the $\rho^w$-coefficient of $\tilde{D}^j_s$ is less than $w$. Hence, if we set $$p_{2l,s}=c_{2l,s}P_{2l,s},\qquad\text{where}\quad
c_{2l,s}:=\prod_{\nu=0}^{l-1}\frac{1}{(l-\nu)(l-\nu-2s+n+1)},$$ then modulo differential operators of Heisenberg order $\le 2l-1$, $$\begin{split}
P_{2l,(n+1+k)/2}
&=\Delta_bP_{2l-2,(n+1+k)/2}-(l-1)(k-l+1)(iT)^2P_{2l-4,(n+1+k)/2}.
\end{split}$$ Since $P^g_{2k}$ equals $P_{2k,(n+1+k)/2}$ by , the proof of is reduced to the next lemma.
\[lem:Masakis\_lemma\] Let $k$ be a fixed nonnegative integer. If we define the polynomials $q_l\in\mathbb{C}[x,y]$ by $$q_0=1,\qquad
q_1=x,\qquad
q_l=xq_{l-1}-(l-1)(k-l+1)y^2q_{l-2},$$ then $q_k=\prod_{j=0}^{k-1}(x+(k-1-2j)y)$.
For the proof of Lemma \[lem:Masakis\_lemma\], see [@Graham_83]\*[2.5. Proposition]{} or [@Graham_84]\*[Section 1]{}.
One can also observe by the proof of Theorem \[thm:dn\_operator\] above that the operator $P_{2k,s}$ is a polynomial with respect to $s$. This allows us to construct $Q^g_\theta$ by the Graham–Zworski argument. Here we do not take this route, as explained in Introduction, and prove the following theorem instead.
\[thm:log\_expansion\_definition\_of\_Q\] Let $g$ be a $C^\infty$-smooth ACH metric on a $(2n+2)$-dimensional $\Theta$-manifold $X$, and $T^{1,0}M$ its CR structure at infinity. We take a contact form $\theta$ on $M$ and $\rho$ denotes the associated model boundary defining function. Then there exists $U\in C^\infty(\mathring{X})$ of the form $$\label{eq:log_expansion}
U=\log\rho+A+B\rho^{2n+2}\log\rho,\qquad
A, B\in C^\infty(X),\qquad
A|_M=0$$ such that $$\label{eq:equation_for_log_expansion}
\Delta U=(n+1)/2\mod O(\rho^\infty).$$ The function $A$ is unique modulo $O(\rho^{2n+2})$ and $B$ is unique modulo $O(\rho^\infty)$. Moreover, we define $Q^g_\theta\in C^\infty(M)$ by $$B|_M=\frac{(-1)^n}{n!(n+1)!}Q^g_\theta.$$ Then, for any two contact forms $\theta$ and $\Hat{\theta}=e^\Upsilon\theta$, the following holds: $$\label{eq:Qg_transformation}
Q^g_{\Hat{\theta}}=e^{-(n+1)\Upsilon}(Q^g_\theta+P^g_{2n+2}\Upsilon).$$
Again we may assume that $g$ is normalized. By Proposition \[prop:ACH\_Laplacian\], $\Delta\log\rho$ is $C^\infty$-smooth up to the boundary and $(\Delta\log\rho)|_M=(n+1)/2$. Moreover, if $a$, $b\in C^\infty(M)$, then $$\label{eq:ACHLaplacianActingOnSmoothTerm}
\Delta(\rho^ja)=-\frac{1}{4}j(j-2n-2)\rho^ja+O(\rho^{j+1})$$ and $$\label{eq:ACHLaplacianActingOnLogTerm}
\Delta(\rho^j\log\rho\cdot b)=
-\frac{1}{2}(j-n-1)\rho^jb+O(\rho^{j+1})-\frac{1}{4}\log\rho\cdot(j(j-2n-2)\rho^jb+O(\rho^{j+1})),$$ where the terms denoted by $O(\rho^{j+2})$ are all $C^\infty$-smooth. By using inductively, we can show that there is a unique finite expansion $$A_{2n+1}=\sum_{j=1}^{2n+1}\rho^ja^{(j)},\qquad a^{(j)}\in C^\infty(M)$$ such that $\Delta(\log\rho+F_{2n+1})=(n+1)/2+O(\rho^{2n+2})$. The next thing to do is to introduce a $(\rho^{2n+2}\log\rho)$-term so that $\Delta$ applied to it kills the $\rho^{2n+2}$-coefficient of the error term. This is possible in view of because $j-n-1$ is nonzero for $j=2n+2$: there uniquely exists $b^{(0)}\in C^\infty(M)$ for which $$\Delta(\log\rho+A_{2n+1}+\rho^{2n+2}\log\rho\cdot b^{(0)})=\frac{n+1}{2}+O(\rho^{2n+3}\log\rho).$$ We set $B_0=b^{(0)}$ while $A_{2n+2}=A_{2n+1}+\rho^{2n+2}a^{(2n+2)}$, where we choose $a^{(2n+2)}\in C^\infty(M)$ arbitrarily. Equation implies that $\Delta(\log\rho+A_{2n+2}+\rho^{2n+2}\log\rho\cdot B_0)=(n+1)/2+O(\rho^{2n+3}\log\rho)$ holds. Then, inductively in $k$, we can determine $a^{(2n+2+k)}$, $b^{(k)}\in C^\infty(M)$ such that $$A_{2n+2+k}=A_{2n+2+k-1}+\rho^{2n+2+k}a^{(2n+2+k)}\qquad\text{and}\qquad
B_k=B_{k-1}+\rho^{2n+2+k}\log\rho\cdot b^{(k)}$$ satisfies $\Delta(\log\rho+A_{2n+2+k}+\rho^{2n+2}\log\rho\cdot B_k)=(n+1)/2+O(\rho^{2n+2+k+1}\log\rho)$. Finally, by Borel’s Lemma, we obtain $A$, $B\in C^\infty(M\times[0,\infty))$ for which $\Delta(2\log\rho+A+\rho^{2n+2}\log\rho\cdot B)=n+1+O(\rho^\infty)$. The construction shows the ambiguity of $A$ and $B$ are as stated.
To show we first remark that $\tilde{\Upsilon}|_M=\Upsilon$ if $\Hat{\rho}=e^{\tilde{\Upsilon}/2}\rho$ is the model boundary defining function for the new contact form $\Hat{\theta}=e^\Upsilon\theta$. Let $\Hat{U}=\log\Hat{\rho}+\Hat{A}+\Hat{B}\Hat{\rho}^{2n+2}\log\Hat{\rho}$ be a solution of associated to $\Hat{\rho}$. Then $$\Hat{U}
=\log\rho+\frac{\tilde{\Upsilon}}{2}+\Hat{A}
+e^{(n+1)\tilde{\Upsilon}}\Hat{B}\rho^{2n+2}\left(\log\rho+\frac{\tilde{\Upsilon}}{2}\right)$$ and hence $$\Hat{U}-U
=\left(\frac{\tilde{\Upsilon}}{2}+\Hat{A}
+\frac{1}{2}e^{(n+1)\tilde{\Upsilon}}\rho^{2n+2}\Hat{B}\tilde{\Upsilon}\right)
+\rho^{2n+2}\log\rho\cdot(e^{(n+1)\tilde{\Upsilon}}\Hat{B}-B).$$ Since $\Hat{U}-U$ solves $\Delta(\Hat{U}-U)=O(\rho^\infty)$, by Theorem \[thm:dn\_operator\], we have $$e^{(n+1)\Upsilon}\Hat{B}|_M-B|_M=2^{-1}c_{n+1}P^g_{2n+2}\Upsilon,$$ or equivalently $e^{(n+1)\Upsilon}Q^g_{\Hat{\theta}}-Q^g_\theta=P^g_{2n+2}\Upsilon$, which is .
Volume expansion
----------------
We relate the integral of $Q^g_\theta$ with the volume expansion of $g$. Since $dV_g$ extends to a nowhere vanishing section of the $\Theta$-volume bundle ${\lvert\det{{\fourIdx{\Theta}{}{}{}{T^*X}}}\rvert}$, it diverges at the rate of $\rho^{-2n-3}$ at ${\partial}X$ as the usual volume density on $X$. Hence if $M$ is compact, for some arbitrarily fixed $\varepsilon_0$, the volume of the subset $\set{\varepsilon\le\rho\le\varepsilon_0}\subset X$ has the following asymptotic behavior when $\varepsilon\to 0$: $$\label{eq:volume_expansion}
\operatorname{Vol}(\set{\varepsilon\le\rho\le\varepsilon_0})
=\sum_{j=-2n-2}^{-1}c_j\varepsilon^j+L\log\frac{1}{\varepsilon}+O(1).$$
\[prop:volume\_log\_term\_and\_Q\] Let $M$ be compact, and $g$ a $C^\infty$-smooth ACH metric normalized with respect to $\theta$. Then the coefficient $L$ in is given by $$\label{eq:VolumeLogTermAndQ}
L=\frac{2\cdot(-1)^{n+1}}{n!^2(n+1)!}\smash{\overline{Q}}^g,\qquad
\smash{\overline{Q}}^g:=\int_MQ^g_\theta\,\theta\wedge(d\theta)^n=n!\int_MQ^g_\theta\,dV_{\theta^2+h}.$$
We use Green’s formula. Let $g=4\rho^{-2}d\rho^2+k_\rho$, and we define the family $\tilde{k}_\rho$ of Riemannian metrics on $M$ by $$\tilde{k}_\rho=\delta_\rho^*k_\rho,\qquad\text{where}\quad
\delta_\rho T=\rho^2 T\quad\text{and}\quad \delta_\rho Y_i=\rho Y_i.$$ Then $\tilde{k}_\rho$ smoothly extends to $\rho=0$ and $\tilde{k}_0=\theta^2+h$. Since the outward unit normal vector field along the hypersurface $M_\varepsilon=\set{\rho=\varepsilon}$ is $-\frac{1}{2}\varepsilon\partial_\rho$, for any $U\in C^2(X)$ $$\int_{\varepsilon\le\rho\le\varepsilon_0}\Delta U\,dV_g
=\int_{M_\varepsilon}\frac{1}{2}\varepsilon(\partial_\rho U) dV_{k_\varepsilon}+O(1)
=\frac{1}{2}\varepsilon^{-2n-1}\int_{M_\varepsilon}(\partial_\rho U)dV_{\tilde{k}_\varepsilon}+O(1).$$ Now let $U$ be a solution of the Dirichlet problem in Theorem \[thm:log\_expansion\_definition\_of\_Q\]. Then since $\Delta U=(n+1)/2+O(\rho^\infty)$, the equality above reads $$\frac{n+1}{2}\operatorname{Vol}(\set{\varepsilon\le\rho\le\varepsilon_0})
=\frac{1}{2}\varepsilon^{-2n-1}\int_{M_\varepsilon}(\partial_\rho U)dV_{\tilde{k}_\varepsilon}+O(1).$$ Comparing the coefficients of $\log(1/\varepsilon)$ we can see $$\frac{n+1}{2}L=-(n+1)\int_M(B|_M)dV_{\theta^2+h}
=\frac{(-1)^{n+1}}{n!^2}\int_MQ^g_\theta\,dV_{\theta^2+h}.$$ Thus we obtain .
GJMS operators and $Q$-curvature
================================
Invariance of $P^g_{2k}$ and $Q^g_\theta$ for approximately Einstein ACH metrics {#subsec:GJMS}
--------------------------------------------------------------------------------
Now we show Theorem \[thm:dn\_operator\_of\_ACHE\] by applying the results of the previous section to approximate ACH-Einstein metrics $g$. In order to carry out this idea, we have to discuss the dependence of $\Delta$ on the ambiguity of $g$ to ensure that it is not problematic for our construction.
\[lem:dependence\_of\_ACH\_laplacian\] Let $(M,T^{1,0}M)$ be a partially integrable CR manifold, and $g$ a $C^\infty$-smooth normalized ACH metric satisfying the approximate Einstein condition . If $F\in C^\infty(X)$, then $\Delta F$ modulo $O(\rho^{2n+3})$ is irrelevant to the ambiguity of $g$. Moreover, if $\rho$ is a $C^\infty$-smooth boundary defining function, then $\Delta(\log\rho)$ is irrelevant modulo $O(\rho^{2n+3}\log\rho)$ to the ambiguity of $g$.
By , $\tensor{(k_\rho^{-1})}{^i^j}\tensor{(k_\rho^{-1})}{^k^l}\tensor{K}{_k_i_j}\tensor{\nabla}{_l}F$ is uniquely determined modulo $O(\rho^{2n+3})$. Since $\tensor{(k_\rho^{-1})}{^i^j}$ is determined modulo $O(\rho^{2n+2})$ and $\nabla_i\nabla_jF$ is $O(\rho)$, $\tensor{(k_\rho^{-1})}{^i^j}\nabla_i\nabla_jF$ is also determined modulo $O(\rho^{2n+3})$. Thus we conclude by Lemma \[lem:parametrized\_laplacian\] that $\Delta^{k_\rho}F$ is determined up to the error of $O(\rho^{2n+3})$. Furthermore, the trace condition imposed on $g$ implies that ${\lvert\det k_\rho\rvert}$ is determined modulo $O(\rho^{2n+3})$, and hence so is $\rho\partial_\rho(\log{\lvert\det k_\rho\rvert})$. Therefore, by Proposition \[prop:ACH\_Laplacian\], $\Delta F$ is determined modulo $O(\rho^{2n+3})$. The second statement is shown similarly.
Recall the Dirichlet-type problem in Theorem \[thm:dn\_operator\]. Lemma \[lem:dependence\_of\_ACH\_laplacian\] and the proof of Theorem \[thm:dn\_operator\] show that, if $k\le n+1$, then the correspondence of $F|_M=f$ and $G|_M$ depends only on $(M,T^{1,0}M)$ and $\theta$, and not on the ambiguity of $g$. Therefore, the operators $P^g_{2k}$ for such $k$ are determined invariantly. Likewise, the function $B|_M$ in Theorem \[thm:log\_expansion\_definition\_of\_Q\] depends only on $(M,T^{1,0}M)$ and $\theta$, and so is $Q^g_\theta$.
First variational formula of $\overline{Q}$ {#subsec:variational_formula}
-------------------------------------------
We prove Theorem \[thm:first\_variational\_formula\] using the characterization of $\overline{Q}$ given in Proposition \[prop:volume\_log\_term\_and\_Q\]. The key to the proof is introducing the first logarithmic term to our $C^\infty$-smooth approximate ACH-Einstein metric $g^\mathrm{smooth}$, namely, one satisfying . Let $g^\mathrm{smooth}=4\rho^{-2}d\rho^2+k_\rho$ be normalized by $\theta$, and $\tensor{\mathcal{O}}{_\alpha_\beta}$ the CR obstruction tensor trivialized by $\theta$. We define the new ACH metric $g$ by correcting $\tensor*{g}{^{\mathrm{smooth}}_\alpha_\beta}$ with the additional term $4(n+1)^{-1}\cdot\tensor{\mathcal{O}}{_\alpha_\beta}\rho^{2n+2}\log\rho$; or, in the matrix form with respect to $\set{\bm{Z}_I}=\set{\bm{Z}_\infty,\bm{Z}_0,\bm{Z}_\alpha,\bm{Z}_{{\overline{\alpha}}}}$, $$g=\begin{pmatrix}
4 & 0 & 0 & 0 \\
0 & 1 & \tensor{(k_\rho)}{_0_\beta} & \tensor{(k_\rho)}{_0_{{\overline{\beta}}}} \\
0 & \tensor{(k_\rho)}{_\alpha_0}
& \tensor{(k_\rho)}{_\alpha_\beta}+\frac{4}{n+1}\tensor{\mathcal{O}}{_\alpha_\beta}\rho^{2n+2}\log\rho
& \tensor{(k_\rho)}{_\alpha_{{\overline{\beta}}}} \\
0 & \tensor{(k_\rho)}{_{{\overline{\alpha}}}_0} & \tensor{(k_\rho)}{_{{\overline{\alpha}}}_\beta}
& \tensor{(k_\rho)}{_{{\overline{\alpha}}}_{{\overline{\beta}}}}
+\frac{4}{n+1}\tensor{\mathcal{O}}{_{{\overline{\alpha}}}_{{\overline{\beta}}}}\rho^{2n+2}\log\rho \\
\end{pmatrix}.$$ Then the vanishing order of the $(\alpha\beta)$-component of $E={\operatorname{Ric}}+\frac{n+2}{2}g$ improves to $O(\rho^{2n+3}\log\rho)$; see [@Matsumoto_14]\*[Equations (8.2) and (8.7)]{}.
Let $T_t^{1,0}$ be a smooth 1-parameter family of partially integrable CR structures that is tangent to the given infinitesimal deformation $\tensor{\psi}{_\alpha_\beta}\in\tensor{\mathcal{E}}{_(_\alpha_\beta_)}(1,1)$. Then the construction of $g$ above shows that we can take a family $g^t$ of such ACH metrics, each of which is associated to $T_t^{1,0}$ and satisfies $E^t=\operatorname{Ric}(g^t)+\frac{n+2}{2}g^t=O(\rho^{2n+3}\log\rho)$, so that the coefficients of each components of $g_t$ smoothly depend on the parameter $t$. Let $\set{Z_\alpha}$ be a local frame of the original partially integrable CR structure $T_0^{1,0}=T^{1,0}M$. If we compute with $\set{\bm{Z}_0,\bm{Z}_\alpha,\bm{Z}_{{\overline{\alpha}}}}$, where $\bm{Z}_\alpha=\rho Z_\alpha$, then the derivatives of the components of $k_\rho^t$ at $t=0$, which we write $k_\rho^\bullet$, are $$\begin{aligned}
{2}
\tensor*{(k_\rho^\bullet)}{_0_0}&=O(\rho),&\qquad
\tensor*{(k_\rho^\bullet)}{_0_\alpha}&=O(\rho),\\
\tensor*{(k_\rho^\bullet)}{_\alpha_{{{\overline{\beta}}}}}&=O(\rho),&\qquad
\tensor*{(k_\rho^\bullet)}{_\alpha_\beta}&=-2\tensor{\psi}{_\alpha_\beta}+O(\rho).
\end{aligned}$$ Now we start with the fact that there is a uniform estimate on the scalar curvature of $g^t$: $$\operatorname{Scal}^t=-(n+1)(n+2)+O(\rho^{2n+3}).$$ From this we can see that $$\int_{\varepsilon\le\rho\le\varepsilon_0}\operatorname{Scal}^\bullet dV_g=O(1)\qquad\text{as $\varepsilon\to 0$}.$$ On the other hand, the well-known formula of the first variation of the scalar curvature implies $$\label{eq:variation_of_scalar_curvature}
\begin{split}
\operatorname{Scal}^\bullet
&=\tensor{(g^\bullet)}{_I_J_,^I^J}-\tensor{(g^\bullet)}{_I^I_,_J^J}
-\tensor{\operatorname{Ric}}{^I^J}\tensor*{g}{^\bullet_I_J}\\
&=\tensor{(g^\bullet)}{_I_J_,^I^J}-\tensor{(g^\bullet)}{_I^I_,_J^J}
+\frac{1}{2}(n+2)\tensor{g}{^I^J}\tensor*{g}{^\bullet_I_J}+O(\rho^{2n+4}\log\rho).
\end{split}$$ Since $dV_g^\bullet=\frac{1}{2}\tensor{g}{^I^J}\tensor*{g}{^\bullet_I_J}dV_g$, integrates to $$\label{eq:integral_of_variation_of_scalar_curvature}
\int_{\varepsilon\le\rho\le\varepsilon_0}
(\tensor{(g^\bullet)}{_I_J_,^I^J}-\tensor{(g^\bullet)}{_I^I_,_J^J})dV_g
+(n+2)\int_{\varepsilon\le\rho\le\varepsilon_0}dV^\bullet_g=O(1).$$ The unit outward normal vector for $g$ along $\set{\rho=\varepsilon}$ is $\nu=-\frac{1}{2}\varepsilon\partial_\rho$. Therefore, implies $$\label{eq:VariationOfVolume}
\begin{split}
(n+2)\operatorname{Vol}(\set{\varepsilon\le\rho\le\varepsilon_0})^\bullet
&=-\int_{\rho=\varepsilon}
(\tensor{(g^\bullet)}{_I_J_,^I}-\tensor{(g^\bullet)}{_I^I_,_J})\tensor{\nu}{^J}d\sigma+O(1)\\
&=
\int_M(\tensor{(g^\bullet)}{_I_J_,^I}-\tensor{(g^\bullet)}{_I^I_,_J})\cdot
\frac{1}{2}\varepsilon\tensor{\delta}{_\infty^J}\cdot\varepsilon^{-2n-2}dV_{k_\varepsilon}+O(1)\\
&=\frac{1}{2}\varepsilon^{-2n-1}\int_M(\tensor{(g^\bullet)}{_I_\infty_,^I}
-\tensor{(g^\bullet)}{_I^I_,_\infty})
dV_{k_\varepsilon}+O(1).
\end{split}$$ We compare the $\log(1/\varepsilon)$-terms of the both sides of . That of the left-hand side is obviously $(n+2)L^\bullet$. As for the right-hand side, we use $$\label{eq:volume_variation_integrand}
\tensor{(g^\bullet)}{_I_\infty_,^I}-\tensor{(g^\bullet)}{_I^I_,_\infty}
=-\frac{1}{2}\tensor{(k_\rho^{-1})}{^i^j}\tensor{(k_\rho^{-1})}{^k^l}
\tensor{{(k'_\rho)}}{_j_l}\tensor{{(k_\rho^\bullet)}}{_i_k}
-(\tensor{(k_\rho^{-1})}{^i^j}\tensor{{(k_\rho^\bullet)}}{_i_j})',$$ where the primes denotes differentiations in $\rho$. Since $(\det k_\rho)^\bullet=(\det k_\rho)\tensor{(k_\rho^{-1})}{^i^j}\tensor{{(k_\rho^\bullet)}}{_i_j}$, we conclude that $\tensor{(k_\rho^{-1})}{^i^j}\tensor{{(k_\rho^\bullet)}}{_i_j}$ contains no $(\rho^{2n+2}\log\rho)$-term, which implies that the $\log(1/\varepsilon)$-term of the variation of the volume expansion may come only from the first term of the right-hand side of . The logarithmic term that appears in the expansion of $\tensor{{(k'_\rho)}}{_j_l}$ is $8\mathcal{O}\cdot\rho^{2n+1}\log\rho$, and hence, modulo logarithm-free terms, $$\begin{split}
&-\frac{1}{2}\tensor{(k_\rho^{-1})}{^i^j}\tensor{(k_\rho^{-1})}{^k^l}
\tensor{{(k'_\rho)}}{_j_l}\tensor{{(k_\rho^\bullet)}}{_i_k}\\
&= -4\rho^{2n+1}\log\rho\cdot
\tensor{(k_\rho^{-1})}{^{{\overline{\alpha}}}^\beta}\tensor{(k_\rho^{-1})}{^{{\overline{\gamma}}}^\sigma}
\tensor{\mathcal{O}}{_\beta_\sigma}\tensor{{(k_\rho^\bullet)}}{_{{\overline{\alpha}}}_{{\overline{\gamma}}}}
+(\text{the complex conjugate})+O(\rho^{2n+3}\log\rho)\\
&= 8\rho^{2n+1}\log\rho\cdot
(\tensor{\mathcal{O}}{^\alpha^\beta}\tensor{\psi}{_\alpha_\beta}
+\tensor{\mathcal{O}}{^{{\overline{\alpha}}}^{{\overline{\beta}}}}
\tensor{\psi}{_{{\overline{\alpha}}}_{{\overline{\beta}}}})+O(\rho^{2n+3}\log\rho).
\end{split}$$ Therefore the $\log(1/\varepsilon)$-coefficient of the right-hand side of is $-4$ times the integral of $(\tensor{\mathcal{O}}{^\alpha^\beta}\tensor{\psi}{_\alpha_\beta}
+\tensor{\mathcal{O}}{^{{\overline{\alpha}}}^{{\overline{\beta}}}}\tensor{\psi}{_{{\overline{\alpha}}}_{{\overline{\beta}}}})dV_{\theta^2+h}$. Thus we conclude that $$L^\bullet=-\frac{4}{n+2}\int_M
(\tensor{\mathcal{O}}{^\alpha^\beta}\tensor{\psi}{_\alpha_\beta}
+\tensor{\mathcal{O}}{^{{\overline{\alpha}}}^{{\overline{\beta}}}}\tensor{\psi}{_{{\overline{\alpha}}}_{{\overline{\beta}}}})
dV_{\theta^2+h}.$$ By combining this with Proposition \[prop:volume\_log\_term\_and\_Q\], we obtain .
Linearized obstruction operator
===============================
Asymptotic Kählerity
--------------------
Although ACH metrics are not Kähler in general, there is an asymptotic Kähler-like phenomenon about them, which introduces some insight into our computation in the next subsection.
Let $g=4\rho^{-2}d\rho^2+k_\rho$ be a $C^\infty$-smooth ACH metric normalized with respect to $\theta$. Let $\set{Z_\alpha}$ be a local frame of $T^{1,0}M$, and take the associated local frame $\set{\bm{Z}_I}=\set{\bm{Z}_\infty,\bm{Z}_0,\bm{Z}_\alpha,\bm{Z}_{{\overline{\alpha}}}}$ of ${{\fourIdx{\Theta}{}{}{}{TX}}}$. We define the complex subbundle $({{\fourIdx{\Theta}{}{}{}{TX}}}|_M)^{1,0}$ of $\mathbb{C}{{\fourIdx{\Theta}{}{}{}{TX}}}|_M$ to be the one spanned by $\bm{Z}_\alpha|_M$ and $\bm{Z}_\tau|_M$, where $$\bm{Z}_\tau:=\frac{1}{2}\bm{Z}_\infty+i\bm{Z}_0\qquad\text{and}\qquad
\bm{Z}_{{\overline{\tau}}}:={\overline{\bm{Z}_\tau}}=\frac{1}{2}\bm{Z}_\infty-i\bm{Z}_0.$$ Then $({{\fourIdx{\Theta}{}{}{}{TX}}}|_M)^{1,0}$ is actually independent of $\theta$ with which we normalize $g$. The complex structure endomorphism of ${{\fourIdx{\Theta}{}{}{}{TX}}}|_M$ with $i$-eigenbundle $({{\fourIdx{\Theta}{}{}{}{TX}}}|_M)^{1,0}$ is denoted by $J|_M$.
In the sequel, $\set{\bm{Z}_P}=\set{\bm{Z}_\tau,\bm{Z}_\alpha,\bm{Z}_{{\overline{\tau}}},\bm{Z}_{{\overline{\alpha}}}}$ is mainly used instead of $\set{\bm{Z}_I}=\set{\bm{Z}_\infty,\bm{Z}_0,\bm{Z}_\alpha,\bm{Z}_{{\overline{\alpha}}}}$. The next lemma shows the efficiency of this approach. From now on, the indices $P$, $Q$, $R$, $\dots$ run $\set{\tau, 1, \dots, n, {\overline{\tau}},{\overline{1}},\dots,{\overline{n}}}$, while $A$, $B$, $C$, $\dots$ run $\set{\tau, 1, \dots, n}$. The barred indices ${\overline{A}}$, ${\overline{B}}$, ${\overline{C}}$, $\dots$ of course run $\set{{\overline{\tau}},{\overline{1}},\dots,{\overline{n}}}$. We also make an agreement that $\tensor{h}{_\alpha_{{\overline{\beta}}}}$ denotes $h(Z_\alpha,Z_{{\overline{\beta}}})$; it is not $h(\bm{Z}_\alpha,\bm{Z}_{{\overline{\beta}}})$ (which does not even make sense). Therefore, $$\tensor{g}{_\tau_{{\overline{\tau}}}}=2,\qquad\tensor{g}{_\tau_{{\overline{\alpha}}}}=0,\qquad
\tensor{g}{_\alpha_{{\overline{\beta}}}}=\tensor{h}{_\alpha_{{\overline{\beta}}}}\qquad\text{on $M$},$$ and $\tensor{g}{_A_B}$ vanishes on the boundary.
\[prop:asymptotic\_kahlerity\] Under the situation above, let $J\in C^\infty(M,\operatorname{End}({{\fourIdx{\Theta}{}{}{}{TX}}}))$ be any $(1,1)$-$\Theta$-tensor on $X$ that extends $J|_M$. Then the $(2,1)$-$\Theta$-tensor $\nabla J$ vanishes on $M$.
Let $\tensor{\omega}{_Q^R}=\tensor{\Gamma}{^R_P_Q}\bm{\theta}^P$ be the connection 1-$\Theta$-forms of $\nabla$ with respect to the local frame $\set{\bm{Z}_P}$, where $\set{\bm{\theta}^P}$ denotes the dual coframe. Then $\tensor{\Gamma}{_R_P_Q}:=\tensor{g}{_R_S}\tensor{\Gamma}{^S_P_Q}$ are given by $$\label{eq:Levi_Civita}
\tensor{\Gamma}{_R_P_Q}
=\frac{1}{2}(\bm{Z}_P\tensor{g}{_Q_R}+\bm{Z}_Q\tensor{g}{_P_R}-\bm{Z}_R\tensor{g}{_P_Q}
+[\bm{Z}_P,\bm{Z}_Q]_R-[\bm{Z}_P,\bm{Z}_R]_Q-[\bm{Z}_Q,\bm{Z}_R]_P),$$ where $[\bm{Z}_P,\bm{Z}_Q]^R=\bm{\theta}^R([\bm{Z}_P,\bm{Z}_Q])$ and $[\bm{Z}_P,\bm{Z}_Q]_R=\tensor{g}{_R_S}[\bm{Z}_P,\bm{Z}_Q]^S$. Since $\bm{Z}_PF=O(\rho)$ for any function $F\in C^\infty(X)$, we have $$\tensor{\Gamma}{_R_P_Q}
=\frac{1}{2}([\bm{Z}_P,\bm{Z}_Q]_R-[\bm{Z}_P,\bm{Z}_R]_Q-[\bm{Z}_Q,\bm{Z}_R]_P)
\qquad\text{on $M$}.$$ By explicit computation we obtain $$\begin{aligned}
{3}
[\bm{Z}_\tau,\bm{Z}_\tau]&=0,&\qquad
[\bm{Z}_\tau,\bm{Z}_\alpha]&=\frac{1}{2}\bm{Z}_\alpha,&\qquad
[\bm{Z}_\alpha,\bm{Z}_\beta]&=0,\\
[\bm{Z}_\tau,\bm{Z}_{{\overline{\tau}}}]&=-(\bm{Z}_\tau-\bm{Z}_{{\overline{\tau}}}),&\qquad
[\bm{Z}_\tau,\bm{Z}_{{\overline{\alpha}}}]&=\frac{1}{2}\bm{Z}_{{\overline{\alpha}}},&\qquad
[\bm{Z}_\alpha,\bm{Z}_{{\overline{\beta}}}]
&=-\frac{1}{2}\tensor{h}{_\alpha_{{\overline{\beta}}}}(\bm{Z}_\tau-\bm{Z}_{{\overline{\tau}}})
\qquad\text{on $M$},
\end{aligned}$$ and we can conclude from this that the Christoffel symbols of the type $\tensor{\Gamma}{_C_P_B}$ all vanish on $M$. Hence $\tensor{\omega}{_B^{{\overline{C}}}}=0$ on $M$, which is equivalent to $\nabla J$ being zero on $M$.
We record here the boundary values of the remaining Christoffel symbols: on $M$,
\[eq:Christoffel\] $$\begin{aligned}
{4}
\tensor{\Gamma}{^\tau_\tau_\tau}&=-1,&\qquad
\tensor{\Gamma}{^\tau_{{\overline{\tau}}}_\tau}&=1,&\qquad
\tensor{\Gamma}{^\tau_\alpha_\tau}&=0,&\qquad
\tensor{\Gamma}{^\tau_{{\overline{\alpha}}}_\tau}&=0,\\
\tensor{\Gamma}{^\gamma_\tau_\tau}&=0,&\qquad
\tensor{\Gamma}{^\gamma_{{\overline{\tau}}}_\tau}&=0,&\qquad
\tensor{\Gamma}{^\gamma_\alpha_\tau}&=-\tensor{\delta}{_\alpha^\gamma},&\qquad
\tensor{\Gamma}{^\gamma_{{\overline{\alpha}}}_\tau}&=0,\\
\tensor{\Gamma}{^\tau_\tau_\beta}&=0,&\qquad
\tensor{\Gamma}{^\tau_{{\overline{\tau}}}_\beta}&=0,&\qquad
\tensor{\Gamma}{^\tau_\alpha_\beta}&=0,&\qquad
\tensor{\Gamma}{^\tau_{{\overline{\alpha}}}_\beta}&=\tfrac{1}{2}\tensor{h}{_\beta_{{\overline{\alpha}}}},\\
\tensor{\Gamma}{^\gamma_\tau_\beta}&=-\tfrac{1}{2}\tensor{\delta}{_\beta^\gamma},&\qquad
\tensor{\Gamma}{^\gamma_{{\overline{\tau}}}_\beta}&=\tfrac{1}{2}\tensor{\delta}{_\beta^\gamma},&\qquad
\tensor{\Gamma}{^\gamma_\alpha_\beta}&=0,&\qquad
\tensor{\Gamma}{^\gamma_{{\overline{\alpha}}}_\beta}&=0.\end{aligned}$$
We give an observation on the curvature tensor of $g$, which can be seen as a good reason for the name “asymptotically complex hyperbolic metric.”
\[prop:asymptotic\_complex\_hyperbolicity\] Let $R$ be the Riemann curvature tensor, regarded as a $\Theta$-tensor, of a normalized $C^\infty$-smooth ACH metric $g$. Among its components with respect to $\set{\bm{Z}_P}$, the only ones that are non-vanishing on $M$ are those of Kähler-curvature type, i.e., $\tensor{R}{_A_{{\overline{B}}}_C_{{\overline{D}}}}$, $\tensor{R}{_{{\overline{B}}}_A_C_{{\overline{D}}}}$, $\tensor{R}{_A_{{\overline{B}}}_{{\overline{D}}}_C}$, and $\tensor{R}{_{{\overline{B}}}_A_{{\overline{D}}}_C}$. Moreover, the boundary value of $\tensor{R}{_A_{{\overline{B}}}_C_{{\overline{D}}}}$ is given as follows: $$\label{eq:curvature_of_ACH}
\left.(\tensor{R}{_A_{{\overline{B}}}_C_{{\overline{D}}}})\right|_M
=-\frac{1}{2}\left.(\tensor{g}{_A_{{\overline{B}}}}\tensor{g}{_C_{{\overline{D}}}}
+\tensor{g}{_A_{{\overline{D}}}}\tensor{g}{_C_{{\overline{B}}}})\right|_M.$$
Lemma \[prop:asymptotic\_kahlerity\] implies that the curvature $R$, seen as an $\operatorname{End}({{\fourIdx{\Theta}{}{}{}{TX}}})$-valued 2-$\Theta$-form, respects $J$ on $M$. Hence $\tensor{R}{_P_Q_C_D}$ and $\tensor{R}{_P_Q_{{\overline{C}}}_{{\overline{D}}}}$ vanish on $M$, and by the curvature symmetry, only the Kähler-curvature type components survive on $M$. The proof of is given by a direct computation using .
Dirichlet-to-Neumann-type characterization of $\mathcal{O}^\bullet$
-------------------------------------------------------------------
Let $\sigma$ be an arbitrary symmetric 2-$\Theta$-tensor on $X$ equipped with a $C^\infty$-smooth ACH metric $g$. Taking the normalization of $g$ with respect to a contact form $\theta$, we obtain a 2-tensor in $\tensor{\mathcal{E}}{_(_\alpha_\beta_)}$ whose components are $\sigma(\bm{Z}_\alpha,\bm{Z}_\beta)|_M$. If we consider another contact form $\Hat{\theta}=e^{\Upsilon}\theta$ and the local frame $\set{\Hat{\bm{Z}}_P}
=\set{\Hat{\bm{Z}}_\tau,\Hat{\bm{Z}}_\alpha,\Hat{\bm{Z}}_{{\overline{\tau}}},\Hat{\bm{Z}}_{{\overline{\alpha}}}}$ associated to the normalization with respect to $\Hat{\theta}$, then since the model boundary defining functions are related as $\log(\Hat{\rho}/\rho)|_M=\Upsilon/2$, it holds that $$\sigma(\Hat{\bm{Z}}_\alpha,\Hat{\bm{Z}}_\beta)|_M=e^\Upsilon\cdot\sigma(\bm{Z}_\alpha,\bm{Z}_\beta)|_M.$$ Therefore, $\sigma(\bm{Z}_\alpha,\bm{Z}_\beta)|_M$ is more naturally interpreted as defining a weighted tensor in $\tensor{\mathcal{E}}{_(_\alpha_\beta_)}(1,1)$, which we write $\tensor{(\sigma|_M)}{_\alpha_\beta}$. Moreover, if $\sigma=O(\rho^{2k})$, then since $$\Hat{\rho}^{-2k}\sigma(\Hat{\bm{Z}}_\alpha,\Hat{\bm{Z}}_\beta)|_M
=e^{(1-k)\Upsilon}\cdot\rho^{-2k}\sigma(\bm{Z}_\alpha,\bm{Z}_\beta)|_M,$$ $\rho^{-2k}\sigma(\bm{Z}_\alpha,\bm{Z}_\beta)|_M$ is considered to define a weighted tensor that belongs to $\tensor{\mathcal{E}}{_(_\alpha_\beta_)}(1-k,1-k)$, which we write $\tensor{(\rho^{-2k}\sigma|_M)}{_\alpha_\beta}$.
\[prop:lichnerowicz\_equation\] Let $g$ be a $C^\infty$-smooth ACH metric on a $(2n+2)$-dimensional $\Theta$-manifold $X$. Then there exists, for any $\tensor{\psi}{_\alpha_\beta}\in\tensor{\mathcal{E}}{_(_\alpha_\beta_)}(1,1)$, a real $C^\infty$-smooth symmetric 2-$\Theta$-tensor $\sigma$ such that $\tensor{(\sigma|_M)}{_\alpha_\beta}=\tensor{\psi}{_\alpha_\beta}$ and $$\label{eq:linearized_Einstein_equation}
(\Delta_\mathrm{L}+n+2)\sigma=O(\rho^{2n+2}).$$ The $\Theta$-tensor $\sigma$ is uniquely determined modulo $O(\rho^{2n+2})$ by these conditions, and it is automatically approximately trace-free and divergence-free: $$\label{eq:transverse_tracelessness}
\operatorname{tr}\sigma=O(\rho^{2n+2}),\qquad
\delta\sigma=O(\rho^{2n+2}).$$ Furthermore, suppose that $g$ satisfies the approximate Einstein condition . In this case, for any $\sigma$ satisfying , if we write $(\Delta_\mathrm{L}+n+2)\sigma=\tilde{\sigma}$, then $\tensor{(\rho^{-2n-2}\tilde{\sigma}|_M)}{_\alpha_\beta}=-\tensor*{\mathcal{O}}{^\bullet_\alpha_\beta}$, where $\tensor*{\mathcal{O}}{^\bullet_\alpha_\beta}=\tensor{(\mathcal{O}^\bullet\psi)}{_\alpha_\beta}$ is the variation of the obstruction tensor.
Note that, if $\sigma$ satisfies and , then $$\label{eq:linearized_Einstein_equation_restated}
\operatorname{Ric}^\bullet\sigma+\frac{1}{2}(n+2)\sigma
=\frac{1}{2}((\Delta_\mathrm{L}+n+2)\sigma-\mathcal{K}\mathcal{B}\sigma)=O(\rho^{2n+2}),$$ where $\operatorname{Ric}^\bullet$ denotes the linearized Ricci operator and $\mathcal{K}$, $\mathcal{B}$ are the Killing and Bianchi operators: $$\tensor{(\mathcal{K}\eta)}{_P_Q}:=2\tensor{\nabla}{_(_P}\tensor{\eta}{_Q_)}\qquad\text{and}\qquad
\tensor{(\mathcal{B}\sigma)}{_P}
:=\tensor{(\delta\sigma)}{_P}+\frac{1}{2}\tensor{\nabla}{_P}(\operatorname{tr}\sigma).$$
To prove Proposition \[prop:lichnerowicz\_equation\], we need the following lemma.
\[lem:Laplacian\_on\_tensors\] Let $g$ be a $C^\infty$-smooth normalized ACH metric and $j\ge 0$ an integer. If $\mu$ is an $O(\rho^j)$ 1-$\Theta$-form, then the components of $(\Delta_\mathrm{H}+n+2)\mu$ with respect to the local frame $\set{\bm{Z}_P}$, where $\Delta_\mathrm{H}$ is the Hodge Laplacian, are
\[eq:Hodge\_Laplacian\_on\_1-forms\] $$\begin{aligned}
(\Delta_\mathrm{H}+n+2)\tensor{\mu}{_\tau}
&=-\tfrac{1}{4}(j+2)(j-2n-4)\tensor{\mu}{_\tau}+O(\rho^{j+1}),\\
(\Delta_\mathrm{H}+n+2)\tensor{\mu}{_\alpha}
&=-\tfrac{1}{4}\left(j^2-(2n+2)j-2n-7\right)\tensor{\mu}{_\alpha}+O(\rho^{j+1}).
\end{aligned}$$
If $\sigma$ is an $O(\rho^j)$ symmetric 2-$\Theta$-form, then the components of $\delta\sigma$ are
\[eq:divergence\_of\_2-form\] $$\begin{aligned}
\tensor{(\delta\sigma)}{_\tau}
&=-\tfrac{1}{4}(j-2n-4)\tensor{\sigma}{_\tau_\tau}-\tfrac{1}{4}(j-2n)\tensor{\sigma}{_\tau_{{\overline{\tau}}}}
-\operatorname{tr}_h\tensor{\sigma}{_\alpha_{{\overline{\beta}}}}+O(\rho^{j+1}),\\
\tensor{(\delta\sigma)}{_\alpha}
&=-\tfrac{1}{4}(j-2n-5)\tensor{\sigma}{_\tau_\alpha}
-\tfrac{1}{4}(j-2n-1)\tensor{\sigma}{_{{\overline{\tau}}}_\alpha}+O(\rho^{j+1}).
\end{aligned}$$
Under the same assumption, the components of $(\Delta_\mathrm{L}+n+2)\sigma$ are
\[eq:Lichnerowicz\_Laplacian\_of\_2-forms\] $$\begin{aligned}
(\Delta_\mathrm{L}+n+2)\tensor{\sigma}{_\tau_\tau}
&=-\tfrac{1}{4}(j+2)(j-2n-4)\tensor{\sigma}{_\tau_\tau}+O(\rho^{j+1}),\\
(\Delta_\mathrm{L}+n+2)\tensor{\sigma}{_\tau_\alpha}
&=-\tfrac{1}{4}\left(j^2-(2n+2)j-2n-7\right)\tensor{\sigma}{_\tau_\alpha}+O(\rho^{j+1}),\\
(\Delta_\mathrm{L}+n+2)\tensor{\sigma}{_\alpha_\beta}
&=-\tfrac{1}{4}j(j-2n-2)\tensor{\sigma}{_\alpha_\beta}+O(\rho^{j+1}),\\
(\Delta_\mathrm{L}+n+2)\tensor{\sigma}{_\tau_{{\overline{\tau}}}}
&=-\tfrac{1}{4}(j+2)(j-2n-4)\tensor{\sigma}{_\tau_{{\overline{\tau}}}}+O(\rho^{j+1}),\\
(\Delta_\mathrm{L}+n+2)\tensor{\sigma}{_\tau_{{\overline{\alpha}}}}
&=-\tfrac{1}{4}\left(j^2-(2n+2)j-2n-7\right)\tensor{\sigma}{_\tau_{{\overline{\alpha}}}}+O(\rho^{j+1}),\\
\operatorname{tr}_h(\Delta_\mathrm{L}+n+2)\tensor{\sigma}{_\alpha_{{\overline{\beta}}}}
&=-\tfrac{1}{4}(j+2)(j-2n-4)\operatorname{tr}_h\tensor{\sigma}{_\alpha_{{\overline{\beta}}}}+O(\rho^{j+1}),\\
\operatorname{tf}_h(\Delta_\mathrm{L}+n+2)\tensor{\sigma}{_\alpha_{{\overline{\beta}}}}
&=-\tfrac{1}{4}\left(j^2-(2n+2)j-8\right)\operatorname{tf}_h\tensor{\sigma}{_\alpha_{{\overline{\beta}}}}+O(\rho^{j+1}).
\end{aligned}$$
Here, $\operatorname{tr}_h$ and $\operatorname{tf}_h$ denote the trace and the trace-free part with respect to $\tensor{h}{_\alpha_{{\overline{\beta}}}}$.
We first explain that it suffices to assume $j=0$. Let $\nu=\rho^j\tilde{\nu}$ be any $O(\rho^j)$ $\Theta$-tensor. Then $\nabla\nu=\rho^j\nabla\tilde{\nu}+j\rho^jd(\log\rho)\otimes\tilde{\nu}$, so we have $$\delta\nu
=\rho^j\delta\tilde{\nu}-\tfrac{1}{4}j\rho^j(\rho\partial_\rho)\contraction\tilde{\nu}
=\rho^j\delta\tilde{\nu}-\tfrac{1}{4}j\rho^j(\bm{Z}_\tau+\bm{Z}_{{\overline{\tau}}})\contraction\tilde{\nu},$$ which implies that follows from the $j=0$ case. Next we compute $$\nabla^2\nu=\rho^j\nabla^2\tilde{\nu}+2j\rho^jd(\log\rho)\otimes\nabla\tilde{\nu}
+j^2\rho^jd(\log\rho)\otimes d(\log\rho)\otimes\tilde{\nu}
+j\rho^j\nabla^2(\log\rho)\otimes\tilde{\nu},$$ and hence $$\nabla^*\nabla\nu
=\rho^j\nabla^*\nabla\tilde{\nu}
-\tfrac{1}{2}j\rho^j\nabla_{\rho\partial_\rho}\tilde{\nu}
-\tfrac{1}{4}j^2\rho^j\tilde{\nu}
+j\rho^j\Delta(\log\rho)\cdot\tilde{\nu}.$$ Proposition \[prop:ACH\_Laplacian\] shows that $\Delta(\log\rho)=(n+1)/2+O(\rho)$, while implies that $\nabla_{\rho\partial_\rho}\tilde{\nu}=O(\rho)$. Since the difference between $\Delta_\mathrm{H}$ and $\nabla^*\nabla$ is just a linear action of the curvature tensor, we obtain $\Delta_\mathrm{H}\nu=\rho^j\Delta_\mathrm{H}\tilde{\nu}-\tfrac{1}{4}j(j-2n-2)\rho^j\tilde{\nu}+O(\rho^{j+1})$.
Now we assume $j=0$, and in the following computation we omit the $O(\rho)$ terms, which is symbolically indicated by $\equiv$. By we obtain $$\begin{aligned}
{4}
\nabla_\tau\tensor{\mu}{_\tau}&\equiv\tensor{\mu}{_\tau},&\qquad
\nabla_{{\overline{\tau}}}\tensor{\mu}{_\tau}&\equiv -\tensor{\mu}{_\tau},&\qquad
\nabla_\beta\tensor{\mu}{_\tau}&\equiv\tensor{\mu}{_\beta},&\qquad
\nabla_{{\overline{\beta}}}\tensor{\mu}{_\tau}&\equiv 0,\\
\nabla_\tau\tensor{\mu}{_\alpha}&\equiv\tfrac{1}{2}\tensor{\mu}{_\alpha},&\qquad
\nabla_{{\overline{\tau}}}\tensor{\mu}{_\alpha}&\equiv -\tfrac{1}{2}\tensor{\mu}{_\alpha},&\qquad
\nabla_\beta\tensor{\mu}{_\alpha}&\equiv 0,&\qquad
\nabla_{{\overline{\beta}}}\tensor{\mu}{_\alpha}
&\equiv -\tfrac{1}{2}\tensor{h}{_\alpha_{{\overline{\beta}}}}\tensor{\mu}{_\tau},
\end{aligned}$$ and hence $$\nabla_\tau\nabla_{{\overline{\tau}}}\tensor{\mu}{_\tau} \equiv 0,\qquad
\nabla_\beta\nabla_{{\overline{\gamma}}}\tensor{\mu}{_\tau} \equiv 0,\qquad
\nabla_\tau\nabla_{{\overline{\tau}}}\tensor{\mu}{_\alpha} \equiv \tfrac{1}{4}\tensor{\mu}{_\alpha},\qquad
\nabla_\beta\nabla_{{\overline{\gamma}}}\tensor{\mu}{_\alpha}
\equiv \tfrac{1}{4}\tensor{h}{_\beta_{{\overline{\gamma}}}}\tensor{\mu}{_\alpha}.$$ Thus we have $\tensor{\nabla}{_B}\tensor{\nabla}{^B}\tensor{\mu}{_\tau}\equiv 0$ and $\tensor{\nabla}{_B}\tensor{\nabla}{^B}\tensor{\mu}{_\alpha}
\equiv\tfrac{1}{8}(2n+1)\tensor{\mu}{_\alpha}$. Hence we obtain in the $j=0$ case, because by Proposition \[prop:asymptotic\_complex\_hyperbolicity\], $$\begin{split}
(\Delta_\mathrm{H}+n+2)\tensor{\mu}{_A}
&\equiv \nabla^*\nabla\tensor{\mu}{_A}+\tensor{\operatorname{Ric}}{_A^B}\tensor{\mu}{_B}
+(n+2)\tensor{\mu}{_A}\\
&\equiv -2\tensor{\nabla}{_B}\tensor{\nabla}{^B}\tensor{\mu}{_A}
-\tensor{R}{_B^B_A^C}\tensor{\mu}{_C}+\tensor{\operatorname{Ric}}{_A^B}\tensor{\mu}{_B}
+(n+2)\tensor{\mu}{_A}\\
&\equiv -2\tensor{\nabla}{_B}\tensor{\nabla}{^B}\tensor{\mu}{_A}+(n+2)\tensor{\mu}{_A}.
\end{split}$$ Similarly, $$\begin{aligned}
{4}
\nabla_\tau\tensor{\sigma}{_\tau_\tau}&\equiv 2\tensor{\sigma}{_\tau_\tau},&\quad
\nabla_{{\overline{\tau}}}\tensor{\sigma}{_\tau_\tau}&\equiv -2\tensor{\sigma}{_\tau_\tau},&\quad
\nabla_\gamma\tensor{\sigma}{_\tau_\tau}&\equiv 2\tensor{\sigma}{_\tau_\gamma},&\quad
\nabla_{{\overline{\gamma}}}\tensor{\sigma}{_\tau_\tau}&\equiv 0,\\
\nabla_\tau\tensor{\sigma}{_\tau_\alpha}&\equiv \tfrac{3}{2}\tensor{\sigma}{_\tau_\alpha},&\quad
\nabla_{{\overline{\tau}}}\tensor{\sigma}{_\tau_\alpha}
&\equiv -\tfrac{3}{2}\tensor{\sigma}{_\tau_\alpha},&\quad
\nabla_\gamma\tensor{\sigma}{_\tau_\alpha}&\equiv \tensor{\sigma}{_\alpha_\gamma},&\quad
\nabla_{{\overline{\gamma}}}\tensor{\sigma}{_\tau_\alpha}
&\equiv -\tfrac{1}{2}\tensor{h}{_\alpha_{{\overline{\gamma}}}}\tensor{\sigma}{_\tau_\tau},\\
\nabla_\tau\tensor{\sigma}{_\alpha_\beta}&\equiv\tensor{\sigma}{_\alpha_\beta},&\quad
\nabla_{{\overline{\tau}}}\tensor{\sigma}{_\alpha_\beta}&\equiv -\tensor{\sigma}{_\alpha_\beta},&\quad
\nabla_\gamma\tensor{\sigma}{_\alpha_\beta}&\equiv 0,&\quad
\nabla_{{\overline{\gamma}}}\tensor{\sigma}{_\alpha_\beta}
&\equiv -\tensor{h}{_(_\alpha_|_{{\overline{\gamma}}}}\tensor{\sigma}{_\tau_|_\beta_)},\\
\nabla_\tau\tensor{\sigma}{_\tau_{{\overline{\tau}}}}&\equiv 0,&\quad
\nabla_{{\overline{\tau}}}\tensor{\sigma}{_\tau_{{\overline{\tau}}}}&\equiv 0,&\quad
\nabla_\gamma\tensor{\sigma}{_\tau_{{\overline{\tau}}}}&\equiv \tensor{\sigma}{_\gamma_{{\overline{\tau}}}},&\quad
\nabla_{{\overline{\gamma}}}\tensor{\sigma}{_\tau_{{\overline{\tau}}}}
&\equiv \tensor{\sigma}{_\tau_{{\overline{\gamma}}}},\\
\nabla_\tau\tensor{\sigma}{_\tau_{{\overline{\alpha}}}}
&\equiv \tfrac{1}{2}\tensor{\sigma}{_\tau_{{\overline{\alpha}}}},&\quad
\nabla_{{\overline{\tau}}}\tensor{\sigma}{_\tau_{{\overline{\alpha}}}}
&\equiv -\tfrac{1}{2}\tensor{\sigma}{_\tau_{{\overline{\alpha}}}},&\quad
\nabla_\gamma\tensor{\sigma}{_\tau_{{\overline{\alpha}}}}
&\equiv \tensor{\sigma}{_\gamma_{{\overline{\alpha}}}}
-\tfrac{1}{2}\tensor{h}{_\gamma_{{\overline{\alpha}}}}\tensor{\sigma}{_\tau_{{\overline{\tau}}}},&\quad
\nabla_{{\overline{\gamma}}}\tensor{\sigma}{_\tau_{{\overline{\alpha}}}}&\equiv 0,\\
\nabla_\tau\tensor{\sigma}{_\alpha_{{\overline{\beta}}}}&\equiv 0,&\quad
\nabla_{{\overline{\tau}}}\tensor{\sigma}{_\alpha_{{\overline{\beta}}}}&\equiv 0,&\quad
\nabla_\gamma\tensor{\sigma}{_\alpha_{{\overline{\beta}}}}
&\equiv -\tfrac{1}{2}\tensor{h}{_\gamma_{{\overline{\beta}}}}\tensor{\sigma}{_\alpha_{{\overline{\tau}}}},&\quad
\nabla_{{\overline{\gamma}}}\tensor{\sigma}{_\alpha_{{\overline{\beta}}}}
&\equiv -\tfrac{1}{2}\tensor{h}{_\alpha_{{\overline{\gamma}}}}\tensor{\sigma}{_\tau_{{\overline{\beta}}}},
\end{aligned}$$ and we obtain for $j=0$. Moreover, $$\begin{aligned}
{2}
\nabla_\tau\nabla_{{\overline{\tau}}}\tensor{\sigma}{_\tau_\tau}
&\equiv -2\tensor{\sigma}{_\tau_\tau},&\qquad
\nabla_\gamma\nabla_{{\overline{\delta}}}\tensor{\sigma}{_\tau_\tau}
&\equiv 0,\\
\nabla_\tau\nabla_{{\overline{\tau}}}\tensor{\sigma}{_\tau_\alpha}
&\equiv -\tfrac{3}{4}\tensor{\sigma}{_\tau_\alpha},&\qquad
\nabla_\gamma\nabla_{{\overline{\delta}}}\tensor{\sigma}{_\tau_\alpha}
&\equiv \tfrac{1}{4}\tensor{h}{_\gamma_{{\overline{\delta}}}}\tensor{\sigma}{_\tau_\alpha}
-\tfrac{1}{2}\tensor{h}{_\alpha_{{\overline{\delta}}}}\tensor{\sigma}{_\tau_\gamma},\\
\nabla_\tau\nabla_{{\overline{\tau}}}\tensor{\sigma}{_\alpha_\beta}
&\equiv 0,&\qquad
\nabla_\gamma\nabla_{{\overline{\delta}}}\tensor{\sigma}{_\alpha_\beta}
&\equiv \tfrac{1}{2}\tensor{h}{_\gamma_{{\overline{\delta}}}}\tensor{\sigma}{_\alpha_\beta},\\
\nabla_\tau\nabla_{{\overline{\tau}}}\tensor{\sigma}{_\tau_{{\overline{\tau}}}}
&\equiv 0,&\qquad
\nabla_\gamma\nabla_{{\overline{\delta}}}\tensor{\sigma}{_\tau_{{\overline{\tau}}}}
&\equiv \tensor{\sigma}{_\alpha_{{\overline{\beta}}}}
-\tfrac{1}{2}\tensor{h}{_\alpha_{{\overline{\beta}}}}\tensor{\sigma}{_\tau_{{\overline{\tau}}}},\\
\nabla_\tau\nabla_{{\overline{\tau}}}\tensor{\sigma}{_\tau_{{\overline{\alpha}}}}
&\equiv \tfrac{1}{4}\tensor{\sigma}{_\tau_{{\overline{\alpha}}}},&\qquad
\nabla_\gamma\nabla_{{\overline{\delta}}}\tensor{\sigma}{_\tau_{{\overline{\alpha}}}}
&\equiv -\tfrac{1}{4}\tensor{h}{_\gamma_{{\overline{\delta}}}}\tensor{\sigma}{_\tau_{{\overline{\alpha}}}}
-\tfrac{1}{2}\tensor{h}{_\gamma_{{\overline{\alpha}}}}\tensor{\sigma}{_\tau_{{\overline{\delta}}}},\\
\nabla_\tau\nabla_{{\overline{\tau}}}\tensor{\sigma}{_\alpha_{{\overline{\beta}}}}
&\equiv 0,&\qquad
\nabla_\gamma\nabla_{{\overline{\delta}}}\tensor{\sigma}{_\alpha_{{\overline{\beta}}}}
&\equiv -\tfrac{1}{2}\tensor{h}{_\gamma_{{\overline{\beta}}}}\tensor{\sigma}{_\alpha_{{\overline{\delta}}}}
+\tfrac{1}{4}\tensor{h}{_\alpha_{{\overline{\delta}}}}\tensor{h}{_\gamma_{{\overline{\beta}}}}
\tensor{\sigma}{_\tau_{{\overline{\tau}}}},
\end{aligned}$$ and therefore
\[eq:partial\_Laplacian\_on\_2-tensors\] $$\begin{aligned}
{2}
\nabla_C\nabla^C\tensor{\sigma}{_\tau_\tau}
&\equiv -\tensor{\sigma}{_\tau_\tau},&\qquad
\nabla_C\nabla^C\tensor{\sigma}{_\tau_\alpha}
&\equiv \tfrac{1}{8}(2n-7)\tensor{\sigma}{_\tau_\alpha},\\
\nabla_C\nabla^C\tensor{\sigma}{_\alpha_\beta}
&\equiv \tfrac{1}{2}n\tensor{\sigma}{_\alpha_\beta},&\qquad
\nabla_C\nabla^C\tensor{\sigma}{_\tau_{{\overline{\tau}}}}
&\equiv -\tfrac{1}{2}n\tensor{\sigma}{_\tau_{{\overline{\tau}}}}+\operatorname{tr}_h\tensor{\sigma}{_\gamma_{{\overline{\delta}}}},\\
\nabla_C\nabla^C\tensor{\sigma}{_\tau_{{\overline{\alpha}}}}
&\equiv -\tfrac{1}{8}(2n+3)\tensor{\sigma}{_\tau_{{\overline{\alpha}}}},&\qquad
\nabla_C\nabla^C\tensor{\sigma}{_\alpha_{{\overline{\beta}}}}
&\equiv -\tfrac{1}{2}\tensor{\sigma}{_\alpha_{{\overline{\beta}}}}
+\tfrac{1}{4}\tensor{h}{_\alpha_{{\overline{\beta}}}}\tensor{\sigma}{_\tau_{{\overline{\tau}}}}.
\end{aligned}$$
Furthermore, by Proposition \[prop:asymptotic\_complex\_hyperbolicity\], $$\begin{split}
(\Delta_\mathrm{L}+n+2)\tensor{\sigma}{_A_B}
&\equiv \nabla^*\nabla\tensor{\sigma}{_A_B}
+2\tensor{\operatorname{Ric}}{_(_A^C}\tensor{\sigma}{_B_)_C}
+2\tensor{R}{_A^C_B^D}\tensor{\sigma}{_C_D}
+(n+2)\tensor{\sigma}{_A_B}\\
&\equiv -2\nabla_C\nabla^C\tensor{\sigma}{_A_B}
-\tensor{R}{_C^C_A^D}\tensor{\sigma}{_D_B}-\tensor{R}{_C^C_B^D}\tensor{\sigma}{_A_D}\\
&\phantom{\equiv}\quad+2\tensor{\operatorname{Ric}}{_(_A^C}\tensor{\sigma}{_B_)_C}
+2\tensor{R}{_A^C_B^D}\tensor{\sigma}{_C_D}+(n+2)\tensor{\sigma}{_A_B}\\
&\equiv -2\nabla_C\nabla^C\tensor{\sigma}{_A_B}+n\tensor{\sigma}{_A_B}
\end{split}$$ and $$\begin{split}
(\Delta_\mathrm{L}+n+2)\tensor{\sigma}{_A_{{\overline{B}}}}
&\equiv \nabla^*\nabla\tensor{\sigma}{_A_{{\overline{B}}}}
+\tensor{\operatorname{Ric}}{_A^C}\tensor{\sigma}{_C_{{\overline{B}}}}
+\tensor{\operatorname{Ric}}{_{{\overline{B}}}^{{\overline{C}}}}\tensor{\sigma}{_A_{{\overline{C}}}}
+2\tensor{R}{_A^C_{{\overline{B}}}^{{\overline{D}}}}\tensor{\sigma}{_C_{{\overline{D}}}}
+(n+2)\tensor{\sigma}{_A_{{\overline{B}}}}\\
&\equiv -2\nabla_C\nabla^C\tensor{\sigma}{_A_{{\overline{B}}}}
-\tensor{R}{_C^C_A^D}\tensor{\sigma}{_D_{{\overline{B}}}}
+\tensor{R}{_C^C^{{\overline{D}}}_{{\overline{B}}}}\tensor{\sigma}{_A_{{\overline{D}}}}\\
&\phantom{\equiv}\quad+\tensor{\operatorname{Ric}}{_A^C}\tensor{\sigma}{_C_{{\overline{B}}}}
+\tensor{\operatorname{Ric}}{_{{\overline{B}}}^{{\overline{C}}}}\tensor{\sigma}{_A_{{\overline{C}}}}
-2\tensor{R}{_A^C^{{\overline{D}}}_{{\overline{B}}}}\tensor{\sigma}{_C_{{\overline{D}}}}
+(n+2)\tensor{\sigma}{_A_{{\overline{B}}}}\\
&\equiv -2\nabla_C\nabla^C\tensor{\sigma}{_A_{{\overline{B}}}}
+\tensor{\sigma}{_A_{{\overline{B}}}}+\tensor{g}{_A_{{\overline{B}}}}\tensor{\sigma}{_C^C}.
\end{split}$$ These equalities and imply the desired result.
Before we go to the proof of Proposition \[prop:lichnerowicz\_equation\], we recall one general formula valid on any Riemannian manifold. Define the 3-($\Theta$-)tensor $D\operatorname{Ric}$ by $$\tensor{(D\operatorname{Ric})}{_P_Q_R}
:=\nabla_P\tensor{\operatorname{Ric}}{_Q_R}-\nabla_Q\tensor{\operatorname{Ric}}{_P_R}-\nabla_R\tensor{\operatorname{Ric}}{_P_Q}$$ and its action on arbitrary symmetric 2-($\Theta$-)tensors by $\tensor{((D\operatorname{Ric})^\circ\sigma)}{_P}:=\tensor{(D\operatorname{Ric})}{_P_Q_R}\tensor{\sigma}{^Q^R}$, where the indices are raised by $g$. Then, by direct calculation, one can show that $$\label{eq:divergence_Laplacian_commutation}
\delta\circ\Delta_\mathrm{L}=\Delta_\mathrm{H}\circ\delta+(D\operatorname{Ric})^\circ.$$
We take the normalization with respect to a contact form $\theta$. We have imposed the condition that $\sigma$ should satisfy $\tensor{\sigma}{_\alpha_\beta}|_M=\tensor{\psi}{_\alpha_\beta}$, and since $\sigma$ is real, $\tensor{\sigma}{_{{\overline{\alpha}}}_{{\overline{\beta}}}}|_M$ must be of course $\tensor{\psi}{_{{\overline{\alpha}}}_{{\overline{\beta}}}}$. Equations show that the boundary values of the other components have to be zero in order that $(\Delta_\mathrm{L}+n+2)\sigma=O(\rho)$ is satisfied. Then by using recursively, it can be shown that $\sigma$ is uniquely determined modulo $O(\rho^{2n+2})$ so that holds.
We can prove that this approximate solution in fact satisfies $\operatorname{tr}\sigma=O(\rho^{2n+2})$ and $\delta\sigma=O(\rho^{2n+2})$ as follows. Firstly, taking the trace of we obtain $$(\Delta+n+2)(\operatorname{tr}\sigma)=O(\rho^{2n+2}).$$ On the other hand, shows that if $F=O(\rho^j)$ is a function then $(\Delta+n+2)F=-\frac{1}{4}(j+2)(j-2n-4)F+O(\rho^{j+1})$. Hence we can conclude that $\operatorname{tr}\sigma$ is actually $O(\rho^{2n+2})$. Secondly, if we take the divergence of , then shows that the 1-$\Theta$-tensor $\delta\sigma$ satisfies $$(\Delta_\mathrm{H}+n+2)(\delta\sigma)
=\delta(\Delta_\mathrm{L}+n+2)\sigma-(D\operatorname{Ric})^\circ\sigma=O(\rho^{2n+2}).$$ Then, by recursively applying , we can show that $\delta\sigma$ must be $O(\rho^{2n+2})$.
Now let $T^{1,0}_t$ be a smooth 1-parameter family of partially integrable CR structures that is tangent to $\tensor{\psi}{_\alpha_\beta}$, and we take an associated $C^\infty$-smooth normalized ACH metric $g^t$ satisfying for each $T^{1,0}_t$ so that $g^t$ is also smooth in $t$. Let $\sigma_\mathrm{normal}:=-\frac{1}{2}(d/dt)g^t|_{t=0}$. Then it of course solves , satisfies $\tensor{(\sigma_\mathrm{normal}|_M)}{_\alpha_\beta}=\tensor{\psi}{_\alpha_\beta}$, and if we write $$2\left(\operatorname{Ric}^\bullet+\frac{n+2}{2}\right)\sigma_\mathrm{normal}=\tilde{\sigma}_\mathrm{normal},$$ then $\tensor{(\rho^{-2n-2}\tilde{\sigma}_\mathrm{normal}|_M)}{_\alpha_\beta}$ equals minus of $\tensor*{\mathcal{O}}{^\bullet_\alpha_\beta}$. We want to prove that we can take a 1-$\Theta$-form $\eta$ satisfying $$\label{eq:modification_of_normalized_linearized_Einstein_equation}
(\Delta_\mathrm{L}+n+2)\sigma=O(\rho^{2n+2}),
\qquad\text{where}\quad \sigma=\sigma_\mathrm{normal}+\mathcal{K}\eta,$$ and moreover, if $\eta^\sharp$ is the dual $\Theta$-vector field of $\eta$, $$\label{eq:property_of_modification}
\tensor{(\mathcal{L}_{\eta^\sharp}E)}{_\alpha_\beta}=O(\rho^{2n+3}),$$ where $\mathcal{L}_{\eta^\sharp}E$ is the Lie derivative of $E=\operatorname{Ric}+\frac{1}{2}(n+2)g$. Suppose that such an $\eta$ exists. Then $\tensor{(\sigma|_M)}{_\alpha_\beta}=\tensor{\psi}{_\alpha_\beta}$ and $$(\Delta_\mathrm{L}+n+2)\sigma-2\left(\operatorname{Ric}^\bullet+\frac{n+2}{2}\right)\sigma_\mathrm{normal}
=(2\operatorname{Ric}^\bullet{}+n+2)\mathcal{K}\eta+\mathcal{K}\mathcal{B}\sigma
=2\mathcal{L}_{\eta^\sharp}E+\mathcal{K}\mathcal{B}\sigma.$$ As $\mathcal{B}\sigma=\delta\sigma+\frac{1}{2}d(\operatorname{tr}\sigma)=O(\rho^{2n+2})$ and hence $\tensor{(\mathcal{K}\mathcal{B}\sigma)}{_\alpha_\beta}=O(\rho^{2n+3})$ by , we can conclude that $\tensor{(\rho^{-2n-2}\tilde{\sigma}|_M)}{_\alpha_\beta}
=\tensor{(\rho^{-2n-2}\tilde{\sigma}_\mathrm{normal}|_M)}{_\alpha_\beta}
=-\tensor*{\mathcal{O}}{^\bullet_\alpha_\beta}$.
Suppose $\eta$ is taken so that $\mathcal{B}(\sigma_\mathrm{normal}+\mathcal{K}\eta)=O(\rho^{2n+2})$. Then since $\operatorname{Ric}^\bullet\mathcal{K}\eta=\mathcal{L}_{\eta^\sharp}\operatorname{Ric}=O(\rho^{2n+2})$, implies that $\sigma=\sigma_\mathrm{normal}+\mathcal{K}\eta$ satisfies . To construct such an $\eta$, the equation to be solved is $$\label{eq:gauge_equation}
\mathcal{B}\mathcal{K}\eta=-\mathcal{B}\sigma_\mathrm{normal}+O(\rho^{2n+2}).$$ The left-hand side is actually rewritten as $$\mathcal{B}\mathcal{K}\eta
=\Delta_\mathrm{H}\xi-2\operatorname{Ric}^\circ\xi=(\Delta_\mathrm{H}+n+2)\eta+O(\rho^{2n+2}).$$ Therefore, by using recursively we can construct a solution of . It remains to show that our solution $\eta$ satisfies . Since the boundary value of $\sigma_\mathrm{normal}$ has no components other than $\tensor{(\sigma_\mathrm{normal})}{_\alpha_\beta}$ and its complex conjugate, shows that $\delta\sigma_\mathrm{normal}=O(\rho)$, and hence $\mathcal{B}\sigma_\mathrm{normal}=O(\rho)$. Therefore, $\eta=O(\rho)$. Let $\eta^\sharp=\rho\xi$, where $\xi$ is a $\Theta$-vector field. Then we obtain $$\begin{split}
\tensor{(\mathcal{L}_{\eta^\sharp}E)}{_\alpha_\beta}
&=(\mathcal{L}_{\eta^\sharp}E)(\bm{Z}_\alpha,\bm{Z}_\beta)
=\eta^\sharp(E(\bm{Z}_\alpha,\bm{Z}_\beta))-E([\eta^\sharp,\bm{Z}_\alpha],\bm{Z}_\beta)
-E(\bm{Z}_\alpha,[\eta^\sharp,\bm{Z}_\beta])\\
&=\rho\xi(E(\bm{Z}_\alpha,\bm{Z}_\beta))-\rho E([\xi,\bm{Z}_\alpha],\bm{Z}_\beta)
-\rho E(\bm{Z}_\alpha,[\xi,\bm{Z}_\beta])=O(\rho^{2n+3}).
\end{split}$$ This finishes the proof.
Heisenberg principal part {#subsec:Heisenberg_principal_part}
-------------------------
As the first application of Proposition \[prop:lichnerowicz\_equation\], we prove the following theorem.
\[thm:linearized\_obstruction\] Let $(M,T^{1,0}M)$ be a partially integrable CR manifold of dimension $2n+1\ge 5$. Then, the linearized obstruction operator $\mathcal{O}^\bullet$ has the following expression for any choice of $\theta$: $$\label{eq:linearized_obstruction}
\mathcal{O}^\bullet
=\mathcal{O}^\bullet_\mathrm{pr}+(\text{a differential operator with Heisenberg order $\le 2n+1$}),$$ where $$\begin{split}
\label{eq:principal_part_of_linearized_obstruction}
\tensor{(\mathcal{O}^\bullet_\mathrm{pr}\psi)}{_\alpha_\beta}
&=\frac{(-1)^{n+1}}{(n!)^2}\left[
\left(\prod_{k=0}^n (\Delta_b+i(n+2-2k)\nabla^\mathrm{TW}_T)\right)\tensor{\psi}{_\alpha_\beta}
\right.\\
&\phantom{\;=\;}
+\frac{4(n+1)}{n+2}\left(\prod_{k=0}^{n-1}(\Delta_b+i(n+2-2k)\nabla^\mathrm{TW}_T)\right)
\tensor*{\nabla}{^{\mathrm{TW}}_(_\alpha}\tensor{(\nabla^\mathrm{TW})}{^\gamma}
\tensor{\psi}{_\beta_)_\gamma}\\
&\phantom{\;=\;}
+\left.\frac{4n}{n+2}\left(\prod_{k=0}^{n-2}(\Delta_b+i(n+2-2k)\nabla^\mathrm{TW}_T)\right)
\tensor*{\nabla}{^{\mathrm{TW}}_\alpha}\tensor*{\nabla}{^{\mathrm{TW}}_\beta}
\tensor{(\nabla^\mathrm{TW})}{^\gamma}\tensor{(\nabla^\mathrm{TW})}{^\delta}
\tensor{\psi}{_\gamma_\delta}\right].
\end{split}$$ Here $\nabla^\mathrm{TW}$ denotes the Tanaka–Webster connection.
We start with some preliminary considerations. Since $\tensor{\mathcal{O}}{_\alpha_\beta}$ has a universal expression in terms of the Tanaka–Webster connection, so does its linearization $\mathcal{O}^\bullet$. The expression of the Heisenberg principal part of $\mathcal{O}^\bullet$ cannot involve $N$, $A$, or $R$, which is shown as follows. Suppose that $$\begin{gathered}
\operatorname{contr}((h^{-1})^{\otimes a}
\otimes(\nabla^\mathrm{TW}_{1,0})^{\otimes b}
\otimes(\nabla^\mathrm{TW}_{0,1})^{\otimes b'}
\otimes(\nabla^\mathrm{TW}_T)^{\otimes b''}
\otimes N^{\otimes c}
\otimes A^{\otimes c'}
\otimes R^{\otimes c''}
\otimes(\text{$\psi$ or ${\overline{\psi}}$}))\end{gathered}$$ is a term in the expression of $\tensor*{\mathcal{O}}{^\bullet_\alpha_\beta}=\tensor{(\mathcal{O}^\bullet\psi)}{_\alpha_\beta}$. Here $N$, $A$, $R$ denote $\tensor{N}{_\alpha_\beta^{{\overline{\gamma}}}}$, $\tensor{A}{_\alpha_\beta}$, $\tensor{R}{_\alpha^\beta_\gamma_{{\overline{\delta}}}}$ or their complex conjugates so that they are invariant under constant scalings of $\theta$. Moreover, each $\nabla^\mathrm{TW}_{1,0}$, $\nabla^\mathrm{TW}_{0,1}$, and $\nabla^\mathrm{TW}_T$ is understood to be applied to one of $N$, $A$, $R$, and $(\text{$\psi$ or ${\overline{\psi}}$})$. Then, since $\tensor{\psi}{_\alpha_\beta}\in\tensor{\mathcal{E}}{_(_\alpha_\beta_)}(1,1)$ and $\tensor*{\mathcal{O}}{^\bullet_\alpha_\beta}\in\tensor{\mathcal{E}}{_(_\alpha_\beta_)}(-n,-n)$, $a+b''$ has to be $n+1$. On the other hand, since the contraction is taken so that two holomorphic indices remain downstairs, counting the number of indices gives $2a=b+b'+c+2c'+2c''$. Hence $(b+b'+2b'')+(c+2c'+2c'')=2n+2$, which shows that this term has Heisenberg order $\le 2n+1$ unless $c=c'=c''=0$. Therefore, to determine the Heisenberg principal part of $\mathcal{O}^\bullet$, it is sufficient to prove the following.
\[prop:linearized\_obstruction\_heisenberg\] Let $M$ be the Heisenberg group of dimension $2n+1\ge 5$. Then, with respect to the standard contact form $\theta$, the right-hand side of gives the exact formula of $\tensor*{\mathcal{O}}{^\bullet_\alpha_\beta}$.
Let the complex hyperbolic metric $g$ be normalized by the standard contact form $\theta$ (see Example \[ex:complex\_hyperbolic\]). The natural complex structure on the complex hyperbolic space induces a section $J\in C^\infty(X,\operatorname{End}({{\fourIdx{\Theta}{}{}{}{TX}}}))$, whose $i$-eigenbundle is actually spanned by $\set{\bm{Z}_\tau,\bm{Z}_\alpha}$. Let $$\label{eq:Tanaka_Webster_expression_of_linearized_obstruction}
\tensor*{\mathcal{O}}{^\bullet_\alpha_\beta}
=\tensor{(P')}{_\alpha_\beta^\gamma^\delta}\tensor{\psi}{_\gamma_\delta}
+\tensor{(P'')}{_\alpha_\beta^{{\overline{\gamma}}}^{{\overline{\delta}}}}\tensor{\psi}{_{{\overline{\gamma}}}_{{\overline{\delta}}}}$$ be the expression of $\tensor*{\mathcal{O}}{^\bullet_\alpha_\beta}$ as a sum of contractions of Tanaka–Webster covariant derivatives of $\tensor{\psi}{_\alpha_\beta}$ and $\tensor{\psi}{_{{\overline{\alpha}}}_{{\overline{\beta}}}}$. By Proposition \[prop:lichnerowicz\_equation\], this is the obstruction to the existence of $C^\infty$-smooth solutions of . Now, since $J$ is parallel in the current case, if $\sigma$ solves then $\sigma(J{\mathord{\cdot}},{\mathord{\cdot}})$ is also a solution to with boundary data $i\tensor{\psi}{_\alpha_\beta}$. Therefore, $$i\tensor*{\mathcal{O}}{^\bullet_\alpha_\beta}
=\tensor{(P')}{_\alpha_\beta^\gamma^\delta}(i\tensor{\psi}{_\gamma_\delta})
+\tensor{(P'')}{_\alpha_\beta^{{\overline{\gamma}}}^{{\overline{\delta}}}}
(-i\tensor{\psi}{_{{\overline{\gamma}}}_{{\overline{\delta}}}})
=i(\tensor{(P')}{_\alpha_\beta^\gamma^\delta}\tensor{\psi}{_\gamma_\delta}
-\tensor{(P'')}{_\alpha_\beta^{{\overline{\gamma}}}^{{\overline{\delta}}}}
\tensor{\psi}{_{{\overline{\gamma}}}_{{\overline{\delta}}}}).$$ Combined with , this shows that $P''=0$, or equivalently, $\tensor*{\mathcal{O}}{^\bullet_\alpha_\beta}$ is a certain sum of covariant derivatives of $\tensor{\psi}{_\alpha_\beta}$.
Note that, again by the parallelity of $J$, we can always take a solution $\sigma$ to that is anti-hermitian, i.e., such that $\tensor{\sigma}{_A_{{\overline{B}}}}=0$. This is because the uniqueness statement of Proposition \[prop:lichnerowicz\_equation\] implies that $\sigma$ must agree with $-\sigma(J{\mathord{\cdot}},J{\mathord{\cdot}})$ modulo $O(\rho^{2n+2})$, which means that $\tensor{\sigma}{_A_{{\overline{B}}}}=O(\rho^{2n+2})$; then, since the higher-order terms are arbitrary, we can set $\tensor{\sigma}{_A_{{\overline{B}}}}=0$. The same reasoning as in the preceding paragraph shows that, if we expand $\tensor{\sigma}{_A_B}$ in the powers of $\rho$, then all the coefficients can be written in terms of $\tensor{\psi}{_\alpha_\beta}$ only.
We prepare a precise version of Lemma \[lem:Laplacian\_on\_tensors\].
\[lem:Laplacian\_on\_complex\_hyperbolic\_space\] Let $\sigma$ be an anti-hermitian symmetric 2-$\Theta$-tensor on the complex hyperbolic space normalized by the standard contact form $\theta$ on the Heisenberg group $\mathcal{H}$. Suppose a local frame $\set{Z_\alpha}$ of $T^{1,0}\mathcal{H}$ is taken so that the Tanaka–Webster connection forms are zero. Then, with respect to the local frame $\set{\bm{Z}_\tau,\bm{Z}_\alpha,\bm{Z}_{{\overline{\tau}}},\bm{Z}_{{\overline{\alpha}}}}$,
\[eq:divergence\_of\_2-tensor\_complex\_hyperbolic\] $$\begin{aligned}
\tensor{(\delta\sigma)}{_\tau}
&=-\tfrac{1}{4}(\rho\partial_\rho-2n-4)\tensor{\sigma}{_\tau_\tau}
+\tfrac{i}{2}\rho^2T\tensor{\sigma}{_\tau_\tau}
-\rho\tensor{Z}{^\alpha}\tensor{\sigma}{_\tau_\alpha},\\
\tensor{(\delta\sigma)}{_\alpha}
&=-\tfrac{1}{4}(\rho\partial_\rho-2n-5)\tensor{\sigma}{_\tau_\alpha}
+\tfrac{i}{2}\rho^2T\tensor{\sigma}{_\tau_\alpha}
-\rho\tensor{Z}{^\beta}\tensor{\sigma}{_\alpha_\beta},
\end{aligned}$$
where $T$ is the Reeb vector field and $Z^\alpha=\tensor{h}{^\alpha^{{\overline{\beta}}}}Z_{{\overline{\beta}}}$. Moreover, under the condition $\delta\sigma=O(\rho^{2n+2})$, $\tilde{\sigma}=(\Delta_\mathrm{L}+n+2)\sigma$ is given by, modulo $O(\rho^{2n+2})$,
\[eq:lichnerowicz\_Laplacian\_of\_2-tensor\_complex\_hyperbolic\] $$\begin{aligned}
\label{eq:lichnerowicz_Laplacian_of_2-tensor_complex_hyperbolic_1}
\tensor{\tilde{\sigma}}{_\tau_\tau}
&\equiv -\tfrac{1}{4}(\rho\partial_\rho-2)(\rho\partial_\rho-2n-4)\tensor{\sigma}{_\tau_\tau}
+\rho^2(\Delta_b+2iT)\tensor{\sigma}{_\tau_\tau}
-\rho^4T^2\tensor{\sigma}{_\tau_\tau},\\
\label{eq:lichnerowicz_Laplacian_of_2-tensor_complex_hyperbolic_2}
\tensor{\tilde{\sigma}}{_\tau_\alpha}
&\equiv -\tfrac{1}{4}(\rho\partial_\rho-1)(\rho\partial_\rho-2n-3)\tensor{\sigma}{_\tau_\alpha}
+\rho^2(\Delta_b+2iT)\tensor{\sigma}{_\tau_\tau}-\rho^4T^2\tensor{\sigma}{_\tau_\tau}
+\rho Z_\alpha\tensor{\sigma}{_\tau_\tau},\\
\label{eq:lichnerowicz_Laplacian_of_2-tensor_complex_hyperbolic_3}
\tensor{\tilde{\sigma}}{_\alpha_\beta}
&\equiv -\tfrac{1}{4}\rho\partial_\rho(\rho\partial_\rho-2n-2)\tensor{\sigma}{_\alpha_\beta}
+\rho^2(\Delta_b+2iT)\tensor{\sigma}{_\alpha_\beta}-\rho^4T^2\tensor{\sigma}{_\alpha_\beta}
+2\rho\tensor{Z}{_(_\alpha_|}\tensor{\sigma}{_\tau_|_\beta_)}.
\end{aligned}$$
In this setting, one can check that gives not only the boundary values of the Christoffel symbols of the Levi-Civita connection of $g$ but the exact formula of them. Then follows by a direct computation. Another long computation shows that
$$\begin{aligned}
\label{eq:lichnerowicz_Laplacian_of_2-tensor_complex_hyperbolic_raw_1}
\begin{split}
\tensor{\tilde{\sigma}}{_\tau_\tau}
&=-\tfrac{1}{4}((\rho\partial_\rho)^2-(2n+2)\rho\partial_\rho-4n-8)\tensor{\sigma}{_\tau_\tau}
+\rho^2(\Delta_b+4iT)\tensor{\sigma}{_\tau_\tau}-\rho^4T^2\tensor{\sigma}{_\tau_\tau}\\
&\phantom{\;=\;}\qquad-4\rho Z^\gamma\tensor{\sigma}{_\tau_\gamma},
\end{split}\\
\label{eq:lichnerowicz_Laplacian_of_2-tensor_complex_hyperbolic_raw_2}
\begin{split}
\tensor{\tilde{\sigma}}{_\tau_\alpha}
&=-\tfrac{1}{4}((\rho\partial_\rho)^2-(2n+2)\rho\partial_\rho-2n-7)\tensor{\sigma}{_\tau_\alpha}
+\rho^2(\Delta_b+3iT)\tensor{\sigma}{_\tau_\tau}-\rho^4T^2\tensor{\sigma}{_\tau_\tau}\\
&\phantom{\;=\;}\qquad+\rho Z_\alpha\tensor{\sigma}{_\tau_\tau}
-2\rho Z^\gamma\tensor{\sigma}{_\alpha_\gamma},
\end{split}\\
\tensor{\tilde{\sigma}}{_\alpha_\beta}
&=-\tfrac{1}{4}\rho\partial_\rho(\rho\partial_\rho-2n-2)\tensor{\sigma}{_\alpha_\beta}
+\rho^2(\Delta_b+2iT)\tensor{\sigma}{_\alpha_\beta}-\rho^4T^2\tensor{\sigma}{_\alpha_\beta}
+2\rho\tensor{Z}{_(_\alpha_|}\tensor{\sigma}{_\tau_|_\beta_)}.
\end{aligned}$$
The last equation is nothing but (this is an exact equality actually). To show and , we use the following identities that follow from and the assumption $\delta\sigma=O(\rho^{2n+2})$: $$\begin{aligned}
\rho Z^\gamma\tensor{\sigma}{_\tau_\gamma}
&=-\tfrac{1}{4}(\rho\partial_\rho-2n-4)\tensor{\sigma}{_\tau_\tau}
+\tfrac{i}{2}\rho^2T\tensor{\sigma}{_\tau_\tau}+O(\rho^{2n+2}),\\
\rho Z^\gamma\tensor{\sigma}{_\alpha_\gamma}
&=-\tfrac{1}{4}(\rho\partial_\rho-2n-5)\tensor{\sigma}{_\tau_\alpha}
+\tfrac{i}{2}\rho^2T\tensor{\sigma}{_\tau_\alpha}+O(\rho^{2n+2}).
\end{aligned}$$ In fact, putting them into and shows and .
Let $\sigma$ be an anti-hermitian solution to and $\tilde{\sigma}=(\Delta_\mathrm{L}+n+2)\sigma$. It holds that $\delta\sigma=O(\rho^{2n+2})$ as stated in Proposition \[prop:lichnerowicz\_equation\]. We may further improve this solution so that the following holds: $$\label{eq:divergence_improved_solution}
\delta\sigma=O(\rho^{2n+4}).$$ In fact, shows that the $\rho^{2n+2}$- and $\rho^{2n+3}$-coefficients of $\delta\sigma$ can be controlled by those of $\tensor{\sigma}{_\tau_\tau}$ and $\tensor{\sigma}{_\tau_\alpha}$ (actually suffices for this purpose). Equation together with implies $(\Delta_\mathrm{H}+2n+4)\tensor{(\delta\sigma)}{_\tau}=O(\rho^{2n+5})$ and $(\Delta_\mathrm{H}+2n+4)\tensor{(\delta\sigma)}{_\alpha}=O(\rho^{2n+4})$. Since $\operatorname{Ric}$ is parallel, shows that $\delta\circ\Delta_\mathrm{L}=\Delta_\mathrm{H}\circ\delta$, which gives us $$\tensor{(\delta\tilde{\sigma})}{_\tau}=O(\rho^{2n+5})\qquad\text{and}\qquad
\tensor{(\delta\tilde{\sigma})}{_\alpha}=O(\rho^{2n+4}).$$ Then, by we conclude that $\tensor{\tilde{\sigma}}{_\tau_\alpha}=O(\rho^{2n+3})$ and $\tensor{\tilde{\sigma}}{_\tau_\tau}=O(\rho^{2n+4})$. Let $\tensor*{k}{^{(2n+3)}_\tau_\alpha}\in\tensor{\mathcal{E}}{_\alpha}$ and $\tensor*{k}{^{(2n+2)}_\alpha_\beta}\in\tensor{\mathcal{E}}{_(_\alpha_\beta_)}$ be defined by $$\tensor{\tilde{\sigma}}{_\tau_\alpha}
=\rho^{2n+3}\tensor*{k}{^{(2n+3)}_\tau_\alpha}+O(\rho^{2n+4})\qquad\text{and}\qquad
\tensor{\tilde{\sigma}}{_\alpha_\beta}
=\rho^{2n+2}\tensor*{k}{^{(2n+2)}_\alpha_\beta}+O(\rho^{2n+3}).$$ Again by , if $\set{Z_\alpha}$ is such that the Tanaka–Webster connection forms vanish,
$$\label{eq:divergence_of_high_coefficient_1}
2Z^\beta\tensor*{k}{^{(2n+2)}_\alpha_\beta}=\tensor*{k}{^{(2n+3)}_\tau_\alpha}$$
and $$\label{eq:divergence_of_high_coefficient_2}
Z^\alpha\tensor*{k}{^{(2n+3)}_\tau_\alpha}=0.$$
We want to compute $\tensor*{k}{^{(2n+2)}_\alpha_\beta}\in\tensor{\mathcal{E}}{_\alpha_\beta}$ explicitly, which by Proposition \[prop:lichnerowicz\_equation\] equals $-\tensor*{\mathcal{O}}{^\bullet_\alpha_\beta}$ trivialized by $\theta$. As a step toward this, we first write $\tensor*{k}{^{(2n+3)}_\tau_\alpha}$ down. By and the discussion preceding Lemma \[lem:Laplacian\_on\_complex\_hyperbolic\_space\], $\tensor*{k}{^{(2n+3)}_\tau_\alpha}$ is a certain sum of contractions of derivatives of $\tensor{\psi}{_\alpha_\beta}$. We divide the terms into two groups depending whether the two indices of $\psi$ are both contracted or one of them is uncontracted; we call those terms that belong to the first group doubly-contracted. Since all the coefficients of $\tensor{\sigma}{_\tau_\tau}$ have to be doubly-contracted, shows that $$\tensor{\tilde{\sigma}}{_\tau_\alpha}
\equiv-\tfrac{1}{4}(\rho\partial_\rho-1)(\rho\partial_\rho-2n-3)\tensor{\sigma}{_\tau_\alpha}
+\rho^2(\Delta_b+2iT)\tensor{\sigma}{_\tau_\alpha}-\rho^4T^2\tensor{\sigma}{_\tau_\alpha}
+O(\rho^{2n+2}),$$ where $\equiv$ means that we omit the doubly-contracted terms. This implies that, omitting doubly-contracted terms, we can write down the expansion of $\tensor{\sigma}{_\tau_\alpha}$ as $$\tensor{\sigma}{_\tau_\alpha}
\equiv\frac{2}{n+2}\sum_{l=0}^{n-1}\rho^{2l+1}c'_{2l+1}P'_{2l+1}
Z^\beta\tensor{\psi}{_\alpha_\beta}+O(\rho^{2n+3}),\qquad
c'_l=\left(l!\prod_{\nu=0}^{l-1}(l-n-1-\nu)\right)^{-1},$$ where we define the differential operators $P'_{2l+1}\colon\tensor{\mathcal{E}}{_\alpha}\longrightarrow\tensor{\mathcal{E}}{_\alpha}$ by $$P'_1=1,\qquad P'_3=\Delta_b,\qquad
P'_{2l+1}=(\Delta_b+2iT)P'_{2l-1}-(l-1)(l-n-2)T^2P'_{2l-3}.$$ Consequently, by Lemma \[lem:Masakis\_lemma\], $$\tensor*{k}{^{(2n+3)}_\tau_\alpha}
\equiv \frac{2}{n+2}c'_nP'_{2n+3}Z^\beta\tensor{\psi}{_\alpha_\beta}
=\frac{2\cdot(-1)^n}{(n+2)\cdot n!^2}
\left(\prod_{l=0}^n(\Delta_b+(n+2-2l)iT)\right)Z^\beta\tensor{\psi}{_\alpha_\beta}.$$ Now we determine the omitted doubly-contracted terms using . There is some polynomial $q'(\Delta_b,T)$ of $\Delta_b$ and $T$ such that $$\tensor*{k}{^{(2n+3)}_\tau_\alpha}
=\frac{2\cdot(-1)^n}{(n+2)\cdot n!^2}
\left(\prod_{l=0}^n(\Delta_b+(n+2-2l)iT)\right)Z^\beta\tensor{\psi}{_\alpha_\beta}
+q'(\Delta_b,T)Z_\alpha Z^\beta Z^\gamma\tensor{\psi}{_\beta_\gamma}.$$ Since holds, by the commutation relations $[Z^\alpha,\Delta_b]=-2iTZ^\alpha$ and $[Z^\alpha,T]=0$ we can compute that $$\begin{split}
Z^\alpha q'(\Delta_b,T)Z_\alpha Z^\beta Z^\gamma\tensor{\psi}{_\beta_\gamma}
&=\frac{2\cdot(-1)^{n+1}}{(n+2)\cdot n!^2}
Z^\alpha\left(\prod_{l=0}^n(\Delta_b+(n+2-2l)iT)\right)Z^\beta\tensor{\psi}{_\alpha_\beta}\\
&=\frac{2\cdot(-1)^{n+1}}{(n+2)\cdot n!^2}
\left(\prod_{l=0}^n(\Delta_b+(n-2l)iT)\right)Z^\alpha Z^\beta\tensor{\psi}{_\alpha_\beta}.
\end{split}$$ This implies that $$\frac{2\cdot(-1)^{n+1}}{(n+2)\cdot n!^2}\prod_{l=0}^n(\Delta_b+(n-2l)iT)
=Z^\alpha q'(\Delta_b,T)Z_\alpha=q'(\Delta_b-2iT,T)Z^\alpha Z_\alpha.$$ Since $Z^\alpha Z_\alpha=-\frac{1}{2}(\Delta_b-inT)$, we obtain $$q'(\Delta_b-2iT,T)=\frac{4\cdot(-1)^n}{(n+2)\cdot n!^2}\prod_{l=0}^{n-1}(\Delta_b+(n-2l)iT),$$ and hence $$\begin{gathered}
\label{eq:expression_tau_alpha}
\tensor*{k}{^{(2n+3)}_\tau_\alpha}
=\frac{2\cdot(-1)^n}{(n+2)\cdot n!^2}\left[
\left(\prod_{l=0}^n(\Delta_b+(n+2-2l)iT)\right)Z^\beta\tensor{\psi}{_\alpha_\beta}
\right.\\
+\left.2\left(\prod_{l=0}^{n-1}(\Delta_b+(n+2-2l)iT)\right)
Z_\alpha Z^\beta Z^\gamma\tensor{\psi}{_\beta_\gamma}\right].
\end{gathered}$$
Finally we compute $\tensor*{k}{^{(2n+2)}_\alpha_\beta}$. It is expressed as a sum of contractions of derivatives of $\tensor{\psi}{_\alpha_\beta}$, and we divide the terms into two groups: those in which at least one of the two indices of $\psi$ is contracted (which we call contracted terms), and those in which the two indices of $\psi$ are both uncontracted. Then, if we omit the contracted terms, shows that $$\tensor{\tilde{\sigma}}{_\alpha_\beta}
\equiv-\tfrac{1}{4}\rho\partial_\rho(\rho\partial_\rho-2n-2)\tensor{\sigma}{_\alpha_\beta}
+\rho^2(\Delta_b+2iT)\tensor{\sigma}{_\tau_\alpha}-\rho^4T^2\tensor{\sigma}{_\tau_\alpha}
+O(\rho^{2n+2}).$$ This implies that, omitting the contracted terms, we can write $$\tensor{\sigma}{_\alpha_\beta}
\equiv\sum_{l=0}^{n-1}\rho^{2l}c'_lP''_{2l}
\tensor{\psi}{_\alpha_\beta}+O(\rho^{2n+3}),$$ where we define $P''_{2l}\colon\tensor{\mathcal{E}}{_(_\alpha_\beta_)}\longrightarrow\tensor{\mathcal{E}}{_(_\alpha_\beta_)}$ by $$P''_0=1,\qquad P''_2=\Delta_b,\qquad
P''_{2l}=(\Delta_b+2iT)P''_{2l-2}-(l-1)(l-n-2)T^2P''_{2l-4}.$$ Consequently, again by using Lemma \[lem:Masakis\_lemma\], we obtain $$\tensor*{k}{^{(2n+2)}_\alpha_\beta}
\equiv c'_nP''_{2n+2}\tensor{\psi}{_\alpha_\beta}
=\frac{(-1)^n}{n!^2}\left(\prod_{l=0}^n(\Delta_b+(n+2-2l)iT)\right)\tensor{\psi}{_\alpha_\beta}.$$ We shall determine the omitted terms. There are some polynomials $q''_1(\Delta_b,T)$ and $q''_2(\Delta_b,T)$ such that $$\begin{gathered}
\tensor*{k}{^{(2n+2)}_\alpha_\beta}
=\frac{(-1)^n}{n!^2}\left(\prod_{l=0}^n(\Delta_b+(n+2-2l)iT)\right)\tensor{\psi}{_\alpha_\beta}\\
+q''_1(\Delta_b,T)\tensor{Z}{_(_\alpha}Z^\gamma\tensor{\psi}{_\beta_)_\gamma}
+q''_2(\Delta_b,T)\tensor{Z}{_\alpha}Z_\beta Z^\gamma Z^\delta\tensor{\psi}{_\gamma_\delta}.
\end{gathered}$$ Then, $$\begin{gathered}
\label{eq:expression_divergence_of_alpha_beta}
Z^\beta\tensor*{k}{^{(2n+2)}_\alpha_\beta}
=\frac{(-1)^n}{n!^2}\left(\prod_{l=0}^n(\Delta_b+(n-2l)iT)\right)Z^\beta\tensor{\psi}{_\alpha_\beta}\\
+q''_1(\Delta_b-2iT,T)Z^\beta\tensor{Z}{_(_\alpha}Z^\gamma\tensor{\psi}{_\beta_)_\gamma}
+q''_2(\Delta_b-2iT,T)Z^\beta Z_\alpha Z_\beta Z^\gamma Z^\delta\tensor{\psi}{_\gamma_\delta}.
\end{gathered}$$ Note that the following holds: $$\begin{aligned}
Z^\beta\tensor{Z}{_(_\alpha}Z^\gamma\tensor{\psi}{_\beta_)_\gamma}
&=-\frac{1}{4}(\Delta_b-i(n+2)T)Z^\beta\tensor{\psi}{_\alpha_\beta}
+\frac{1}{2}Z_\alpha Z^\beta Z^\gamma\tensor{\psi}{_\beta_\gamma},\\
Z^\beta Z_\alpha Z_\beta Z^\gamma Z^\delta\tensor{\psi}{_\gamma_\delta}
&=-\frac{1}{2}(\Delta_b-inT)Z_\alpha Z^\beta Z^\gamma\tensor{\psi}{_\beta_\gamma}.
\end{aligned}$$ Therefore, by , , and , $$\begin{gathered}
\frac{(-1)^n}{(n+2)\cdot n!^2}\prod_{l=0}^n(\Delta_b+(n+2-2l)iT)\\
=\frac{(-1)^n}{n!^2}\prod_{l=0}^n(\Delta_b+(n-2l)iT)
-\frac{1}{4}(\Delta_b-i(n+2)T)q''_1(\Delta_b-2iT,T)
\end{gathered}$$ and $$\frac{2\cdot(-1)^n}{(n+2)\cdot n!^2}\prod_{l=0}^{n-1}(\Delta_b+(n+2-2l)iT)
=\frac{1}{2}Q''_1(\Delta_b-2iT,T)-\frac{1}{2}(\Delta_b-inT)q''_2(\Delta_b-2iT,T).$$ Hence $$\begin{aligned}
q''_1(\Delta_b,T)
&=\frac{(-1)^n}{(n!)^2}\cdot\frac{4(n+1)}{n+2}\prod_{l=0}^{n-1}(\Delta_b+(n+2-2l)iT)\\
\intertext{and}
q''_2(\Delta_b,T)
&=\frac{(-1)^n}{(n!)^2}\cdot\frac{4n}{n+2}\left(\prod_{l=0}^{n-2}(\Delta_b+(n+2-2l)iT)\right),
\end{aligned}$$ which show that $\tensor*{k}{^{(2n+2)}_\alpha_\beta}$ is minus of the right-hand side of .
Further properties of the linearized obstruction operator {#subsec:further_properties}
---------------------------------------------------------
We now consider the case in which $M$ is an integrable CR manifold and prove Theorem \[thm:obstruction\_operator\_observation\]. By formal embedding, we may assume that $M$ is the boundary of a domain $\Omega$ in $\mathbb{C}^{n+1}$. In this case we can take $X$ to be the square root of $\overline{\Omega}$ in the sense of Epstein, Melrose, and Mendoza (see Example \[ex:Bergman\_type\]). As described in Subsection \[subsec:approximate\_ACHE\], Fefferman’s approximate solution to the complex Monge–Ampère equation defines a Bergman-type metric $g$ that satisfies if considered as an ACH metric on $X$. The complex structure $J$ on $\overline{\Omega}$, with respect to which $g$ is Kähler, is naturally regarded as a section of $\operatorname{End}({{\fourIdx{\Theta}{}{}{}{TX}}})$. The $i$-eigenbundle is denoted by $({{\fourIdx{\Theta}{}{}{}{TX}}})^{1,0}$.
Since $g$ is Kähler, we can apply the argument for the complex hyperbolic metric in Subsection \[subsec:Heisenberg\_principal\_part\] also in this case, and we conclude that $\tensor*{\mathcal{O}}{^\bullet_\alpha_\beta}=\tensor{(\mathcal{O}^\bullet\psi)}{_\alpha_\beta}$ is written as a sum of contractions of Tanaka–Webster local invariants (i.e. covariant derivatives of $N$, $A$, and $R$) and covariant derivatives of $\tensor{\psi}{_\alpha_\beta}$, which means that $\tensor*{\mathcal{O}}{^\bullet_\alpha_\beta}$ is complex-linear in $\tensor{\psi}{_\alpha_\beta}$.
In view of the fact that $\mathcal{O}^\bullet$ has a universal expression in terms of the Tanaka–Webster connection, to prove the formal self-adjointness of $\mathcal{O}^\bullet$, it suffices to consider the case in which $M$ is compact. We use Theorem \[thm:first\_variational\_formula\] as follows. Let $\tensor{\chi}{_\alpha_\beta}$, $\tensor{\psi}{_\alpha_\beta}\in\tensor{\mathcal{E}}{_(_\alpha_\beta_)}(1,1)$ and, for sufficiently small $\varepsilon>0$, we define $T^{1,0}_{s,t}$ to be the partially integrable CR structure spanned by $\set{Z_\alpha+\tensor{\varphi}{_\alpha^{{\overline{\beta}}}}Z_{{\overline{\beta}}}}$, where $s$, $t\in(-\varepsilon,\varepsilon)$ and $\tensor{\varphi}{_\alpha_\beta}=s\tensor{\chi}{_\alpha_\beta}+t\tensor{\psi}{_\alpha_\beta}$. If $\smash{\overline{Q}}^{s,t}$ is the total CR $Q$-curvature of $(M,T^{1,0}_{s,t})$, then it is smooth in $s$, $t$ and hence $\partial_s\partial_t\smash{\overline{Q}}^{s,t}=\partial_t\partial_s\smash{\overline{Q}}^{s,t}$. Evaluated at $s=t=0$, this implies that $$\int_M \operatorname{Re}\braket{\mathcal{O}^\bullet\chi,\psi}=\int_M \operatorname{Re}\braket{\mathcal{O}^\bullet\psi,\chi},$$ where the bracket denotes the Hermitian inner product. The right-hand side is the same as the integral of $\operatorname{Re}\braket{\chi,\mathcal{O}^\bullet\psi}$. By replacing $\psi$ with $i\psi$, we also obtain the equality between the integrals of the imaginary parts of $\braket{\mathcal{O}^\bullet\chi,\psi}$ and $\braket{\chi,\mathcal{O}^\bullet\psi}$, thereby showing the self-adjointness of $\mathcal{O}^\bullet$.
Next we give a direct proof of the second equality of . Here we abandon the previous notation of local frames and introduce a new one. We first define the $(1,0)$-vector field $\xi$ on $\overline{\Omega}$ near the boundary by the requirement $$\partial r(\xi)=1,\qquad\text{and}\qquad \xi\contraction\partial{\overline{\partial}}r=\kappa{\overline{\partial}}r$$ for some function real-valued function $\kappa$, which is called the *transverse curvature* of $r$ [@Graham_Lee_88]. Then we take a local frame $\set{\zeta_\alpha}$ of $\ker\partial r\subset T^{1,0}\overline{\Omega}$ and set $\bm{\xi}=r\xi$ and $\bm{\zeta}_\alpha=\sqrt{r/2}\,\zeta_\alpha$, so that $\set{\bm{\xi},\bm{\zeta}_\alpha}$ spans $({{\fourIdx{\Theta}{}{}{}{TX}}})^{1,0}$. The index notation in this subsection is for this frame; the index $\tau$ is associated to $\bm{\xi}$, which we also write $\bm{\zeta}_\tau$.
We define $\tilde{\theta}=\frac{i}{2}(\partial r-{\overline{\partial}}r)$ and write $h(Z,{\overline{W}})=-i\,d\tilde{\theta}(Z,{\overline{W}})$ for vectors $Z$, $W$ in $T^{1,0}\overline{\Omega}$. Then, $$\label{eq:ddbar_r}
\partial{\overline{\partial}}r
=\kappa\partial r\wedge{\overline{\partial}}r
-\tensor{h}{_\alpha_{{\overline{\beta}}}}\theta^\alpha\wedge\theta^{{\overline{\beta}}},$$ where $\tensor{h}{_\alpha_{{\overline{\beta}}}}=h(\zeta_\alpha,\zeta_{{\overline{\beta}}})$, and $\theta^\alpha$ are taken so that $\set{\partial r,\theta^\alpha}$ is the dual coframe of $\set{\xi,\zeta_\alpha}$. Consequently, we have $$\label{eq:Bergman_type}
g=4(1-\kappa r)\bm{\theta}^\tau\bm{\theta}^{{\overline{\tau}}}
+2\tensor{h}{_\alpha_{{\overline{\beta}}}}\bm{\theta}^\alpha\bm{\theta}^{{\overline{\beta}}},$$ where $\bm{\theta}^\tau=\partial r/r$ and $\bm{\theta}^\alpha=\theta^\alpha/\sqrt{r/2}$.
Equation implies that $D\operatorname{Ric}=O(\rho^{2n+4})$. Moreover, we remark that $$\label{eq:differentiated_Ricci_tensor_Kahler}
\tensor{(D\operatorname{Ric})}{_\tau_{{\overline{\alpha}}}_{{\overline{\beta}}}}=O(\rho^{2n+5}).$$ This can be seen as follows. Since $\operatorname{Ric}$ has hermitian-type components only, $\tensor{(D\operatorname{Ric})}{_\tau_{{\overline{\alpha}}}_{{\overline{\beta}}}}$ equals $-2\tensor{\nabla}{_(_{{\overline{\alpha}}}_|}\tensor{\operatorname{Ric}}{_\tau_|_{{\overline{\beta}}}_)}$, which is $-2\tensor{\nabla}{_(_{{\overline{\alpha}}}_|}\tensor{E}{_\tau_|_{{\overline{\beta}}}_)}$ if we set $E={\operatorname{Ric}}+\frac{n+2}{2}g$. Then, because $$\tensor{\nabla}{_{{\overline{\alpha}}}}\tensor{E}{_\tau_{{\overline{\beta}}}}
=\bm{\zeta}_{{\overline{\alpha}}}\tensor{E}{_\tau_{{\overline{\beta}}}}
-\tensor{\Gamma}{^\tau_{{\overline{\alpha}}}_\tau}\tensor{E}{_\tau_{{\overline{\beta}}}}
-\tensor{\Gamma}{^\gamma_{{\overline{\alpha}}}_\tau}\tensor{E}{_\gamma_{{\overline{\beta}}}}
-\tensor{\Gamma}{^{{\overline{\tau}}}_{{\overline{\alpha}}}_{{\overline{\beta}}}}\tensor{E}{_\tau_{{\overline{\tau}}}}
-\tensor{\Gamma}{^{{\overline{\gamma}}}_{{\overline{\alpha}}}_{{\overline{\beta}}}}\tensor{E}{_\tau_{{\overline{\gamma}}}},$$ by and we obtain .
\[lem:divergence\_on\_KE\] Let $\sigma$ be an $O(\rho^j)$ anti-hermitian symmetric 2-$\Theta$-tensor on $(X,g)$, where $X$ is the square root of ${\overline{\Omega}}$ and $g$ is the Bergman-type metric given by a $C^\infty$-smooth boundary defining function $r$ of ${\overline{\Omega}}$, which is regarded as a $\Theta$-metric. Let $\rho=\sqrt{r/2}$. Then, with respect to a local frame $\set{\bm{\zeta}_\tau,\bm{\zeta}_\alpha,\bm{\zeta}_{{\overline{\tau}}},\bm{\zeta}_{{\overline{\alpha}}}}$,
\[eq:divergence\_of\_2\_tensor\_KE\] $$\begin{aligned}
\tensor{(\delta\sigma)}{_\tau}
&=-\tfrac{1}{4}(\rho\partial_\rho-2n-4)\tensor{\sigma}{_\tau_\tau}
-\rho\tensor{(\nabla^\mathrm{TW})}{^\alpha}\tensor{\sigma}{_\tau_\alpha}+O(\rho^{j+2}),\\
\tensor{(\delta\sigma)}{_\alpha}
&=-\tfrac{1}{4}(\rho\partial_\rho-2n-5)\tensor{\sigma}{_\tau_\alpha}
-\rho\tensor{(\nabla^\mathrm{TW})}{^\beta}\tensor{\sigma}{_\alpha_\beta}+O(\rho^{j+2}),
\end{aligned}$$
where $\nabla^\mathrm{TW}$ is the Tanaka–Webster connection of the contact form $\theta=\tilde{\theta}|_{TM}$, which acts on $\tensor{\sigma}{_\tau_\alpha}$ and $\tensor{\sigma}{_\alpha_\beta}$ by interpreting them as tensors in $\tensor{\mathcal{E}}{_\alpha}$ and $\tensor{\mathcal{E}}{_(_\alpha_\beta_)}$ with parameter $r$.
We compute the Christoffel symbols $\tensor{\Gamma}{^R_P_Q}$ of the Levi-Civita connection of $g$ modulo $O(\rho^2)$, i.e., modulo $O(r)$, which are considered as functions on $M$ with parameter $r$ rather than as functions on $X$. Recall that we can take such a local frame $\set{\zeta_\alpha}$ that the Tanaka–Webster connection forms for $\theta$ with respect to $\set{\zeta_\alpha|_M}$ vanish at a prescribed point $p\in M$ (see [@Lee_88]\*[Lemma 2.1]{}). We compute using such a frame, and the following equalities are to be understood as equalities at $p$. Note that $(\zeta_\gamma\tensor{h}{_\alpha_{{\overline{\beta}}}})|_M$ and $(\zeta_{{\overline{\gamma}}}\tensor{h}{_\alpha_{{\overline{\beta}}}})|_M$ vanish at $p$ for such a local frame. First, $\tensor{g}{_\tau_{{\overline{\tau}}}}=2(1-\kappa r)$, $\tensor{g}{_\tau_{{\overline{\alpha}}}}=0$, and $\tensor{g}{_\alpha_{{\overline{\beta}}}}=\tensor{h}{_\alpha_{{\overline{\beta}}}}$ imply that $\bm{\zeta}_{{\overline{C}}}\tensor{g}{_A_{{\overline{B}}}}$ are $O(r)$. On the other hand, $$d\bm{\theta}^\tau=d\left(\frac{\partial r}{r}\right)
=(1-\kappa r)\bm{\theta}^\tau\wedge\bm{\theta}^{{\overline{\tau}}}
+\frac{1}{2}\tensor{h}{_\alpha_{{\overline{\beta}}}}\bm{\theta}^\alpha\wedge\bm{\theta}^{{\overline{\beta}}}$$ and $$\label{eq:str_equation_Graham_Lee}
d\bm{\theta}^\gamma=d\left(\frac{\theta^\gamma}{\sqrt{r/2}}\right)
=-\frac{1}{2}(1-\kappa r)(\bm{\theta}^\tau+\bm{\theta}^{{\overline{\tau}}})\wedge\bm{\theta}^\gamma
+\bm{\theta}^\beta\wedge\tensor{\varphi}{_\beta^\gamma}
-ir\tensor{A}{_{{\overline{\beta}}}^\gamma}\bm{\theta}^\tau\wedge\bm{\theta}^{{\overline{\beta}}}+O(r^{3/2}),$$ where $\tensor{\varphi}{_\beta^\gamma}$ is Graham–Lee’s connection forms [@Graham_Lee_88]\*[Proposition 1.1]{} for $r$ and $\tensor{A}{_{{\overline{\beta}}}^\gamma}$ is, if restricted to each hypersurface $M_\varepsilon=\set{r=\varepsilon}$, the Tanaka–Webster torsion for $\theta_\varepsilon=\tilde{\theta}|_{TM_\varepsilon}$. We do not need the details about Graham–Lee’s connection; the point here is that it restricts to the Tanaka–Webster connection on $M$. Therefore, because of our choice of frame, $\tensor{\varphi}{_\beta^\gamma}(\zeta_\alpha)|_M$ and $\tensor{\varphi}{_\beta^\gamma}(\zeta_{{\overline{\alpha}}})|_M$ are both zero at $p$. Consequently we obtain $$\begin{aligned}
{3}
[\bm{\zeta}_\tau,\bm{\zeta}_\tau]&=0,&\qquad
[\bm{\zeta}_\tau,\bm{\zeta}_\alpha]&=\frac{1}{2}\bm{\zeta}_\alpha+O(r),&\qquad
[\bm{\zeta}_\alpha,\bm{\zeta}_\beta]&=O(r),\\
[\bm{\zeta}_\tau,\bm{\zeta}_{{\overline{\tau}}}]&=-(\bm{\zeta}_\tau-\bm{\zeta}_{{\overline{\tau}}})+O(r),&\qquad
[\bm{\zeta}_\tau,\bm{\zeta}_{{\overline{\alpha}}}]&=\frac{1}{2}\bm{\zeta}_{{\overline{\alpha}}}+O(r),&\qquad
[\bm{\zeta}_\alpha,\bm{\zeta}_{{\overline{\beta}}}]
&=-\frac{1}{2}\tensor{h}{_\alpha_{{\overline{\beta}}}}(\bm{\zeta}_\tau-\bm{\zeta}_{{\overline{\tau}}})+O(r).
\end{aligned}$$ These results imply that the formulae of the Christoffel symbols remain to hold in a stronger sense, that is, modulo $O(r)$. Hence also holds in this case if we omit $O(r)$ times the components of $\sigma$, which are $O(\rho^{j+2})$.
By the same argument as in Subsection \[subsec:Heisenberg\_principal\_part\], we may take an anti-hermitian solution $\sigma$ to such that $\delta\sigma=O(\rho^{2n+4})$, and hence implies that $\tensor{(\Delta_\mathrm{H}+n+2)(\delta\sigma)}{_\tau}=O(\rho^{2n+5})$ and $\tensor{(\Delta_\mathrm{H}+n+2)(\delta\sigma)}{_\alpha}=O(\rho^{2n+4})$. Because of and , if we set $\tilde{\sigma}=(\Delta_\mathrm{L}+n+2)\sigma$, then $$\label{eq:divergence_freeness_Kahler}
\tensor{(\delta\tilde{\sigma})}{_\tau}=O(\rho^{2n+5})\qquad\text{and}\qquad
\tensor{(\delta\tilde{\sigma})}{_\alpha}=O(\rho^{2n+4}).$$ Since $\tilde{\sigma}$ is an $O(\rho^{2n+2})$ anti-hermitian symmetric $2$-$\Theta$-tensor for which $\delta\tilde{\sigma}=O(\rho^{2n+4})$, shows that $\tensor{\tilde{\sigma}}{_\tau_\alpha}=O(\rho^{2n+3})$ and $\tensor{\tilde{\sigma}}{_\tau_\tau}=O(\rho^{2n+4})$. Let $\tensor*{k}{^{(2n+3)}_\tau_\alpha}\in\tensor{\mathcal{E}}{_\alpha}$ and $\tensor*{k}{^{(2n+2)}_\alpha_\beta}\in\tensor{\mathcal{E}}{_(_\alpha_\beta_)}$ be defined by $$\tensor{\tilde{\sigma}}{_\tau_\alpha}=\rho^{2n+3}\tensor*{k}{^{(2n+3)}_\tau_\alpha}+O(\rho^{2n+4})
\qquad\text{and}\qquad
\tensor{\tilde{\sigma}}{_\alpha_\beta}=\rho^{2n+2}\tensor*{k}{^{(2n+2)}_\alpha_\beta}+O(\rho^{2n+3}).$$ We shall write down what means in terms of $\tensor*{k}{^{(2n+3)}_\tau_\alpha}$ and $\tensor*{k}{^{(2n+2)}_\alpha_\beta}$. As in the proof of the previous lemma, we take $\set{\zeta_\alpha}$ so that the Tanaka–Webster connection forms with respect to $\set{\zeta_\alpha|_M}$ vanish at a point $p\in M$, and conduct the computation at this point. Since $$\begin{aligned}
\tensor{\nabla}{_{{\overline{\tau}}}}\tensor{\tilde{\sigma}}{_\tau_\tau}
&=\bm{\zeta}_{{\overline{\tau}}}\tensor{\tilde{\sigma}}{_\tau_\tau}
-2\tensor{\Gamma}{^\tau_{{\overline{\tau}}}_\tau}\tensor{\tilde{\sigma}}{_\tau_\tau}
-2\tensor{\Gamma}{^\alpha_{{\overline{\tau}}}_\tau}\tensor{\tilde{\sigma}}{_\tau_\alpha},\\
\tensor{\nabla}{_{{\overline{\beta}}}}\tensor{\tilde{\sigma}}{_\tau_\alpha}
&=\bm{\zeta}_{{\overline{\beta}}}\tensor{\tilde{\sigma}}{_\tau_\alpha}
-\tensor{\Gamma}{^\tau_{{\overline{\beta}}}_\tau}\tensor{\tilde{\sigma}}{_\tau_\alpha}
-\tensor{\Gamma}{^\gamma_{{\overline{\beta}}}_\tau}\tensor{\tilde{\sigma}}{_\gamma_\alpha}
-\tensor{\Gamma}{^\tau_{{\overline{\beta}}}_\alpha}\tensor{\tilde{\sigma}}{_\tau_\tau}
-\tensor{\Gamma}{^\gamma_{{\overline{\beta}}}_\alpha}\tensor{\tilde{\sigma}}{_\tau_\gamma},\\
\tensor{\nabla}{_{{\overline{\tau}}}}\tensor{\tilde{\sigma}}{_\tau_\alpha}
&=\bm{\zeta}_{{\overline{\tau}}}\tensor{\tilde{\sigma}}{_\tau_\alpha}
-\tensor{\Gamma}{^\tau_{{\overline{\tau}}}_\tau}\tensor{\tilde{\sigma}}{_\tau_\alpha}
-\tensor{\Gamma}{^\beta_{{\overline{\tau}}}_\tau}\tensor{\tilde{\sigma}}{_\beta_\alpha}
-\tensor{\Gamma}{^\tau_{{\overline{\tau}}}_\alpha}\tensor{\tilde{\sigma}}{_\tau_\tau}
-\tensor{\Gamma}{^\beta_{{\overline{\tau}}}_\alpha}\tensor{\tilde{\sigma}}{_\tau_\beta},\\
\tensor{\nabla}{_{{\overline{\gamma}}}}\tensor{\tilde{\sigma}}{_\beta_\alpha}
&=\bm{\zeta}_{{\overline{\gamma}}}\tensor{\tilde{\sigma}}{_\beta_\alpha}
-\tensor{\Gamma}{^\tau_{{\overline{\gamma}}}_\beta}\tensor{\tilde{\sigma}}{_\tau_\alpha}
-\tensor{\Gamma}{^\delta_{{\overline{\gamma}}}_\beta}\tensor{\tilde{\sigma}}{_\delta_\alpha}
-\tensor{\Gamma}{^\tau_{{\overline{\gamma}}}_\alpha}\tensor{\tilde{\sigma}}{_\tau_\beta}
-\tensor{\Gamma}{^\delta_{{\overline{\gamma}}}_\alpha}\tensor{\tilde{\sigma}}{_\tau_\delta},
\end{aligned}$$ the fact that is true modulo $O(\rho^2)$ shows that
$$\begin{aligned}
\label{eq:divergence_freeness_KE_meaning}
O(\rho^{2n+5})
&=\tensor{(\delta\tilde{\sigma})}{_\tau}
=-\rho\tensor{h}{^\alpha^{{\overline{\beta}}}}\zeta_{{\overline{\beta}}}\tensor{\tilde{\sigma}}{_\tau_\alpha}
+\rho\tensor{h}{^\alpha^{{\overline{\beta}}}}
\tensor{\Gamma}{^\gamma_{{\overline{\beta}}}_\tau}\tensor{\tilde{\sigma}}{_\gamma_\alpha}+O(\rho^{2n+5}),\\
O(\rho^{2n+4})
&=\tensor{(\delta\tilde{\sigma})}{_\alpha}
=-\rho\tensor{h}{^\beta^{{\overline{\gamma}}}}\zeta_{{\overline{\gamma}}}\tensor{\tilde{\sigma}}{_\alpha_\beta}
+\frac{1}{2}\tensor{\tilde{\sigma}}{_\tau_\beta}+O(\rho^{2n+4}).
\end{aligned}$$
The latter equality implies that $\tensor{(\nabla^\mathrm{TW})}{^\beta}\tensor*{k}{^{(2n+2)}_\alpha_\beta}
=\frac{1}{2}\tensor*{k}{^{(2n+3)}_\tau_\alpha}$. To squeeze out the meaning of the former, we need the $\rho^2$-coefficient of $\tensor{\Gamma}{^\gamma_{{\overline{\beta}}}_\tau}$. By , $[\bm{\zeta}_\tau,\bm{\zeta}_{{\overline{\beta}}}]=ir\tensor{A}{_{{\overline{\beta}}}^\gamma}\bm{\zeta}_\gamma+O(r^{3/2})
=2i\rho^2\tensor{A}{_{{\overline{\beta}}}^\gamma}\bm{\zeta}_\gamma+O(\rho^3)$. Thus, using the symmetry of the Tanaka–Webster torsion, we obtain $$\begin{split}
\tensor{\Gamma}{_{{\overline{\gamma}}}_{{\overline{\beta}}}_\tau}
&=\frac{1}{2}(\bm{\zeta}_{{\overline{\beta}}}\tensor{g}{_\tau_{{\overline{\gamma}}}}
-\bm{\zeta}_{{\overline{\gamma}}}\tensor{g}{_\tau_{{\overline{\beta}}}}
-[\bm{\zeta}_{{\overline{\beta}}},\bm{\zeta}_\tau]_{{\overline{\gamma}}}
+[\bm{\zeta}_{{\overline{\beta}}},\bm{\zeta}_{{\overline{\gamma}}}]_\tau
+[\bm{\zeta}_\tau,\bm{\zeta}_{{\overline{\gamma}}}]_{{\overline{\beta}}})\\
&=-\frac{1}{2}([\bm{\zeta}_{{\overline{\beta}}},\bm{\zeta}_\tau]_{{\overline{\gamma}}}
-[\bm{\zeta}_{{\overline{\beta}}},\bm{\zeta}_{{\overline{\gamma}}}]_\tau
-[\bm{\zeta}_\tau,\bm{\zeta}_{{\overline{\gamma}}}]_{{\overline{\beta}}})
=2i\rho^2\tensor{A}{_{{\overline{\beta}}}_{{\overline{\gamma}}}}+O(\rho^3).
\end{split}$$ Therefore, implies that $\tensor{(\nabla^\mathrm{TW})}{^\alpha}\tensor*{k}{^{(2n+3)}_\tau_\alpha}
=2i\tensor{A}{^\alpha^\beta}\tensor*{k}{^{(2n+2)}_\alpha_\beta}$. Thus we have $$\tensor{(\nabla^\mathrm{TW})}{^\alpha}\tensor{(\nabla^\mathrm{TW})}{^\beta}
\tensor*{k}{^{(2n+2)}_\alpha_\beta}
-i\tensor{A}{^\alpha^\beta}\tensor*{k}{^{(2n+2)}_\alpha_\beta}=0,$$ which means that $D^*\tensor*{\mathcal{O}}{^\bullet_\alpha_\beta}=0$.
In [@Matsumoto_14] the author showed that, for partially integrable CR manifolds in general, $$\label{eq:double_divergence_free_for_partially_integrable}
\operatorname{Im}(\tensor{\nabla}{^\alpha}\tensor{\nabla}{^\beta}\tensor{\mathcal{O}}{_\alpha_\beta}
-i\tensor{A}{^\alpha^\beta}\tensor{\mathcal{O}}{_\alpha_\beta}
-\tensor{N}{^\gamma^\alpha^\beta}\tensor{\nabla}{_\gamma}\tensor{\mathcal{O}}{_\alpha_\beta}
-(\tensor{\nabla}{_\gamma}\tensor{N}{^\gamma^\alpha^\beta})\tensor{\mathcal{O}}{_\alpha_\beta})=0.$$ Then, by differentiating it, we obtain $\operatorname{Im}(D^*\tensor*{\mathcal{O}}{^\bullet_\alpha_\beta})=0$ for integrable CR manifolds; what we proved above is an improvement on this. Why are not we able to improve equation itself? Recall that comes out of the contracted second Bianchi identity of a normalized ACH metric $g$ satisfying . Namely, if we set $E={\operatorname{Ric}}+\frac{n+2}{2}g$ then $\mathcal{B}E=\delta E+\frac{1}{2}d(\operatorname{tr}E)=0$ holds, so we combine the identities $(\mathcal{B}E)(\bm{Z}_0)=0$ and $(\mathcal{B}E)(\bm{Z}_\alpha)=0$ to get . A natural idea to refine this argument is to use the remaining identity: $(\mathcal{B}E)(\bm{Z}_\infty)=0$. Nevertheless, this does not work by the following reason. Let $\tensor*{E}{^{(2n+3)}_I_J}$ be the $\rho^{2n+3}$-coefficient of $\tensor{E}{_I_J}$. Then $\operatorname{Re}\tensor{(\nabla^\mathrm{TW})}{^\alpha}\tensor*{E}{^{(2n+3)}_\infty_\alpha}$ appears in the $\rho^{2n+4}$-coefficient of $(\mathcal{B}E)(\bm{Z}_\infty)$, while $\operatorname{Re}\tensor{(\nabla^\mathrm{TW})}{^\alpha}\tensor*{E}{^{(2n+3)}_0_\alpha}$ appears in the $\rho^{2n+4}$-coefficient of $(\mathcal{B}E)(\bm{Z}_0)$. But by the lack of Kähler structure, there is no means to compare $\operatorname{Re}\tensor{(\nabla^\mathrm{TW})}{^\alpha}\tensor*{E}{^{(2n+3)}_\infty_\alpha}$ and $\operatorname{Re}\tensor{(\nabla^\mathrm{TW})}{^\alpha}\tensor*{E}{^{(2n+3)}_0_\alpha}$, and consequently we cannot make a good use of these two equalities. In the proof of Theorem \[thm:obstruction\_operator\_observation\] (2) above, can be seen as the infinitesimal version of $\mathcal{B}E=0$. In this case however we have the extra equality $\tensor{\tilde{\sigma}}{_{{\overline{\tau}}}_\alpha}=0$, which is an outcome of the Kählerity of the bulk metric $g$. (This comparison is not very parallel because we define $E$ by the *normalized* approximate solution to the Einstein equation. However, even if we use the Bianchi gauge solution to define $E$, the same problem for improving still happens.)
To prove the last part of the theorem, recall the Weitzenböck formula for $(0,q)$-forms on a Kähler manifold $X$ with values in a holomorphic vector bundle $E$ (see, e.g., [@Moroianu_07]\*[Theorem 20.2]{}). In the case where $E=T^{1,0}X$ is the holomorphic tangent bundle, the formula reads as follows: if $\eta=\tensor{\eta}{_{{\overline{j}}_1}_{\dotsb}_{{\overline{j}}_q}^k}$ is a $(T^{1,0}X)$-valued $(0,q)$-form, then the Dolbeault Laplacian $\Delta_{{\overline{\partial}}}=\smash{{\overline{\partial}}}^*{\overline{\partial}}+{\overline{\partial}}\smash{{\overline{\partial}}}^*$ acts on $\eta$ as $$2\tensor{(\Delta_{{\overline{\partial}}}\eta)}{_{{\overline{i}}_1}_{\dotsb}_{{\overline{i}}_q}^j}
=\tensor{(\nabla^*\nabla\eta)}{_{{\overline{i}}_1}_{\dotsb}_{{\overline{i}}_q}^j}
+\sum_{s=1}^q
\tensor{\operatorname{Ric}}{^{{\overline{k}}}_{{\overline{i}}_s}}
\tensor{\eta}{_{{\overline{i}}_1}_{\dotsb}_{{\overline{k}}}_{\dotsb}_{{\overline{i}}_q}^j}
-\tensor{\operatorname{Ric}}{_l^j}\tensor{\eta}{_{{\overline{i}}_1}_{\dotsb}_{{\overline{i}}_q}^l}
+2\sum_{s=1}^q\tensor{R}{^{{\overline{k}}}_{{\overline{i}}_s}_l^j}
\tensor{\eta}{_{{\overline{i}}_1}_{\dotsb}_{{\overline{k}}}_{\dotsb}_{{\overline{i}}_q}^l}.$$ By taking the complex conjugate, we obtain a similar formula of $\Delta_\partial$ acting on $({\overline{T^{1,0}X}})$-valued $(p,0)$-forms. In particular, if $\sigma=\tensor{\sigma}{_j_k}$ is a symmetric tensor of type $(2,0)$, which is identified with a $({\overline{T^{1,0}X}})$-valued $(1,0)$-form by raising an index, then $$\label{eq:Weitzenbock_in_2_tensors}
2\tensor{(\Delta_\partial\sigma)}{_j_k}
=\tensor{(\Delta_\mathrm{L}\sigma)}{_j_k}-2\tensor{\operatorname{Ric}}{_k^l}\tensor{\sigma}{_j_l}.$$ Therefore, on a Kähler–Einstein manifold with $\operatorname{Ric}=\lambda g$, the operator $\Delta_\mathrm{L}-2\lambda$ on type $(2,0)$ symmetric tensors is equivalent to $2\Delta_\partial$, as described in [@Besse_87]\*[Equation (12.93$'$)]{}.
Let us go back to our situation, in which $X$ is the square root of $\overline{\Omega}$. The Weitzenböck formula in particular implies that if $\eta$ is a $({{\fourIdx{\Theta}{}{}{}{TX}}})^{1,0}$-valued $p$-$\Theta$-form of type $(p,0)$ then so is $\Delta_\partial\eta$, because $\nabla$ is a $\Theta$-connection and $\operatorname{Ric}$, $R$ are $\Theta$-tensors. Moreover, since holds, implies that $$2\Delta_\partial\sigma=(\Delta_\mathrm{L}+n+2)\sigma+O(\rho^{2n+4}).$$ Therefore, if $\sigma$ is a solution of , then it solves $$\Delta_\partial\sigma=O(\rho^{2n+2}).$$ Since $\mathcal{V}_\Theta=C^\infty(X,{{\fourIdx{\Theta}{}{}{}{TX}}})$ is closed under the Lie bracket, one can check that $\partial\Delta_\partial\sigma=O(\rho^{2n+2})$. Therefore, $$\Delta_\partial(\partial\sigma)=O(\rho^{2n+2}).$$ Thus we are led to studying $\Delta_\partial$ acting on what we call *Nijenhuis-type* 3-$\Theta$-tensors, i.e., real 3-$\Theta$-tensors $\nu$ whose components other than $\tensor{\nu}{_A_B_C}$ and $\tensor{\nu}{_{{\overline{A}}}_{{\overline{B}}}_{{\overline{C}}}}$ are zero that satisfy $$\tensor{\nu}{_(_A_B_)_C}=0\qquad\text{and}\qquad
\tensor{\nu}{_(_A_B_C_)}=0.$$
\[lem:Laplacian\_on\_Nijenhuis\_type\_tensors\] Suppose $\nu$ is a Nijenhuis-type 3-$\Theta$-tensor. Then, if $\nu=O(\rho^j)$, $$\begin{aligned}
2\tensor{(\Delta_\partial\nu)}{_\tau_(_\alpha_\beta_)}
&=-\frac{1}{4}j(j-2n-2)\tensor{\nu}{_\tau_(_\alpha_\beta_)}+O(\rho^{j+1}),\\
2\tensor{(\Delta_\partial\nu)}{_\tau_[_\alpha_\beta_]}
&=-\frac{1}{4}(j^2-(2n+2)j-8)\tensor{\nu}{_\tau_[_\alpha_\beta_]}+O(\rho^{j+1}),\\
2\tensor{(\Delta_\partial\nu)}{_\alpha_\beta_\gamma}
&=-\frac{1}{4}(j-1)(j-2n-1)\tensor{\nu}{_\alpha_\beta_\gamma}+O(\rho^{j+1}).
\end{aligned}$$
By the Weitzenböck formula, we have $$\begin{gathered}
2\tensor{(\Delta_\partial\nu)}{_A_B_C}
=\nabla^*\nabla\tensor{\nu}{_A_B_C}+\tensor{\operatorname{Ric}}{_A^D}\tensor{\nu}{_D_B_C}
+\tensor{\operatorname{Ric}}{_B^D}\tensor{\nu}{_A_D_C}-\tensor{\operatorname{Ric}}{_C^D}\tensor{\nu}{_A_B_D}\\
+2\tensor{R}{_A^D_C^E}\tensor{\nu}{_D_B_E}+2\tensor{R}{_B^D_C^E}\tensor{\nu}{_A_D_E}.
\end{gathered}$$ Then and show that, under the assumption $\nu=O(\rho^j)$, $$2\tensor{(\Delta_\partial\nu)}{_A_B_C}=\left(\nabla^*\nabla-\frac{n+8}{2}\right)\tensor{\nu}{_A_B_C}
+O(\rho^{j+1}).$$ The argument in the beginning of the proof of Lemma \[lem:Laplacian\_on\_tensors\] shows that, if $\nu=\rho^{j}\tilde{\nu}$, then $\nabla^*\nabla\nu=\rho^j\nabla^*\nabla\tilde{\nu}-\frac{1}{4}j(j-2n-2)\rho^j\tilde{\nu}+O(\rho^{j+1})$. Therefore, it suffices to consider the $j=0$ case and check that the following hold on $M$: $$\tensor{(\nabla^*\nabla\nu)}{_\tau_(_\alpha_\beta_)}
=\frac{n+8}{2}\tensor{\nu}{_\tau_(_\alpha_\beta_)},\qquad
\tensor{(\nabla^*\nabla\nu)}{_\tau_[_\alpha_\beta_]}
=\frac{n+12}{2}\tensor{\nu}{_\tau_[_\alpha_\beta_]},\qquad
\tensor{(\nabla^*\nabla\nu)}{_\alpha_\beta_\gamma}
=\frac{15}{4}\tensor{\nu}{_\alpha_\beta_\gamma}.$$ These are obtained by a straightforward calculation using .
If $\sigma$ is a solution of with boundary data $\tensor{\psi}{_\alpha_\beta}$, then $\nu=\partial\sigma$ is given by $$\tensor{\nu}{_A_B_C}=\tensor{\nabla}{_[_A}\tensor{\sigma}{_B_]_C}.$$ We first compute the lowest coefficients of $\tensor{\nu}{_A_B_C}$. The boundary values of the Christoffel symbols show that $$\tensor{\nabla}{_\tau}\tensor{\sigma}{_\alpha_\beta}=\tensor{\sigma}{_\alpha_\beta}+O(\rho)
\qquad\text{and}\qquad
\tensor{\nabla}{_\alpha}\tensor{\sigma}{_\tau_\beta}=\tensor{\sigma}{_\alpha_\beta}+O(\rho),$$ and hence $\tensor{\nu}{_\tau_\alpha_\beta}=O(\rho)$, while $$\tensor{\nabla}{_\alpha}\tensor{\sigma}{_\beta_\gamma}
=\rho\tensor*{\nabla}{^{\mathrm{TW}}_\alpha}\tensor{\psi}{_\beta_\gamma}+O(\rho^2),$$ which implies that $\tensor{\nu}{_\alpha_\beta_\gamma}=\frac{1}{2}\rho\tensor{(N^\bullet\psi)}{_\alpha_\beta_\gamma}+O(\rho^2)$. Then Lemma \[lem:Laplacian\_on\_Nijenhuis\_type\_tensors\] shows that, if we write $$\tensor{\nu}{_\tau_\alpha_\beta}
=\sum_{j=1}^{2n+1}\rho^j\tensor*{n}{^{(j)}_\tau_\alpha_\beta}+O(\rho^{2n+2})\qquad\text{and}\qquad
\tensor{\nu}{_\alpha_\beta_\gamma}
=\rho\tensor{(N^\bullet\psi)}{_\alpha_\beta_\gamma}
+\sum_{j=2}^{2n+1}\rho^j\tensor*{n}{^{(j)}_\alpha_\beta_\gamma}+O(\rho^{2n+1}),$$ then $\tensor*{n}{^{(1)}_\tau_\alpha_\beta}$, $\dotsc$, $\tensor*{n}{^{(2n+1)}_\tau_\alpha_\beta}$ and $\tensor*{n}{^{(1)}_\alpha_\beta_\gamma}$, $\dotsc$, $\tensor*{n}{^{(2n)}_\alpha_\beta_\gamma}$ have universal expression in terms of the Tanaka–Webster connection and $N^\bullet\psi$. To determine $\tensor*{n}{^{(2n+1)}_\alpha_\beta_\gamma}$, we use the fact that $\partial\nu=\partial^2\sigma=0$. If $\nu=O(\rho^j)$, then we can compute $$\tensor{(\partial\nu)}{_\tau_\alpha_\beta_\gamma}
=\tensor{\nabla}{_[_\tau}\tensor{\nu}{_\alpha_]_\beta_\gamma}
=\frac{1}{4}(j+1)\tensor{\nu}{_\alpha_\beta_\gamma}+O(\rho^{j+1}).$$ Since this should be zero, $\tensor*{n}{^{(2n+1)}_\alpha_\beta_\gamma}$ is again written in terms of the Tanaka–Webster connection and $N^\bullet\psi$.
Now we compute the $\rho^{2n+2}$-coefficient of $\tilde{\sigma}=(\Delta_\mathrm{L}+n+2)\sigma=\Delta_\partial\sigma+O(\rho^{2n+4})$. If we choose $\sigma$ for which $\delta\sigma=O(\rho^{2n+4})$, then $$\tilde{\sigma}=\partial^*\partial\sigma+O(\rho^{2n+4})=\partial^*\nu+O(\rho^{2n+4}).$$ If $\nu=O(\rho^j)$, then $$\tensor{(\partial^*\nu)}{_(_\alpha_\beta_)}
=-\tensor{\nabla}{^\tau}\tensor{\nu}{_\tau_(_\alpha_\beta_)}
-\tensor{\nabla}{^\gamma}\tensor{\nu}{_\gamma_(_\alpha_\beta_)}
=-\frac{1}{4}(j-2n-2)\tensor{\nu}{_\tau_(_\alpha_\beta_)}+O(\rho^{j+1}),$$ which means that the $\rho^{2n+2}$-coefficient of $\tensor{\tilde{\sigma}}{_\alpha_\beta}$ is expressed by the coefficients of $\nu$ up to $(2n+1)$st order. To sum up, $\tensor*{\mathcal{O}}{^\bullet_\alpha_\beta}$ can be written as a universal polynomial of contractions of Tanaka–Webster local invariants and covariant derivatives of $\tensor{(N^\bullet\psi)}{_\alpha_\beta_\gamma}$.
[^1]: The other possible approach, which utilizes the Fefferman metric, is not discussed in this article. There is a work of Leitner [@Leitner_10] on a “gauged” Fefferman construction on partially integrable CR manifolds, and this direction should also be pursued further.
[^2]: The approximate Einstein condition is slightly modified from what we imposed in [@Matsumoto_14] to make the exposition simpler and easier to understand.
|
---
author:
- Abhishek Badki
- Alejandro Troccoli
- Kihwan Kim
- Jan Kautz
- Pradeep Sen
- Orazio Gallo
title: '[Bi3D]{}: [S]{}tereo Depth Estimation via Binary Classifications'
---
@affilsepx
Introduction {#sec:intro}
============
Related Work {#sec:related}
============
Method {#sec:method}
======
Implementation {#sec:implementation}
==============
Evaluation and Results {#sec:results}
======================
Conclusions {#sec:conclusions}
===========
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors thank Zhiding Yu for the continuous discussions and Shoaib Ahmed Siddiqui for help with distributed training. UCSB acknowledges partial support from NSF grant IIS 16-19376 and an NVIDIA fellowship for A. Badki.
Additional Details on Training {#sec:training}
==============================
Supplementary Video {#sec:adadepth}
===================
|
---
abstract: 'Instance Segmentation is an interesting yet challenging task in computer vision. In this paper, we conduct a series of refinements with the Hybrid Task Cascade (HTC) Network, and empirically evaluate their impact on the final model performance through ablation studies. By taking all the refinements, we achieve 0.47 on the COCO test-dev dataset and 0.47 on the COCO test-challenge dataset.'
author:
- |
Dongdong Yu, Zehuan Yuan, Jinlai Liu, Kun Yuan, Changhu Wang\
ByteDance AI Lab, China\
bibliography:
- 'egbib.bib'
title: Towards Good Practices for Instance Segmentation
---
Introduction
============
Instance segmentation is an interesting yet challenging task in computer vision. The goal is to label each pixel into foreground or backbone for each object instance detection bounding-box. It is important for many applications, such as autonomous driving, virtual reality, human-computer interaction and activity recognition.
Recently, the problem of instance segmentation has been greatly improved by the involvement of deep convolutional neural networks [@he2016deep; @chen2019mmdetection; @chen2019hybrid; @oksuz2019imbalance]. Existing approaches are mostly follows the Mask-RCNN method [@he2017mask], which extends Faster R-CNN by adding a branch for predicting an object mask in parallel with the existing branch for bounding box recognition. For example, Hybrid Task Cascade (HTC) Network [@chen2019hybrid] interweaves the box detection task and instance segmentation task for a joint multi-stage refinement processing and devils spatial context information to help distinguishing hard foreground from cluttered background, which ranks 1st in the COCO 2018 Challenge Object Detection Task. There are many tricks to improve the performance of instance segmentation, such as, strong backbone, balanced learning strategy, effective sampling strategy, multi-scale training, multi-scale testing and model ensemble.
In this paper, we follow the Hybrid Task Cascade Network pipeline and conduct a series of refinements(new backbone, nms strategy and re-compute the mask score) based on the Hybrid Task Cascade Network and evaluate their impact on the final model performance through ablation studies. Finally, we achieve 0.47 on the COCO test-dev dataset and 0.47 on the COCO test- challenge dataset.
Method
======
To handle the instance segmentation, we follow the Hybrid Task Cascade Network pipeline to detect each instance in the image and then classify the foreground and backbone pixels for each detected instance.
The HTC network can adopts the ResNet, ResNext or SENet as the backbone of the feature encoder. In our work, we propose a new backbone, named RefineNet [@yu2019towards], which can well handle the scale variant cases. Different than most algorithms, we proposed a post-processing method to re-compute the mask score, instead of directly using the bounding-box score.
Experiments
===========
Datasets
--------
The training dataset only includes the COCO train2017 dataset [@lin2014coco; @caesar2018coco], we do not use any other dataset. The final results are reported on the COCO test-dev dataset and the COCO test-challenge dataset.
Results
-------
### Ablation Study
In this subsection, we will step-wise decompose our model to reveal the effect of each component, including our new backbone, our re-compute mask score strategy and nms strategy. In the following experiments, we evaluate all comparisons on the COCO val2017 dataset.
[**Effect of Backbone**]{} In our paper, we modify the Residual bottleneck and propose a new backbone, named RefineNet, which is similar to ResNet, ResNext or SENet. Based on the HTC Network [@chen2019mmdetection], we implement the SENet154 HTC and SE-RefineNet154 HTC. As shown in Table \[table:table1\], we implement the SENet154 in mmdetection, the AP performance is obvious lower than the author’s implement, unfortunately. Based on our implement SENet154, we replace the bottleneck with our modified bottleneck, the AP is improved from 0.426 to 0.434.
[**Effect of Mask Score**]{} In our experiments, we find that it is not the best choice to use the bounding-box score as the mask score. We re-compute the score which takes the mask confidence score into account. As shown in Table \[table:table2\], by using the re-compute score strategy, the AP is improved from 0.434 to 0.445.
[**Effect of NMS**]{} Besides, we also try the soft-nms strategy. As shown in Table \[table:table3\], the AP is improved from 0.445 to 0.447.
### Development and Challenge Results
In this subsection, we ensemble four RefineNet models, and take the multi-scale testing strategy and flip testing strategy for the instance segmentation. As shown in Table \[table:table4\], the AP of test-dev is 0.473, and the AP of test-cha is 0.47.
Conclusion
==========
In this paper, we conduct a series of refinements with the Hybrid Task Cascade (HTC) Network, and empirically evaluate their impact on the final model performance through ablation studies. By taking all the refinements, we achieve 0.47 on the COCO test-dev dataset and 0.47 on the COCO test-challenge dataset.
|
---
abstract: 'We describe explicit formulas relevant to the F-theory/heterotic string duality that reconstruct from a specific Jacobian elliptic fibration on the Shioda-Inose surface covering a generic Kummer surface the corresponding genus-two curve using the level-two Satake coordinate functions. We derive explicitly the rational map on the moduli space of genus-two curves realizing the algebraic correspondence between a sextic curve and its Satake sextic. We will prove that it is not the original sextic defining the genus-two curve, but its corresponding Satake sextic which is manifest in the F-theory model, dual to the $\mathfrak{so}(32)$ heterotic string with an unbroken $\mathfrak{so}(28)\oplus \mathfrak{su}(2)$ gauge algebra.'
address:
- 'Department of Mathematics and Statistics, Utah State University, Logan, UT 84322'
- 'Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309'
author:
- 'A. Malmendier'
- 'T. Shaska'
bibliography:
- 'ref.bib'
title: The Satake sextic in elliptic fibrations on K3
---
Introduction
============
Constructing equations of algebraic curves from a given point in the moduli space or a given Jacobian has always been interesting to both mathematicians and physicists. The only case where such constructions can be made explicit is the case of genus-two curves. There have been attempts by other authors before where equations of the genus-two curve is written in terms of the thetanulls of the Jacobian; see [@psw] and [@sw].
By a sextic curve we mean a projective curve of degree six. To each sextic curve one can associate another sextic curve, called the *Satake sextic*. The algebraic correspondence between these two sextics is quite complicated, and we give explicit formulas for its construction. In fact, starting with a plane curve, for example in Rosenhain normal form, the computation of the Igusa invariants provides an effective method for computing the corresponding Satake sextic. Conversely, starting with the roots of the Satake sextic we will derive explicit formulas for the reconstruction of the original sextic up to equivalence. This will allow us to explicitly determine the rational map on the moduli space of genus-two curves realizing the correspondence between a sextic curve and its Satake sextic.
For a generic genus-two curve $\mathcal{C}$ the Jacobian variety $\operatorname{Jac}(\mathcal{C})$ is principally polarized abelian surface, and the minimal resolution of the quotient by the involution automorphism is a special K3 surface. called the *Kummer surface* $\mathrm{Kum}(\operatorname{Jac}\mathcal{C})$. There is a closely related K3 surface, called the *Shioda-Inose surface* $\mathrm{SI}(\operatorname{Jac}\mathcal{C})$, which carries a *Nikulin involution*, i.e., an automorphism of order two preserving the holomorphic two-form, such that quotienting by this involution and blowing up the fixed points recovers the Kummer surface. By using the Shioda-Inose surface $\mathrm{SI}(\operatorname{Jac}\mathcal{C})$ that covers the Kummer surface, one establishes a one-to-one correspondence between two different types of surfaces with the same Hodge-theoretic data, principally polarized abelian surfaces and algebraic K3 surfaces polarized by a special lattice, which is known as *geometric two-isogeny*.
In string theory, compactifications of the so-called type-IIB string in which the complex coupling varies over a base are generically referred to as F-theory. The simplest such construction corresponds to a Jacobian elliptic fibration on a K3 surface. By taking this K3 surface to be the Shioda-Inose surface $\mathrm{SI}(\operatorname{Jac}\mathcal{C})$ a phenomenon called *F-theory/heterotic string duality* is manifested as the aforementioned geometric two-isogeny. An important question is whether the original genus-two curve $\mathcal{C}$ is still manifest in this F-theoretic description of non-geometric heterotic string backgrounds. We will prove that it is not the original sextic defining the genus-two curve $\mathcal{C}$, but the corresponding Satake sextic which is manifest in the F-theoretic data. In fact, the ramification locus of the Satake sextic is the genus-two component of the fixed point set of the Nikulin involution on $\mathrm{SI}(\operatorname{Jac}\mathcal{C})$.
This article is structured as follows: in Section 2 we give a brief review of principally polarized abelian surfaces, the thetanulls for genus two, and the Satake coordinate functions, as well as their relations to the Igusa invariants and Siegel modular forms. We then prove a Picard like result, which gives the Rosenhein roots of a genus-two curve in terms of the thetanulls and also in terms of the Satake coordinate functions. These explicit formulas are instrumental in computing the rational map on the moduli space of genus-two curves that realizes the algebraic correspondence between the sextic and its corresponding Satake sextic. In Section 3 we describe the construction of the Kummer surface $\mathrm{Kum}(\operatorname{Jac}\mathcal{C})$ and Shioda-Inose surface $\mathrm{SI}(\operatorname{Jac}\mathcal{C})$, as well as the Jacobian elliptic fibrations on them which are relevant for the F-theory/heterotic string duality. We then prove that the positions of 7-branes with string charge $(1,0)$ in the F-theory model dual to the $\mathfrak{so}(32)$ heterotic string with an unbroken $\mathfrak{so}(28)\oplus \mathfrak{su}(2)$ gauge algebra and only one non-vanishing Wilson line form the ramification locus of the Satake sextic which is in algebraic correspondence with the genus-two curve $\mathcal{C}$.
The correspondence between a sextic and its Satake sextic
=========================================================
In this section we give a brief review of principally polarized abelian surfaces, the thetanulls, the Satake coordinate functions, and their relations to the Igusa invariants and Siegel modular forms. We then prove a Picard like result, which gives the Rosenhein roots of genus two curve in terms of the thetanulls and also in terms of the Satake coordinate functions. We also compute the rational map on the moduli space of genus-two curves realizing the correspondence between a sextic curve and its Satake sextic.
Abelian surfaces
----------------
The *Siegel upper-half space* is the set of two-by-two symmetric matrices over $\mathbb{C}$ whose imaginary part is positive definite, i.e., $$\label{Siegel_tau}
\mathbb{H}_2 = \left. \left\lbrace \tau = \left( \begin{array}{cc} \tau_1 & z \\ z & \tau_2\end{array} \right) \right|
\tau_1, \tau_2, z \in \mathbb{C}\,,\; {\textnormal{Im}\;\;\!\!\!}{(\tau_1)} \, {\textnormal{Im}\;\;\!\!\!}{(\tau_2}) > {\textnormal{Im}\;\;\!\!\!}{(z)}^2\,, \; {\textnormal{Im}\;\;\!\!\!}{(\tau_2)} > 0 \right\rbrace .$$ The Siegel three-fold is a quasi-projective variety of dimension three obtained from the Siegel upper half plane when quotienting out by the action of the modular transformations $\Gamma_2:=
{\operatorname{Sp}_4(\Z)}$, i.e., $$\mathcal{A}_2 = \mathbb{H}_2 / \Gamma_2 \;.$$ For each $\tau \in \mathbb{H}_2$ the columns of the matrix $\left[ \, \mathbb{I}_2 | \tau \right]$ form a lattice $\Lambda$ in $\C^2$ and determine a principally polarized complex abelian surface $\mathbf{A}_{\tau} = \C^2/\Lambda$. Two abelian surfaces $\mathbf{A}_{\tau}$ and $\mathbf{A}_{\tau'}$ are isomorphic if and only if there is a symplectic matrix $M \in \Gamma_2$ such that $\tau' = M (\tau)$. It follows that the Siegel three-fold $\mathcal{A}_2$ is also the set of isomorphism classes of principally polarized abelian surfaces. The even Siegel modular forms of $\mathcal{A}_2$ are a polynomial ring in four free generators of degrees $4$, $6$, $10$ and $12$ that will be denoted by $\psi_4, \psi_6, \chi_{10}$ and $\chi_{12}$, respectively. Igusa showed in [@MR0229643] that for the full ring of modular forms, one needs an additional generator $\chi_{35}$ which is algebraically dependent on the others. We also define $\Gamma_2(2n) = \lbrace M \in \Gamma_2 | \, M \equiv \mathbb{I} \mod{2n}\rbrace$ with corresponding Siegel modular threefold $\mathcal{A}_2(2)$ such that $ \Gamma_2/\Gamma_2(2)\cong S_6$ where $S_6$ is the permutation group of order $720$.
If $\mathcal{C}$ is an irreducible nonsingular projective curve with self-intersection $\mathcal{C}\cdot \mathcal{C}=2$ then $\mathcal{C}$ is a smooth curve of genus two. We choose a symplectic homology basis for $\mathcal{C}$, say $ \{ A_1, A_2, B_1, B_2 \},$ such that the intersection products $A_i \cdot A_j = B_i \cdot B_j =0$ and $A_i \cdot B_j= \d_{i j},$ where $\d_{i j}$ is the Kronecker delta. We choose a basis $\{ w_i\}$ for the space of holomorphic one-forms such that $\int_{A_i} w_j = \d_{i j}$. The matrix $$\tau = \left[ \int_{B_i} w_j\right]$$ is the *period matrix* of $\mathcal{C}$ and $\J\!(\mathcal{C}) = \mathbf{A}_{\tau}$ is the Jacobian of $\mathcal{C}$. Moreover, the map $\jmath_\mathcal{C}: \mathcal{C} \to \operatorname{Jac}(\mathcal{C})$ is an embedding of the moduli space of genus-two curves $\mathcal{M}_2$ into the space of principally polarized abelian surfaces, i.e., $$\M_2 \embd \A_2 \;,$$ where the hermitian form associated to the divisor class $[\mathcal{C}]$ is the principal polarization $\rho$ on $\operatorname{Jac}(\mathcal{C})$. Moreover, a curve $\mathcal{C}$ of genus-two is called generic if the Néron-Severi lattice is generated by $[\mathcal{C}]$, i.e., $\mathrm{NS}(\operatorname{Jac} \mathcal{C})=\mathbb{Z}[\mathcal{C}]$. Since we have $\rho^2=2$, the transcendental lattice is $T(\operatorname{Jac}\mathcal{C}) = H \oplus H \oplus \langle -2 \rangle$ in this case.[^1] Conversely, one can always regain $\mathcal{C}$ from the pair $(\operatorname{Jac}\mathcal{C}, \rho)$ where $\rho$ is a principal polarization.
The Humbert surface $H_{\Delta}$ with invariant $\Delta$ is the space of principally polarized abelian surfaces admitting a symmetric endomorphism of discriminant $\Delta$. The discriminant $\Delta$ is always a positive integer $\equiv 0, 1\mod 4$. In fact, $H_{\Delta}$ is the image of the equation $$a \, \tau_1 + b \, z + c \, \tau_3 + d\, (z^2 -\tau_1 \, \tau_2) + e = 0 \;,$$ with integers $a, b, c, d, e$ satisfying $\Delta=b^2-4\,a\,c-4\,d\,e$ and $\tau
= \bigl(\begin{smallmatrix}
\tau_1&z\\ z&\tau_2
\end{smallmatrix} \bigr) \in \mathbb{H}_2$. Therefore, inside of $\mathcal{A}_2$ sit the Humbert surfaces $H_1$ and $H_4$ that are defined as the images of the rational divisors associated to $z=0$ and $\tau_1=\tau_2$, respectively. In fact, $H_1$ and $H_4$ form the two connected components of the singular locus of $\mathcal{A}_2$, and their formal sum $H_1 + H_4$ is the vanishing divisor of $\chi_{35}$.
Furthermore, Torelli’s theorem states that the map sending a curve $\mathcal{C}$ to its Jacobian variety $\mathrm{Jac}(C)$ induces a birational map from the moduli space $\mathcal{M}_2$ of genus-two curves to the complement of the Humbert surface $H_1$ in $\mathcal{A}_2$. This locus is expressed in terms of modular forms as $\mathcal{A}_2 \backslash \, {\textnormal{supp}}{ (\chi_{10})}_0$. That is, a period point $\tau$ is equivalent to a point with $z=0$, i.e., $\tau \in H_1$, if and only if $\chi_{10}(\tau)=0$, if and only if the principally polarized abelian surface $\mathbf{A}_{\tau} $ is a product of two elliptic curves $\mathbf{A}_{\tau} =\mathcal{E}_{\tau_1} \times \mathcal{E}_{\tau_2}$. In turn, the transcendental lattice is $T(\mathbf{A}_{\tau}) = H \oplus H$.
On the other hand, it is known that the vanishing divisor of $Q= 2^{12} \, 3^9 \, \chi_{35}^2 /\chi_{10}$ is the Humbert surface $H_4$ [@MR1438983], that is, a period point $\tau$ is equivalent to a point with $\tau_1=\tau_2$, i.e., $\tau \in H_4$, if and only if $Q(\tau)=0$. In turn, the transcendental lattice degenerates to $T(\mathbf{A}_{\tau}) = H \oplus \langle 2 \rangle \oplus \langle -2 \rangle$. Bolza [@MR1505464] described the possible automorphism groups of genus-two curves defined by sextics. In particular, he proved that a sextic curve $Y^2=F(X)$ defining the genus-two curve $\mathcal{C}$ with $\mathbf{A}_{\tau}=\mathrm{Jac}(C)$ has an extra involution, which then can be represented as $(X,Y) \mapsto (-X,Y)$, if and only if $Q(\tau)=0$.
Thetanulls for genus two
------------------------
For any $z \in \C^2$ and $\t \in \H_2$ *Riemann’s theta function* is defined as $$\theta (z , \t) = \sum_{u\in \Z^2} e^{\pi i ( u^t \t u + 2 u^t z ) }$$ where $u$ and $z$ are two-dimensional column vectors and the products involved in the formula are matrix products. The fact that the imaginary part of $\t$ is positive makes the series absolutely convergent over any compact sets. Therefore, the function is analytic. The theta function is holomorphic on $\C^2\times
\H_2$ and satisfies $$\theta(z+u,\tau)=\theta(z,\tau),\quad \theta(z+u\tau,\tau)=e^{-\pi i( u^t \tau u+2z^t u )}\cdot
\theta(z,\tau),$$ where $u\in \Z^2$; see [@MR2352717] for details. Any point $e \in \operatorname{Jac}(\mathcal{C})$ can be written uniquely as $e = (b,a) \begin{pmatrix} \mathbb{I}_2 \\ \tau \end{pmatrix}$, where $a,b \in \R^2$. We shall use the notation $[e] =
{\begin{bmatrix}
a \\
b\\
\end{bmatrix}}$ for the characteristic of $e$. For any $a, b \in \Q^2$, the theta function with rational characteristics is defined as $$\theta {\begin{bmatrix}
a \\
b\\
\end{bmatrix}} (z , \t) = \sum_{u\in \Z^2} e^{\pi i ( (u+a)^t \t (u+a) + 2 (u+a)^t (z+b) ) }.$$ When the entries of column vectors $a$ and $b$ are from the set $\{ 0,\frac{1}2\}$, then the characteristics $ {\begin{bmatrix}
a \\
b\\
\end{bmatrix}} $ are called the *half-integer characteristics*. The corresponding theta functions with rational characteristics are called *theta characteristics*. A scalar obtained by evaluating a theta characteristic at $z=0$ is called a *theta constant*. Points of order $n$ on $\operatorname{Jac}(\mathcal{C})$ are called the $\frac 1 n$-*periods*. Any half-integer characteristic is given by $$\m = \frac{1}2m = \frac{1}2
\begin{pmatrix} m_1 & m_2 \\ m_1^{\prime} & m_2^{\prime} \end{pmatrix} \;,$$ where $m_i, m_i^{\prime} \in \Z.$ For $\gamma = {\begin{bmatrix}
\gamma ^\prime \\
\gamma^{\prime \prime}\\
\end{bmatrix}} \in
\frac{1}2\Z^4/\Z^4$ we define $e_*(\gamma) = (-1)^{4 (\gamma ^\prime)^t \gamma^{\prime \prime}}.$ Then, $$\theta [\gamma] (-z , \t) = e_* (\gamma) \theta [\gamma] (z , \t) \;.$$ We say that $\gamma$ is an **even** (resp. **odd**) characteristic if $e_*(\gamma) = 1$ (resp. $e_*(\gamma) = -1$).
For any genus-two curve we have six odd theta characteristics and ten even theta characteristics; see [@psw] for details. The following are the sixteen theta characteristics, where the first ten are even and the last six are odd. We denote the even theta constants by
$$\begin{split}
\theta_1 = {\begin{bmatrix}
0 & 0\\[3pt]
0 & 0
\end{bmatrix}} , \,
\theta_2 = {\begin{bmatrix}
0 & 0\\[3pt]
\frac{1}2 & \frac{1}2
\end{bmatrix}} , \,
\theta_3 &= {\begin{bmatrix}
0 & 0\\[3pt]
\frac{1}2 & 0
\end{bmatrix}} , \,
\theta_4 = {\begin{bmatrix}
0 & 0\\[3pt]
0 & \frac{1}2
\end{bmatrix}} , \, \;
\theta_5 = {\begin{bmatrix}
\frac{1}2 & 0\\[3pt]
0 & 0
\end{bmatrix}} ,\\[5pt]
\theta_6 = {\begin{bmatrix}
\frac{1}2 & 0\\[3pt]
0 & \frac{1}2
\end{bmatrix}} , \,
\theta_7 = {\begin{bmatrix}
0 & \frac{1}2\\[3pt]
0 & 0
\end{bmatrix}} , \,
\theta_8 &= {\begin{bmatrix}
\frac{1}2 & \frac{1}2\\[3pt]
0 & 0
\end{bmatrix}} , \,
\theta_9 = {\begin{bmatrix}
0 & \frac{1}2\\[3pt]
\frac{1}2 & 0
\end{bmatrix}} , \,
\theta_{10} = {\begin{bmatrix}
\frac{1}2 & \frac{1}2\\[3pt]
\frac{1}2 & \frac{1}2
\end{bmatrix}} ,
\end{split}$$
where we write $$\label{Eqn:theta_short}
\theta_i(z) \quad \text{instead of} \quad \theta {\begin{bmatrix}
a^{(i)} \\
b^{(i)}\\
\end{bmatrix}} (z , \tau) \quad \text{with $i=1,\dots ,10$,}$$ and $\theta_i =\theta_i(0)$. Similarly, the odd theta functions correspond to the following characteristics $${\begin{bmatrix}
0 & \frac{1}2\\[3pt]
0 & \frac{1}2
\end{bmatrix}} , \,
{\begin{bmatrix}
0 & \frac{1}2\\[3pt]
\frac{1}2 & \frac{1}2
\end{bmatrix}} , \,
{\begin{bmatrix}
\frac{1}2 & 0\\[3pt]
\frac{1}2 & 0
\end{bmatrix}} , \, \,
{\begin{bmatrix}
\frac{1}2 & \frac{1}2\\[3pt]
\frac{1}2 & 0
\end{bmatrix}} , \,
{\begin{bmatrix}
\frac{1}2 & 0\\[3pt]
\frac{1}2 & \frac{1}2
\end{bmatrix}} , \,
{\begin{bmatrix}
\frac{1}2 & \frac{1}2\\[3pt]
0 & \frac{1}2
\end{bmatrix}} \;.$$ Thetanulls are modular forms of $\mathcal{A}_2(2)$, and the even theta fourth powers define a compactification of $\mathcal{A}_2(2)$ by $\operatorname{Proj}[\T^4_1: \dots : \T^4_{10}]$, known as the *Satake compactfication*. $\theta_1, \dots , \theta_4$ are called *fundamental thetanulls*; see [@psw] for details. They are determined via the Göpel systems. We have the following Frobenius identities relating the remaining theta constants to the fundamental thetanulls $$\label{Eq:FrobeniusIdentities}
\begin{array}{lllclll}
\theta_5^2 \theta_6^2 & = & \theta_1^2 \theta_4^2 - \theta_2^2 \theta_3^2 \,, &\qquad
\theta_5^4 + \theta_6^4 & =& \theta_1^4 - \theta_2^4 - \theta_3^4 + \theta_4^4 \,, \\[0.2em]
\theta_7^2 \theta_9^2 & = & \theta_1^2 \theta_3^2 - \theta_2^2 \theta_4^2 \,, &\qquad
\theta_7^4 + \theta_9^4 &= & \theta_1^4 - \theta_2^4 + \theta_3^4 - \theta_4^4 \, , \\[0.2em]
\theta_8^2 \theta_{10}^2 & = & \theta_1^2 \theta_2^2 - \theta_3^2 \theta_4^2 \, , &\qquad
\theta_8^4 + \theta_{10}^4 & = & \theta_1^4 + \theta_2^4 - \theta_3^4 - \theta_4^4 \,,
\end{array}$$ as well as the following mixed relations $$\label{Eq:FrobeniusIdentitiesb}
\begin{array}{lllclll}
\theta_5^2 \theta_9^2 & = & \theta_3^2 \theta_8^2 - \theta_4^2 \theta_{10}^2 \,,&\qquad
\theta_5^2 \theta_7^2 & = & \theta_1^2 \theta_8^2 - \theta_2^2 \theta_{10}^2 .
\end{array}$$
Let a genus-two curve $\mathcal{C}$ be given by $$\label{Rosen2}
Y^2= F(X) = X(X-1)(X-\lambda_1)(X-\lambda_2)(X-\lambda_3).$$ The ordered tuple $(\lambda_1, \lambda_2, \lambda_3)$ where the $\lambda_i$ are all distinct and different from $0, 1, \infty$ determines a point in $\mathcal{M}_2(2)$, the moduli space of genus-two curves together with a level-two structure, and, in turn, a level-two structure on the corresponding Jacobian variety, i.e., a point in the moduli space of principally polarized abelian surfaces with level-two structure $\mathcal{A}_2(2)$. As functions on $\mathcal{M}_2(2)$, the Rosenhain invariants generate its coordinate ring $\mathbb{C}(\lambda_1, \lambda_2, \lambda_3)$ and hence generate the function field of $\mathcal{A}_2(2)$.
The three $\lambda$-parameters in the Rosenhain normal (\[Rosen2\]) can be expressed as ratios of even theta constants by Picard’s lemma. There are 720 choices for such expressions since the forgetful map $\mathcal{M}_2(2) \to \mathcal{M}_2$ is a Galois covering of degree $720 = |S_6|$ since $S_6$ acts on the roots of $F$ by permutations. Any of the $720$ choices may be used, we picked the one from [@MR2367218]:
If $\mathcal{C}$ is a genus-two curve with period matrix $\tau$ and $\chi_{10}(\tau)\not =0$, then $\mathcal{C}$ is equivalent to the curve (\[Rosen2\]) with Rosenhain parameters $\lambda_1, \lambda_2, \lambda_3$ given by $$\label{Picard}
\lambda_1 = \frac{\theta_1^2\theta_3^2}{\theta_2^2\theta_4^2} \,, \quad \lambda_2 = \frac{\theta_3^2\theta_8^2}{\theta_4^2\theta_{10}^2}\,, \quad \lambda_3 =
\frac{\theta_1^2\theta_8^2}{\theta_2^2\theta_{10}^2}\,.$$ Conversely, given three distinct complex numbers $(\lambda_1, \lambda_2, \lambda_3)$ different from $0, 1, \infty$ there is complex abelian surface $\mathbf{A}_{\tau}$ with period matrix $[\mathbb{I}_2 | \tau]$ such that $\mathbf{A}_{\tau}=\operatorname{Jac} (\mathcal{C})$ where $\mathcal{C}$ is the genus-two curve with period matrix $\tau$.
Igusa functions and Siegel modular forms
----------------------------------------
Let $I_2, \dots , I_{10}$ denote Igusa invariants of the binary sextic $Y^2=F(X)$ as defined in [@SV]\*[Eq. 9]{} and explicitly given by Equations (\[IgRos\]) in the appendix for a curve in Rosenhain normal form (\[Rosen2\]). The Igusa functions or absolute invariants are defined as $$(j_1, j_2, j_3 ) = \left( \frac {I_2^5} {I_{10}}, \frac {I_4 I_2^3} {I_{10}}, \frac {I_6 I_2^2} {I_{10}} \right) \;.$$ Two genus-two curves $\mathcal{C}$ and $\mathcal{C}^\prime$ are isomorphic if and only if $$\left( j_1, j_2, j_3 \right) = \left( j_1^\prime, j_2^\prime, j_3^\prime \right)\;.$$ Moreover, $j_1, j_2, j_3$ are given as rational functions of fourth powers of the fundamental theta functions $\theta_1, \dots , \theta_4$.
The even Siegel modular forms of $\mathcal{A}_2$ form a polynomial ring in four free generators of degrees $4$, $6$, $10$ and $12$ denoted by $\psi_4, \psi_6, \chi_{10}$ and $\chi_{12}$, respectively. Igusa [@MR0229643]\*[p. 848]{} proved that the relation between the Igusa invariants of the binary sextic $Y^2=F(X)$ defining a genus-two curve $\mathcal{C}$ with period matrix $\tau$ and the even Siegel modular forms are as follows: $$\label{invariants}
\begin{split}
I_2(F) & = -2^3 \cdot 3 \, \dfrac{\chi_{12}(\tau)}{\chi_{10}(\tau)} \;, \\
I_4(F) & = \phantom{-} 2^2 \, \psi_4(\tau) \;,\\
I_6(F) & = -\frac{2^3}3 \, \psi_6(\tau) - 2^5 \, \dfrac{\psi_4(\tau) \, \chi_{12}(\tau)}{\chi_{10}(\tau)} \;,\\
I_{10}(F) & = -2^{14} \, \chi_{10}(\tau) \;.
\end{split}$$ Notice that the Igusa invariant $I_{10}$ is the discriminant of the sextic $Y^2=F(X)$, i.e., $\Delta_F = I_{10}(F)$. Conversely, for $r\not = 0$ the point $[I_2 : I_4 : I_6 : I_{10}]$ in weighted projective space equals
$$\label{IgusaClebschProjective}
\begin{split}
\Big[ 2^3 \, 3 \, (3r\chi_{12})\, : \, 2^2 3^2 \, \psi_4 \, (r\chi_{10})^2 \, : \, 2^3\, 3^2\, \Big(4 \psi_4 \, (3r\chi_{12})+ \psi_6 \,(r\chi_{10}) \Big)\, (r\chi_{10})^2: 2^2 \, (r\chi_{10})^6 \Big].
\end{split}$$
Furthermore, Igusa showed in [@MR0229643] that for the full ring of modular forms of $\mathcal{A}_2$, one needs an additional generator $\chi_{35}$ which is algebraically dependent on the others. In fact, its square can be written as follows:
$$\label{chi35sqr}
\begin{split}
\chi_{35}^2 & = \frac{1}{2^{12} \, 3^9} \; \chi_{10} \, \Big(
2^{24} \, 3^{15} \; \chi_{12}^5 - 2^{13} \, 3^9 \; \psi_4^3 \, \chi_{12}^4 - 2^{13} \, 3^9\; \psi_6^2 \, \chi_{12}^4 + 3^3 \; \psi_4^6 \, \chi_{12}^3 \\
& - 2\cdot 3^3 \; \psi_4^3 \, \psi_6^2 \, \chi_{12}^3 - 2^{14}\, 3^8 \; \psi_4^2 \, \psi_6 \, \chi_{10} \, \chi_{12}^3 -2^{23}\, 3^{12} \, 5^2\, \psi_4 \, \chi_{10}^2 \, \chi_{12}^3 + 3^3 \, \psi_6^4 \, \chi_{12}^3\\
& + 2^{11}\,3^6\,37\,\psi_4^4\,\chi_{10}^2\,\chi_{12}^2+2^{11}\,3^6\,5\cdot 7 \, \psi_4 \, \psi_6^2\, \chi_{10}^2 \, \chi_{12}^2 -2^{23}\, 3^9 \, 5^3 \, \psi_6\, \chi_{10}^3 \, \chi_{12}^2 \\
& - 3^2 \, \psi_4^7 \, \chi_{10}^2 \, \chi_{12} + 2 \cdot 3^2 \, \psi_4^4 \, \psi_6^2 \, \chi_{10}^2 \, \chi_{12} + 2^{11} \, 3^5 \, 5 \cdot 19 \, \psi_4^3 \, \psi_6 \, \chi_{10}^3 \, \chi_{12} \\
& + 2^{20} \, 3^8 \, 5^3 \, 11 \, \psi_4^2 \, \chi_{10}^4 \, \chi_{12} - 3^2 \, \psi_4 \, \psi_6^4 \, \chi_{10}^2 \, \chi_{12} + 2^{11} \, 3^5 \, 5^2 \, \psi_6^3 \, \chi_{10}^3 \, \chi_{12} - 2 \, \psi_4^6 \, \psi_6 \, \chi_{10}^3 \\
& - 2^{12} \, 3^4 \, \psi_4^5 \, \chi_{10}^4 + 2^2 \, \psi_4^3 \, \psi_6^3 \, \chi_{10}^3 + 2^{12} \, 3^4 \, 5^2 \, \psi_4^2 \, \psi_6^2 \, \chi_{10}^4 + 2^{21} \, 3^7 \, 5^4 \, \psi_4 \, \psi_6 \, \chi_{10}^5 \\
& - 2 \, \psi_6^5 \, \chi_{10}^3 + 2^{32} \, 3^9 \, 5^5 \, \chi_{10}^6 \Big) \;.
\end{split}$$
Hence, $Q= 2^{12} \, 3^9 \, \chi_{35}^2 /\chi_{10}$ is a polynomial of degree $60$ in the even generators.
The Satake coordinate functions
-------------------------------
For a symplectic matrix $T \in \mathrm{Sp}_4(\Z)$, there is an induced action on the characteristics of the theta constants $\m \mapsto T \cdot \m$ such that the characteristic $T \cdot \m$ has the same parity as $\m$ and $T \cdot \m =\m$ if $T \equiv \mathbb{I} (2)$. The latter property implies that $\Gamma_2 /\Gamma_2(2) \cong \operatorname{Sp}_4(\mathbb{F}_2)$ acts on the characteristics. It turns out that this action is transitive on the six odd characteristics and gives an isomorphism between the permutation group $S_6$ and $\operatorname{Sp}_4(\mathbb{F}_2)$ [@MR2744034].
On any function $f: \mathbb{H}_2 \to \mathbb{C}$, a right action of $T = \bigl(\begin{smallmatrix}
a&b\\ c&d
\end{smallmatrix} \bigr) \in \mathrm{Sp}_4(\R)$ is given by setting $f\circ[T](\tau):=\det(c\tau+d)^{-2} \, f(T\cdot \tau)$. It then follows that $\T^4_\m\circ[T^{-1}]=\pm \T^4_{T \cdot \m}$ for all $T \in \mathrm{Sp}_4(\Z)$ with $\Gamma_2(2)$ acting trivially. Thus, $S_6$ acts on the vector space $\mbox{Mat}_2 \left( \Gamma (2) \right)$ spanned by the ten even theta fourth powers. The vector space $\mbox{Mat}_2 \left( \Gamma (2) \right)$ is a five-dimensional vector space, and we will use the set of theta functions $$\left\{ \T_1^4, \T_2^4, \T_3^4 , \T_4^4, \T_5^4 \right\}$$ as a basis for the space $\mbox{Mat}_2 \left( \Gamma_2(2) \right)$. In fact, the other even theta fourth powers are represented in terms of this basis as $$\label{theta_reduction}
\begin{split}
\T_6^4 & = \T_1^4 - \T_2^4 - \T_3^4 + \T_4^4 - \T_5^4, \\
\T_7^4 & = \T_3^4 - \T_4^4 + \T_5^4 ,\\
\T_8^4 & = \T_2^4 - \T_4^4 + \T_5^4 ,\\
\T_9^4 & = \T_1^4 - \T_2^4 - \T_5^4 ,\\
\T_{10}^4 & = \T_1^4 - \T_3^4 - \T_5^4 .
\end{split}$$ If we set $u_k = \sum_\m \T_\m^{4k}$ it can be checked using the Frobenius identities (\[Eq:FrobeniusIdentities\]) that $u_2^2 = 4u_4$; see [@MR2744034]. Therefore, Equations (\[theta\_reduction\]) realize the Satake compactification of $\mathcal{A}_2(2)$ as the quartic threefold $u_2^2 = 4u_4$ in $\operatorname{Proj}[\T^4_1: \dots : \T^4_5]$.
The following functions are linear combinations of the fourth-powers of even generators and are called *level-two Satake coordinate functions* $$\label{Satake2Theta}
\begin{split}
x_1 & = -\T_1^4 + 2 \, \T_2^4 +2 \, \T_3^4 - \T_4^4+ 3 \, \T_5^4,\\
x_2 & = -\T_1^4 + 2 \, \T_2^4 - \, \T_3^4 - \T_4^4, \\
x_3 & = -\T_1^4-\T_2^4 -\T_3^4+ 2\, \T_4^4,\\
x_4 & = 2 \, \T_1^4-\T_2^4 -\T_3^4- \T_4^4, \\
x_5 & = - \T_1^4-\T_2^4 + 2\,\T_3^4- \T_4^4, \\
x_6 & = 2 \, \T_1^4 - \T_2^4 - \T_3^4 + 2 \, \T_4^4- 3 \, \T_5^4.
\end{split}$$ It is obvious that $\sum_i x_i =0$. A direct computation also shows that the group $S_6$ acts on $(x_1, \dots, x_6)$ by permutation [@MR2744034].
We have the following lemma:
For $\tau \in \mathcal{A}_2(2)$ with $\chi_{10}(\tau) \not =0$ the level-two Satake coordinate functions $x_1, \dots x_6$ determine a curve in Rosenhain normal from (\[Rosen2\]) with $\Delta_F = -2^{14} \, \chi_{10}(\tau)$ and Igusa invariants (\[invariants\]) where the Rosenhain roots $(\lambda_1, \lambda_2, \lambda_3)$ are given by relations $$\label{Picard2}
\lambda_1 = \frac{1}2 + \frac{\T_1^4\T_3^4-\T_7^4\T_9^4}{2 \, \T_2^4\T_4^4}, \quad
\lambda_2 = \frac{1}2 + \frac{\T_3^4\T_8^4-\T_5^4\T_9^4}{2\, \T_4^4\T_{10}^4},\quad
\lambda_3 = \frac{1}2 + \frac{\T_1^4\T_8^4-\T_5^4\T_7^4}{2 \, \T_2^4\T_{10}^4},$$ and $$\label{Theta2Satake}
\begin{array}{lllclll}
\T_1^4 & = & - \frac{1}3 \, \left( x_2 + x_3 + x_5\right), &\qquad
\T_2^4 & = & - \frac{1}3 \, \left( x_3 + x_4 + x_5\right) , \\[0.2em]
\T_3^4 & = & - \frac{1}3 \, \left( x_2 + x_3 + x_4\right) , &\qquad
\T_4^4 & = & - \frac{1}3 \, \left( x_2 + x_4 + x_5\right) , \\[0.2em]
\T_5^4 & = & \phantom{-} \frac{1}3 \, \left( x_1 + x_3 + x_4\right) , &\qquad
\T_6^4 & = & - \frac{1}3 \, \left( x_1 + x_2 + x_5\right) , \\[0.2em]
\T_7^4 & = & \phantom{-} \frac{1}3 \, \left( x_1 + x_4 + x_5\right) , &\qquad
\T_8^4 & = & \phantom{-} \frac{1}3 \, \left( x_1 + x_2 + x_4\right) , \\[0.2em]
\T_9^4 & = & - \frac{1}3 \, \left( x_1 + x_2 + x_3\right) , &\qquad
\T_{10}^4 & = & - \frac{1}3 \, \left( x_1 + x_3 + x_5\right) .
\end{array}$$
The proof follows when using the Frobenius identities (\[Eq:FrobeniusIdentities\]) to re-write the Rosenhain roots in terms of fourth powers of theta functions and solving Equations (\[Satake2Theta\]) for $\T_1^4, \dots, \T_{10}^4$.
Define the $j$-th power sums $s_j$ are defined by $$s_j = \sum_{i=1}^6 x_i^j \;.$$ Apart from the obvious identity $s_1=0$, the relation $u_2^2 = 4 \, u_4$ implies the only other relation $s_2^2 = 4 \, s_4$. Therefore, Equations (\[Satake2Theta\]) define an embedding of the Satake compactfication of $\mathcal{A}_2(2)$, into $\mathbb{P}^5 \ni [x_1:x_2:x_3:x_4:x_5:x_6]$. The image in $\mathbb{P}^5$, known as the *Igusa quartic*, is the intersection of the hyperplane $s_1=0$ and the quartic hypersurface $s_2^2 = 4 \, s_4$.
We have the following lemma describing the descent to the Igusa invariants:
We have the following relations between $s_2, s_3, s_5, s_6$ and the Igusa invariants $$\label{Satake2Igusa}
\begin{split}
s_2 & = 3 \, I_4, \\
s_3 & = \frac32 I_2 I_4 - \frac{9}2 I_6, \\
s_5 & = \frac{15}{8} I_2 I_4^2 - \frac{45}{8} I_4 I_6 + 1215 \, I_{10}, \\
s_6 & = \frac{27}{16} I_4^3 + \frac3{8} I_2^2 I_4^2 - \frac{9}4 I_2 I_4 I_6 + \frac{27}{8} I_6^2 + \frac{729}4 I_2 I_{10} \;,\qquad
\end{split}$$ and, conversely, $$\begin{split}
I_2 & = \frac53 \frac{3 \, s_2^3 + 8 s_3^2 - 48 s_6}{5 s_2 s_3 - 12 s_5}, \\
I_4 & = \frac{1}3 s_2, \\
I_6 & = \frac{1}{27} \frac{15 \, s_2^4 + 10 s_2 s_3^2-240 s_2 s_6+ 72 s_3 s_5}{5 \, s_2 s_3 - 12 s_5},\\
I_{10} & = - \frac{1}{2916} s_2 s_3 + \frac{1}{1215} s_5 .
\end{split}$$
Using the definition of the Igusa invariants we prove the lemma by explicit computation.
The Satake sextic
-----------------
We combine the level-two Satake functions in another plane sextic curve, called the *Satake sextic*, given by $$f(x) = \prod_{i=1}^6 (x -x_i)$$ The coefficients of the Satake sextic are polynomials in $\Z \left[ \frac 1 2 , \frac 1 3 , s_2, s_3, s_4, s_6 \right]$. In fact, we obtain $$f(x) = x^6 + \sum_{i=1}^6 \frac{(-1)^i}{i!} \mathcal{B}_i(Z) \, x^{6-i} \,$$ where $Z= \lbrace s_1, -s_2, 2! \, s_3, - 3!\, s_4, 4! \, s_5, -5! \, s_6 \rbrace$ and $\mathcal{B}_i(Z)$ is the complete Bell polynomial of order $i$ in the variables contained in $Z$. The following proposition follows:
\[Satake6\] For $\tau \in \mathcal{A}_2$ the level-two Satake coordinate functions $x_1, \dots , x_6$ are the roots of the Satake polynomial $f \in \Z \left[ \psi_4, \psi_6, \chi_{10}, \chi_{12} \right] [x] $ given by $$\label{SatakeSextic}
f(x) = \, \left(x^3- \frac{s_2}{4} \, x - \frac{s_3}{6}\right)^2 + \left( \frac{s_2 s_3}{12} - \frac{s_5}{5} \right) x + \frac{s_2^3}{96} + \frac{s_3^2}{36} - \frac{s_6}{6} ,$$ or, equivalently, $$\label{SatakeSextic_3}
\begin{split}
f(x) = & \, \left(x^3 - 3 \, \psi_4 \, x - 2 \, \psi_6 \right)^2 + 2^{14}3^5 \, \left( \chi_{10} \, x -3 \chi_{12} \right) .
\end{split}$$ A genus-two curve $\mathcal{S}$ is defined by $y^2=f(x)$ iff the discriminant does not vanish, i.e., $$\Delta_f = I_{10}(f) = 2^{52} 3^{21} \, Q(\tau) \not = 0 \;.$$
The proof follows from explicit computation of the Bell polynomials and using the relations $s_1=0$ and $s_2^2 = 4 \, s_4$. One then checks that the discriminant of the polynomial in Equation (\[SatakeSextic\]) after using relations (\[Satake2Igusa\]) and (\[invariants\]) coincides up to factor with $Q= 2^{12} \, 3^9 \, \chi_{35}^2 /\chi_{10}$ where $\chi_{35}^2$ was given in Equation (\[chi35sqr\]).
A genus-two curve $\mathcal{C}$ with period matrix $\tau$ determines a Satake sextic polynomial (\[SatakeSextic\_3\]) in Proposition \[Satake6\]. Therefore, we get a map $\Phi: \mathcal C \mapsto \mathcal S$ by mapping the genus-two curve $\mathcal{C}$ to the Satake sextic $y^2=f(x)$ with $f$ given in Equation (\[SatakeSextic\]). Note that this map is defined as a map on the moduli space $\M_2$ if and only if $\chi_{10}(\tau)\not=0$ and $\Delta_f= - 2^{14} \chi_{10}(\tau') = 2^{52} 3^{21} Q(\tau)\not=0$ because only then do we have genus-two curves in the domain and range with period matrixes $\tau$ and $\tau'$, respectively. Though not by explicit formulas it was proved in [@MR2069800] that this map is a rational map of degree 16. The following lemma provides the explicit formulas:
Let $(j_1, j_2, j_3)$ and $(j_1^\prime, j_2^\prime, j_3^\prime)$ be the absolute invariants of $\mathcal{C}$ and $\mathcal{S}$, respectively. The map $\Phi: \M_2 \backslash {\textnormal{supp}}{ (\chi_{35})}_0 \to \M_2 $ with $\mathcal C \mapsto \mathcal S$ is given by $$\label{Transfo}
\begin{split}
j_1^\prime = \frac{64}{729} \, \frac{g^{(1)}(j_1,j_2,j_3)}{q(j_1,j_2,j_3)},\quad
j_2^\prime = \frac{4}{729} \, \frac{g^{(2)}(j_1,j_2,j_3)}{q(j_1,j_2,j_3)}, \quad
j_3^\prime = \frac{1}{729} \, \frac{g^{(3)}(j_1,j_2,j_3)}{q(j_1,j_2,j_3)},
\end{split}$$ where the polynomials $g^{(n)}$ for $n=1,2,3$ with integer coefficients are given in Equations (\[Components\_Phi\]) in the appendix, and the denominator is related to the modular form $Q= 2^{12} \, 3^9 \, \chi_{35}^2 /\chi_{10}$ by the equation $$Q(\tau) \, I_2^{-30} = 2^{-63} \, j_1^{-15} \, q(j_1,j_2,j_3).$$
Let $\mathcal{C}$ be a genus-two curve with period matrixes $\tau$. Similarly, let $\mathcal{S}$ be a genus-two curve with period matrixes $\tau'$ defined by the sextic $y^2=f(x)$ with $f$ given in Equation (\[SatakeSextic\_3\]). We compute the Igusa invariants $I_n(f)$ for $n=2, 3, 6, 10$ and use Equations (\[invariants\]) to compute $\psi_4(\tau'), \psi_6(\tau'), \chi_{10}(\tau')$ and $\chi_{12}(\tau')$. It follows that $$\begin{split}
\psi_4(\tau') & = \phantom{-} 2^4 3^6 \, K(\tau) ,\\
\psi_6(\tau') & = - 2^6 3^9 \, L(\tau) ,\\
\chi_{10}(\tau') & = - 2^{38} 3^{21} \, Q(\tau) ,\\
\chi_{12}(\tau') & = \phantom{-} 2^{40} 3^{23} \, Q(\tau) \, M(\tau),
\end{split}$$ where $K(\tau) = \psi_4(\tau)^6 + \dots$ and $L(\tau) = \psi_6(\tau)^6 + \dots$ are polynomials in the four even generators of degree $24$ and $36$, respectively, $Q(\tau) = 2^{32} 3^9 5^5 \chi_{10}(\tau)^6 + \dots $ is a polynomial of degree $60$, and $$M(\tau) = \psi_4(\tau)^3 - \psi_6(\tau)^2 + 2^{13} \, 3^4 \, 5 \, \chi_{12}(\tau) \;.$$ We also find $$Q(\tau') = 2^{210} 3^{132} \, Q(\tau)^3 \, N(\tau)^2 \;,$$ where $N(\tau)=\psi_4(\tau)^{15} \chi_{10}(\tau)^3 + \dots$ is a polynomial of degree $90$. Then, the Igusa functions $(j_1^\prime, j_2^\prime, j_3^\prime)$ of the genus-two curve $\mathcal{S}$ are given by $$\begin{split}
j_1^\prime &= \frac{64}{729} \, \frac{m(j_1,j_2,j_3)^5}{q(j_1,j_2,j_3)},\\
j_2^\prime &= \frac{4}{729} \, \frac{m(j_1,j_2,j_3)^3 \, k(j_1,j_2,j_3)}{q(j_1,j_2,j_3)}, \\
j_3^\prime &= \frac{1}{2187} \, \frac{m(j_1,j_2,j_3)^2 \, \big( l + 4 \, k \,m\big)(j_1,j_2,j_3)}{q(j_1,j_2,j_3)} ,
\end{split}$$ where the polynomials $m, k, l, q$ with integer coefficients are given by $$\begin{split}
M(\tau) \, I_2^{-6} &= 2^{-6} \, j_1^{-3} \, m(j_1,j_2,j_3) ,\\
K(\tau) \, I_2^{-12} &= 2^{-12} \, j_1^{-6} \, k(j_1,j_2,j_3) ,\\
L(\tau) \, I_2^{-18} &= 2^{-18} \, j_1^{-9} \, l(j_1,j_2,j_3) ,\\
Q(\tau) \, I_2^{-30}&= 2^{-63} \, j_1^{-15} \, q(j_1,j_2,j_3).
\end{split}$$
The roots of the Satake sextic $y^2=f(x)$ determine by means of Equation and Equation the Rosenhaim roots of an equivalent genus-two curve in Rosenhain normal form (\[Rosen2\]), with the original Igusa invariants, that is an element in $\M_2(2)$. The permutation group $S_6$ acts on the roots $(x_1, \dots, x_6)$ by permutation. It is clear that the corresponding action on the sextic curve in Rosenhain normal form (\[Rosen2\]) is the transition between the $720 = |S_6|$ choices of ratios of even theta constants which Picard’s lemma allows for the three $\lambda$-parameters.
Jacobian elliptic Kummer and Shioda-Inose surfaces
==================================================
In this section we describe the construction of the Kummer surface $\mathrm{Kum}(\operatorname{Jac}\mathcal{C})$ and the Shioda-Inose surface $\mathrm{SI}(\operatorname{Jac}\mathcal{C})$ together with the Jacobian elliptic fibrations on these surfaces which are relevant to the F-theory/heterotic string duality.
Jacobian elliptic fibrations
----------------------------
A surface is called Jacobian elliptic fibration if it is a (relatively) minimal elliptic surface $\pi: \mathcal{X} \to \mathbb{P}^1$ over $\mathbb{P}^1$ with a distinguished section $S_0$. The complete list of possible singular fibers has been given by Kodaira [@MR0184257]. It encompasses two infinite families $(I_n, I_n^*, n \ge0)$ and six exceptional cases $(II, III, IV, II^*, III^*, IV^*)$. To each Jacobian elliptic fibration $\pi: \mathcal{X} \to \mathbb{P}^1$ there is an associated Weierstrass model $\bar{\pi}: \bar{\mathcal{X}\,}\to \mathbb{P}^1$ with a corresponding distinguished section $\bar{S}_0$ obtained by contracting all components of fibers not meeting $S_0$. The fibers of $\bar{\mathcal{X}\,}$ are all irreducible whose singularities are all rational double points, and $\mathcal{X}$ is the minimal desingularization. If we choose $t \in \mathbb{C}$ as a local affine coordinate on $\mathbb{P}^1$, we can present $\bar{\mathcal{X}\,}$ in the Weierstrass normal form $$\label{Eq:Weierstrass}
Y^2 = 4 \, X^3 - g_2(t) \, X - g_3(t) \;,$$ where $g_2$ and $g_3$ are polynomials in $t$ of degree four and six, respectively, because $\mathcal{X}$ is a K3 surface. Type of singular fibers can then be read off from the orders of vanishing of the functions $g_2$, $g_3$ and the discriminant $\Delta= g_2^3 - 27 \, g_3^2$ at the various singular base values. Note that the vanishing degrees of $g_2$ and $g_3$ are always less or equal three and five, respectively, as otherwise the singularity of $\bar{X}$ is not a rational double point.
For a family of Jacobian elliptic surfaces $\pi: \mathcal{X} \to \mathbb{P}^1$, the two classes in Néron-Severi lattice $\mathrm{NS}(\mathcal{X})$ associated with the elliptic fiber and section span a sub-lattice $\mathcal{H}$ isometric to the standard hyperbolic lattice $H$ with the quadratic form $q=x_1x_2$, and we have the following decomposition as a direct orthogonal sum $$\mathrm{NS}(\mathcal{X}) = \mathcal{H} \oplus \mathcal{W} \;.$$ The orthogonal complement $T(\mathcal{X}) = \mathrm{NS}(\mathcal{X})^{\perp} \in H^2(\mathcal{X},\mathbb{Z})\cap H^{1,1}(\mathcal{X})$ is called the transcendental lattice and carries the induced Hodge structure.
The Kummer surface
------------------
For the Jacobian variety $\operatorname{Jac}(\mathcal{C})$ of a genus-two curve $\mathcal{C}$, let $\imath$ be the involution automorphism on $\operatorname{Jac}(\mathcal{C})$ given by $\imath: a \mapsto -a$. The quotient, $\operatorname{Jac}(\mathcal{C})/\lbrace \mathbb{I}, \imath \rbrace$, is a singular surface with sixteen ordinary double points. Its minimum resolution, $\operatorname{Kum}(\operatorname{Jac} \mathcal{C})$, is a special K3 surface called the *Kummer surface* associated to $\operatorname{Jac} \mathcal{C}$.
On the Kummer surface $\operatorname{Kum}(\operatorname{Jac} \mathcal{C})$, there are two sets of sixteen $(-2)$-curves, called nodes and tropes, which are either the exceptional divisors corresponding to the blowup of the 16 two-torsion points of the Jacobian $\operatorname{Jac}(\mathcal{C})$ or they arise from the embedding of $\mathcal{C}$ into $\operatorname{Jac}(\mathcal{C})$ as symmetric theta divisors. These two sets of smooth rational curves have a rich symmetry, the so-called $16_6$-configuration, where each node intersects exactly six tropes and vice versa [@MR1097176].
Using curves and symmetries in the $16_6$-configuration one can define various elliptic fibrationson $\operatorname{Kum}(\operatorname{Jac} \mathcal{C})$, since all irreducible components of a reducible fiber in an elliptic fibration are $(-2)$-curves [@MR0184257]. In fact, for the Kummer surface of a generic curve of genus two all inequivalent elliptic fibrations were determined explicitly by Kumar in [@MR3263663]. In particular, Kumar computed elliptic parameters and Weierstrass equations for all twenty five different fibrations that appear, and analyzed the reducible fibers and Mordell-Weil lattices.
### A first elliptic fibration on $\mathrm{Kum}(\operatorname{Jac}\mathcal{C})$
Given a genus-two curve $\mathcal{C}$ defined by a sextic $Y^2 =F(X)$, the Jacobian variety $\operatorname{Jac}(\mathcal{C})$ is birational to the symmetric product of two copies of $\mathcal{C}$, i.e., $(\mathcal{C}\times\mathcal{C})/\{ \mathbb{I}, \pi \}$, where we have set $\pi(X_1)=X_2$ and $\pi(Y_1)=Y_2$. The function field is the sub-field of $\mathbb{C}[X_1,X_2,Y_1,Y_2]$ such that $Y_i^2 =F(X_i)$ for $i=1,2$ which is fixed under $\pi$.
The Kummer surface $\mathrm{Kum}(\operatorname{Jac}\mathcal{C})$ is birational to the quotient $\operatorname{Jac}(\mathcal{C})/\{ \mathbb{I}, \imath \}$ with $\imath(X_i)=X_i$ and $\imath(Y_i)=-Y_i$ for $i=1,2$. Its function field is the sub-field of $\mathbb{C}[X_1,X_2,Y_1,Y_2]$ with $Y_i^2 =F(X_i)$ for $i=1,2$ which is fixed under both $\pi$ and $\imath$. Thus, the function field of $\mathrm{Kum}(\operatorname{Jac}\mathcal{C})$ is generated by $Y=Y_1Y_2$, $t=X_1X_2$, and $X=X_1+X_2$. We have the following lemma:
The function field of the Kummer surface $\mathrm{Kum}(\operatorname{Jac}\mathcal{C})$ for the genus-two curve $\mathcal{C}$ given by the sextic (\[Rosen2\]) is generated by $Y, X, t$ subject to the relation $\mathcal{K}(Y,X,t)=0$ with $$\label{kummer2}
\mathcal{K}(Y,X,t) = Y^2 - t \, \big( 1 - X + t\big) \, \big( \lambda_1^2 - \lambda_1 \, X + t\big) \, \big( \lambda_2^2 - \lambda_2 \, X + t\big) \, \big( \lambda_3^2 - \lambda_3 \, X + t\big) .$$
Equation (\[kummer2\]) defines a Jacobian elliptic fibration $\bar{\pi}: \bar{\mathcal{X}\,} \to \mathbb{P}^1$ with a distinguished section $\bar{S}_0$ on $\mathcal{X}=\mathrm{Kum}(\operatorname{Jac}\mathcal{C})$ by choosing $t$ as the elliptic parameter and the point at infinity in each fiber for the section. In fact, this fibration is well-known and labeled fibration ‘1’ in [@MR3263663]. The following lemma is immediate and follows by comparison with the explicit results in [@MR3263663].
\[lem:EllLeft\] Equation (\[kummer2\]) is a Jacobian elliptic fibration $\bar{\pi}: \bar{\mathcal{X}} \to \mathbb{P}^1$ with 6 singular fibers of type $I_2$, two singular fibers of type $I_0^*$, and the Mordell-Weil group of sections $\operatorname{MW}(\bar{\pi})=(\mathbb{Z}/2)^2\oplus \langle 1 \rangle$.
The sextic curve is recovered directly from the Jacobian elliptic fibration by the following corollary:
A sextic associated with the genus-two curve $\mathcal{C}$ is recovered from the Jacobian elliptic fibration (\[kummer2\]) on $\mathrm{Kum}(\operatorname{Jac}\mathcal{C})$ by letting $X \to \infty$ while keeping $t/X=\xi$, $Y^2/X^5=\eta^2$ fixed, i.e., $$\lim_{\epsilon \to 0} \epsilon^{10} \, \mathcal{K}\left(Y=\frac{\eta}{\epsilon^5}, X= \frac{1}{\epsilon^2}, t= \frac{\xi}{\epsilon^2} \right) = \eta^2 - F(\xi) \;.$$
The proof follows from an explicit computation.
### A second elliptic fibration on $\mathrm{Kum}(\operatorname{Jac}\mathcal{C})$
There is another elliptic fibration $\mathcal{X} \to \mathbb{P}^1$ on $\mathrm{Kum}(\operatorname{Jac}\mathcal{C})$ which is more relevant for us, labeled fibration ‘23’ in [@MR3263663]. Kumar proved the following [@MR2427457]:
A Jacobian elliptic fibration $\bar{\pi}: \bar{\mathcal{X}} \to \mathbb{P}^1$ with 6 singular fibers of type $I_2$, one fiber of type $I_5^*$, one fiber of type $I_1$, and a Mordell-Weil group of sections $\operatorname{MW}(\bar{\pi})=\mathbb{Z}/2$ is given by the Weierstrass equation $$\label{KumFib2}
\begin{split}
Y^2 = & \, X^3 - 2 \, \left( t^3 - \frac{I_4}{12} t+ \frac{I_2 I_4-3 I_6}{108} \right) \,X^2 \\
& + \left( \left(t^3 - \frac{I_4}{12} t + \frac{I_2 I_4 - 3 I_6}{108}\right)^2 + I_{10} \left(t- \frac{I_2}{24}\right)\right) \, X \;.
\end{split}$$
An immediate corollary is the following:
The positions of the $I_2$ fibers in the elliptic fibration (\[KumFib2\]) on the Kummer surface $\mathrm{Kum}(\operatorname{Jac}\mathcal{C})$ are given by the roots of the polynomial $$\label{SatakeSextic_a}
\left(t^3 - \frac{I_4}{12} t + \frac{I_2 I_4 - 3 I_6}{108}\right)^2 + I_{10} \left(t- \frac{I_2}{24}\right) =0 \;,$$ or equivalently by $$\label{SatakeSextic_b}
\left(t^3 - \frac{\psi_4}3 t + \frac{2 \, \psi_6}{27}\right)^2 -2^{14} \, \left(\chi_{10} \, t + \chi_{12}\right) =0 \;.$$ Equivalently, the loci of $I_2$ fibers form the ramification locus of the Satake sextic (\[SatakeSextic\]) if we set $t=-x/3$.
The Shioda-Inose surface
------------------------
A K3 surface $\mathcal{Y}$ has a Shioda-Inose structure if it admits an involution fixing the holomorphic two-form, such that the quotient is the Kummer surface $\operatorname{Kum}(\mathbf{A})$ of a principally polarized abelian surface $\mathbf{A}$ and the rational quotient map $p: \mathcal{Y} \dashrightarrow \operatorname{Kum}(\mathbf{A})$ of degree two induces a Hodge isometry[^2] between the transcendental lattices $T(\mathcal{Y})(2)$[^3] and $T(\operatorname{Kum} \mathbf{A})$. Morrison proved that a K3 surface $\mathcal{Y}$ admits a Shioda-Inose structure if and only if there exists a Hodge isometry between the following transcendental lattices $T(\mathcal{Y}) \cong T(\mathbf{A})$.
The Shioda-Inose $\mathcal{Y}$ of the Kummer surface $\operatorname{Kum}(\operatorname{Jac} \mathcal{C})$ for a generic genus-two curve $\mathcal{C}$ is a K3 surface of Picard-rank 17 and has a transcendental lattice isomorphic to $H \oplus H \oplus \langle -2 \rangle$ by Morrison’s criterion. It was shown in [@MR2306633] that for fixed $\mathcal{C}$ this K3 surface $\mathcal{Y}$ is in fact unique. In the following, we always let $\mathcal{Y}=\mathrm{SI}(\operatorname{Jac}\mathcal{C})$ be this K3 surface with Shioda-Inose structure.
Clingher and Doran proved in [@MR2824841] that as the genus-two curve $\mathcal{C}$ varies the K3 surface $\mathcal{Y}$ admits a birational model isomorphic to a quartic surface with canonical $H \oplus E_8(-1)\oplus E_7(-1)$[^4] lattice polarization[^5] that fits into the following four-parameter family in $\mathbb{P}^3$ [@MR2824841]\*[Eq. (3)]{} given in terms of the variables $[\mathbf{W}:\mathbf{X}:\mathbf{Y}:\mathbf{Z}]\in \mathbb{P}^3$ by the equation $$\label{Inose}
\mathbf{Y}^2\mathbf{ZW}-4\, \mathbf{X}^3\mathbf{Z}+3\, \alpha \, \mathbf{XZW}^2 + \beta \, \mathbf{ZW}^3 + \gamma \, \mathbf{XZ}^2 \mathbf{W} -\frac{1}2(\delta \,
\mathbf{Z}^2\mathbf{W}^2+\mathbf{W}^4)=0.$$ They also found the parameters $(\alpha,\beta,\gamma,\delta)$ in terms of the standard even Siegel modular forms $\psi_4, \psi_6, \chi_{10}, \chi_{12}$ (cf. [@MR0141643]) given by $$(\alpha,\beta,\gamma,\delta) = \left(\psi_4, \psi_6, 2^{12}3^5 \, \chi_{10}, 2^{12}3^6 \, \chi_{12}\right) \;,$$ or, equivalently, in terms of the Igusa-Clebsch invariants using Equations (\[invariants\]) by $$(\alpha,\beta,\gamma,\delta) = \left(\frac{1}4 I_4, \frac{1}{8} I_2 \, I_4 - \frac3{8} I_6, - \frac{243}4 I_{10}, \frac{243}{32} I_2\, I_{10}\right) \;,$$ where $I_n$ for $n=2,4,6,10$ are the Igusa invariants of the sexic curve (\[Rosen2\]) defining the genus-two curve $\mathcal{C}$ if $I_{10} \not = 0$. The Shioda-Inose surface $\mathcal{Y}=\mathrm{SI}(\operatorname{Jac}\mathcal{C})$ of a generic genus-two curve $\mathcal{C}$ admits two Jacobian elliptic fibrations realizing the two inequivalent ways of embedding of $H$ into the lattice $H \oplus E_8(-1)\oplus E_7(-1)$. These two elliptic fibrations were described in [@MR2427457; @MR2935386]. A similar picture was developed in earlier work for the case of a $H \oplus E_8(-1) \oplus E_8(-1)$ lattice polarization in [@MR2369941; @MR2279280] that generalized a special two-parameter family of K3 surfaces introduced by Inose in [@MR578868].
From the point of view of K3 geometry, if the periods are preserved by a reflection of $\delta$ with $\delta^2=-2$, then $\delta$ must belong to the Néron-Severi lattice of the K3 surface. That is, the lattice $H \oplus E_8(-1)\oplus E_7(-1)$ must be enlarged by adjoining $\delta$. It is not hard to show (using methods of [@MR525944], for example), that there are only two ways this enlargement can happen (if we have adjoined a single element only): either the lattice is extended to $H \oplus E_8(-1)\oplus E_8(-1)$ or it is extended to $H\oplus E_8(-1)\oplus E_7(-1) \oplus
\langle -2 \rangle$.
### The alternate fibration
The first Jacobian elliptic fibration on (\[Inose\]), called the *alternate fibration*, has two disjoint sections and a singular fiber of Kodaira-type $I_{10}^*$. For convenience, let us introduce the parameters $(a,b,c,d,e)$ given by $$\label{moduli_abcde}
a =- \dfrac{I_4}{12}, \; b=\dfrac{I_2 \, I_4 - 3 \, I_6}{108},\; c=-1, \; d=\dfrac{I_2}{24}, \; e=\dfrac{I_{10}}4.$$ The alternate fibration is obtained by setting $$\label{transfo_alt}
\mathbf{X} = \dfrac{t \, x^3}{2^{29} \, 3^5} \;, \quad \mathbf{Y}=\dfrac{\sqrt6\, i \, x^2 y}{2^{29} \, 3^5} \;,\quad \mathbf{W}=-\dfrac{x^3}{2^{28} \, 3^6} \;, \quad \mathbf{Z}= \dfrac{x^2}{2^{28} \, 3^9} \;,$$ in Equation (\[Inose\]), and given in Weierstrass form by $$\label{WEq.bak.alt}
y^2 = x^3 + (t^3 + a t + b) \, x^2 + e \, (ct+d) \, x \;.$$ The discriminant of Equation (\[WEq.bak.alt\]) is given by $$\Delta = 16\, e^2 \, \left( ct+d \right)^2 \, \Big( (t^3 + a t + b)^2 - 4 \, e \, (ct+d) \Big) \;.$$ The fibration (\[WEq.bak.alt\]) has special fibers of Kodaira-types $I_{10}^*$ and $I_2$, and six fibers of Kodaira-type $I_1$, and a second two-torsion section $\bar{S}_1: (x,y)=(0,0)$. This proves the following:
\[Prop:alternate\] For a generic genus-two curve $\mathcal{C}$ there is a Jacobian elliptic fibration $\bar{\pi}_{\text{alt}}: \bar{\mathcal{Y}\,} \to \mathbb{P}^1$ on $\mathcal{Y}=\mathrm{SI}(\operatorname{Jac}\mathcal{C})$ given by Equation (\[WEq.bak.alt\]) with 6 singular fibers of type $I_1$, one fiber of type $I_{10}^*$, and one fiber of type $I_2$, and a Mordell-Weil group of sections $\operatorname{MW}(\bar{\pi})=\mathbb{Z}/2$ with elliptic parameter $t=t_{\text{alt}}$.
The following corollary is crucial:
\[cor:position\_nodes\] The positions of the $I_1$ fibers in the Jacobian elliptic fibration (\[KumFib2\]) are given by the roots of the polynomial $$\label{Satake_b}
f(t) = (t^3 + a t + b)^2 - 4 \, e \, (ct+d) = 0 \;.$$ Equivalently, the loci of $I_1$ fibers form the ramification locus of the Satake sextic (\[SatakeSextic\]) if we set $t=-x/3$.
Given the discussion at the end of previous section, the following corollaries are immediate:
\[cor:Qvanish\] For the Jacobian elliptic fibration (\[WEq.bak.alt\]) two $I_1$ fibers coalesce and form a fiber of type $I_2$ if and only if the discriminant of the Satake sextic (\[Satake\_b\]) vanishes, i.e., $\Delta_{f} = Q = 0$, or, equivalently, $$\begin{split}
0 = {e}^3 \Big( 16\,{a}^{7}{c}^2d-16\,{a}^6b{c}^3+16\,{a}^{
5}{c}^4e+16\,{a}^6{d}^3\\+216\,{a}^4{b}^2{c}^2d+888\,{a}^4
{c}^2{d}^2e-216\,{a}^3{b}^3{c}^3-3420\,{a}^3b{c}^3de+
2700\,{a}^2{b}^2{c}^4e\\+4125\,{a}^2{c}^4d{e}^2-5625\,ab{c}^
5{e}^2+3125\,{c}^6{e}^3+216\,{a}^3{b}^2{d}^3+864\,{a}^{3
}{d}^4e\\-2592\,{a}^2bc{d}^3e+729\,a{b}^4{c}^2d-5670\,a{b}^2
{c}^2{d}^2e+16200\,a{c}^2{d}^3{e}^2-729\,{b}^5{c}^3\\+6075
\,{b}^3{c}^3de-13500\,b{c}^3{d}^2{e}^2+729\,{b}^4{d}^3-
5832\,{b}^2{d}^4e+11664\,{d}^5{e}^2 \Big).
\end{split}$$ Equivalently, $\Delta_{f} =0$ iff the lattice polarization $H \oplus E_8(-1)\oplus E_7(-1)$ of the family (\[WEq.bak.alt\]) extends to $H\oplus E_8(-1)\oplus E_7(-1) \oplus \langle -2 \rangle$.
We remark that an $I_1$ fiber will coalesce with the $I_2$ fiber to form a fiber of type $III$ if and only if $$e^3 \, (a \, c^2 \, d-b \, c^3+d^3) = 2 \, \psi_6 \, \chi_{10}^3 +9 \, \psi_4 \, \chi_{10}^2 \, \chi_{12} -27 \, \chi_{12}^3
=0.$$ However, it is easy to show that this does not change the lattice polarization of the family (\[WEq.bak.alt\]).
If we use a normalization consistent with F-theory [@MR3366121] and set $$\mathbf{X} = \dfrac{t \, x^3}{2^9 \, 3^5} \;, \quad \mathbf{Y}=\dfrac{x^2 \, y}{2^{15/2} \, 3^{9/2}} \;,\quad \mathbf{W}=\dfrac{x^3}{2^{10} \, 3^6} \;, \quad \mathbf{Z}= \dfrac{x^2}{2^{16} \, 3^9} \;,$$ and obtain from Equation (\[Inose\]) the Jacobian elliptic fibration $$\label{WEq.bak.alt2}
y^2 = x^3 + \left( t^3 - \frac{\psi_4}{48} \, t - \frac{\psi_6}{864} \right) \, x^2 - \Big( 4 \, \chi_{10} \, t - \chi_{12} \Big) \, x\;,$$ which is equivalent to Equation (\[WEq.bak.alt2\]) for $\chi_{10}(\tau)\not =0$. Equation (\[WEq.bak.alt2\]) remains well-defined for $\chi_{10}(\tau)=0$ when the principally polarized abelian surface $\mathbf{A}_{\tau} $ degenerates to a product of two elliptic curves $\mathbf{A}_{\tau} =\mathcal{E}_{\tau_1} \times \mathcal{E}_{\tau_2}$. It follows:
\[cor:Chi10vanish\] For the Jacobian elliptic fibration (\[WEq.bak.alt2\]) the two fibers of type $I_2$ and $I_{10}^*$ coalesce and form a fiber of type $I_{12}^*$ if and only if the discriminant of the sextic (\[Rosen2\]) vanishes, i.e., $$\Delta_{F} = \chi_{10} = e = 0 \;.$$ Equivalently, $\Delta_{F} =0$ iff the lattice polarization $H \oplus E_8(-1)\oplus E_7(-1)$ of the family (\[WEq.bak.alt2\]) extends to $H\oplus E_8(-1)\oplus E_8(-1)$.
Using fibration (\[WEq.bak.alt\]) and fibration (\[KumFib2\]), the rational quotient map $$p: \mathcal{Y}=\mathrm{SI}(\operatorname{Jac}\mathcal{C}) \dashrightarrow \mathcal{X}=\operatorname{Kum}(\operatorname{Jac}\mathcal{C})$$ is realized as a fiberwise two-isogeny between elliptic surfaces, also known as a *Van Geemen-Sarti involution*. Together with the dual isogeny one obtains a chain of rational maps $\mathcal{Y} \dashrightarrow \mathcal{X} \dashrightarrow \mathcal{Y}$ called a *Kummer sandwich* in [@MR2279280]. In fact, the translation of the elliptic fiber $\mathcal{E}=\mathcal{Y}_t$ in Equation (\[WEq.bak.alt\]) by a two-torsion point $\bar{S}_1: (x,y)=(0,0)$ yields the two-isogeneous fiber $\mathcal{E}'= \mathcal{E}/\lbrace S_0, S_1 \rbrace$ given by $$\label{KumFib2b}
Y^2 = X^3 - 2 (t^3 + a t + b) \, X^2 + \Big( (t^3 + a t + b)^2 - 4 \, e \, (ct+d)\Big) \, X,$$ which is precisely the fibration (\[KumFib2\]). The fibrewise isogeny $\mathcal{E} \to \mathcal{E}'=\mathcal{X}_t$ is given by $$\label{Isogeny}
(x,y) \mapsto (X,Y)=\left( \frac{y^2}{x^2}, \frac{y \, (e \, (ct+d) -x^2)}{x^2} \right),$$ and the dual isogeny $\mathcal{E}' \to \mathcal{E}$ is given by $$\label{Isogeny_dual}
(X,Y) \mapsto (x,y)=\left( \frac{Y^2}{4 X^2}, \frac{Y \, \Big( (t^3 + a t + b)^2 - 4 e (ct+d) -X^2\Big)}{X^2} \right) \;.$$ The resulting *Nikulin involution* $\varphi$ on the K3 surface $\mathcal{Y}$, i.e., the automorphism of order two preserving the holomorphic two-form, in this case the two-form $dt \wedge dx/y$, is given by adding to a point the two-torsion section in each generic fiber $\mathcal{Y}_t$, i.e., $$\label{eq:Nik_alt}
(x,y) \mapsto (x,y) \overset{.}{+} \bar{S}_1 = \left( \frac{e (ct + d)}{x}, - \frac{y \,e (ct + d)}{x^2}\right) .$$ A fiber of type $I_1$ is a rational curve with a node, whereas a fiber of type $I_2$ looks like two copies of $\mathbb{P}^1$ intersecting in two distinct points. The involution $\varphi$ is free on the generic fibers and has exactly $8$ fixed points, namely the nodal points on the $I_1$ fibers and the intersecting points on the $I_2$ fiber. We have the following corollary:
\[cor:position\_nodes2\] The positions of the $I_1$ fibers in the Jacobian elliptic fibration (\[KumFib2\]) are contained in the fixed point set of the Nikulin involution $\varphi$.
### The standard fibration
The second elliptic fibration, called the *standard fibration*, is the Jacobian elliptic fibration with two distinct special fibers of Kodaira-types $II^*$ and $III^*$, respectively. By setting $$\label{transfo_std}
\mathbf{X} = - \dfrac{2^7\, \chi_{10}^3 \, t \, x}{3^5} \;, \quad \mathbf{Y}=\dfrac{2^7 \, \sqrt6 \, i \, \chi_{10}^3 \, y}{3^5} \;,\quad \mathbf{W}=\dfrac{2^8 \, \chi_{10}^3 \, t^3}{3^6} \;, \quad \mathbf{Z}= \dfrac{\chi_{10}^2 \, t^2}{2^4 \, 3^9} \;,$$ in Equation (\[Inose\]) we obtain $$\label{WEq.bak}
y^2 = x^3 + t^3 \, (a \, t + c) \, x + t^5 \, (e \, t^2 + b \, t +d) \;.$$ The fibration (\[WEq.bak\]) was investigated in [@MR2427457]\*[Theorem 11]{}. The birational transformation between the standard and the alternate fibration is given by $$\Big( t, x, y \Big)_{\text{std}} = \Big( \frac{x}{e}, \frac{t x^2}{e^2}, - \frac{x^2 y}{e^3} \Big)_{\text{alt}} \;,$$ and combining it with Equation (\[eq:Nik\_alt\]) recovers the Nikulin involution for the standard elliptic fibration given by (cf. [@MR2427457]\*[Theorem 11]{}) $$\Big( t, x, y \Big) \mapsto \left( \frac{cx + d t^2}{e t^3}, \frac{x (cx + d t^2)^2}{e^2 t^8}, - \frac{y (cx + d t^2)^3}{e^3 t^{12}} \right).$$
It then follows:
\[Prop:standard\] For a generic genus-two curve $\mathcal{C}$ there is a Jacobian elliptic fibration $\bar{\pi}_{\text{std}}: \bar{\mathcal{Y}\,} \to \mathbb{P}^1$ on $\mathcal{Y}=\mathrm{SI}(\operatorname{Jac}\mathcal{C})$ given by Equation (\[WEq.bak\]) with 5 singular fibers of type $I_1$, one fiber of type $II^*$, one fiber of type $III^*$, and a trivial Mordell-Weil group with elliptic parameter $t=t_{\text{std}}$.
For the standard fibration, there are statements analogous to Corollary \[cor:Qvanish\] or Corollary \[cor:Chi10vanish\] when two $I_1$ fibers are coalescing to form a fiber of type $II$ or a fiber from type $III^*$ goes to type $II^*$, respectively [@MR3366121].
Relation to string theory
-------------------------
In string theory a nontrivial connection appears as the eight-dimensional manifestation of a phenomenon called *F-theory/heterotic string duality*. This correspondence leads to a geometric picture that links together moduli spaces for two seemingly distinct types of geometrical objects: Jacobian elliptic fibrations on K3 surfaces and flat bundles over elliptic curves [@MR1675162].
In string theory compactifications of the so-called type-IIB string in which the complex coupling varies over a base are generically referred to as F-theory. The simplest such construction corresponds to K3 surfaces that are elliptically fibered over $\mathbb{P}^1$ with a section, in physics equivalent to type-IIB string theory compactified on $\mathbb{P}^1$ and hence eight-dimensional in the presence of 7-branes [@MR1416354]. In this way, a Jacobian elliptic K3 surface with elliptic fibers $\mathcal{E}_{\tau }=\mathbb{C}/(\mathbb{Z}\oplus \mathbb{Z}\tau )$ defines an F-theory vacuum in eight dimensions where the complex-valued scalar field $\tau$ of the type-IIB string theory is now allowed to be multi-valued. The Kodaira-table of singular fibers gives a precise dictionary between the characteristics of the Jacobian elliptic fibrations and the content of the 7-branes present in the physical theory.
To make contact with the F-theory/heterotic string duality one considers Jacobian elliptic fibrations on a special K3 surface, namely the Shioda-Inose surface $\mathrm{SI}(\operatorname{Jac}\mathcal{C})$ of the principally polarized abelian surface $\operatorname{Jac}(\mathcal{C})$ where $\mathcal{C}$ is a generic genus-two curve. The K3 surface $\mathcal{Y}=\mathrm{SI}(\operatorname{Jac}\mathcal{C})$ carries a Nikulin involution $\varphi$ such that the quotient $\mathcal{Y}/\lbrace\mathbb{I}, \varphi\rbrace$ is birational to the Kummer surface $\mathrm{Kum}(\operatorname{Jac}\mathcal{C})$ and we have a Hodge-isometry between the transcendental lattices $T(\mathcal{Y}) \cong T(\operatorname{Jac}\mathcal{C})$. In this way, a one-to-one correspondence between two different types of surfaces with the same Hodge-theoretic data is established: the principally polarized abelian surfaces $\operatorname{Jac}\mathcal{C}$ and the algebraic K3 surfaces $\mathcal{Y}$ polarized by the rank-17 lattice $H \oplus E_8(-1)\oplus E_7(-1)$.
To see the connection to the heterotic string theory, let us first consider the limit as the Jacobian variety degenerates to a product of two elliptic curves $\mathcal{E}_{1}\times \mathcal{E}_2$ and the involved K3 surfaces have Picard-rank 18, obtained by letting $\chi_{10}\to 0$. This limit describes a well-understood case of the F-theory/heterotic string duality in the absence of any additional bundle data given by so-called Wilson lines. In fact, the moduli space of Jacobian elliptic K3 surfaces with $H\oplus E_8(-1)\oplus E_8(-1)$ lattice polarization is identified with the moduli space of the heterotic string vacua with gauge algebra $\mathfrak{e}_8 \oplus \mathfrak{e}_8$ and $\mathfrak{so}(32)$, respectively, compactified on a two-torus $T^2$ (cf. [@MR1416960]). If any flat connection on $T^2$ is assumed to be trivial, the only two moduli of such a string theory, i.e, the Kähler metric and the $B$-field of $T^2$, identify the torus with the elliptic curves $\mathcal{E}_{1}$ and $\mathcal{E}_2$, respectively. Notice that the existence of two inequivalent elliptic fibrations, the standard and the alternate fibration, is essential and corresponds to the two possible gauge groups of the heterotic string.
The first author together with David Morrison studied in [@MR3366121] the *non-geometric* heterotic string compactified on $T^2$ that produces an eight-dimensional effective theory corresponding to the Jacobian elliptic K3 surfaces with Picard-rank 17 when $\chi_{10}\not=0$. The corresponding heterotic models were called non-geometric because the Kähler and complex structures on $T^2$, and the Wilson line values, are not distinguished but instead are mingled together. The fibration in Equation (\[WEq.bak\]) then describes a model dual to the $\mathfrak{e}_8\oplus \mathfrak{e}_8$ heterotic string, with an unbroken gauge algebra of $\mathfrak{e}_8\oplus \mathfrak{e}_7$ ensuring that only a single Wilson line expectation value is nonzero and all remaining Wilson lines values associated to the $E_8(-1)\oplus E_7(-1)$ sublattice be trivial. Similarly, the fibration in Equation (\[WEq.bak.alt\]) gives the analogous story for the $\mathfrak{so}(32)$ heterotic string: the fibration describes a model dual to the $\mathfrak{so}(32)$ heterotic string, with an unbroken gauge algebra of $\mathfrak{so}(28)\oplus \mathfrak{su}(2)$. By a result of Vinberg [@MR3235787] and its interpretation in string theory given in [@MR3366121], the function field of the Narain moduli space of these heterotic theories turns out to be generated by the ring of Siegel modular forms of [*even weight.*]{} This is the physical manifestation of why the fibrations (\[WEq.bak\]) and (\[WEq.bak.alt\]) only depend on the polynomial ring in the four free generators of degrees $4$, $6$, $10$ and $12$ given by the even Siegel modular forms.[^6]
Generic non-geometric compactification constructed from the family of lattice-polarized K3 surfaces in Equation (\[Inose\]) will have two types of five-branes analogous to a single D7-brane in F-theory. From the heterotic side, these five-brane solitons are easy to see: in the situation of Corollary \[cor:Qvanish\] we have an additional gauge symmetry enhancement by a factor of $\mathfrak{su}(2)$, and the parameters of the theory will include a Coulomb branch with Weyl group $W_{\mathfrak{su}(2)}=\mathbb{Z}_2$. Therefore, there is a five-brane solution with a $\mathbb{Z}_2$ ambiguity when encircling the location in the moduli space of enhanced gauge symmetry. The other five-brane solution is similar: in the situation of Corollary \[cor:Chi10vanish\] the gauge group enhances to $\mathfrak{so}(32)$ gauge symmetry with a similar $\mathbb{Z}_2$ ambiguity.
Further specializations of the multi-parameter family of K3 surfaces in Equation (\[Inose\]) are obtained from degenerations of the underlying genus-two curves. As we have seen, the parameters in Equation (\[Inose\]) are Siegel modular forms of even degree or, equivalently, the Igusa-Clebsch invariants of a binary sextic. Namikawa and Ueno gave a geometrical classification of all (degenerate) fibers in pencils of curves of genus two in [@MR0369362]. For each such pencil allowed by their classification one can now apply the heterotic/F-theory duality map to express the heterotic background in terms of F-theory. Each resulting F-theory compactification will be a family of Jacobian elliptic K3 surfaces. Notice that any such degenerating pencil of genus-two curves is not the description of a heterotic model itself, but rather a computational tool for providing an interesting class of degenerations and their associated five-branes. Moreover, the F-theory background dual to a given five-brane defect on the heterotic side will in general be highly singular. For some of theses cases the singularities can be resolved by performing a finite number of blow-ups in the base, and the resulting smooth geometry was constructed in [@Font:2016aa].
Conversely, the combination of Proposition \[Satake6\] and Corollary \[cor:position\_nodes\] give a computational recipe for how a degenerating pencil of genus-two curves is obtained from an F-theory background dual to the $\mathfrak{so}(32)$ string with only one non-vanishing Wilson line. In comparison, the work of the authors in [@Malmendier:2016aa] always allows for the construction of an explicit pencil of sextic curves given any family of Igusa invariants over a quadratic extension of the full ring of modular forms. However, this construction does not use the F-theoretic data of the $\mathfrak{so}(32)$ string background, i.e., the Jacobian elliptic fibration, and requires lifting of the family to a covering space of the moduli space. In contrast, Corollary \[cor:position\_nodes\] shows that the Satake sextic is inherently manifest in the Jacobian elliptic fibration (\[WEq.bak.alt\]).
We rephrase Corollary \[cor:position\_nodes\] according to the discussion in this section as follows:
\[cor:position\_nodes\_b\] The positions of the 7-branes with string charge $(1,0)$ in the F-theory model, dual to the $\mathfrak{so}(32)$ heterotic string with an unbroken gauge algebra of $\mathfrak{so}(28)\oplus \mathfrak{su}(2)$ and only a single non-vanishing Wilson line expectation value and no additional gauge-extension, are given by the loci of $I_1$ fibers in the Jacobian elliptic fibration (\[WEq.bak.alt\]) on the Shioda-Inose surface $\mathrm{SI}(\operatorname{Jac}\mathcal{C})$ of a generic genus-two curve $\mathcal{C}$ and form the ramification locus of the Satake sextic (\[SatakeSextic\_a\]) corresponding to $\mathcal{C}$, or, equivalently, the genus-two component of the fixed point set of the Nikulin involution $\varphi$ on $\mathrm{SI}(\operatorname{Jac}\mathcal{C})$.
The section $(x,y)=(0,0)$ defines an element of order $2$ in the Mordell-Weil group of the Jacobian elliptic fibration (\[WEq.bak.alt\]). It follows as in [@MR1416960; @MR1643100] that the actual gauge group of this heterortic model is $(\operatorname{Spin}(28)\times SU(2))/\mathbb{Z}_2$.
In turn, the roots of the Satake sextic then determine a sextic curve (\[Rosen2\]) with full level-two structure by using Equation (\[Theta2Satake\]) and Equation (\[Picard2\]).
Appendix
========
The Igusa-Clebsch invariants for the curve (\[Rosen2\]) in Rosenhain normal form are given by the following expressions: @size[7]{}@mathfonts@@@\#1 $$\begin{aligned}
I_2 & = 40\,\lambda_1\lambda_2\lambda_3-16\, \left( 1+ \lambda_1+ \lambda_2 + \lambda_3 \right) \left( \lambda_1\lambda_2\lambda_3+\lambda_2\lambda_1+\lambda_3
\lambda_1+\lambda_2\lambda_3 \right) +6\, \left( \lambda_2\lambda_1+\lambda_3\lambda_1+\lambda_2\lambda_3+\lambda_1+\lambda_2+\lambda_3 \right) ^2,\\
I_4 &=
-12\, \left( \lambda_1+\lambda_2+\lambda_3 \right) ^3
\lambda_1\lambda_2\lambda_3+4\, \left( \lambda_1+
\lambda_2+\lambda_3 \right) ^2 \left( \lambda_2\lambda_1+\lambda_3\lambda_1+\lambda_2\lambda_3 \right) ^2-4\, \left( \lambda_1+\lambda_2+\lambda_3 \right) ^2
\left( \lambda_2\lambda_1+\lambda_3\lambda_1+\lambda_2\lambda_3 \right) \lambda_1\lambda_2\lambda_3\\
& +4\, \left( \lambda_1+\lambda_2+\lambda_3 \right) ^2\lambda_1^2\lambda_2^2\lambda_3^2+12\, \left( \lambda_1+\lambda_2
+\lambda_3 \right) ^2\lambda_1\lambda_2\lambda_3-4\, \left( \lambda_1+\lambda_2+\lambda_3 \right) \left( \lambda_2\lambda_1
+\lambda_3\lambda_1+\lambda_2\lambda_3 \right) ^2\\
&+44\, \left( \lambda_1+\lambda_2+\lambda_3 \right) \left( \lambda_2\lambda_1
+\lambda_3\lambda_1+\lambda_2\lambda_3 \right) \lambda_1\lambda_2\lambda_3-12\, \left( \lambda_2\lambda_1+\lambda_3\lambda_1
+\lambda_2\lambda_3 \right) ^3+12\, \left( \lambda_2\lambda_1+\lambda_3\lambda_1+\lambda_2\lambda_3 \right) ^2\lambda_1\lambda_2
\lambda_3\\
&-12\, \left( \lambda_2\lambda_1+\lambda_3\lambda_1+\lambda_2\lambda_3 \right) \lambda_1^2\lambda_2^2\lambda_3^2-12\, \left( \lambda_1+
\lambda_2+\lambda_3 \right) \lambda_1\lambda_2\lambda_3+4\, \left( \lambda_2\lambda_1+\lambda_3\lambda_1
+\lambda_2\lambda_3 \right) ^2-72\,\lambda_1^2\lambda_2^2\lambda_3^2,
\stepcounter{equation}\tag{\theequation}\label{IgRos}\\
I_6 &=
-24\, \left( \lambda_1+\lambda_2+\lambda_3 \right) ^3
\lambda_1\lambda_2\lambda_3+10\, \left( \lambda_1+
\lambda_2+\lambda_3 \right) ^2\lambda_1^2\lambda_2^2\lambda_3^2+32\, \left( \lambda_2\lambda_1+
\lambda_3\lambda_1+\lambda_2\lambda_3 \right) ^2
\lambda_1\lambda_2\lambda_3+150\, \left( \lambda_2
\lambda_1+\lambda_3\lambda_1+\lambda_2\lambda_3
\right) \lambda_1^2\lambda_2^2\lambda_3^2\\
&+8\,
\left( \lambda_1+\lambda_2+\lambda_3 \right) ^2 \left(
\lambda_2\lambda_1+\lambda_3\lambda_1+\lambda_2
\lambda_3 \right) ^2\lambda_1^2\lambda_2^2\lambda_3^2+118\, \left( \lambda_1+\lambda_2+\lambda_3 \right) ^3 \left( \lambda_2\lambda_1+\lambda_3
\lambda_1+\lambda_2\lambda_3 \right) \lambda_1\lambda_2\lambda_3\\
&-194\, \left( \lambda_1+\lambda_2+\lambda_3 \right) ^2 \left( \lambda_2\lambda_1+\lambda_3
\lambda_1+\lambda_2\lambda_3 \right) \lambda_1^2\lambda_2^2\lambda_3^2+118\, \left( \lambda_1+
\lambda_2+\lambda_3 \right) \left( \lambda_2\lambda_1
+\lambda_3\lambda_1+\lambda_2\lambda_3 \right) ^3
\lambda_1\lambda_2\lambda_3\\
&-66\, \left( \lambda_1+
\lambda_2+\lambda_3 \right) \left( \lambda_2\lambda_1
+\lambda_3\lambda_1+\lambda_2\lambda_3 \right) ^2\lambda_1^2\lambda_2^2\lambda_3^2+76\, \left(
\lambda_1+\lambda_2+\lambda_3 \right) \left( \lambda_2\lambda_1+\lambda_3\lambda_1+\lambda_2\lambda_3
\right) \lambda_1^3\lambda_2^3\lambda_3^3\\
&-194\, \left( \lambda_1+\lambda_2+\lambda_3 \right) \left(
\lambda_2\lambda_1+\lambda_3\lambda_1+\lambda_2
\lambda_3 \right) ^2\lambda_1\lambda_2\lambda_3+412
\, \left( \lambda_1+\lambda_2+\lambda_3 \right) \left(
\lambda_2\lambda_1+\lambda_3\lambda_1+\lambda_2
\lambda_3 \right) \lambda_1^2\lambda_2^2\lambda_3^2\\
&+20\, \left( \lambda_1+\lambda_2+\lambda_3
\right) ^4 \left( \lambda_2\lambda_1+\lambda_1\lambda_3+\lambda_2\lambda_3 \right) \lambda_1\lambda_2
\lambda_3-36\, \left( \lambda_1+\lambda_2+\lambda_3
\right) ^3 \left( \lambda_2\lambda_1+\lambda_1\lambda_3+\lambda_2\lambda_3 \right) ^2\lambda_1\lambda_2
\lambda_3\\
&+20\, \left( \lambda_1+\lambda_2+\lambda_3
\right) ^3 \left( \lambda_2\lambda_1+\lambda_1\lambda_3+\lambda_2\lambda_3 \right) \lambda_1^2\lambda_2^2\lambda_3^2-8\, \left( \lambda_1+\lambda_2+
\lambda_3 \right) ^2 \left( \lambda_2\lambda_1+\lambda_1\lambda_3+\lambda_2\lambda_3 \right) ^3\lambda_1
\lambda_2\lambda_3\\
&+8\, \left( \lambda_1+\lambda_2+\lambda_3 \right) ^2 \left( \lambda_2\lambda_1+\lambda_1\lambda_3+\lambda_2\lambda_3 \right) ^2
-252\,\lambda_1^3\lambda_2^3\lambda_3^3-36\,\lambda_1^4\lambda_2^4\lambda_3^4-24\, \left( \lambda_2\lambda_1+\lambda_1\lambda_3+\lambda_2\lambda_{3
} \right) ^5+48\, \left( \lambda_2\lambda_1+\lambda_1
\lambda_3+\lambda_2\lambda_3 \right) ^4\\
&-24\, \left( \lambda_2\lambda_1+\lambda_1\lambda_3+\lambda_2
\lambda_3 \right) ^3+8\, \left( \lambda_1+\lambda_2+
\lambda_3 \right) ^4 \left( \lambda_2\lambda_1+\lambda_1\lambda_3+\lambda_2\lambda_3 \right) ^2-8\, \left(
\lambda_1+\lambda_2+\lambda_3 \right) ^3 \left( \lambda_2\lambda_1+\lambda_1\lambda_3+\lambda_2\lambda_3 \right) ^3\\
&+8\, \left( \lambda_1+\lambda_2+\lambda_3
\right) ^2 \left( \lambda_2\lambda_1+\lambda_1\lambda_3+\lambda_2\lambda_3 \right) ^4 -8\, \left( \lambda_1+
\lambda_2+\lambda_3 \right) ^3 \left( \lambda_2\lambda_1+\lambda_1\lambda_3+\lambda_2\lambda_3 \right) ^2-36\, \left( \lambda_1+\lambda_2+\lambda_3 \right) ^2
\left( \lambda_2\lambda_1+\lambda_1\lambda_3+\lambda_2\lambda_3 \right) ^3\\
&+20\, \left( \lambda_1+\lambda_{2
}+\lambda_3 \right) \left( \lambda_2\lambda_1+\lambda_1\lambda_3+\lambda_2\lambda_3 \right) ^4+20\, \left(
\lambda_1+\lambda_2+\lambda_3 \right) \left( \lambda_2\lambda_1+\lambda_1\lambda_3+\lambda_2\lambda_3
\right) ^3-36\,\lambda_1^2\lambda_2^2\lambda_3^2-24\, \left( \lambda_1+\lambda_2+\lambda_3 \right) ^5\lambda_1\lambda_2\lambda_3\\
&+48\, \left( \lambda_1+
\lambda_2+\lambda_3 \right) ^4\lambda_1^2\lambda_2^2\lambda_3^2-24\, \left( \lambda_1+\lambda_2+
\lambda_3 \right) ^3\lambda_1^3\lambda_2^3\lambda_3^3+24\, \left( \lambda_1+\lambda_2+\lambda_3 \right) ^4\lambda_1\lambda_2\lambda_3-136\, \left(
\lambda_1+\lambda_2+\lambda_3 \right) ^3\lambda_1^2\lambda_2^2\lambda_3^2\\
&+32\, \left( \lambda_1+
\lambda_2+\lambda_3 \right) ^2\lambda_1^3\lambda_2^3\lambda_3^3+24\, \left( \lambda_2\lambda_1+
\lambda_1\lambda_3+\lambda_2\lambda_3 \right) ^4 \lambda_1\lambda_2\lambda_3-24\, \left( \lambda_2
\lambda_1+\lambda_1\lambda_3+\lambda_2\lambda_3\right) ^3\lambda_1^2\lambda_2^2\lambda_3^2
+150\, \left( \lambda_1+\lambda_2+\lambda_3 \right) \lambda_1^3\lambda_2^3\lambda_3^3\\
&-136\, \left(
\lambda_2\lambda_1+\lambda_1\lambda_3+\lambda_2
\lambda_3 \right) ^3\lambda_1\lambda_2\lambda_3+10\,
\left( \lambda_2\lambda_1+\lambda_1\lambda_3+\lambda_2\lambda_3 \right) ^2\lambda_1^2\lambda_2^2\lambda_3^2-42\, \left( \lambda_2\lambda_1+\lambda_{{1}
}\lambda_3+\lambda_2\lambda_3 \right) \lambda_1^3\lambda_2^3\lambda_3^3\\
&-42\, \left( \lambda_1+
\lambda_2+\lambda_3 \right) \lambda_1^2\lambda_2^2\lambda_3^2+76\, \left( \lambda_1+\lambda_2+
\lambda_3 \right) \left( \lambda_2\lambda_1+\lambda_1\lambda_3+\lambda_2\lambda_3 \right) \lambda_1\lambda_2
\lambda_3-66\, \left( \lambda_1+\lambda_2+\lambda_3 \right) ^2 \left( \lambda_2\lambda_1+\lambda_1\lambda_3
+\lambda_2\lambda_3 \right) \lambda_1\lambda_2\lambda_3, \\
I_{10} & =\lambda_1^2\lambda_2^2\lambda_3^2
\left( \lambda_3-1 \right) ^2 \left( \lambda_2-1 \right) ^2 \left( -\lambda_3+\lambda_2 \right) ^2 \left( \lambda_1-1 \right) ^2 \left( -\lambda_3+\lambda_1 \right) ^2
\left( -\lambda_2+\lambda_1 \right) ^2.\end{aligned}$$
The components of the rational map $\Phi: \M_2 \backslash {\textnormal{supp}}{ (\chi_{35})}_0 \to \M_2$ with $(j_1, j_2, j_3) \mapsto (j'_1, j'_2, j'_3)$ are given by $$\begin{split}
j_1^\prime = \frac{64}{729} \, \frac{g^{(1)}(j_1,j_2,j_3)}{q(j_1,j_2,j_3)},\quad
j_2^\prime = \frac{4}{729} \, \frac{g^{(2)}(j_1,j_2,j_3)}{q(j_1,j_2,j_3)}, \quad
j_3^\prime = \frac{1}{729} \, \frac{g^{(3)}(j_1,j_2,j_3)}{q(j_1,j_2,j_3)},
\end{split}$$ with @size[7]{}@mathfonts@@@\#1 $$\begin{aligned}
q(j_1,j_2,j_3) & = j_1^5 \, \Big(j_2^4j_1^3-12\,j_1^3j_2^3j_3+54\,j_1^3j_2^2j_3^2-108\,j_1^3j_2j_3^3+81\,j_1^3j_3^4
+78\,j_2^5j_1^2-1332\,j_1^2j_2^4j_3+8910\,j_1^2j_2^3j_3^2-29376\,j_1^2j_2^2j_3^3+47952\,j_1^2j_2j_3^4\\
&-31104\,j_1^2j_3^5-159\,j_1j_2^6+1728\,j_1j_2^5j_3-6048\,j_1j_2^4j_3^2+6912\,j_1j_2^3j_3^3+80\,j_2^{7}-384\,j_2^6j_3
-972\,j_1^4j_2^2+5832\,j_1^4j_2j_3-8748\,j_1^4j_3^2-77436\,j_1^3j_2^3\\
&+870912\,j_1^3j_2^2j_3-3090960\,j_1^3j_2j_3^2 +3499200\,j_1^3j_3^3+592272\,j_2^4j_1^2-4743360\,j_1^2j_2^3j_3+9331200\,j_1^2j_2^2j_3^2-41472\,j_1j_2^5
+236196\,j_1^5\\
&+19245600\,j_2j_1^4-104976000\,j_1^4j_3-507384000\,j_2^2j_1^3+2099520000\,j_1^3j_2j_3+125971200000\,j_1^4\Big) ,\\
g^{(1)}(j_1,j_2,j_3) & = \Big(- j_2^2j_1+6\,j_2j_3j_1-9\,j_3^2j_1+j_2^3+540\,j_1^2 \Big)^5 ,\\
g^{(2)}(j_1,j_2,j_3) & = \Big( j_2^4j_1^2-12\,j_1^2j_2^3j_3+54\,j_1^2j_2^2j_3^2-108\,j_1^2j_2j_3^3+81\,j_1^2j_3^4
-2\,j_1j_2^5+12\,j_1j_2^4j_3-18\,j_1j_2^3j_3^2+j_2^6-756\,j_2^2j_1^3+4536\,j_1^3j_2j_3-6804\,j_1^3j_3^2\\
&+5130\,j_1^2j_2^3-17496\,j_1^2j_2^2j_3+131220\,j_1^4-2332800\,j_2j_1^3 \Big) \Big( -j_2^2j_1+6\,j_2j_3j_1-9\,j_3^2j_1+j_2^3+540\,j_1^2 \Big)^3 ,
\stepcounter{equation}\tag{\theequation}\label{Components_Phi}\\
g^{(3)}(j_1,j_2,j_3) & = \Big( -j_1^3j_2^6+18\,j_1^3j_2^5j_3-135\,j_1^3j_2^4j_3^2+540\,j_1^3j_2^3j_3^3
-1215\,j_1^3j_2^2j_3^4+1458\,j_1^3j_2j_3^5-729\,j_1^3j_3^6+3\,j_1^2j_2^{7}-36\,j_1^2j_2^6j_3
+162\,j_1^2j_2^5j_3^2-324\,j_1^2j_2^4j_3^3\\
& +243\,j_1^2j_2^3j_3^4-3\,j_1j_2^{8}+18\,j_1j_2^{7}j_3
-27\,j_1j_2^6j_3^2+j_2^{9}+1350\,j_1^4j_2^4-16200\,j_1^4j_2^3j_3+72900\,j_1^4j_2^2j_3^2-145800\,j_1^4j_2j_3^3
+109350\,j_1^4j_3^4-6345\,j_1^3j_2^5\\
& +52650\,j_1^3j_2^4j_3-144585\,j_1^3j_2^3j_3^2+131220\,j_1^3j_2^2j_3^3
+4995\,j_1^2j_2^6-14580\,j_1^2j_2^5j_3-599724\,j_1^5j_2^2+3598344\,j_1^5j_2j_3-5397516\,j_1^5j_3^2
+4175226\,j_1^4j_2^3\\
& -15390648\,j_1^4j_2^2j_3+4898880\,j_1^4j_2j_3^2-1961496\,j_1^3j_2^4+87392520\,j_1^6
-881798400\,j_1^5j_2-1259712000\,j_1^5j_3 \Big) \\
&\times \Big( -j_1j_2^2+6\,j_1j_2j_3-9\,j_1j_3^2+j_2^3+540\,j_1^2 \Big)^2 .\\
\end{aligned}$$
[^1]: $H$ is the standard hyperbolic lattice with the quadratic form $q=x_1x_2$.
[^2]: A Hodge isometry between two transcendental lattices is an isometry preserving the Hodge structure.
[^3]: The notation $T(\mathcal{Y})(2)$ indicates that the bilinear pairing on the transcendental lattice $T(\mathcal{Y})$ is multiplied by $2$.
[^4]: Here, $E_8(-1)$ and $E_7(-1)$ are the negative definite lattice associated with the exceptional root systems of $E_8$ and $E_7$, respectively.
[^5]: A lattice polarization of a K3 surface $\mathcal{Y}$ is a primitive embedding of a lattice $L' \hookrightarrow L = H_2(\mathcal{Y}, \mathbb{Z})$ such that the image of $L'$ lies in the Néron-Severi group $\operatorname{NS}(\mathcal{Y}) = L \cap H^{1,1}(\mathcal{Y})$ and contains a pseudo-ample class.
[^6]: In contrast, Igusa showed in [@MR0229643] that for the full ring of modular forms, one needs an additional generator $\chi_{35}$ which is algebraically dependent on the others.
|
---
abstract: 'Stochastic gravitational waves (GW) associated with unresolved astrophysical sources at frequency bands of the ongoing GW interferometers LIGO/VIRGO and LISA are studied. We show that GW noise from rotating galactic neutron stars with low magnetic fields may reach the advanced LIGO sensitivity level at frequency $f\sim 100$ Hz. Within LISA frequency band ($10^{-4}-10^{-1}$ Hz), the GW background from galactic binary stars is shown to mainly contribute up to a frequency of $3\times 10^{-2}$ Hz, depending on the galactic rate of binary white dwarf mergers. To be detectable by LISA, relic GW backgrounds should be as high as $\Omega_{GW}h_{100}^2>10^{-8}$ at $10^{-2}$ Hz.'
---
= 14pt = 6.0in = 8.5in -0.25truein 0.30truein 0.30truein \#1 \#1
[**Astrophysical Sources of Stochastic Gravitational Radiation\
in the Universe**]{}\
K.A.Postnov\
[*Sternberg Astronomical Institute, Moscow State University,\
Moscow 119899, Russia*]{}
Introduction
============
In a few years, with the completion of construction of gravitational wave detectors of high sensitivity, a new window into the Universe will be open (see Thorne 1995, Schutz 1996 for a recent review of gravitaitonal wave astronomy). In this connection, extensive studies of possible sources of gravitationa radiation are now being conducted. The most promising targets for the initial laser interferometers with the rms sensitivity level $h_{rms}\approx 10^{-21}$ at $f=100$ Hz are coalescing binary neutron stars and/or black holes, which can be observable from distances up to 100 Mpc (Lipunov, Postnov & Prokhorov 1997).
Stars are most numerous baryonic objects in the Universe ($\sim
10^{11}$ within the Galaxy, $\sim 10^{21}$ within the Hubble radius $R_H\sim 3000h_{100}^{-1}$ Mpc, where $h_{100}=H_0/100$ km/s/Mpc is the present value of the Hubble constant). Among them the most significant sources of GW are rotating triaxial neutron stars (NS) and binary stars. Only a small fraction of galactic NS (about 700) is observed as radiopulsars, so when trying to search for GW from them we have an advantage of knowing the precise spin period and the position on the sky. The same argument relates to the binary stars with known orbital periods. But the vast majority of them are unresolved sources and will form a stochastic background. A stochastic GW background is commonly measured in terms of the energy density per logarithmic frequency interval related to the critical energy density to close the Universe, $\Omega_{GW}=dE_{GW}/d\ln
f/\rho_{cr} c^2$ ($\rho_{cr}=3H_0^2/8\pi G\approx 1.9\times 10^{-29}
h_{100}^2$ g cm$^{-3}$ where $c$ is the speed of light). For comparison with dimensionless detector’s sensitivity $h$, one commonly uses the equivalent characteristic strain $h_c(f)=(1/2\pi)(H_0/f)\Omega_{GW}^{1/2}$.
Being interesting by themselves, astrophysical backgrounds, however, are viewed as a noise burying a possible cosmological gravitational wave background (CGWB), which bears the unique imprint of physical processes occurring at the very early (near-Plankian) age of the Universe (see e.g. Grishchuk 1988 for a review). Exact value of CGWB is still very controversial (see Grishchuk 1996 for fresh estimates). What is more reliable (however, not completely parameter-free) are GWBs formed by known astrophysical sources (old neutron stars, binary stars), and here we address the question how much astrophysical GW noises contribute at LIGO/VIRGO (10-1000 Hz) and LISA ($10^{-4}-10^{-1}$ Hz) frequencies.
GW noise from sources with changing frequency
=============================================
To calculate GW noise produced by some unresolved sources we need to know the number of sources per logarithmic frequency interval. At the first glance, this would require knowing precise formation and evolution of sources. However, when only GW carries away angular momentum from the emitting source, the problem becomes very simple and physically clear.
GW energy loss leads to changing the frequency $\omega=2\pi f$ of emitting objects. In the case of a rotating triaxial body, the positive rotational energy $E_{rot}=I\omega_{rot}^2/2$ (here $I$ is the moment of inertia) is being lost and the spin frequency (hence, GW freqency) decreases. In contrast, in the case of a binary star with masses of the components $M_1$ AND $m_2$ in a circular orbit ofr adius $a$ the negative orbital energy, $E_{orb}= -M_1M_2/2a\sim \M c^2
f_{orb}^{2/3}$ (here $\M=(M_1+M_2)^{2/5}(M_1M_2)^{3/5}$ is the so-called “chirp mass” of the system) is being lost and the orbital frequency increases.
To a very good approximation, the conditions of star formation and evolution of astrophysical objects in our Galaxy may be viewed as stationary. This is true at least for last 5 billion years. Let the formation rate of GW sources be $\R$. For example, the mean formation rate of massive stars ($>$10 M$_\odot$ to produce NS) is about 1 per 30 years.
The stationarity implies that the number of sources per unit logarithmic frequency interval is $$dN/d\ln f \equiv N(f) = \R \times (f/\dot f)\,.
\label{contin}$$ The total energy emitted in GW per second per unit logarithmic frequency interval at $f$ by all such sources in the galaxy is $$dE_{GW}/(dt\,d\ln f) \equiv L(f)_{GW}= \widetilde L(f)_{GW} N(f) =
\widetilde L(f)_{GW} \R \times (f/\dot f)\,,$$ where $\widetilde L(f)_{GW}$ is the GW luminosity of the typical source at frequency $f$ ($\widetilde L(f)_{GW}\propto f^6$ for non-axisymmetric neutron stars, $\widetilde L(f)_{GW}\propto f^{10/3}$ for binary systems).
Finally, for an isotropic background we have $$\Omega_{GW}(f)\rho_{cr} c^2 = L(f)_{GW}/(4\pi c \langle r \rangle^2)
\label{omega}$$ where $\langle r \rangle$ is the inverse-square average distance to the typical source. Strictly speaking, this distance (as well as the binary chirp mass $\cal M$ and moment of inertia $I$ of NS) may be a function of frequency since the binaries characterized by different $\cal M$ may be differently distributed in the galaxy. We are highly ignorant about the real distribution of old NS and binaries in the galaxy, but taking the mean photometric distance for a spheroidal distribution in the form $dN\propto \exp[-r/r_0]\,\exp[-(z/z_0)^2]$ ($r$ is the radial distance to the galactic center and $z$ the hight above the galactic plane) with $r_0=5$ kpc and $z_0=4.2$ kpc with $\langle r \rangle
\approx 7.89$ kpc is sufficient for our purposes.
For the cases considered the energy reservoir radiated in GW is either rotational energy (neutron stars) or orbital energy (binary systems), both depending as some power of the corresponding frequency: $E\propto
f^\alpha$, $\alpha_{NS}=2$, $\alpha_{bs}=2/3$. Hence the frequecy change $(f/\dot f)$ may be found from the equation $
dE/dt=\alpha E (\dot f/f)
$ By energy conservation law $
dE/dt=(dE/dt)_{GW}+(dE/dt)_{EM}+(dE/dt)_{\ldots}
$ where index EM stands for elecromagnetic losses and $\ldots$ means other possible losses of energy. Finally, we obtain $$(f/\dot f)= \alpha E / ((dE/dt)_{GW}+(dE/dt)_{EM}+(dE/dt)_{\ldots})$$ and $$L(f)_{GW}= \R \alpha \widetilde E \frac{1}{
1+\frac{(dE/dt)_{EM}}{(dE/dt)_{GW}}+
\frac{(dE/dt)_{\ldots}}{(dE/dt)_{GW}}}$$ The remarkable result is that if GW is the dominant source of energy removal, the resulting GW stochastic background depends only on the source formation rate: $$\Omega_{GW}(f)\rho_{cr} c^2 = \R \alpha \widetilde E /(4\pi c \langle
r \rangle^2)
\label{omega_R}$$
GWB from old neutron stars at LIGO/VIRGO frequencies
====================================================
Spin evolution of rotating non-axisymmetric NS with ellipticity $\epsilon$ and magnetic moment $\mu$ may be driven by GW or elecgtromagnetic losses. The condition that a stochastic signal appears within the detector band deoends on the rate of the frequency change. The upper frequency of the stochastic background for pure electromagnetic energy losses is $ f^{EM}_0\approx
10^3(\hbox{Hz})\R_{30}^{1/2}I_{45}^{1/2}\mu_{30}^{-1} $ where $\mu_{30}=\mu/(10^{30}\hbox{G cm}^3)$ is NS magnetic moment. For pure GW losses this upper frequency is $ f^{GW}_0\approx 1.4\times
10^4(\hbox{Hz}) \R_{30}^{1/4}I_{45}^{-1/4}\epsilon_{-6}^{-1/2} $ where $\epsilon_{-6}=\epsilon/10^{6}$ is the NS ellipticity.
For plausible values of the NS magnetic fields ($\mu_{30}=10^{-4}$–$10^2$) and ellipticities ($\epsilon_{-6}=10^{-3}$–$10^2$), at any frequency $<10^3$ Hz we deal with stochastic backgrounds from galactic NS. Physically, this is due to the inability of old NS to leave frequency interval $\Delta\omega \sim \omega$ during the typical time between consecutive supernova explosions.
For purely GR-driven NS spin-down the resulting spectrum is independent of the unknown value of $\epsilon$ in the NS population. Any additional braking mechanism always lowers the resulting signal. Taking typical values $I=10^{45}$ g cm$^2$, $\R=1/30$ yr$^{-1}$ we obtain from Eq. (\[omega\_R\]) $$\Omega_{NS}\approx 10^{-7}\R_{30}^{1/2}
I_{45}(f/100 Hz)^2h_{100}^{-2}(r/10 kpc)^{-2}$$ or in terms of $h_c$ $$h_c \approx \frac{1}{\tilde r} \sqrt{GI\R/c^3}\,
\approx 10^{-24}\left(\frac{10\hbox{kpc}}{\tilde r}\right)
\R_{30}^{1/2}I^{1/2}_{45}
\label{h_lim}$$ (here we assumed the characteristic distance to NS population of order 10 kpc). Remarkably, this limit does not depend on frequency. The GR background of such strength could be detected by the advanced LIGO/VIRGO interferometers in one year integration (Thorne 1987; Giazotto 1997).
For realistic NS parameters the ratio of electromagnetic to GW losses $x=\dot E_{EM}/\dot E_{GW}$ is $$x\approx 4000 \mu_{30}^2\epsilon_{-6}^{-2}
\left(\frac{100 \hbox{Hz}}{f}\right)^2
\label{x}$$ and electormagnetic losses becomes insignificant ($x\ll 1$) only at high frequencies $
f >f_{cr}\approx 6.3(\hbox{kHz})\, \frac{\mu_{30}}{\epsilon_{-6}}
$ If we would take $\epsilon_{-6}=10^{-3}$ and $\mu_{30}=10^{-4}$ as in millisecond pulsars, we would obtain $f_{cr}\approx 630$ Hz, however millisecond pulsars are spun up by accretion in binary systems and are not considered here.
Therefore, for realistic NS we must consider the case $x\gg 1$. Then the stochastic background from old NS becomes $$h_c(f)\approx 5\times 10^{-28}
\left(\frac{10\hbox{kpc}}{\tilde r}\right)
\R_{30}^{1/2}I^{1/2}_{45}\epsilon_{-6}\mu_{30}^{-1}f
\label{h(nu)_em}$$ and is lis below even advanced LIGO sensitivity at $f\sim 100$ Hz.
We have shown that if the NS form ellipticity is present, the stochastic GR background produced by old NS population is naturally formed due to NS rotation braking. In the limiting case when only GR angular momentum loss causes NS spin-down, this background is [*independent*]{} on both exact value of the NS form ellipticity $\epsilon$ and frequency and can be detected by advanced LIGO/VIRGO interferometers. In reality, the magnetic field of NS causes more effective electromagnetic energy loss: to be insignificant, the magnetic field of a NS should be less than (see Eq. (\[x\])) $
\mu < 1.5\times 10^{26} (\hbox{G cm}^3) \epsilon_{-6}\,f
$
According to Urpin & Muslimov (1992), the magnetic field can decay very fastly provided that the field was initially concentrated in the outer crust layers with the density $<10^{10}- 10^{11}$ g cm$^{-3}$, and such very low magnetic field for old NS may be possible. In the limiting case that the NS magnetic field does not decay at all (for example, if only accretion-induced field decay is possible in binary systems (Bisnovatyi–Kogan & Komberg 1974)), old NS should lose their energy through electromagnetic losses and be very slow rotators with periods of about a few seconds. Then the initial magnetic field distribution becomes crucial. If it is centered at $\sim 10^{12}$ G (as implied by radipulsar $P$–$\dot P$ measurements), we have little chances to detect the old NS population at 10–100 Hz frequency band unless close mean distances ($<$10 kpc) are assumed (Giazotto et al. 1997). However, if nature prefers a scale-free law (like $f(\mu)\propto 1/\mu$), the fraction of low-field NS could amount to a few $10\%$ and they can contribute to the frequency-independent GR background. Then Eq. (\[h\_lim\]) implies that such a background can be detected by the advanced LIGO/VIRGO interferometer in the frequency band 10–1000 Hz in one-year integration even if the formation rate of such NS is as small as 1 per 300 years and the characteristic distance to them is 100 kpc.
GW noise from unresolved binary stars at LISA frequencies
=========================================================
Inside LISA frequency range, $10^{-4}-10^{-1}$ Hz, only coalescing binary white dwarfs (WD) and binary neutron stars contribute. Even if binary neutron stars coalesce at a rate of 1/10000 yr in the Galaxy, their number still should be much smaller than the white dwarf binaries, and in this section we restrict ourselves to considering only binary WD.
Substituting $E=E_{orb}\sim {\M}c^2({\M}f)^{2/3}$ into equation (\[omega\_R\]) we obtain $$\Omega_{WD}(f)\approx 2\times 10^{-8} \R_{100} (f/10^{-3}
\hbox{Hz})^{2/3}(\widetilde {\M}/M_\odot)^{5/3}( \langle r
\rangle/10\, \hbox{kpc})^{-2}h_{100}^{-2}\,,$$ where $\R_{100}=\R /(0.01$ yr$^{-1})$ is the galactic rate of binary WD mergers.
In terms of the characteristic dimensionless amplitude of the noise background that determines the signal-to-noise ratio when cross-correlating outputs of two independent interferometers we have $$h_c(f)\approx 7.5 \times 10^{-20} \R_{100}^{1/2}
(f/10^{-3} \hbox{Hz})^{-2/3}(\widetilde {\M}/M_\odot)^{5/6}
( \langle r \rangle/10\,\hbox{kpc})^{-1}
\label{h_c}$$
Equation (\[h\_c\]) shows that at high frequencies of interest here the GW background is fully determined by the galactic rate of binary WD mergers and is independent of (complicated) details of binary evolution at lower frequencies (the examples of calculated spectra at lower frequencies see in Lipunov & Postnov 1987; Lipunov, Postnov & Prokhorov 1987; Hils et al. 1990).
But the real galactic merger rate of close binary WD is unknown. One possible way to recover it is searching for close white dwarf binaries. A recent study (Marsh et al. 1995), revealed a larger fraction of such systems than had previously been thought. Still, the statistics of such binaries in the Galaxy remains very poor.
If coalescing binary WD are associated with SN Ia explosions, as proposed by Iben & Tutukov (1984) and further investigated by many authors (for a recent review of SN Ia progenitors see Branch et al. 1995), their coalescence rate can be constrained using much more representative SN Ia statistics. Branch et al. (1995) concluded that coalescing CO-CO binary WD remain the most plausible candidates mostly contributing to the SN Ia explosions. The galactic rate of SN Ia is estimated $4\times 10^{-3}$ per year (Tamman et al. 1994; van den Bergh and McClure 1994), which is close to the calculated rate of CO-CO coalescences ($\sim (1-3)\times 10^{-3}$). The coalescence rate for He-CO WD and He-He WD (other possible progenitors of SN Ia) falls ten times short of that for CO-CO WD (Branch et al. 1995). As SN Ia explosions may well be triggered by other mechanisms, we conclude that the observed SN Ia rate provides a secure [*upper limit*]{} to the double WD merger rate regardless of the evolutionary considerations.
The upper limit (\[h\_c\]) is plotted in Fig. 1 for different rates of binary WD mergers $\R_{100}=1, 1/3, 1/10, 1/30$ assuming the chirp mass ${\M}\approx 0.52 M_\odot$ (as for two CO white dwarfs with equal masses $M_1=M_2=0.6 M_\odot$). These lines intersect the proposed LISA rms sensitivity at $
f>f_{lim}\approx 0.03-0.07 \hbox{Hz}\,.
\label{lim}
$ This means that at frequencies higher than $0.07$ Hz no continuous GW backgrounds of galactic origin are presently known to contribute above the rms-level of LISA space laser interferometer. The contribution from extragalactic binaries is still lower regardless of the poorly known binary WD merging rate (at least in the limit of no strong source evolution with $z$). Other possible sources could be extragalactic massive BH binary systems (e.g. Hils and Bender 1995). Their number in the Universe can be fairly high (e.g. Rees 1997), but no reliable estimates of their contribution are available at present. The lower limit (\[lim\]) is already close to the LISA sensitivity limit at 0.1 Hz, but we stress that the assumptions used in its derivation are upper limits, so the actual frequency beyond which no binary stochastic backgrounds contribute may be three times lower. This precise limit depends on the details of binary WD formation and evolution which are still poorly known.
Fig. 1 demonstrates that the calculated GW background intersects LISA sensitivity curve at frequencies $\sim 0.05$ Hz, and Bender et al’s curve at even lower frequencies $\sim 0.01$ Hz. The latter is probably due to Bender et al’s curve being derived from observational estimate of double WD galactic density in the solar neighborhoods; we stress once more that once formed, the binary WD will evolve until the less massive companion fills its Roche lobe; unless the mass ratio is sufficiently far from one (cf. Webbink 1984), the merger should occur. Therefore Bender et al’s curve provides a secure [*lower limit*]{} to the galactic binary stochastic GW background.
Presently, we cannot rule out the high galactic double WD merger rate (1/300-1/1000 yr$^{-1}$), and therefore can consider $f_{lim}$ to lie within the frequency range $0.01-0.07$ Hz. We conclude that no GW background of galactic origin above this frequencies should contribute at the rms-noise level of LISA interferometer, and hence the detection of an isotropic stochastic signal at frequencies $0.03-0.1$ Hz with an appreciable signal-to-noise level (which possibly may be done using one interferometer) would strongly indicate its cosmological origin. To be detectable by LISA, the power of relic GW background should be $\Omega_{GW}h_{100}^2>10^{-8}$ in this frequency range.
=0.6
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author:
- A1
bibliography:
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title: 'CMB constraints on K-mouflage models'
---
Introduction
============
In the last few decades, cosmological probes have been able to measure the parameters describing our Universe with increasing accuracy, confirming that the six-parameter $\Lambda$CDM model based on the General Theory of Relativity (GR), provides an adequate description of observations, the latest of which is the *Planck* probe . Nevertheless, this model assumes the presence of a cosmological constant $\Lambda$ associated with the vacuum energy of the Universe, which accounts for almost 70% of the total energy of the observable universe – the dark energy. It is responsible for the late-time cosmic acceleration of the Universe [@1998AJ....116.1009R; @1999ApJ...517..565P], but its nature remains so far unexplained. Understanding the nature of dark matter and testing GR are two of the major challenges in cosmology, and both space [@2011PhRvD..83b3012M] and ground based [@2009arXiv0912.0201L] missions have been developed to make measurements in this direction. The cosmological constant represents the simplest explanation of dark energy, but it has no compelling physical motivation and moreover its numerical value is in disagreement with the zero-point energy suggested from quantum field theory (the *cosmological constant problem*). This has prompted theorists to look for more natural alternative explanations, some of which involve the presence of additional degrees of freedom through the introduction of scalar fields.
GR has been extensively tested within the Solar System to high accuracy, but on cosmological scales, independent measurements and tests are more difficult to perform and thus deviations from GR may be present, but they must incorporate a screening mechanism such that this modifications are hidden on small scales. The simplest way to modify GR is to introduce an additional scalar field, but this has to satisfy stability and no-ghost conditions (see *e.g.* Ref. [@2016RPPh...79d6902K] for a review). In the case of scalar field theories that are conformally coupled to matter, there are three possibilities of screening: chameleon [@2004PhRvL..93q1104K; @2004PhRvD..69d4026K], Vainshtein [@1972PhLB...39..393V] and K-mouflage [@2009IJMPD..18.2147B; @2014PhRvD..90b3507B; @2014PhRvD..90b3508B].
Many of the models developed over the years successfully explain the observations, and the *Planck* probe has placed stringent constraints on some of them . However, as the models deal with the late-time evolution of the Universe, the Cosmic Microwave Background (CMB) may be unaffected by the modifications of gravity and then the constraints will be weak and other types of probes, such as *Euclid* [@2011PhRvD..83b3012M] may be more suitable for investigating them.
One model which presents interesting features and opportunities to be constrained with the CMB is the K-mouflage model, first proposed in Ref. [@2009IJMPD..18.2147B] and which is the focus of this paper. We use the Effective Field Theory formulation of this model [@2016JCAP...01..020B] and we implement it into the <span style="font-variant:small-caps;">EFTCAMB</span> code [@2014PhRvD..89j3530H] two different variants of the model. The first one is a five-function parametrisation of the model where the background expansion history is left free to vary, the second one is a three-function parametrization of the model that is required to reproduce a $\Lambda$CDM-like background expansion. We determine the CMB anisotropies in the two cases and we compare them to the $\Lambda$CDM result. We show that in both cases the CMB power spectrum substantially changes when the parameters of the model are away from the $\Lambda$CDM limit, and therefore the CMB is a powerful tool to constrain such models. We use a Markov-Chain-Monte-Carlo (MCMC) approach [@Raveri2014] to place constraints on the parameters of the model using the *Planck* likelihood . We use a Fisher Matrix formalism to show that these constraints can be improved in the future by around one order of magnitude when using the COrE probe [@2016arXiv161200021D]. We describe this model in the formalism of Horndeski theories and we thus show that gravitational waves travel at the speed of light. Moreover, we show from the constraints that the ISW signal is enhanced, but negligibly due to the values of the parameters.
Our paper is structured as follows. In Section \[sec:km\] we present the K-mouflage model, its features and screening mechanism. In Section \[sec:eftcamb\] we discuss the model in the formalism of the effective field theory of cosmic acceleration, including the parametrisation required for the numerical implementation. In Section \[sec:ps\] we analyse the CMB power spectra obtained in the K-mouflage model for different choices of parameters and the allowed ranges of these parameters from simple Solar System tests. In Section \[sec:constr\] we derive constraints on the parameters of the model using a MCMC method and we compute forecasts for future CMB probes with a Fisher Matrix method. In Section \[sec:interpret\] we discuss the physical consequences of the constraints in the framework of the general Horndeski theories and we conclude in Section \[sec:concl\].
The K-mouflage model {#sec:km}
====================
The K-mouflage class of models with one scalar field $\varphi$ is defined by the action [@2016JCAP...01..020B; @2014PhRvD..90b3507B]
$$\begin{aligned}
S &= \int d^4 x \sqrt{-\tilde{g}} \left[ \frac{\tilde{M}_{\rm Pl}^2}{2} \tilde{R} + {\cal M}^4 K(\tilde{\chi}) \right]+ \nonumber \\
& \int d^4 x \sqrt{-g} L_{\rm m}(\psi_i, g_{\mu\nu})
+ \int d^4 x \sqrt{-g} \frac{1}{4\alpha} F^{\mu\nu}F_{\mu\nu} ,
\label{eq:actkm}\end{aligned}$$
where $\tilde{M}_{\rm Pl}$ is the bare Planck mass, ${\cal M}^4$ is the energy scale of the scalar field, $g_{\mu \nu}$ is the Jordan frame metric, $\tilde{g}_{\mu \nu}$ is the Einstein frame metric, $g_{\mu \nu}=A^2(\varphi)\tilde{g}_{\mu \nu}$ and $$\tilde{\chi} = - \frac{\tilde{g}^{\mu\nu} \partial_{\mu}\varphi\partial_{\nu}\varphi}{2 {\cal M}^4} \,$$ and the metric signature is $(-,+,+,+)$. $\mathcal{L}_{m}$ is the Lagrangian of the matter fields $\psi_{m}^{(i)}$ while $F^{\mu\nu}$ is the electromagnetic tensor and $\alpha$ is the fine structure constant . We consider a background of an expanding universe with scale factor $a$. Then in the Jordan frame the space-averaged Klein-Gordon equation yields $$\frac{d}{dt}\left[\bar{A}^{-2} a^3 \frac{d \bar{\varphi}}{dt}\bar{K}' \right]=-a^3 \bar{\rho} \frac{d \ln \bar{A}}{d \varphi} \, ,$$ where $\bar{\rho}_{\varphi}$ is the scalar field energy density. Considering linear perturbations, the metric in the Newtonian gauge can be expressed as $$ds^2=a^2 \left[-(1+2 \Phi) d\tau^2+(1-2 \Psi)d{\textbf{x}}^2 \right] \,.$$ Then, in the non-relativistic and small-scale limit, the two potentials $\Phi$ and $\Psi$ satisfy $$\begin{aligned}
\Phi=(1+\epsilon_1)\Phi_N \, ,\\
\Psi=(1-\epsilon_1)\Phi_N \, ,\end{aligned}$$ where $\Phi_N$ is the $\Lambda$CDM Newtonian potential and $$\begin{aligned}
\epsilon_1&=\frac{2 \beta^2}{\bar{K}'} \, ,\\
\beta &\equiv \tilde{M}_{\rm Pl} \frac{d\ln \bar{A}}{d\varphi} = \epsilon_2
\tilde{M}_{\rm Pl} \left( \frac{d\bar{\varphi}}{d\ln a} \right)^{-1} \, ,
\label{beta-a} \\
\epsilon_2 &\equiv \frac{d \ln \bar{A}}{d \ln a} \, , \label{eps2_func}\end{aligned}$$ where $\beta$ is the coupling strength and the two (related) functions $\epsilon_1$ and $\epsilon_2$ encode the deviation of K-mouflage from standard $\Lambda$CDM. Indeed, the $\epsilon_2$ function causes a running of the Jordan-frame Planck mass $M_{\rm Pl}$, which is given by $$M_{\rm Pl} = \frac{\tilde{M}_{\rm Pl}}{A} , \;\;\; \mbox{hence} \;\;\;
\epsilon_2 = - \frac{d\ln M_{\rm Pl}}{d\ln a} ,
\label{MPlanck-Jordan}$$ while $\epsilon_1$ is responsible for the appearance of a fifth force, or equivalently for a change in the effective gravitational constant $ G_{N, \rm eff}$, which is related to the standard one $G_N$, as $$G_{N, \rm eff} = G_N (1+ \epsilon_1) \, ,$$ showing that gravity is modified. Here and in the following we normalize the Planck mass to its current value at $a=1$, $$A_0 \equiv A(a=1) = 1.
\label{A0-def}$$
In these models, screening occurs in a similar fashion to the Vainshtein mechanism, with the important difference that the radius of the screened region depends on scale. By considering perturbations $\delta \varphi \equiv \varphi - \varphi_0$ of the scalar field around the background value $\varphi_0$, the Lagrange density corresponding to $\delta \varphi$ is given by $$\mathcal L = - \frac{Z(\varphi_0)}{2}(\partial \delta \varphi)^2-\frac{m^2(\varphi_0)}{2} (\delta \varphi)^2 -\beta(\varphi_0)\frac{\delta \varphi}{\tilde{M}_{\rm Pl}}\delta \rho_m \, .$$ Screening occurs when $$Z(\varphi_0) \gtrsim 1 \, .$$ To leading order, $$Z(\varphi)=1+a(\varphi)\frac{(\partial \varphi)^2}{\mathcal{M}^4}+b(\varphi)L^2 \frac{\Box\varphi}{\tilde{M}_{\rm Pl}} \, ,
\label{eq:Z}$$ where $a$ and $b$ are functions of order one and $L$ is a typical length scale. Screening in the Vainshtein scenario occurs when $a=0$ and hence $$\frac{|\nabla^2 \varphi|}{M_{\rm Pl}} \gtrsim L^{-2} \, ,$$ while screening in the K-mouflage scenario corresponds to $b=0$, $$|\nabla \varphi|\gtrsim \mathcal{M}^2$$ or equivalently $$|\nabla \Psi_N|\gtrsim \frac{\mathcal{M}^2}{2 \beta \tilde{M}_{\rm Pl}}$$ where $$\nabla^2 \Psi_N=4 \pi A(\varphi_0) \frac{1}{8 \pi M_{\rm Pl}^2}\delta\rho_m \, .$$
Action (\[eq:actkm\]) can be used to derive the Klein-Gordon equation for the $\varphi$ and thus the equation of motion of the scalar field. The evolution of the background and of linear perturbations have been studied in Refs. [@2014PhRvD..90b3507B; @2014PhRvD..90b3508B]. To make computations easier, we choose to switch to an Effective Field Theory (EFT) approach to compute CMB anisotropies from this model.
In the Jordan frame, the Friedmann equations read $$3 \tilde{M}_{\rm Pl}^2 H^2 = \frac{A^2}{(1-\epsilon_2)^2} \;
(\rho+\rho_{\gamma}+\rho_{\varphi}) ,
\label{E00}$$ and $$\begin{aligned}
&& -2 \tilde{M}_{\rm Pl}^2 \frac{dH}{dt} = \frac{A^2}{1-\epsilon_2} \;
(\rho+\rho_{\gamma}+\rho_{\varphi}+p_{\gamma}+p_{\varphi}) \nonumber \\
&& + \frac{2 A^2}{3 (1-\epsilon_2)^2} \left( \epsilon_2 - \frac{1}{1-\epsilon_2}
\frac{d\epsilon_2}{d\ln a} \right) (\rho+\rho_{\gamma}+\rho_{\varphi}) ,
\;\;\;
\label{Eii}\end{aligned}$$ where $\rho_{\varphi}$ and $p_{\varphi}$ are the scalar field energy density and pressure.\
The matter and radiation energy densities satisft the standard $\Lambda$CDM continuity equations as, but the scalar field energy density and pressure are now given by $$\rho_{\varphi} = \frac{{\cal M}^4}{A^4} (2 \tilde\chi K' - K) , \;\;\;
p_{\varphi} = \frac{{\cal M}^4}{A^4} K .
\label{rho-phi-def}$$ In particular, at the background level the equation of motion of the scalar field is equivalent to its continuity equation, $$\frac{d\rho_{\varphi}}{dt} = - 3 H (\rho_{\varphi}+p_{\varphi}) - \epsilon_2 H
(\rho_{\varphi}-3p_{\varphi}+\rho) ,
\label{continuity-phi}$$ with $$\left| \frac{d\varphi}{d\ln a} \right| = \sqrt{ \frac{2{\cal M}^4 \tilde\chi}{A^2 H^2} } .
\label{eps2_func}$$
In a fashion similar to the $\Lambda$CDM case, we can check that the continuity equation (\[continuity-phi\]) and the two Friedmann equations (\[E00\])-(\[Eii\]) are not independent. Thus, we can also derive the non-conservation equation (\[continuity-phi\]) from the two Friedmann equations (\[E00\])-(\[Eii\]). This means that, at the background level, we can discard the equation of motion of the scalar field and only keep track of the two Friedmann equations.
Because the matter and radiation follow standard continuity equations, we can again write $$\frac{\rho}{3 \tilde{M}_{\rm Pl}^2 H_0^2} = \frac{\Omega_{\rm m 0}^{\rm K}}{a^3} , \;\;\;
\frac{\rho_{\gamma}}{3 \tilde{M}_{\rm Pl}^2 H_0^2} =
\frac{\Omega_{\gamma 0}^{\rm K}}{a^4} ,
\label{continuity}$$ where the cosmological parameters $\Omega^{\rm K}_*$ may differ from the $\Lambda$CDM reference. For the scalar field energy density, we define at $z=0$ $$\frac{\rho_{\varphi 0}}{3 \tilde{M}_{\rm Pl}^2 H_0^2} = \Omega_{\varphi 0}^{\rm K} .
\label{Omega-phi0-def}$$ Then, the Friedmann equations (\[E00\])-(\[Eii\]) can be written as $$\frac{H^2}{H_0^2} = \frac{A^2}{(1-\epsilon_2)^2} \left[
\frac{\Omega_{\rm m 0}^{\rm K}}{a^3} + \frac{\Omega_{\gamma 0}^{\rm K}}{a^4}
+ \Omega_{\varphi 0}^{\rm K} \frac{\rho_{\varphi}}{\rho_{\varphi 0}} \right] ,
\label{E00-1}$$ and $$\begin{aligned}
- \frac{2}{3 H_0^2} \frac{dH}{dt} & = & \frac{A^2}{1-\epsilon_2} \;
\left[ \frac{\Omega_{\rm m 0}^{\rm K}}{a^3} + \frac{4 \Omega_{\gamma 0}^{\rm K}}{3 a^4}
+ \Omega_{\varphi 0}^{\rm K} \frac{\rho_{\varphi}+p_{\varphi}}{\rho_{\varphi 0}} \right]
\nonumber \\
&& + \frac{2 A^2}{3 (1-\epsilon_2)^2}
\left( \epsilon_2 - \frac{1}{1-\epsilon_2} \frac{d\epsilon_2}{d\ln a} \right)
\nonumber \\
&& \times \left[ \frac{\Omega_{\rm m 0}^{\rm K}}{a^3}
+ \frac{\Omega_{\gamma 0}^{\rm K}}{a^4} + \Omega_{\varphi 0}^{\rm K}
\frac{\rho_{\varphi}}{\rho_{\varphi 0}} \right] .
\label{Eii-1}\end{aligned}$$ The cosmological factors $\Omega^{\rm K}$ defined in this fashion do not add up to unity because of the prefactor $(1-\epsilon_2)^{-2}$.
Full K-mouflage models {#sec:params}
----------------------
We first consider a parametrisation of the model in terms of five parameters, $\alpha_U$, $\gamma_U$, $m$, $\epsilon_{2,0}$, $\gamma_A$, with full details on the derivation of these functions given in Ref. [@2016JCAP...01..020B]. We start by considering as a background an expanding Universe with scale factor $a$. The Friedmann and the Klein-Gordon equations yield $$\begin{aligned}
\frac{d \bar{\varphi}}{d \ln a}&=-\frac{\sqrt{2 \bar{\tilde{\chi}}{\cal M}^4}}{\bar{A}H} \, , \label{eq:dphida} \\
\frac{d}{d \ln a}\left[\bar{A}^{-3} a^3 \sqrt{2 \bar{\tilde{\chi}}{\cal M}^4} K' \right]&=-\frac{\epsilon_2 \bar{A} \bar{\rho}_0}{\sqrt{2 \bar{\tilde{\chi}}{\cal M}^4}} \,.\end{aligned}$$ Defining $U(a) \equiv a^3 \sqrt{\bar{\tilde{\chi}}}K'$, and a normalisation of $A(a=1)=1$ today, solutions for functions $\bar{A}$ and $U$ are given by $$\begin{aligned}
\bar{A}(a)&=1+\alpha_A- \alpha_A \left[ \frac{a (\gamma_A+1)}{a+\gamma_A} \right]^{\nu_A} \label{A_def} \, , \\
U(a)& \propto \frac{a^2 \ln(\gamma_U+a)}{(\sqrt{a_{\mathrm{eq}}}+\sqrt{a})\ln(\gamma_U+a)+\alpha_U a^2} \,\end{aligned}$$ where $a_{\mathrm{eq}}$ represents the scale factor at radiation-matter equality and $$\begin{aligned}
\nu_A&=\frac{3 (m-1)}{2m-1} \, , \\
\alpha_A&=-\frac{\epsilon_{2,0}(\gamma_A+1)}{\gamma_A \nu_A} \, .\end{aligned}$$ Hence, the kinetic term is given by $$\sqrt{\tilde{\chi}}=-\frac{\bar{\rho}_0}{{\cal M}^4} \frac{\epsilon_2 \bar{A}^4}{2 U(-3 \epsilon_2+\frac{d \ln U}{d \ln a})} \, . \label{eq:chi}$$
The mass scale of the scalar field $M^4$ is fixed by the choice of the cosmological parameters: $$\frac{M^4}{\bar{\rho}_0}= \frac{\Omega_{\varphi,0}}{\Omega_{m,0}}+ \frac{\epsilon_{2,0}}{-3 \epsilon_{2,0} + \frac{{\mathrm{d}}ln U}{{\mathrm{d}}ln a}\vert_{a=1}}.$$ After obtaining $\bar{\tilde{\chi}}$ one can compute the kinetic function by integrating the equation $$\frac{{\mathrm{d}}\bar{K}(a)}{{\mathrm{d}}\ln a }= \frac{{\mathrm{d}}\bar{K}}{{\mathrm{d}}\bar{\tilde{\chi}}} \frac{{\mathrm{d}}\bar{\tilde{\chi}}}{{\mathrm{d}}\ln a}=\frac{U}{a^3 \sqrt{\bar{\tilde{\chi}}}} \frac{{\mathrm{d}}\bar{\tilde{\chi}}}{{\mathrm{d}}\ln a}, \label{eq:dK/dlna}$$ with the condition $\bar{K}(a=1)=\bar{K}_0 =-1$.
Reproducing the $\Lambda$CDM expansion history: K-mimic models. {#sec:mapping}
---------------------------------------------------------------
Although the background expansion history in K-mouflage models is a priori different from that of $\Lambda$CDM, the model has enough freedom to reproduce the same $H(a)$ of $\Lambda$CDM. We can do this opportunely fixing the kinetic function. The K-mouflage universe reproduces the $\Lambda$CDM expansion history if the right-hand side of the K-mouflage Friedmann equations (\[E00-1\])-(\[Eii-1\]) is equal to the right-hand side of the $\Lambda$CDM Friedmann equations (\[E00-LCDM-1\])-(\[Eii-LCDM-1\]). This yields
$$\begin{aligned}
\Omega_{\varphi 0}^{\rm K} \frac{\rho_{\varphi}}{\rho_{\varphi 0}} & = &
\frac{(1-\epsilon_2)^2}{A^2} \left[ \frac{\Omega_{\rm m 0}}{a^3}
+ \frac{\Omega_{\gamma 0}}{a^4} + \Omega_{\Lambda 0} \right] \nonumber \\
&& - \left( \frac{\Omega_{\rm m 0}^{\rm K}}{a^3} + \frac{\Omega_{\gamma 0}^{\rm K}}{a^4}
\right) ,\end{aligned}$$
and $$\begin{aligned}
&& \Omega_{\varphi 0}^{\rm K} \frac{p_{\varphi}}{\rho_{\varphi 0}} =
- \frac{1-\epsilon_2}{A^2} \Omega_{\Lambda 0} + \frac{1-\epsilon_2}{A^2}
\frac{\Omega_{\gamma 0}}{3 a^4} - \frac{\Omega_{\gamma 0}^{\rm K}}{3 a^4}
+ \frac{1-\epsilon_2}{3 A^2} \nonumber \\
&& \times \left( \epsilon_2 + \frac{2}{1-\epsilon_2}
\frac{d\epsilon_2}{d\ln a} \right) \left( \frac{\Omega_{\rm m 0}}{a^3}
+ \frac{\Omega_{\gamma 0}}{a^4} + \Omega_{\Lambda 0} \right) .\end{aligned}$$ From Eqs.(\[rho-phi-def\]) this can be written as $$\begin{aligned}
\Omega_{\varphi 0}^{\rm K} \frac{2 \tilde\chi K'-K}{2\tilde\chi_0 K'_0 - K_0} & = & u(a) ,
\label{u-a}
\\
\Omega_{\varphi 0}^{\rm K} \frac{K}{2\tilde\chi_0 K'_0 - K_0} & = & v(a) ,
\label{v-a}\end{aligned}$$ where we defined the functions of time $$\begin{aligned}
u(a) & = & A^2(1-\epsilon_2)^2 \left[ \frac{\Omega_{\rm m 0}}{a^3}
+ \frac{\Omega_{\gamma 0}}{a^4} + \Omega_{\Lambda 0} \right] \nonumber \\
&& - A^4 \left( \frac{\Omega_{\rm m 0}^{\rm K}}{a^3}
+ \frac{\Omega_{\gamma 0}^{\rm K}}{a^4}
\right) ,
\label{u-def}\end{aligned}$$ and $$\begin{aligned}
&& v(a) = - A^2 (1-\epsilon_2) \Omega_{\Lambda 0} + A^2 (1-\epsilon_2)
\frac{\Omega_{\gamma 0}}{3 a^4} - A^4 \frac{\Omega_{\gamma 0}^{\rm K}}{3 a^4}
\nonumber \\
&& + \frac{A^2(1-\epsilon_2)}{3} \left( \epsilon_2 + \frac{2}{1-\epsilon_2}
\frac{d\epsilon_2}{d\ln a} \right) \left( \frac{\Omega_{\rm m 0}}{a^3}
+ \frac{\Omega_{\gamma 0}}{a^4} + \Omega_{\Lambda 0} \right) .
\nonumber \\
\label{v-def}\end{aligned}$$ Here we used the normalization $A_0=1$.
From Eqs.(\[u-a\])-(\[v-a\]), we obtain $$\Omega_{\varphi 0}^{\rm K} = u_0 \equiv u(a=1) ,
\label{Omega-phi0-u0}$$ and $$\Omega_{\varphi 0}^{\rm K} \frac{2 \tilde\chi K'}{2\tilde\chi_0 K'_0 - K_0}
= u(a)+v(a) .
\label{u+v-a}$$ At low redshift, $z \simeq 0$, the scalar field approximately behaves as a cosmological constant, with $\tilde\chi \ll 1$ and $K \simeq K_0 \simeq -1$. Therefore, we can choose $$N_0 \equiv 2\tilde\chi_0 K'_0 - K_0 = 1 .
\label{N_0}$$ The choice $N_0=1$ corresponds to a choice of normalization for the nonlinear kinetic function $K(\tilde\chi)$, which is slightly different from the one used in Eq. (\[eq:dK/dlna\]). Then, Eq.(\[v-a\]) gives the background kinetic function $K$ as a function of the scale factor, $$K(a) = \frac{v(a)}{u_0} .
\label{K-a}$$ This determines in turns the background scalar kinetic energy $\tilde\chi$ as a function of the scale factor through Eq.(\[u+v-a\]), $$\frac{{\mathrm{d}}\ln\tilde\chi}{{\mathrm{d}}\ln a} = \frac{2 u_0}{u(a)+v(a)} \frac{d K}{{\mathrm{d}}\ln a} .
\label{chi-a}$$ Choosing the normalization $K_0'=1$, we obtain from Eq.(\[u+v-a\]) the initial condition at $z=0$, $\tilde\chi_0=(u_0+v_0)/(2u_0)$, and the integration of Eq.(\[chi-a\]) provides $\tilde\chi(a)$ at all times. Together with Eq.(\[K-a\]), this gives a parametric definition of the kinetic function $K(\tilde\chi)$.
To complete the definition of the K-mouflage model, we implicitly define the conformal coupling through a given function $A(a)$. This directly yields the factor $\epsilon_2(a)$ from Eq.(\[eps2-def\]), and we obtain $\varphi(a)$ by integrating Eq.(\[dphidlna\]), with the initial condition $\varphi(a=0)=0$. This provides a parametric definition of the coupling $A(\varphi)$. We choose the sign of $\varphi$ to be the same as that of $\epsilon_2$, so that the coupling strength $\beta$ is positive.
To obtain a cosmology that is close to the $\Lambda$CDM reference (\[sec:LCDM\]), we set $\Omega^{\rm K}_{\rm b0}=\Omega_{\rm b0}$ and $\Omega^{\rm K}_{\gamma 0}=\Omega_{\gamma 0}$. This means that the baryon and radiation densities are the same in both universes. We keep the dark matter densities slightly different, in order to accommodate the constraints that must be satisfied by the functions $u(a)$ and $v(a)$. Indeed, as we require $\tilde\chi > 0$ and $K'>0$ we can see from Eq. (\[u+v-a\]) that we must have $$0 \leq a \leq 1 : \;\;\; u+v > 0 .
\label{u+v-a-constraint}$$ At low redshift we have $K \simeq -1$ whereas at high redshift we have $K \gg 1$. Then Eq. (\[v-a\]) implies $$v_0 < 0 , \;\;\; \mbox{and for} \;\;\; a \ll 1 : \;\;\; v > 0 .
\label{v-a-constraint}$$ Then, for a coupling function $A(a)$ with the same form as in Eq. (\[A\_def\]), these requirements constrain the dark matter $\Omega^{\rm K}_{\rm dm}$ to be: $$\Omega^{\rm K}_{\rm dm}\lesssim \Omega_{\rm dm} -\frac{\epsilon_{2,0} (4 + 3 \Omega_{m0} + 4 \Omega_{\gamma 0} +
\gamma_A (2 - 2 \nu_{A} + 3 \Omega_{m0} + 4 \Omega_{\gamma 0}))}{3(1 + \gamma_A)}.
\label{v-a-constraint}$$
Allowed ranges of the parameters
--------------------------------
The five parameters of the model are arbitrary, but they must satisfy constraints that ensure the stability of the solutions, as well as the Solar System and cosmological constraints [@2015PhRvD..91l3522B]. These have been discussed in Ref. [@2016JCAP...01..020B]:
- $|\epsilon_{2,0}| \lesssim 0.01$. The $\Lambda$CDM limit is recovered when $\epsilon_{2,0} \to 0$, independent of the values of the other four parameters. The sign of this parameter can be arbitrarily chosen for full-K-mouflage models, when the kinetic function is left free to vary. We choose it to be negative for full K-mouflage models, adopting the same convention as [@2016JCAP...01..020B]. Conversely in the case of K-mimic models $\epsilon_{2,0}$ has to be positive in order to match the stability requirements.
- $m>1$; describes the large $\chi$ power-law behaviour of the kinetic function: .
- $\gamma_A>0$, describes the transition to the dark energy dominated epoch in the $A(a)$ coupling function.
- $\gamma_U \ge 1$; sets the transition to the dark energy dominated epoch in the $K(a)$ kinetic function.
- $\alpha_U>0$. describes the transition from to the dark energy dominated epoch in the $K(a)$ kinetic function.
Although there are no a priori upper bounds on the last four parameters, by investigating the numerical behaviour of the solutions to the equations, we have checked that if too high values of these parameters are taken, either there are negligible changes in the results or ghosts appear. Therefore, we have taken the parameters to be of order one.
K-mouflage in the Effective Field Theory of Dark Energy {#sec:eftcamb}
=======================================================
The EFT of dark energy represents a general framework for describing dark energy and modified gravity that includes all single field models [@2013JCAP...02..032G; @2013JCAP...08..010B; @2013JCAP...08..025G; @2013JCAP...12..044B; @2014JCAP...05..043P]. It is built in the unitary gauge in analogy to the EFT of inflation [@2006JHEP...12..080C; @2008JHEP...03..014C] by using operators represented by perturbations of quantities which are invariant under spatial diffeomorphisms: $g^{00}$, the extrinsic curvature tensor $K^{\mu}_{\; \nu}$ and the Riemann tensor $R_{\mu \nu \rho \sigma}$.
The action given in Eq. (\[eq:actkm\]) can be expressed in the framework of the Effective Field Theory (EFT) of dark energy [@2016JCAP...01..020B]. In the case of K-mouflage only operations involving perturbations of $g^{00}$ appear, of the form $(\delta g^{00})^n$. In this case, the Jordan-frame EFT action can be expressed as
$$\begin{aligned}
S = & \int d^4 x \sqrt{-g} \left[ \frac{M_{\rm Pl}^2}{2} R
- \Lambda(\tau) - c(\tau) g^{00} \right. \nonumber \\
& \left. + \sum_{n=2}^{\infty}
\frac{M_n^4(\tau)}{n!} (\delta g^{00})^n \right] \nonumber \\
& + S_{\rm m}(\psi_i, g_{\mu\nu}) + S_{\rm rad}(F_{\mu\nu}, g_{\mu\nu}) ,\end{aligned}$$
with $$\begin{aligned}
M_{\rm Pl}^2&= \bar{A}^{-2} \tilde{M}_{\rm Pl}^2 \, , \\
\Lambda&= - \bar{A}^{-4} {\cal M}^4 (\bar{K}+ \chi_{*} \bar{K}' \tilde{g}^{00}) \, , \\
c&=A^{-2} \tilde{c} -\frac{3}{4} M_{\rm Pl}^2 \left(\frac{d \ln (\bar{A}^{-2})}{d \tau} \right)^2 \\
M_n^4 &= \bar{A}^{-2(2-n)} {\cal M}^4 (-\chi_{*})^n \bar{K}^{(n)}\end{aligned}$$ where $\bar{K}^{(n)} \equiv \frac{d^n \bar{K}}{d \tilde{\chi}^n}$, $\chi_{*}=\bar{A}^{-2}\chi$ and the overbar denotes background quantities . We have incorporated the K-mouflage model into the EFTCAMB code [@Lewis:2002ah; @Lewis:1999bs; @2014PhRvD..89j3530H] in order to compute the CMB anisotropies in this theory. To constrain the parameters of the model, we have modified the EFTCosmoMC code [@Lewis:2002ah; @Raveri2014] to include K-mouflage.
Power Spectra {#sec:ps}
=============
We have used the code described in Sec. \[sec:eftcamb\] to determine the CMB power spectrum in the K-mouflage model, for different values of the parameters. The CMB peaks are shifting on the $l$-axis as the parameters of the models are varied. This feature of the K-mouflage model allows tight constraints to be placed on its parameters through the CMB. As discussed in Sec. \[sec:eftcamb\], the $\Lambda$CDM model is recovered when $\epsilon_2 \to 0$, independent of the values of the other parameters. In Fig. \[fig:eps\] we have plotted the temperature anisotropy power spectrum when $\epsilon_2 =-10^{-8}$ ($\Lambda$CDM limit) and $\epsilon_2 =-10^{-2}$ (limit imposed by the Solar System constraint).\
Because all models with $\epsilon_2 \to 0$ converge to the standard $\Lambda$CDM cosmology, in order to investigate the impact of modifying other parameters, we fix $\epsilon_2 = -10^{-2}$ and we we have varied them one by one. $\alpha_U$ and $\gamma_U$ do not change the temperature power spectrum, but $\gamma_A$ and $m$ do. This behaviour is illustrated in Fig. \[fig:gam\]
![Temperature power spectrum with $\Lambda$CDM parameters and K-mouflage parameters $\alpha_U=0.2$, $\gamma_U=1$, $m=3$, $\gamma_A=0.2$, and $\epsilon_2=-10^{-8}$ and $\epsilon_2=-10^{-2}$ respectively.[]{data-label="fig:eps"}](plots/TT.pdf){width="6in"}
Parameter constraints {#sec:constr}
=====================
The parameters of the K-mouflage model can be constrained by current and future CMB data. In this section we present the formalism for constraining the parameters of the model by performing Fisher Matrix forecasts, as well as a full MCMC analysis using EFTCosmoMC [@Raveri2014].
Fisher Matrix Forecasts
-----------------------
In the following paragraphs we give a brief description of the Fisher Matrix formalism [@1997ApJ...482....6S; @1997PhRvD..55.1830Z; @1997PhRvL..78.2058K; @1999ApJ...518....2E] for determining parameter forecasts from the CMB in the context of the K-mouflage model. We consider a parameter space consisting of the standard $\Lambda$CDM parameters together with the K-mouflage parameters, $$\label{eq:par}
\textbf{P}=\{ \Omega_b h^2, \Omega_c h^2, H_0, n_s, \tau, A_s\} \cup \{\alpha_U, \gamma_U, m, \epsilon_2, \gamma_A\} \, .$$ We determine the CMB power spectrum in multipole space ($C_l$’s) in the K-moulflage model with the extension to the EFTCAMB code discussed in Sec. \[sec:eftcamb\]. We consider the following temperature and polarisation channels for the power spectra: $TT$, $EE$, $TE$, $dd$, $dT$ and $dT$, where $T$ is the temperature, $E$ – the $E$-mode polarisation and $d$ – the deflection angle.
Assuming Gaussian perturbations and Gaussian noise,the Fisher matrix is then calculated as $$\label{eq:fisher}
F_{ij}=\sum_l \sum_{X,Y} \frac{\partial C_l^X}{\partial p_i} (\text{Cov}_l)_{XY}^{-1} \frac{\partial C_l^Y}{\partial p_j} \, ,$$ where the indices $i$ and $j$ span the parameter space **P** from Eq. (\[eq:par\]), $X$ and $Y$ represent the channels considered and $\text{Cov}_l$ is the covariance matrix for multipole $l$. In calculating the covariance matrix, the instrumental noise must be considered. Given the instrumental noise for the temperature and $E$-polarisation channel, the noise corresponding to the deflection angle can be determined through lensing reconstruction using the minimum variance estimator [@2002ApJ...574..566H]. The covariance matrix is discussed in detail in Ref. [@1999ApJ...518....2E], where its elements are given explicitly \[Eqs. (4)-(11)\].
Once the Fisher matrix has been computed the standard deviation on parameter $p_i$ is given by $$\label{eq:sigma}
\sigma(p_i)=\sqrt{\left(F^{-1}\right)_{ii}} \, .$$
The error bars on the parameters depend on the fiducial values considered. We consider the *Planck* values as the fiducial values to the $\Lambda$CDM parameters, while for the K-mouflage we test a few scenarios.
We consider two space probes, *Planck* and *COrE* [@2011arXiv1102.2181T] and we show that the K-mouflage models can be tightly constrained with existing CMB data from *Planck* and that the constraints can be significantly improved in the future with *COrE*, by around one order of magnitude. The noise specifications used for the two experiments are given in Table \[tab:cmbspec\].
[ c c c c c]{}\
Experiment & Frequency & $\theta_{\text{beam}}$ & $\sigma_T$ & $\sigma_P$\
Planck & & & &\
& & & &\
COrE & & & &\
& & & &\
\[tab:cmbspec\]
When considering a fiducial value of $\epsilon_2=-10^{-8}$, the other four K-mouflage parameters are almost unconstrained, and in the *Planck* scenario the $\sigma_{(\epsilon_2)} \sim 10^{-3}$. Full forecasts for the two probes are presented in Table \[tab:fc1\].
[ c c c c]{}\
Parameter & Fiducial value & $\sigma_{\text{Planck}}$ & $\sigma_{\text{COrE}}$\
$\alpha_U$ & 0.1 & 10912 & 25.18\
$\gamma_U$ & 1 & 54562 & 194\
$m$ & 3 & 5411 & 378\
$\epsilon_2$ & $-10^{-8}$ & $1.68 \times 10^{-3}$ & $1.03 \times 10^{-4}$\
$\gamma_A$ & 0.2 & 39.52 & 17.23\
$\Omega_bh^2$ & 0.0226 & $2.12 \times 10^{-4}$ & $2.58 \times 10^{-5}$\
$\Omega_ch^2$ & 0.112 & $1.48 \times 10^{-3}$ & $4.99 \times 10^{-4}$\
$H_0$ & 70 & 2.51 & 0.227\
$n_s$ & 0.96 & $5.91 \times 10^{-3}$ & $1.41 \times 10^{-3}$\
$\tau$ & 0.09 & $4.23 \times 10^{-3}$ & $1.91 \times 10^{-3}$\
$A_s$ & $2.10 \times 10^{-9}$ & $1.83 \times 10^{-11}$ & $8.30 \times 10^{-12}$\
\[tab:fc1\]
Markov-Chain-Monte-Carlo constraints
------------------------------------
The MCMC exploration of the K-mouflage model parameter space is carried out using the CosmoMC code. We use data from the CMB temperature power spectrum and CMB lensing potential power spectrum from Planck (temperature, polarization and lensing) as well as BAO measurements from SDSS DR12 and Supernovae from JLA. We place separate constraints on the K-mouflage and K-mimic models. For all these models we vary the following cosmological parameters
![Marginalised $1 \sigma$ (light colours) and $2 \sigma $ (dark colours) contours and one-dimensional posterior in the $(\gamma_{A}, \epsilon_{2,0}, m)$ parameter space. The parameters $gamma_U, alpha_U$ have been fixed to 1 and 0 respectively.[]{data-label="fig:epstest"}](plots/KM_m_eps2_gammaA.pdf "fig:"){width="3.2in"}\
Physical interpretation of K-mouflage models {#sec:interpret}
============================================
The EFT approach, discussed in previous sections, is a powerful and universal way of describing dark energy and modified gravity models. The operator expansion contains however redundant degrees of freedom. In Ref. [@Bellini2014] it has been shown that in the case of Horndeski class of actions [@1974IJTP...10..363H] (which is the most general action describing single-field models, which has a most second order derivatives in the resulting equation of motion), only four functions of time are required to fully describe linear perturbation theory. These functions are: $\alpha_K$ – kineticity, $\alpha_B$ – braiding, $\alpha_M$ – running of the Planck mass, $\alpha_T$ – tensor excess speed.
We aim to discuss the properties of the perturbations of the K-mouflage models in this general framework, by expressing Eq. \[eq:actkm\] in the Jordan frame and matching the terms to the general form $$S = \int d^4 x \sqrt{-g} \left[\sum_{i=2}^5{\cal L}_{i}+{\cal L}_{m}[g_{\mu \nu}] \right] \, ,$$ with $$\begin{aligned}
{\cal L}_{2} & = K(\varphi,\, X)\,,\\
{\cal L}_{3} & = -G_{3}(\varphi,\, X)\Box\varphi\,,\\
{\cal L}_{4} & = G_{4}(\varphi,\, X)R+G_{4X}(\varphi,\, X)\left[\left(\Box\varphi\right)^{2}-\varphi_{;\mu\nu}\varphi^{;\mu\nu}\right]\,,\\
{\cal L}_{5} & = G_{5}(\varphi,\, X)G_{\mu\nu}\varphi^{;\mu\nu}-\frac{1}{6}G_{5X}(\varphi,\, X) \times \nonumber \\
&\left[\left(\Box\varphi\right)^{3}+2{\phi_{;\mu}}^{\nu}{\varphi_{;\nu}}^{\alpha}{\varphi_{;\alpha}}^{\mu}-3\phi_{;\mu\nu}\varphi^{;\mu\nu}\Box\varphi\right]\,.\end{aligned}$$
Hence, in the K-mouflage theories, the terms appearing in the action (Eq. \[eq:actkm\]) of the Horndeski action are given by $$\begin{aligned}
K_H &= \frac{{\cal M}^4}{\bar{A}^4} K \left(\frac{\bar{A}^2 X}{{\cal M}^4} \right) +6 \tilde{M}_{\rm Pl}^2 \left(2 \frac{\bar{A}'^2}{\bar{A}^4}-\frac{\bar{A}''}{\bar{A}^3} \right) \, ,\\
G_3 &=-3 \tilde{M}_{\rm Pl}^2 \frac{\bar{A}'}{\bar{A}^3} \, ,\\
G_4 &=\frac{1}{2}\tilde{M}_{\rm Pl}^2 \frac{1}{\bar{A}^2} \, ,\\
G_5 &=0 \, ,\end{aligned}$$ where $K_H$ is the Horndeski function $K$. The variable $X$ satisfies $\tilde{\chi}=\frac{A^2}{{\cal M}^4} X$. Based on this mapping, we derive the coefficients $\alpha_i$ of Ref. [@Bellini2014] for the general K-mouflage model, $$\begin{aligned}
\alpha_K &= \frac{2X}{H^2} \left[-6 \frac{\bar{A}'^2}{\bar{A}^2}+\frac{\bar{K}'( \frac{\bar{A}^2 X}{{\cal M}^4})}{\tilde{M}_{\rm Pl}^2}+\frac{2\bar{A}^2 \bar{K}''(\frac{A^2 X}{{\cal M}^4})}{{\cal M}^4 \tilde{M}_{\rm Pl}^2}X \right] \, , \\
\alpha_B &= \frac{2}{H} \frac{\bar{A}'}{\bar{A}}\dot{\varphi} \, , \\
\alpha_M &= -\frac{2}{H} \frac{\bar{A}'}{\bar{A}}\dot{\varphi} = - \alpha_B \, , \\
\alpha_T &= 0 \, . \label{eq:alphaT}\end{aligned}$$
Using the solutions of Sec. \[sec:params\], these general functions can be expressed in terms of the explicit parametrisation of [@2016JCAP...01..020B] $$\alpha_B = 2 \epsilon_2 = - \alpha_M \, , \label{eq:alphaB2}$$ while $\alpha_K$ can be calculated using Eqs. (\[eq:dphida\]), (\[eq:eps2\]) and (\[eq:chi\]) in terms of $U$, $\epsilon_2$ and their derivatives with respect to the scale factor. Eq. (\[eq:alphaT\]) shows that gravitational waves travel at the speed of light, while Eq. (\[eq:alphaB2\]) shows that braiding is small and negative, $|\alpha_B| \lesssim {\cal O}\left(10^{-2}\right)$, while the running of the effective Planck mass is small and positive. The kineticity is not expected to significantly modify the growth of matter or of matter perturbations [@2016JCAP...07..040R] with respect to standard GR. The running of the effective Planck mass has an effect of increasing the effective gravitational constant at late times, clustering matter more than in the usual scenario and thus enhancing the matter power spectrum on scales $k \lesssim 10^{-3} h/\mathrm{Mpc}$. The negative braiding is expected to increase the Integrated Sachs-Wolfe (ISW) effect, but insignificantly as $\alpha_B$ remains small. We have checked that this is indeed the case by increasing the magnitude of $\epsilon_{2,0}$ to a value higher than the Solar System constraint limit, to $\epsilon_{2,0}=-0.3$ and calculating the CMB power spectrum. This is illustrated in Fig. \[fig:epstest\] through a comparison to a $\Lambda$CDM-like model. The plots shows a significant enhancement of the late-time ISW effect for a high negative braiding.
![Temperature power spectrum with $\Lambda$CDM parameters and K-mouflage parameters $\alpha_U=0.2$, $\gamma_U=1$, $\gamma_A=0.2$, $m=3$ and $\epsilon_{2,0}=-10^{-8}$ ($\Lambda$CDM-like) and $\epsilon_{2,0}=-0.3$. A significant enhancement of the late-time ISW effect is observed for a high magnitude of $\epsilon_{2,0}$.[]{data-label="fig:epstest"}](plots/fig_largeeps.pdf "fig:"){width="3.2in"}\
Conclusions and Outlook {#sec:concl}
=======================
In this paper we have investigated the effects of K-mouflage models on modified gravity on the CMB. We have employed a effective field theory description of these models to implement two parametrisations of K-mouflage in the EFTCAMB code in order to determine the CMB power spectra that they generate. The former is based on five parameters, where the expansion history of the Universe is free to vary, while the latter (K-mimic) has three free parameters and has a $\Lambda$CDM-like background expansion. By varying the parameters of the models, we have shown that the CMB power spectrum can be significantly modified with respect to the [*Planck*]{} measurements and therefore it is expected that the models would be tightly constrained by CMB probes. We have confirmed this by using a Fisher matrix forecast method, for [*Planck*]{} and COrE, showing that future CMB experiments would be promising in reducing the size of the parameter space for this class of models. We have implemented the model in the MCMC EFTCosmoMC code, and using a [*Planck*]{} likelihood, we have shown the allowed parameter space for the two K-mouflage models, confirming the Fisher Matrix predictions
By mapping the K-mouflage models into the Horndeski class of actions, we have shown that the gravitational waves travel at the speed of light and that the braiding and the Planck mass run rate have equal magnitudes, but opposite signs. Due to the constraints on the parameters, a negligible enhancement of the ISW with respect to $\Lambda$CDM can be expected.
This work shows that the K-mouflage models can be strongly constrained by CMB data. In the future, we plan to generalise the study of K-moulflage to assess the impact of CMB-LSS cross correlations on the constraints. In the case of the models not reproducing the $\Lambda$CDM expansion, the constraints are expected to get even tighter.
Acknowledgements
================
K-mouflage implementation in the EFTCAMB code
=============================================
In this paper we investigate cosmological perturbations at the linear level in K-mouflage scenarios using the EFTCAMB patch of the public Einstein-Boltzmann solver CAMB. For the implementation of the model in the EFTCAMB code we adopted the so called “full-mapping” approach. In these scheme the mapping relations between the K-mouflage and the EFT action, along with the cosmological and model parameters, are fed to a module that solves the cosmological background equations, for the specific theory, and outputs the time evolution of the EFT functions. These functions are then used to evolve the full perturbed Einstein-Boltzmann equations and compute cosmological observables. EFTCAMB evolves the full equations for linear perturbations without relying on any quasi-static approximation.\
For our purposes, we implemented two different versions of the model in the EFTCAMB solver. In one case the background expansion history is left free to deviate from the $\Lambda$CDM and the user has to fix both the $\bar{A}(a)$ and $K(a)$ functions. A model of K-mouflage is then completely specified by the choice of the standard cosmological parameters (in particular $H_0$, $\Omega_{m,0}$, $\Omega_{b,0}$, $n_s$, $A_s$, $\tau$) and by the five additional parameters: $\alpha_U , \ \gamma_U , \ m, \ \epsilon_{2,0} , \ \gamma_A$ introduced in Section \[sec:params\].\
Otherwise the user can switch on a flag that forces the model to reproduce exactly the $\Lambda$CDM background expansion history. In this case one has to specify, apart from the standard cosmological parameters, only the three parameters related to the background coupling function $\bar{A}(a)$, i.e. $\ m, \ \epsilon_{2,0} , \ \gamma_A$. Then the code solves Eq. (\[chi-a\]) that allows to determine the kinetic function. Once the form of $U(a)$ and $\bar{A}(a)$ has been fixed, the code computes the cosmological background expansion by solving the Klein-Gordon equation and the Friedmann equation Eq. where $\bar{\rho}_0$ is the matter density today and $\bar{\tilde{\chi}}$ is given by Eq. (2.8) of Ref. [@2016JCAP...01..020B] while $M^4$ is the mass scale of the scalar field. Once the kinetic function has been computed from the integration of the Klein-Gordon equation the code solves the Friedman equation Eq. (\[E00\]), using the standard $\Lambda$CDM solution as initial condition at $a \ll 1$. The density parameter $\Omega_{\varphi,0}$ is related to the dark energy density parameter today by the following: $$\Omega_{\varphi,0}=(1-\epsilon_{2,0})^2 \Omega_{de,0} - (2 \epsilon_{2,0} -\epsilon_{2,0}^2)(\Omega_{m,0}+\Omega_{rad,0}).$$ The mapping of the K-mouflage action in terms of EFT functions given e.g. in Eq. (6.5) of Ref. [@2016JCAP...01..020B] cannot be used directly for our purpose, since there is some difference w.r.t. the functions used by EFTCAMB, that are given in Ref. [@Hu2014].\
Comparing the K-mouflage action in unitary gauge and Jordan frame, Eq. (3.11) of Ref. [@2016JCAP...01..020B], with Eq. (1) of Ref. [@Hu2014], we can identify the following correspondences between the EFTCAMB functions and K-mouflage:
$$\begin{aligned}
\Omega(a)=& \bar{A}^{-2} -1 \\
\Omega'(a)=& -2\bar{A}^{-3} \bar{A}' \\
\Omega''(a)=& 6\bar{A}^{-4} (\bar{A}')^2 -2\bar{A}^{-3} \bar{A}'' \\
\Omega'''(a)=& -24\bar{A}^{-5} (\bar{A}')^3 +18 \bar{A}^{-4} \bar{A}' \bar{A}'' -2\bar{A}^{-3} \bar{A}'''\end{aligned}$$
$$\frac{\Lambda(a) a^2}{m_{0}^{2}}= \frac{a^2 \mathcal{M}^4 \bar{K}}{m_{0}^{2} \bar{A}^4 }-\frac{3 \mathcal{H}^2 \epsilon_{2}^2}{\bar{A}^{2} }$$
$$\begin{aligned}
&\frac{\dot{\Lambda}(a) a^2}{m_{0}^{2}} = \frac{\mathcal{H}}{m_{0}^{2} \bar{A}^5} \left(-4 a^3 M^4 \bar{A}' \bar{K}+a^3 M^4 \bar{A} \bar{\tilde{\chi}}' \frac{{\mathrm{d}}\bar{K}}{{\mathrm{d}}\bar{\tilde{\chi}}} \right. \nonumber
\\ &+ \left. 6 a m_{0}^{2} \bar{A}^2 \epsilon_{2}^{2} \mathcal{H}^{2} \bar{A}'+6 m_{0}^{2} \bar{A}^3 \epsilon_{2} \mathcal{H} \left(\epsilon_{2} \left(\mathcal{H}-a \mathcal{H}'\right)-a \mathcal{H} \epsilon_{2}'\right. \right)\end{aligned}$$
$$\frac{c(a) a^2}{m_{0}^{2}}=\frac{a^2 M^4 \bar{\tilde{\chi}} \frac{{\mathrm{d}}\bar{K}}{{\mathrm{d}}\bar{\tilde{\chi}}}}{m_{0}^{2} \bar{A}^4}-\frac{3 \epsilon_{2}^{2} \mathcal{H}^{2}}{\bar{A}^2}$$
$$\begin{aligned}
&\frac{\dot{c}(a) a^2}{m_{0}^{2}}= \frac{c'(a) \mathcal{H} a^3}{m_{0}^{2}} \nonumber \\
&=\frac{\mathcal{H} \left(-4 a^3 M^4 \bar{\tilde{\chi}} \bar{A}' \frac{{\mathrm{d}}\bar{K}}{{\mathrm{d}}\bar{\tilde{\chi}}} + a^3 M^4 \bar{A} \bar{\tilde{\chi}}' \left(\bar{\tilde{\chi}} \frac{{\mathrm{d}}^2 \bar{K}}{{\mathrm{d}}\bar{\tilde{\chi}}^2}+ \frac{{\mathrm{d}}\bar{K}}{{\mathrm{d}}\bar{\tilde{\chi}}}\right)+6 a m_{0}^2 \bar{A}^2 \epsilon_{2}^{2} \mathcal{H}^2 \bar{A}'+6 m_{0}^2 \bar{A}^3 \epsilon_{2} \mathcal{H} \left(\epsilon_{2} \left(\mathcal{H}-a \mathcal{H}'\right)-a \mathcal{H} \epsilon_{2}'\right)\right)}{m_{0}^{2} \bar{A}^5}\end{aligned}$$
$$\begin{aligned}
&\gamma_1(a)= \frac{M^{4}_{2}}{m_{0}^{2} H_{0}^{2}}= \frac{\mathcal{M}^4 A^{-4} \bar{\tilde{\chi}}^2 \frac{{\mathrm{d}}^2 \bar{K}}{{\mathrm{d}}\bar{\tilde{\chi}}^2}}{m_{0}^{2} H_{0}^{2}}\\
&\gamma_1'(a)= \gamma_1(a) \left(-4 \frac{\bar{A}'}{\bar{A}} + \frac{2 \chi'}{\chi} + \frac{ \chi' \frac{{\mathrm{d}}^3 \bar{K}}{{\mathrm{d}}\bar{\tilde{\chi}}^3}}{\frac{{\mathrm{d}}^2 \bar{K}}{{\mathrm{d}}\bar{\tilde{\chi}}^2}} \right) \\
&\frac{{\mathrm{d}}^2 \bar{K}}{{\mathrm{d}}\bar{\tilde{\chi}}^2}=\frac{6 a^3 M^4 \bar{\tilde{\chi}} \bar{A}' \frac{{\mathrm{d}}\bar{K}}{{\mathrm{d}}\bar{\tilde{\chi}}}-a^3 M^4 \bar{A} \bar{\tilde{\chi}}' \frac{{\mathrm{d}}\bar{K}}{{\mathrm{d}}\bar{\tilde{\chi}}}-6 a^2 M^4 \bar{A} \bar{\tilde{\chi}} \frac{{\mathrm{d}}\bar{K}}{{\mathrm{d}}\bar{\tilde{\chi}}}-\rho_{m,0} \bar{A}^4 \bar{A}'}{2 a^3 M^4 \bar{A} \bar{\tilde{\chi}} \bar{\tilde{\chi}}'}\end{aligned}$$
$$\begin{aligned}
\frac{{\mathrm{d}}^3 \bar{K}}{{\mathrm{d}}\bar{\tilde{\chi}}^3}&=\frac{-6 a^4 M^4 \bar{\tilde{\chi}}^2 (\bar{A}')^2 \bar{\tilde{\chi}}' \frac{{\mathrm{d}}\bar{K}}{{\mathrm{d}}\bar{\tilde{\chi}}}+6 a^4 M^4 \bar{A} \bar{\tilde{\chi}}^2 \left(\bar{A}' (\bar{\tilde{\chi}}')^2 \frac{{\mathrm{d}}^2 \bar{K}}{{\mathrm{d}}\bar{\tilde{\chi}}^2}+\frac{{\mathrm{d}}\bar{K}}{{\mathrm{d}}\bar{\tilde{\chi}}} \left(\bar{A}'' \bar{\tilde{\chi}}'-\bar{A}' \bar{\tilde{\chi}}''\right)\right)}{2 a^4 M^4 \bar{A}^2 \bar{\tilde{\chi}}^2 (\bar{\tilde{\chi}}')^2}+ \nonumber \\
&+\frac{a^2 M^4 \bar{A}^2 \left(\frac{{\mathrm{d}}\bar{K}}{{\mathrm{d}}\bar{\tilde{\chi}}} \left(a^2 (\bar{\tilde{\chi}}')^3+6 (\bar{\tilde{\chi}}')^2 \left(a \bar{\tilde{\chi}}''+\bar{\tilde{\chi}}'\right)\right)-a \bar{\tilde{\chi}} (\bar{\tilde{\chi}}')^2 \left(a \bar{\tilde{\chi}}'+6 \bar{\tilde{\chi}}\right)\frac{{\mathrm{d}}^2 \bar{K}}{{\mathrm{d}}\bar{\tilde{\chi}}^2}\right)}{2 a^4 M^4 \bar{A}^2 \bar{\tilde{\chi}}^2 (\bar{\tilde{\chi}}')^2} \nonumber \\
&-\frac{3 a \rho_{m,0} \bar{A}^4 \bar{\tilde{\chi}} (\bar{A}')^2 \bar{\tilde{\chi}}'+\rho_{m,0} \bar{A}^5 \left(a \bar{\tilde{\chi}} \bar{A}' \bar{\tilde{\chi}}''+\bar{\tilde{\chi}}' \left(\bar{A}' \left(a \bar{\tilde{\chi}}'+3 \bar{\tilde{\chi}}\right)-a \bar{\tilde{\chi}} \bar{A}''\right)\right)}{2 a^4 M^4 \bar{A}^2 \bar{\tilde{\chi}}^2 (\bar{\tilde{\chi}}')^2}\end{aligned}$$
The overdot represents derivation with respect to conformal time while the prime represents derivation with respect to the scale factor $a$ and $\mathcal{H}(a)=a H(a)$.
|
---
abstract: 'Power spectrum densities for the number of tick quotes per minute (market activity) on three currency markets (USD/JPY, EUR/USD, and JPY/EUR) for periods from January 1999 to December 2000 are analyzed. We find some peaks on the power spectrum densities at a few minutes. We develop the double-threshold agent model and confirm that stochastic resonance occurs for the market activity of this model. We propose a hypothesis that the periodicities found on the power spectrum densities can be observed due to stochastic resonance.'
author:
- |
Aki-Hiro Sato\
Department of Applied Mathematics and Physics,\
Graduate School of Informatics, Kyoto University,\
Kyoto 606-8501, Japan
title: 'Characteristic time scales of tick quotes on foreign currency markets: an empirical study and agent-based model'
---
tick quotes, foreign currency market, power spectrum density, double-threshold agent model, stochastic resonance\
[**PACS**]{} 89.65.Gh, 87.15.Ya, 02.50.-r
Introduction {#sec:introduction}
============
In the past few years there has been increasing interest in the investigation of financial markets as complex systems in statistical mechanics. The empirical analysis of the high frequency financial data reveals nonstationary statistics of market fluctuations and several mathematical models of markets based on the concept of the nonequilibrium phenomena have been proposed [@Mantegna:00; @Dacrogna:00].
Recently Mizuno [*et al.*]{} investigate high frequency data of the USD/JPY exchange market and conclude that dealers’ perception and decision are mainly based on the latest 2 minutes data [@Mizuno]. This result means that there are feedback loops of information in the foreign currency market.
As microscopic models of financial markets some agent models are proposed [@Aki:98; @Lux:99; @Challet:01; @Jefferies:01]. Specifically Ising-like models are familiar to statistical physicists and have been examined in the context of econophysics. The analogy to the paramagnetic-ferromagnetic transition is used to explain crashes and bubbles. Krawiecki [*et al.*]{} consider the effect of a weak external force acting on the agents in the Ising model of the financial market and conclude that apparently weak stimuli from outside can have potentially effect on financial market due to stochastic resonance [@Krawiecki]. This conclusion indicates that it is possible to observe the effect of the external stimuli from the market fluctuations.
Motivated by their conclusion we investigate high-frequency financial data and find a potential evidence that stochastic resonance occurs in financial markets. In this article the results of data analysis are reported and the agent-based model is proposed in order to explain this phenomenon.
Analysis {#sec:analysis}
========
We analyze tick quotes on three foreign currency markets (USD/JPY, EUR/USD, and JPY/EUR) for periods from January 1999 to December 2000 [@CQG]. This database contains time stamps, prices, and identifiers of either ask or bid. Since generally market participants (dealers) must indicate both ask and bid prices in foreign currency markets the nearly same number of ask and bid offering are recorded in the database. Here we focus on the ask offering and regard the number of ask quotes per unit time (one minute) $A(t)$ as the market activity. The reason why we define the number of ask quotes as the market activity is because this quantity represents amount of dealers’ responses to the market.
In order to elucidate temporal structure of the market activity power spectrum densities of $A(t)$, estimated by $$S(f) =
\frac{1}{2\pi}\lim_{T\rightarrow\infty}\frac{1}{T}
\Bigl\langle|\int_{0}^{T}A(\tau)e^{-2\pi i f \tau}d\tau|^2 \Bigr\rangle,$$ where $f$ represents frequency, and $T$ a maximum period of the power spectrum density, are calculated. Figs. \[fig:power-spectrum-usdjpy\], \[fig:power-spectrum-eurusd\], and \[fig:power-spectrum-eurjpy\] show the power spectrum densities for three foreign currency markets (USD/JPY, UER/USD, and EUR/JPY) from January 1999 to December 2000. It is found that they have some peaks at the high frequency region.
There is a peak at 2.5 minutes on the USD/JPY market, at 3 minutes on the EUR/USD market, and there are some peaks on the JPY/EUR. We confirm that these peaks appear and disappear depending on observation periods. On the USD/JPY market there is the peak for periods of January 1999–July 1999, March 2000–April 2000, and August 2000–November 2000; on the EUR/USD market July 1999–September 1999; and on the EUR/JPY market January 1999–March 1999, April 1999–June 1999, November 1999, and July 2000–December 2000.
These peaks mean that market participants offer quotes periodically and in synchronization. The possible reasons for these peaks to appear in the power spectrum densities of the market activity are follows:
1. The market participants are affected by common periodic information.
2. The market participants are spontaneously synchronized.
In the next section the double-threshold agent model is introduced and explain this phenomenon on the basis of the reason (1).
![Semi-log plots of power spectrum densities for time series of the number of ask quotes per minute on the USD/JPY market on 1999 (left) and 2000 (right). These power spectrum densities are estimated by averaging power spectrum densities for intraday time series of the number of ask quotes per minute over day for each year.[]{data-label="fig:power-spectrum-usdjpy"}](SatoFig1a.ps "fig:") ![Semi-log plots of power spectrum densities for time series of the number of ask quotes per minute on the USD/JPY market on 1999 (left) and 2000 (right). These power spectrum densities are estimated by averaging power spectrum densities for intraday time series of the number of ask quotes per minute over day for each year.[]{data-label="fig:power-spectrum-usdjpy"}](SatoFig1b.ps "fig:")
![Semi-log plots of power spectrum densities for time series of the number of ask quotes per minute on the EUR/USD market on 1999 (left) and 2000 (right).[]{data-label="fig:power-spectrum-eurusd"}](SatoFig2a.ps "fig:") ![Semi-log plots of power spectrum densities for time series of the number of ask quotes per minute on the EUR/USD market on 1999 (left) and 2000 (right).[]{data-label="fig:power-spectrum-eurusd"}](SatoFig2b.ps "fig:")
![Semi-log plots of power spectrum densities for time series of the number of ask quotes per minute on the EUR/JPY market on 1999 (left) and 2000 (right).[]{data-label="fig:power-spectrum-eurjpy"}](SatoFig3a.ps "fig:") ![Semi-log plots of power spectrum densities for time series of the number of ask quotes per minute on the EUR/JPY market on 1999 (left) and 2000 (right).[]{data-label="fig:power-spectrum-eurjpy"}](SatoFig3b.ps "fig:")
Double-threshold agent model {#sec:model}
============================
Here we consider a microscopic model for financial markets in order to explain the dependency of the peak height on observation periods. We develop the double-threshold agent model based on the threshold dynamics.
In foreign exchange markets the market participants attend the markets with utilizing electrical communication devices, for example, telephones, telegrams, and computer networks. They are driven by both exogenous and endogenous information and determine their investment attitudes. Since the information to motivate buying and one of selling are opposite to each other we assume that the information is a scaler variable. Moreover the market participants perceive the information and determine their investment attitude based on the information. The simplest model of the market participant is an element with threshold dynamics.
We consider a financial market consisting of $N$ market participants having three kinds of investment attitudes: buying, selling, and doing nothing. Recently we developed an array of double-threshold noisy devices with a global feedback [@Sato:05]. Applying this model to the financial market we construct three decision model with double thresholds and investigate the dependency of market behavior on an exogenous stimuli.
The investment attitude of the $i$th dealer $y_i(t)$ at time $t$ is determined by his/her recognition for the circumstances $x_i(t) = s(t) + z_i(t)$, where $s(t)$ represents the investment environment, and $z_i(t)$ the $i$th dealer’s prediction from historical market data. $y_i(t)$ is given by $$y_i(t) =
\left\{
\begin{array}{lll}
1 & (x_i(t)+\xi_i(t) \geq B_i(t)) & : \mbox{\it buy}\\
0 & (B_i(t) > x_i(t)+\xi_i(t) > S_i(t)) & : \mbox{\it inactive}\\
-1 & (x_i(t)+\xi_i(t) \leq S_i(t)) & : \mbox{\it sell}
\end{array}
\right.,$$ where $B_i(t)$ and $S_i(t)$ represent threshold values to determine buying attitude and selling attitude at time $t$, respectively. $\xi_i(t)$ is the uncertainty of the $i$th dealer’s decision-making. For simplicity it is assumed to be sampled from an identical and independent Gaussian distribution, $$p(\xi_i) =
\frac{1}{\sqrt{2\pi}\sigma_{\xi}}\exp\Bigl(-\frac{\xi_i^2}{2\sigma_{\xi}^2}\Bigr),$$ where $\sigma_{\xi} (>0)$ is a standard deviation of $\xi_i(t)$. Of course this assumption can be weakened. Namely we can extend the uncertainty in the case of non-Gaussian noises and even correlated noises.
The excess demand is given by the sum of investment attitudes over the market participants, $$r(t) = N^{-1}\sum_{i=1}^{N}y_i(t),$$ which can be an order parameter. Furthermore the market price $P(t)$ moves to the direction to the excess demand $$\ln P(t+\Delta t) = \ln P(t) + \gamma r(t),$$ where $\gamma$ represents a liquidity constant and $\Delta t$ is a sampling period. $r(t)$ may be regarded as an order parameter.
The dealers determine their investment attitude based on exogenous factors (fundamentals) and endogenous factors (market price changes). Generally speaking, the prediction of the $i$th dealer $z_i(t)$ is determined by a complicated strategy described as a function with respect to historical market prices, $F_i(s,P(t),P(t-\Delta t),\ldots)$. Following the Takayasu’s first order approximation [@Takayasu:99] we assume that $z_i(t)$ is given by $$z_i(t) = a_i(t)(\ln P(t) - \ln P(t-\Delta t)) = \gamma a_i(t) r(t-\Delta
t),$$ where $a_i(t)$ is the $i$th dealer’s response to the market price changes.
It is assumed that the dealers’ response can be separated by common and individual factors, $$a_i(t) = \zeta(t) + \eta_i(t),$$ where $\zeta(t)$ denotes the common factor, and $\eta_i(t)$ the individual factor. Generally these factors are time-dependent and seem to be complicated functions of both exogenous and endogenous variables.
For simplicity it is assumed that these factors vary rapidly in the limit manner. Then this model becomes well-defined in the stochastic manner. We assume that $\zeta(t)$ and $\eta_i(t)$ are sampled from the following identical and independent Gaussian distributions, respectively: $$\begin{aligned}
P_{\zeta}(\zeta) &=&
\frac{1}{\sqrt{2\pi}\sigma_{\zeta}}\exp\Bigl(-\frac{(\zeta-a)^2}{2\sigma_{\zeta}^2}\Bigr),
\\
P_{\eta}(\eta_i) &=& \frac{1}{\sqrt{2\pi}\sigma_{\eta}}\exp\Bigl(-\frac{\eta_i^2}{2\sigma_{\eta}^2}\Bigr),\end{aligned}$$ where $a$ represents a mean of $\zeta(t)$, $\sigma_{\zeta} (> 0)$ a standard deviation of $\zeta(t)$, and $\sigma_{\eta}$ a standard deviation of $\eta (> 0)$.
Since we regard the market activity as the number of tick quotes per unit time it should be defined as the sum of dealers’ actions: $$q(t) = \frac{1}{N}\sum_{i=1}^N |y_i(t)|.$$ The market activity $q(t)$ may be regarded as an order parameter.
Numerical Simulation {#sec:simulation}
====================
This agent model has nine model parameters. We fix $N=100$, $B_i=0.01$, $S_i=-0.01$, $\gamma=0.1$, $\sigma_{\eta}=0.01$, and $a=0.0$ throughout all numerical simulations. It is assumed that an exogenous periodic information to the market is subject to $s(t)=q_0 \sin(2\pi \Delta t f
t)$ at $q_0=0.001$, $f=0.8$ and $\Delta t = 1$.
We calculate the signal-to-noise ratio (SNR) of the market activity as a function of $\sigma_{\xi}$. The SNR is defined as $$SNR = \log_{10}\frac{S}{N},$$ where $S$ represents a peak height of the power spectrum density, and $N$ noise level.
From the numerical simulation we find non-monotonic dependency of the SNR of $q(t)$ on $\sigma_{\xi}$. Fig. \[fig:SNR\] shows a relation between the SNR and the noise strength $\sigma_{\xi}$. It has an extremal value around $\sigma_{\xi}=0.0035$. Namely the uncertainty of decision-making plays a constructive role to enhance information transmission. If there are exogenous periodic information and the uncertainty of decision-making we can find the peak on power spectrum densities at appropriate uncertainty of decision-making due to stochastic resonance [@Gammaitoni].
![Signal-to-noise ratio (SNR) obtained from power spectrum densities of $q(t)$ for the double-threshold agent model is plotted against the uncertainty of decision-making of agents at $N=100$, $B_i=0.01$, $S_i=-0.01$, $\gamma=0.1$, $\sigma_{\eta} = 0.01$, $a=0.0$, $\sigma_{\zeta}=0.3$, $q_0=0.001$, $f=0.8$, and $\Delta t = 1$.[]{data-label="fig:SNR"}](SatoFig4.ps)
Conclusion {#sec:conclusion}
==========
We analyzed time series of the number of tick quotes (market activity) and found there are short-time periodicities in the time series. The existence and positions of these peaks of the power spectrum densities depend on foreign currency markets and observation periods. The power spectrum densities have a peak at 2.5 minutes on the USD/JPY market, 3 minutes on the EUR/USD. There are some peaks at a few minutes on the JPY/EUR.
We developed the double-threshold agent model for financial markets where the agents choose three kinds of states and have feedback strategies to determine their decision affected by last price changes. From the numerical simulation we confirmed that the information transmission is enhanced due to stochastic resonance related to the uncertainty of decision-making of the market participants. We propose a hypothesis that the periodicities of the market activity can be observed due to stochastic resonance.
Appearance and disappearance of these peaks may be related to the efficiency of the markets. The efficiency market hypothesis [@Fama:91] says that prices reflect information. Because quotes make prices tick frequency can reflect information. If the peaks of the power spectrum densities come from exogenous information then SNR is related to the efficiency of the market. Namely the market may be efficient when the peaks appear.
The author thanks Prof. Dr. T. Munakata for stimulative discussions and useful comments. This work is partially supported by the Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research \# 17760067.
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|
---
author:
- 'C.A.Caretta, J.M.Islas-Islas, J.P.Torres-Papaqui, R.Coziol, H.Bravo-Alfaro'
- 'H.Andernach'
title: Galaxy activity influenced by the environment in the cluster of galaxies Abell 85
---
Introduction
============
It is widely accepted that the environment has strong influence on the formation and evolution of galaxies. One of the most established examples is the morphology-density relation [e.g., @Dre80]. The question then is: which are the mechanisms that cause this influence, and specifically, how do these mechanisms affect the activity of the galaxies – probably one of the most important characteristics related to galaxy evolution? Here we present some results on a case study, the cluster Abell 85, of the influence of the environment on the star formation (SF) and AGN activity of its member galaxies.
The Cluster Abell 85
====================
The cluster of galaxies A85 is a moderately rich (R = 1), relatively nearby (z = 0.055) cluster. It has been extensively studied over all the electromagnetic spectrum, especially in the radio, optical and X-rays. Many works have revealed a dynamically young system undergoing the merging of smaller clumps of galaxies. In Fig. \[Fdelt\] we show the results of the application of the $\Delta$-test [@DeS88] to the 3D distribution of 367 member galaxies in A85 \[projected positions from SuperCOSMOS data [@Ham01] and radial velocities from an updated version of the compilation by @And05\]. This test searches for deviations of the local kinematics (based on the 10 nearest neighbors for each galaxy) and the global one (called $\delta$). We identify five substructures in this cluster based on the concentration of galaxies with deviant kinematics (displayed in magenta in Fig. \[Fdelt\]):
[**C2**]{}: this substructure is composed of two clumps: the northern one, which is centered on the 2$^{nd}$-brightest galaxy; and the southern one, identified in a Chandra X-ray image [@Kem02];
[**SB**]{} (South Blob): identified originally from X-ray imaging [@Kem02]; its kinematics is not particularly deviant but it represents the second highest density peak of the cluster;
[**F**]{}: X-ray filament: also identified in X-rays [@Dur98];
[**SE**]{}: this substructure is projected in the direction of the background cluster A87 (at z = 0.130);
[**W**]{}: probably a clump that started interaction with A85 most recently.
Details of this analysis are presented in (Bravo-Alfaro et al. 2008, in prep.).
Classification of Galaxies
==========================
For the analysis of activity (SF and AGN) we used an homogeneous sample of 232 spectra of A85 galaxy members (63% of known members) available from SDSS [e.g., @Ade07]. Using the <span style="font-variant:small-caps;">starlight</span> routine [@Cid05] we obtained the population synthesis of these spectra. Subtracting the respective synthetic stellar spectrum from each observed one we could fit and measure the emission lines. Only lines with S/N $\ge 3$ were used. Two diagnostic diagrams [@Coz98; @BPT81] and an exhautive visual inspection were used in order to separate the emission line galaxies into HII and AGN domains (Figs. \[dCoz\] and \[BPT\]). Our results show that 17% of A85 members are HII, 17% are LLAGNs, 5% are LINERs and 2.6% are high-luminosity AGNs, the remaining being galaxies without emission-lines (noEL).
This AGN fraction in a cluster is much higher than previously reported, especially before the era of large spectroscopic surveys, when the estimates showed a decrease of the number of AGNs in clusters with respect to the field [e.g., @Gis78]. A recent X-ray survey of A85 [@Siv08] detected 4 high luminosity AGNs in a sample of 170 member galaxies of A85 (2.4%), in accordance with our results.
Environmental Effects
=====================
We correlate the distribution of our sample of galaxies spectroscopically classified with the substructures in A85. In Fig. \[Fclas2\] one can see that the noEL present a fairly homogeneous distribution, concentrated toward the center as expected. On the other hand, the HII galaxies show some concentration in the SE quadrant. This is the region comprising the majority of the substructures found in A85, namely the SE, the X-ray filament and the SB. Since this ridge of clumps is poblably the most dynamically active part of the cluster in recent times, representing a sequence of future merging structures to the main body of the cluster, we are lead to conclude that there is some stimulation of SF activity by this dynamical status. More than that, the AGN activity (Fig. \[Fclas3\]), both the low energy phenomenon (LLAGNs and LINERs) and high energy one (Sy2), present a more enhanced concentration around the ridge. All of the Sy2 galaxies are located on the eastern side of A85, and two of them are members of the SE substructure. This relative overdensity of active galaxies inside the substructure SE suggests that activity is favored inside substructures that are beginning to interact with the cluster. We have also started to study the complete SF history of our sample galaxies using <span style="font-variant:small-caps;">starlight</span>, which will be presented elsewhere.
Conclusions
===========
The main results of this work are: (a) we found a very large number of LLAGNs [@Ho93] in A85, with a similar abundance to the one found in compact groups of galaxies [@Coz98]; this suggests the LLAGN phenomenon is very common in dense environments; (b) there is some evidence of enhanced activity, both SF and AGN, in substructures in the early stage on their process of merging with the cluster; (c) we suggest that the distribution of active galaxies may be used to search for the presence of subtructures in clusters of galaxies.
We are grateful to CONACyT and Universidad de Guanajuato for supporting this project. We also acknowledge the use of SDSS data (the SDSS Web Site is http://www.sdss.org/).
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|
---
abstract: |
In an investigation of the cause of the cosmic microwave background decrement in the field of the $z = 3.8$ quasar pair PC1643+4631, we have carried out a study to photometrically estimate the redshifts of galaxies in deep multi-colour optical images of the field taken with the WHT.
To examine the possibility that a massive cluster of galaxies lies in the field, we have attempted to recover simulated galaxies with intrinsic colours matching those of the model galaxies used in the photometric redshift estimation. We find that when such model galaxies are added to our images, there is considerable scatter of the recovered galaxy redshifts away from the model value; this scatter is larger than that expected from photometric errors and is the result of confusion, simply due to ground-based seeing, between objects in the field.
We have also compared the likely efficiency of the photometric redshift technique against the colour criteria used to select $z
\gtrsim 3$ galaxies via the strong colour signature of the Lyman-limit break. We find that these techniques may significantly underestimate the true surface density of $z \gtrsim 3$, due to confusion between the high–redshift galaxies and other objects near the line of sight. We argue that the actual surface density of $z\approx3$ galaxies may be as much as 6 times greater than that estimated by previous ground-based studies, and note that this conclusion is consistent with the surface density of high–redshift objects found in the HDF. Finally, we conclude that all ground–based deep field surveys are inevitably affected by confusion, and note that reducing the effective seeing in ground–based images will be of paramount importance in observing the distant universe.
author:
- |
Garret Cotter$^{\! 1}$, Toby Haynes$^{\! 1}$,\
Joanne C. Baker$^{\! 2}$, Michael E. Jones$^{\! 1}$, Richard Saunders$^{\! 1}$\
\
$^1$ Astrophysics, Cavendish Laboratory,\
Madingley Road, Cambridge, CB3 0HE, UK\
$^2$ Astronomy Department, University of California,\
Berkeley CA 94720, USA
bibliography:
- 'general.bib'
nocite: '[@HCB98]'
title: |
Deep optical imaging of the field of PC1643+4631A&B,\
II: Estimating the colours and redshifts of faint galaxies
---
Introduction {#sec:print}
============
The deep five–colour photometry (Paper I) of PC1643+4631 allow an attempt to estimate the redshifts of objects in the field. To make the best use of our images, we must have an equivalent set of colours of galaxies derived from either observations or models. Composite galaxy spectra, such as those compiled by [@CWW] (hereafter known as CWW), can be convolved with the telescope transmission functions through each filter so that the colours of such a galaxy at various redshifts can be modelled, providing us with a “no–evolution mode” set of colours. The original CWW spectra cover wavelengths from 1500Å to 10000Å, while the filters effectively sample from 3000Å to 9000Å, so such an approach would provide model colours for $z<1$. Extending these spectra further into the UV allows models for higher redshifts to be calculated and this can be achieved by careful matching of UV spectra, eg from the [@KBC93] atlas, with the blue end of the optical spectra. The different morphological types of galaxies must also be taken into account – spiral galaxies tend, for example, to be considerably bluer than ellipticals – so different sample spectra must be compiled for each morphological group.
The main restriction encountered with these composite spectra is the difficulty in adapting these spectra to account for evolution. This is not really a problem at low redshifts, but to extend these models back to the possible formation redshifts at $z_f=5$ one must include the evolution of these galaxies. The best one can do at present is to use the existing models, in which typically there is a burst of star formation at some formation redshift, $z_f$, followed by exponentially decreasing star-formation, and calculate the consequent colour evolution.
Modelling the galaxy colours over [$0<z<4$]{}
=============================================
The composite spectra from actual galaxies provide the most representative colours for the low redshift galaxies in the field (see, for example [@SG98]). At higher redshifts there are significant differences between the colours derived from empirical and simulated (in this case using the Bruzual & Charlot (B&C hereafter, [@BC93]) code) spectra: this is the result of including evolution in the simulated spectra. In order to achieve a consistent set of colours for each morphological type for redshifts between 0 and 4, one cannot simply rely on either the B&C simulations or the CWW spectra – the former do not match the low–redshift galaxy spectra so well, while the latter do not include any evolution and can not, in any case, be used beyond $z\approx1.5$ since the UV end of the spectrum starts to move out of the $U$ filter at this redshift.
Colours derived from both sets of galaxy spectra are shown in Figure \[fig:galcomp\]. Note that all the CWW spectra are much redder in both $U-G$ and $G-R$ at $z=1.5$ than the B&C spectra. At this redshift, the star formation rate in the B&C spectra is producing considerably more flux in the UV than is seen in local galaxies, consistent with the apparent star formation history (see, eg. [@MPD98]). The result is that across the optical observing band (3000Å – 8500Å), the spectra of star–forming galaxies are close to flat spectrum for $1.2<z<2.3$. The success of the Lyman–limit imaging searches for $z\gtrsim3$ galaxies using B&C–derived spectra indicates that at least some galaxies match these simulated spectra at high redshift.
The approach adopted by Gwyn ([@SG98]) has changed since his 1996 paper ([@GH96]) on photometric redshifts in the Hubble Deep Field (HDF). Previously, he relied on using spectra derived entirely from B&C simulations. Now, where HDF galaxies are found to be at $z\le1.5$ using the B&C spectra, he derives new photometric redshifts from colours based on the CWW spectra and reports improved accuracy when compared against spectroscopic redshifts. However, this results in a large hiatus between the colours of a $z=1.6$, B&C–derived galaxy and a $z=1.5$, CWW–derived galaxy, which is unsatisfactory, as can be seen in Figure \[fig:galcomp\].
To attempt to make use of the CWW spectra to more accurately estimate the colours of low–redshift galaxies, we have used the following approach:\
-------------------- -------------------------------------------------------
for $z\le0.5$, galaxy colours are derived only from the
CWW empirical spectra;
for $0.5<z\le1.5$, galaxy colours are interpolated smoothly between the
two sets of models, with colours at $z\sim0.6$ being
closest to the CWW galaxy colours, and the colours at
$z\sim1.5$ being closest to the B&C colours;
for $z>1.5$, galaxy colours are derived only from the
B&C simulated spectra.
-------------------- -------------------------------------------------------
The final galaxy models used are illustrated in Figure \[fig:finalcols\]. The CWW spectra are over-plotted to show the differences between the models.
Four morphological types were used, corresponding to E/S0, Sbc, Scd and Irr, and spectra were produced at redshifts from 0.0 to 4.0 at 0.1 intervals. To cover the colour space more completely, these four models were interpolated linearly between the model types and in redshift, to create 13 models at redshift intervals of 0.025.
Matching the model spectra against the observations
===================================================
Since the number of counts received even in a small faint aperture is large due to the plates being background limited, one can treat all errors as gaussian about the mean flux. Fluxes (unnormalised) for the model galaxies are derived from the model spectra by convolution with the telescope response for each filter, including atmospheric absorption and the effect of scattering within the telescope light path.
To compare the models against the observed fluxes, we used a $\chi^2$ test, with
$$\chi^{2} = \sum_i \Big(\frac {f\!_i - \alpha m_i}{\sigma_i}\Big)^2,$$
where $f_i$ is the observed flux in filter $i$, $\sigma_i$ is the error in observed flux in filter $i$, and $m_i$ is the model flux in filter $i$. To obtain the optimum fit between the model and the observed fluxes, we minimised $\chi^2$ with respect to $\alpha$.
However, given the range of models available, it is important to determine the range of best–fit models which are consistent with the observations, and for this we used a relative likelihood method. The likelihood of obtaining an observed set of fluxes $f_i$ with errors $\sigma_i$ from model fluxes $m_i$ (where $m_i$ is a function of redshift, $z$, and morphological type, T) is
$$L(z,T) = \prod_i \exp \bigg[ -\frac{1}{2}\Big(\frac {f\!_i - \alpha
m_i(z,T)}{\sigma_i}\Big)^2 \bigg].$$
This assumes that the errors in the observed fluxes are described by a Gaussian distribution – a good assumption. From this likelihood distribution, the modal redshift can be determined by identifying the model with the highest likelihood.
Since one can apportion a probability to each model, one can use this to determine not only the most likely model, but also the mean redshift and the range of acceptable redshifts which are consistent with the observations within some relative confidence level. This is important, since for any set of models the distribution of likely models may not be mono–modal. We can use the difference between the mean and modal redshifts as a simple discriminator to identify cases where the redshift is not well constrained.
Redshift distribution
=====================
Overview {#sec:over}
--------
The two redshift distribution histograms, based on mean and most probable redshift, are plotted in Figure \[fig:zhist\]. The main features of both these diagrams are the two peaks: one at $0<z<0.6$, and the other at $z\approx2.4$, with only a small number of objects lying in the range $1.0<z<2.2$. This is reinforced by studying the distributions of objects in space against redshift – these are plotted in Figure \[fig:zdist\].
It is clear that the apparent lack of objects in this redshift range is not a real effect – there cannot really be so few objects in this region. Identifying the cause of this effect is important for understanding the limitations of the photometric redshift technique.
From the model colour tracks plotted against the actual objects colours (Figure \[fig:uggrcomp\]), it is clear that the close–packing of the model colours in the redshift range $1.0<z<2.1$ is a problem. This arises because the continuum of the galaxy spectra is almost flat for all spectral types at wavelengths between 1400Åand 2500Å. Only the $R-I$ colours provide any redshift sensitivity for objects with $1.0<z<1.5$. The model colours effectively act as an ’attractor’ for objects with similar colours – ’attracted’ objects are labelled with the redshift and type of the model they end up with. Thus closely spaced models are less effective at distinguishing objects, unless there are significantly more objects concentrated in that area. Even then, the similarity of the objects is sufficient to make the accurate determination of the redshift impossible in this redshift range given these filters. The result is that objects with redshifts $1.0<z<2.1$ end up being pushed to lower ($z\approx0.8$) or higher ($z\approx2.4$) estimated redshifts.
Beyond the limitations imposed by the filter selection, it is also notable that the number of objects that are close to the models SED colours is small in the redshift range $1.0<z<2.0$. This suggests that the model colours being applied here do not directly correspond to observed colours in the field. This raises several possibilities:
(1) the model colours are inaccurate;
(2) there are significant systematic errors in the catalogue calibration;
(3) the model colours are correct, but the observed colours differ due to reddening caused by the presence of dust in these objects.
The accuracy of a catalogue’s photometric measurements can be qualified by examining the accuracy of the photometric calibration, both as determined from the original calibrators and in comparison with other published results. All the calibrators used in the creation of the catalogue have high signal–to–noise, and even including the possibility of poor background level determination, the calibrations are at worst inaccurate by $\pm 0.05$ magnitudes in all filters. This is not sufficient to bias the results as dramatically as observed here. In comparison with other published photometric measurements for the field, such as Hu & Ridgway’s multi–filter imaging around Quasar A ([@HuR94]), the results are all consistent within the photometric errors.
Since the model colours here are based on unreddened spectra, it seems reasonable to suggest that the lack of objects with such blue colours is due to reddening of these objects by dust. This would have the greatest effect in the $U$ & $G$ colours, since these are the ones with the shortest wavelengths. If we examine the spread of colours of the objects in $G-R$ and $R-I$ (Figure \[fig:grricomp\]) we can see there is better coverage of the model tracks in $R-I$ than in $G-R$. The objects at $R-I<-0.5$ are all within 2-$\sigma$ of $R-I=0$; ie this is most probably photometric scatter. This is particularly noticeable in $R-I$ because of the lower depth of the $I$ image compared with the rest of the broadband colours.
Most likely or mean redshift?
-----------------------------
Comparing the two estimates shows several interesting features, as illustrated in Figure \[fig:modeormeanz\]. In 1486 out of 1665 objects with $R<26.0$ , there is less than 0.1 difference between the modal and mean redshift. 63 objects differ by 0.1 – 0.2 in redshift, and 116 differ by more than 0.2. For those objects whose redshift determinations differ by greater than 0.2, these divide into three major groups: those which have a modal redshift $0<\textrm{modal~}z<0.6$ but have a large mean redshift, a set with $2.5<\textrm{modal~}z<3.0$ but a mean redshift of $1.8<\textrm{mean~}z<2.7$, and a set with $\textrm{modal~}z\approx4.0$ but a lower mean redshift.
In Figure \[fig:r.vs.zdiff\], we plot the precision of the redshift estimates as a function of $R$ magnitude. The objects which show the greatest differences are all faint ($R>24.0$); the $\chi^2$ fitting is looser due to the greater flux uncertainties. This in turn leads to more models sharing similar model–fitting likelihoods and a greater spread of redshifts with which the models are statistically acceptable. Examination of the precision of the redshift estimates against the modal redshift (Figure \[fig:modezerr\]) shows that the majority of objects with $0<\textrm{modal~}z<0.8$ have small standard deviations, while the fitting is generally less constrained at higher redshifts ($z>2.0$) although the average standard deviations even at $z\approx3$ is still $\Delta z\approx 0.1$.
The low–redshift group ($0<\textrm{modal~}z<0.6$) consists of 102 objects, all of which have $24<R<26$. Some examples of the object spectra and redshift likelihood distributions are shown in Figure \[fig:lowzconf\], and an explanation of the information on these graphs precedes in Figure \[fig:graphdescr\]
There appear to be two groups of objects which show significant differences between mean and modal redshift: faint blue galaxies which, by the nature of their flat and relatively featureless continuum, are difficult to accurately determine the redshift for without good photometric constraints; and objects with weak constraints on the $U$ magnitude which can mimic both $z\approx0$ galaxies and spirals at $z\approx1.0$.
The intermediate ($2.5<\textrm{modal~}z<3.0$) and high ($3.8<\textrm{modal~}z<4.0$) redshift galaxies also show a fairly flat spectrum in $G$,$V$,$R$ & $I$, and red $U-G>$ colours (see Figures \[fig:medzconf\] & \[fig:highzconf\] for examples). Again this ambiguity between a low redshift E/S0/Sa galaxy with a strong 4000Å break and a high–redshift starforming galaxy with absorption due to the Ly$-\alpha$ forest and extinction below 912Å causes problems for the photometric redshift models. $J$, $H$ and other IR imaging would break this degeneracy, as would significantly deeper $U$ imaging.
In summary, the modal redshift appears to give the better redshift determination. When ambiguities arise in the determination of the redshift, the standard deviation of the redshift is sufficiently sensitive to mark the problem cases.
Low–redshift galaxies – $z<1.0$
-------------------------------
The accuracy of the photometric redshifts is likely to be best in this sub–sample, especially in the range $0<z<0.5$. In a blind test carried out by Hogg et al, the rms deviation of the photometric redshift from the spectroscopic determination was routinely $\Delta z<0.1$, using similar methods and a similar range of wavelengths covered by the filters used here.
Examples of redshift estimations are shown in Figures \[fig:lowzES0\] – \[fig:lowzIrr\]. These also include estimations of galaxy morphological type, made possible by the wide spread of colour space covered by the photometric models at low redshift.
A few spectroscopic redshifts for some of the galaxies in this field are already available, obtained by one of us with the WHT allowing one to check the accuracy of the photometric predictions. Table \[tab:compsp\] shows that the photometric redshifts are accurate to $\Delta z\approx0.15$, although the systematic offset suggests that the models can be further improved. A similar overestimate is seen in [@LYF96] where the photometric models are based on redshifted CWW spectra alone, where the rms scatter was reported as $\Delta z\approx0.15$. Comparing the published redshift for HR10 ([@GD96]), it is clear that the reddening of the spectrum in this $z>1$ galaxy has resulted in an erroneous estimate.
Intermediate–redshift galaxies – $1.0<z<2.5$ {#sec:intz}
--------------------------------------------
As discussed in section \[sec:over\] there are too few objects identified in the range $1.0<z<2.5$. It is important to determine what redshifts objects in this redshift range have in fact been given. Assuming that the models are at least a reasonable pointer to the correct colours, possibly affected by reddening, objects at these redshifts will have been mis-classified towards those models which have colours similar to the $1.0<z<2.1$ models. The most likely cases are therefore the low–redshift Irregular galaxies, which have relatively flat spectra in the visible wavelengths, and those models at redshifts $z\approx 0.8$ and $z\approx2.3$ which are adjacent region we are interested in. This can actually be seen in the histogram of objects against modal redshifts (Figure \[fig:zhist\]) – there is an apparent excess of low–redshift objects at $z=0$ along with another peak in the distribution around $z=2.4$. No excess is seen at $z\approx0.8$ but any increase here would be likely to be a small fraction of the real objects at this redshift.
Those objects which are identified at $1.0<z<2.5$ – examples are shown in Figure \[fig:intzspectra\] – are only weakly distinguished in morphological type. Additionally there is a degeneracy between morphological type and redshift estimate for galaxies with redshifts $1.0<z<1.6$ – at higher redshifts there are few major differences between the galaxy colour tracks and hence no significant discrimination between morphology based on colour evidence. To remove these degeneracies would require more IR imaging to detect the 4000Å break – this would also improve the accuracy of the redshift detections in this redshift range.
High–redshift galaxies – $2.5<z<4.0$
------------------------------------
In Paper I, we have tried to identify galaxies at $z\approx3$ via the 912Å break. Photometric redshifts should prove effective at determining which galaxies lie at these redshifts since they make use of all the photometric measurements together in all filters, rather than just the $U$,$G$ & $R$.
Examples are shown in Figure \[fig:highzspectra\]. It is not possible to differentiate between different morphological types at these redshifts – the colours are too similar between the model galaxies. This is partly a result of similar star–formation histories between the model galaxies at these redshifts; this would be invalid if, for example, there were several phases of galaxy formation.
By selecting galaxies with $2.8<\textrm{modal~}z<3.5$, $R<25.5$ and $G>2\sigma$, we get 46 Ly–break candidates, almost twice as many as with the colour–criteria. The distribution of these galaxies (see Figure \[fig:zapprox3photz\]) also (cf Paper I) shows an apparent hole at a point mid–way between the quasars, despite the increase in surface density of candidates.
Clustering of objects
=====================
We can make use of the redshift determinations to look at the correlation functions for redshift subsets of the catalogue. The results of these are plotted in Figure \[fig:compcorrel\].
The lowest redshift range ($0<z<0.5$) has 558 objects, and there are 352 objects in the range $0.5<z<1.0$. As a result, the statistics for these redshift ranges are sufficiently good to show that there is measurable clustering in this field at low redshifts. Fitting the standard power–law curves ($A_\omega\theta^{-\beta}$) to angular scales $0<\theta<150~\textrm{arcsec}$ gives $A_\omega\approx0.82$ and $\beta\approx1.05$ for $0.0<z<0.5$, and $A_\omega\approx1.88$ and $\beta\approx1.74$ for $0.5<z<1.0$. These are plotted in Figures \[fig:0to0.5correl\] & \[fig:0.5to1correl\]. Also plotted in these figures are the correlation functions for subsets of the sample ordered in redshift. While the noise in these subsets is worse and no strong conclusions can be drawn, the apparent clustering of the $0.7<z<0.9$ objects at small angular scales is interesting.
In the rest of the $z$ bins there is little significant signal, with the least insignificant being 1.5-$\sigma$ clustering at $240''$ in $2.0<z<2.5$. Necessarily, there is no coverage of $1.0<z<2.0$.
Artificial clusters
===================
We now test the ability of this sort of analysis to recover real features from the field. We use these redshift colour models to place a realistic spatial distribution of artificial galaxies into the images and examine the likelihood of detecting a cluster towards PC1643+4631 A&B using colour methods. We can find no similar experiment in the published literature on any field.
Theoretical magnitude and spatial distributions
-----------------------------------------------
To mimic a cluster of galaxies, one needs distributions for both the magnitudes and positions of the galaxies on the sky. For the former we assummed a Schechter luminosity function ([@Schechter]), while for the latter we assumed a Hubble distribution, where the surface density is described by $\sigma_0/(1+(R/R_c)^2)$ ([@WH93]). We used a core radius, $R_c=200$kpc for the spatial distribution (Jones et al., 1997, found, for PC1643, a core radius of $\sim300$kpc), and used half–light radii of 10kpc for the galaxies.
$$I_{\textrm{AB}} = M_{\textrm{AB}} + 5 \log (D_L/10pc) - 2.5 \log (1+z)
- (B-I_z),$$
$D_L$ is the luminosity distance, $I_z$ is the $I$ band shifted to shorter wavelengths by a factor of $(1+z)$, and $B$ is the magnitude in the rest–frame $B$ band. The value of $(B-I_z)$ is taken from the photometric SED models, and hence the value of $B$ must be calculated by interpolation between the $U$, $G$ and other broadband filters. From this, observed magnitudes can be calculated for the redshift range $0<z<1.5$. For $z>1.5$, $B$ moves to longer wavelengths than $I$. A simple extrapolation of the $R-I$ colour can be used to obtain a first–order estimate of the rest frame $B$ band magnitude. This should prove to be a reasonable approximation of the real value given that the 4000Å break is weak in galaxies which are strongly star–forming, which is the case for the SED models used at $z>1$.
We do not attempt to model the evolution of the luminosity function. While this is a shortcoming of this method, at $z<0.9$, the maximum likely error is $\pm0.5$ mag (see [@LTH95]). Beyond $z\approx1$, the evolution of the luminosity function is poorly constrained, but it is reassuring that the luminosity function of $z\approx3$ galaxies have a luminosity function best fitted by $M_*\approx-21$ ([@MD97]).
At low redshifts, there are significant colour differences between different morphological galaxy types, with spirals having much bluer colours than ellipticals. While the ratio of ellipticals to spiral galaxies in general is about 60:40, in the cluster environment the proportion of elliptical galaxies is often much lower, closer to 40:60 (E+S0:S+Irr) ([@DOC97]). To simulate this spread of colours, we opted to create frames with just one morphological type in each. To keep the processing as simple as possible two plates were made, with one having E/S0 type galaxies and the other Scd/Irr types.
Method
------
The processes required to simulate each cluster at each redshift and make photometric measurements of the simulated cluster galaxies after superimposition onto the PC1643 are illustrated in Figure \[fig:simclus\]. Simulated clusters were made at redshifts from 0.1 to 4.0 in steps of 0.1.
We used the ARTGAL package in IRAF to create artificial clusters of galaxies with the required properties. For each ‘cluster’ only two images were made, one each for elliptical and spiral galaxies, made to have the same zero point and exposure time as the $I$ image, and zero background level. No additional noise was simulated because the real images are all background limited.
Recent publications discussing the distributions of the different morphological types suggest that there is some segregation, with steeper velocity dispersions for later–type galaxies (see, eg [@ABM98]). We have not attempted to mimic these results here, since our aim is to examine the ability of the tests previously carried out on this field to recover cluster members, rather than to produce a completely accurate cluster distribution.
Having generated the $I$–band spiral image at each redshift, we used the Sbc model SEDs to determine the colours of the simulated spirals in $R$,$V$,$G$ & $U$, and IMARITH was used to scale the $I$ image to mimic the correct colours in the other broadband images. We used a similar procedure to produce $R$,$V$,$G$ & $U$ images from the $I$–band elliptical image using the E/S0 model SEDs. These $I$, $R$, $V$, $G$ & $U$ spiral and elliptical images were summed together to create composite images, and a catalogue was made from the $R$ composite image using FOCAS, which we will refer to as the matching composite catalogue (MCC).
These composite images were then added onto the real PC1643 images. The resulting $R$ image was processed using FOCAS to produce a catalogue based on $R$ isophotal apertures: catalogues were then made in $I$, $V$, $G$ & $U$ using the $R$ isophotes and the five resulting catalogues were then matched using MATCH against the MCC. To extract the simulated galaxies from these newly created FOCAS catalogues, ‘stripped’ catalogues were made including objects only if they corresponded to the positions indicated by the MCC - ie these catalogues should contain only the information on the measureable simulated cluster galaxy members. These stripped catalogues were then given the same photometric redshift analysis as the real objects in the PC1643 field.
Results
-------
Full–colour images showing the artificial cluster at various redshifts are shown in Figures \[fig:0.2-0.8.images\] & \[fig:1.0-4.0.images\], both in isolation and superimposed on the PC1643 field. While it is easy to recognise the cluster on its own, it becomes impossible to differentiate the cluster against the other objects in the field at $z>1.0$, and the contrast is poor even at $z=0.6$.
More importantly, it is the ability of the catalogue creation and analysis software to detect a cluster which is of greatest interest here, particularly at faint magnitudes $R\gtrsim24$. We have plotted the $U-G$ vs $G-R$ colours for each of the simulation images illustrated in Figures \[fig:0.2-0.8.images\] & \[fig:1.0-4.0.images\]. Even at $z=0.2$ the fainter members of the simulated galaxies show significant photometric scatter away from the expected colours, and this is reflected in the estimates of modal redshift: there is a peak at $z=0.2$ as expected, but over a third of the sample are more than 0.1 away from this redshift, and 9 of the simulated galaxies are mistaken for high–redshift galaxies at $z\gtrsim3$.
At $z\lesssim1.0$, the photometric–redshift technique still produces redshift estimates around the expected value, with increased scatter as the simulated cluster becomes fainter with increasing redshift (see Figures \[fig:modalzexp\] & \[fig:meanzexp\]). This is also reflected in the increased scatter of the $U-G$ and $G-R$ colours, which, it should be stressed, is often greater than 3-$\sigma$, where $\sigma$ is the photometric error based on the noise for the object. This is the result of pollution of the isophotal apertures by nearby neighbours, which is inevitable in a deep, ground–based field like this one. Given that the density of objects in the field including all detections down to the surface brightness limits of 3-$\sigma$ in an area of six pixels is approximately one object in every $5''\times5''$ box, and that the average isophotal area of a object with $R=25.0$ is some 5 arcmin$^2$, this means that at least one in five faint objects will overlap another faint object, so that in this set of deep images, objects *are being confused*.
To determine whether a significantly brighter cluster of galaxies would be visible in this field, we repeated the simulation with all the magnitudes brightened by one magnitude. Bearing in mind that this is the most extreme error likely in the simulation, this is also a good test of the sensitivity of these simulations to the real magnitude distribution. Even with many more members, such a cluster would still be difficult to distinguish using colour–criteria or by excess surface density.
At intermediate redshifts, $1.0<z<2.0$, the photometric redshift technique fails to correctly determine the redshift in almost all cases. Despite starting from simulated galaxies with the correct colours, the colours of the galaxies as recovered from the field are systematically scattered to redder colours. There may be a systematic effect resulting from selecting objects based on their $R$ magnitudes — for example, objects which are polluted to fainter magnitudes in $R$ are missed from the sample, whereas objects which recover fainter $G$ magnitudes than originally simulated appear redder than they really are. There may also be a systematic effect from the different surface densities of objects in $R$ and $G$ at similar magnitudes – there are 763 objects with $25<R<26$, and only 537 with $25<G<26$ – so objects with $G-R\approx0$ stand proportionally more chance of being brightened in $R$ than in $G$. Additionally, if the position of a simulated galaxy does coincide with that of a real object, it is likely that the real object has $G-R\gtrsim0.5$, and hence the pollution of the simulated object will result in redder measured colours.
This failure to accurately detect galaxies in the range $1.0<z<2.0$ is a function of the filters used. Because there are no strong continuum features over the wavelengths spanned by the optical filters used here at these redshifts (approximately 1200Å – 4000Å) the photometric redshift are poorly determined. Additionally, because of the scatter of the objects colours away from the simulated colours, objects with redshifts in the range $1.0<z<2.0$ are scattered to lower and higher redshifts. This is clearly seen in Figures \[fig:modalzexp\] & \[fig:meanzexp\]. The result is that any histogram of galaxies over the redshifts estimated from photometric techniques using a similar range of filters to those used here will show a deficit of galaxies with $1.0<z<2.0$ and peaks above and below this range against the true population of galaxies. Precisely this behaviour is seen in the redshift distribution calculated from the HDF four–colour photometry (in F300W, F450W, F606W and F814W) using photometric redshifts (see [@GH96] and to a lesser extent [@SLY97]). The deficit is not quite so marked as it is in the photometric redshifts estimates of the field of PC1643, due to slight differences in modelling but the effect is still the same. It is notable that the photometric redshift estimates of [@LYF96] (LYFS) do not suffer from this effect, and appear to produce a smooth redshift histogram: the photometric models used by LYFS are based entirely on the CWW spectra extended into the UV without including the effects of evolution, with the effects of the [$\textrm{Ly--}\alpha$]{} forest and 912Å break taken into account. However, the galaxies at $z\approx3$ are predicted to have $U_{300}-V_{450}\approx5$ and $V_{450}-R{606}\approx1.0$ by LYFS, which is considerably redder in $U_{300}$ than those detected by [@StHDF96] which have $U_{300}-V_{450}\approx2$ and $V_{450}-R_{606}\approx0.5$, and therefore casts doubt onto their photometric models.
To test just how much brighter a distant cluster would have to be to be clearly evident, we increased the magnitudes of every member of the simulated cluster at $z=2.0$ until the cluster just became evident in the images. We emphasize that significantly more than two magnitudes of brightening would be required to make the cluster stand out from the field. We stressed that this is far more than the largest expected error in the brightness of these simulations.
At $z\gtrsim2.3$, the 912Å break starts to extinguish a $U$–band observation, and by $z\approx3$ provides a clear colour signature, ie $U-G>2.0$. This feature has been successfully used to detect high–redshift galaxies both with custom filters and in the HDF (eg [@StII], [@StHDF96]). Most of the galaxies previously detected at $z\approx3$ have $R\gtrsim24.5$ [@SGP96], which is consistent with the brightest members of the simulated cluster at the same redshift (Figure \[fig:rmag.zeq3.0\]). Intriguingly, while the photometric redshift estimations do peak at around their expected value, there is also a significant proportion which are mistaken for low–redshift galaxies (again, see Figures \[fig:modalzexp\] & \[fig:meanzexp\]). From the 123 simulated galaxies with colours consistent with $2.5<z<3.0$ and measured $R<26.0$, 52 are identified as having $z>2.5$ and 70 are identified as low redshift galaxies with $z<0.8$. If this mimics the real situation for detecting $z\approx3.0$ galaxies, then it suggests that a significant number of high–redshift galaxies are missed if redshifts are estimated using photometric techniques. Note also that the limiting magnitudes of [@StII] etc. are similar to those of the PC1643 images.
To investigate the efficiency of selecting $z\approx3$ galaxies using the colour–criteria described in Paper I, we concatenated the simulated catalogues with $2.5<z<3.5$ and selected objects with $R<25.5$ and $G>2\sigma$. While the simulations were carried out to mimic a cluster of galaxies, there is negligible overlap between the individual simulated galaxies in each catalogue. Therefore, while there is clustering of the galaxies in the field, it is important to note that the photometric spread and the results are similar to that which would be obtained for a random uniform spatial distribution.
Figure \[fig:2.5to3.5uggr\] shows the measured $U-G$ vs $G-R$ colours of these 104 objects. We note that the simulated objects are scattered from the simulated colours to such an extent that only 12 of the objects show measured $U-G$ and $G-R$ colours consistent with objects at $z\approx3$. Of these same 104 objects, 48 are identified as being at $z>2$ by the photometric redshift estimator (see Figure \[fig:2.5to3.5hist\]). Even if we take just the catalogue for the simulated cluster at $z=3.0$, of the 12 objects with $R<25.5$ and $G>2\sigma$, only two meet the colour–criteria, whereas eight are identified as having estimated $z>2$. Therefore, five–colour photometric redshifts appear to provide a more effective method of detecting high–redshift galaxies than the three–colour $UGR$ method used by Steidel et al. This is not surprising since the extra photometric measurements provide extra constraints, which reduce the effects of individual photometric measurement errors.
Comparing the angular sizes of the simulated objects against Ly-break galaxies observed in the Hubble Deep Field suggests that these galaxies should be about 2–3kpc in size ([@MD97]), which is considerably smaller than the sizes used in this simulation, which are $\approx 10$kpc in size. We repeated the $z=3.0$ simulation to assess whether the angular sizes used had a significant effect, with the angular sizes of the simulated galaxies reduced by a factor of four. This small angular size simulation had 15 objects with $R<25.5$ and $G>2\sigma$, of which three fulfilled the colour–criteria, whereas using photometric redshifts, 8 were identified at $z>2$. Figure \[fig:z=3.0comp\] compares the $U-G$ vs $G-R$ colours for the original and revised simulation.
The increase in numbers of galaxies detected is due to the increase in the peak flux of galaxies due to smaller half–light radii – this makes it easier for FOCAS to detect these galaxies as their peak fluxes are more likely to be greater than the $3\sigma$ detection threshold. It is important to note that decreasing the angular size of the simulated galaxies has not had a significant effect on the fraction of galaxies which fulfill the colour criteria for $z\approx3$ galaxies: one might expect reducing the angular size to reduce the level of confusion observed in these simulations; however, while the peak flux increases and the area covered by the object is decreased for all objects, this also promotes simulated objects which were previously too faint to be detected above the detection threshold and it is these faint objects which are most likely to be significantly affected by confusion with the many real faint objects in the field. The effect of confusion on objects with faint (ie $R>24$) magnitudes is to “average” the simulated object’s colours with that of the confusing object, with the result that a large proportion of the sample is moved towards the average colours of the real objects in the field, ie $U-G\approx1.2$ and $G-R\approx1$.
In [@MD97], the luminosity function of $z\approx3$ galaxies is illustrated, but the HDF count of $z\approx3$ galaxies have been renormalised to the numbers seen in the ground–based images. We suggest that the surface density seen in the HDF is a better estimate of the real surface density of $z\approx3$ galaxies, and there are approximately 6 times more $z\approx3$ galaxies than currently thought. This has important consequences for the understanding of the star–formation history of the universe and would suggest that the peak rate of star–formation occurred at an earlier epoch than currently thought (see [@MPD98]).
To definitively identify the majority of Ly–break candidates and determine the real surface–density of these galaxies, observations with much improved seeing will be necessary; either by further HST observations to increase the area of sky observed and thereby reduce the effects of cosmic variance, or by deep ground–based observations with adaptic optics to reduce the point–spread–function in the images. The effects of confusion can be quantified using similar simulations to those demonstrated here.
Conclusions
===========
In using the multicolour photometric redshift technique on the objects found in the deep imaging of PC1643+4631, we have shown the following.
(1) [The technique is efficient at identifying objects at $z\lesssim1$ where the photometric errors are small, ie at $R\lesssim24$, because the $\chi^2$ fitting is well constrained because the colours change strongly with redshift.]{}
(2) [These photometric redshifts are in broad agreement ($\Delta
z\approx0.15$) with the 4 spectroscopic redshifts in the range $0.62<z<0.81$ in the PC1643+4631 field, whereas HR10 at $z=1.44$ has a colour–estimated redshift of $z=0.65$, indicating that either significant reddening can distort the redshift determination or that at $1<z<2$ the photometric technique fails.]{}
(3) [Ground–based morphological classification is possible on low–redshift ($z\lesssim0.7$) objects with high signal–to–noise photometry (ie $R<24$ in the deep PC1643 images), although this work does not investigate its accuracy, due to the difficulties in obtaining secure morphological classifications for a significant number of galaxies in this field by manual or automatic means.]{}
(4) [There are degeneracies between the morphological type and redshift in objects with $1.0\lesssim z \lesssim 1.6$, due to the models with different morphologies having similar colours at differing redshifts.]{}
(5) [No constraints can be placed on the morphology of high redshift ($z\gtrsim3$) galaxies using these models since there is no significant difference between the colours of the model galaxies with different morphologies at $z\gtrsim2$.]{}
(6) [46 Ly-break candidates have been identified in the PC1643 images with $2.8<z<3.5$, $R<25.5$ and detected at $>2\sigma$ in $G$; the distribution of these candidates still shows an underdensity mid–way between the quasars.]{}
(7) [The histogram of photometric redshift estimates shows two peaks, at $z\approx0.7$ and $z\approx2.3$, and few objects with $1.0<z<2.0$. This is probably an artifact of the photometric redshift technique, arising from the close–spacing of the model colours at these redshifts, and the histogram is therefore not indicative of the real distribution of redshifts.]{}
(8) [By calculating both modal and mean photometric redshifts, as well as a standard deviation for the likelihood distribution, for each object, ambiguous photometric redshift estimates can easily be flagged using $|\textrm{modal } z - \textrm{mean } z|>0.2$, giving an advantage over previous modal–only methods.]{}
(9) [There is evidence of clustering in the field at low ($z\lesssim0.9$) redshift which is consistent with other published results.]{}
We have carried out simulations to examine the ability of the photometric redshift technique to accurately recover simulated galaxies at various redshifts. These show the following.
(10) [For simulated galaxies at $0.1<z<0.3$, faint E/S0 galaxies are mistakenly identified as $z\gtrsim3$ galaxies because of the red $U-G$ colours. An E/S0 with $R\approx24$ will often have no detected flux in $U$, and be mistaken for a Ly–break candidate.]{}
(11) [For simulated galaxies at $0.1<z<1.0$, the average estimate of redshift is consistent with the simulated redshift, but there is a spread of $\Delta z\approx 0.2$ in the distribution. This is partly due to the photometric errors causing the colours of the object to move away from the simulated colours, but also due to contamination of the object colours by other objects in the field.]{}
(12) [For simulated galaxies at $1.0<z<2$, the photometric redshift is unreliable and misclassifies these galaxies to lower ($z\approx0.6$) and higher ($z\approx2.2$) redshifts, resulting in peaks in the photometric redshift distribution which are not representative of the real redshift distribution. This is a fundamental problem with photometric redshift estimates based only on optical imaging. A similar problem is seen in several publications in the literature but has not been appreciated (see [@GH96] and [@SLY97], who both have a decrease in numbers of galaxies between $1<z<2$ but considered it a real observation), which has resulted in erroneous conclusions being drawn.]{}
(13) [For simulated galaxies at $z>2.5$, the photometric technique is effective in detecting about half the sample as being at $z>2.5$, with the rest of the sample being mistaken for low redshift galaxies.]{}
(14) [Photometric estimates may prove effective in selecting high redshift galaxies at $z\approx3$, where the $U-G>2$ colours and fairly flat spectrum in $G$,$V$,$R$ and $I$ provide additional constraints to the three colour $U$,$G$,$R$ method currently used for successfully selecting $z\approx3$ galaxies.]{}
(15) [The scatter of the simulated galaxies away from their simulated colours is two to three times greater than that expected from the photometric errors alone and arises as a result of confusion between the simulated galaxies and the real objects in the field. Since the amount of confusion is directly related to the resolution of the images, the WHT images will be far more affected by confusion than equivalently deep HST images.]{}
(16) [Because of the effects of confusion, the current published estimates of the surface densities of Ly–break galaxies based on ground–based imaging are under–estimates by a factor of $\approx6$; such an increase in the surface density of these galaxies is consistent with the surface densities seen in the HDF.]{}
(17) [Any third image of a single quasar causing the PC1643 pair is extremely likely to be confused and therefore unrecognisable.]{}
(18) [Deep ground–based CCD imaging without adaptive optics is *inevitably confusion–limited* at $R>26.0$. Imaging with smaller point–spread–functions is the *only* way to alleviate this.]{}
Simulations have been carried out to investigate the difficulties in identifying a cluster at various redshifts in the PC1643 field. These show the following.
(19) [Even with our many optical colours, it is difficult to visually identify a cluster of galaxies at $z\approx0.4$ against the other objects in any deep field. A cluster at $z=1$ with galaxy luminosities similar to those seen locally would be lost against the other objects in the field.]{}
(20) [Even a very rich cluster at $z\approx1.0$ would be impossible to distinguish from the field on the basis of this $U$, $G$, $V$, $R$ and $I$ imaging and hence the evident absence of a cluster in these $1.1''$ resolution images is entirely consistent with a distant cluster producing the CMB decrement seen.]{}
(21) [A cluster at $z=1.0$ in which each member was 1 magnitude brighter than expected would still be difficult to detect; however, if the members were more than 2 magnitudes brighter, the cluster would be evident in the images.]{}
Acknowledgements
================
GC acknowledges a PPARC Postdoctoral Research Fellowship. TH acknowledges a PPARC studentship. We thank Steve Rawlings for obtaining the redshift of galaxy 4 in Table 1.
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-------- ----------------- ---------------
Spectroscopic Photometric
Object redshift redshift
1 $0.659\pm0.002$ $0.55\pm0.03$
2 $0.670\pm0.002$ $0.57\pm0.04$
3 $0.670\pm0.002$ $0.52\pm0.05$
4 $0.810\pm0.003$ $0.62\pm0.08$
HR10 $1.44\pm0.01$ $0.65\pm0.08$
-------- ----------------- ---------------
: Comparison of spectroscopic and photometric redshifts in PC1643\[tab:compsp\]
This figure is avaliable at ftp.mrao.cam.ac.uk:/pub/PC1643/paper2.figure50.eps
This figure is avaliable at ftp.mrao.cam.ac.uk:/pub/PC1643/paper2.figure51.eps
|
---
abstract: 'In recent years, many DHT-based P2P systems have been proposed, analyzed, and certain deployments have reached a global scale with nearly one million nodes. One is thus faced with the question of which particular DHT system to choose, and whether some are inherently more robust and scalable. Toward developing such a comparative framework, we present the reachable component method (RCM) for analyzing the performance of different DHT routing systems subject to [*random failures*]{}. We apply RCM to five DHT systems and obtain analytical expressions that characterize their *routability* as a continuous function of system size and node failure probability. An important consequence is that in the large-network limit, the routability of certain DHT systems go to zero for [*any*]{} non-zero probability of node failure. These DHT routing algorithms are therefore *unscalable*, while some others, including Kademlia, which powers the popular eDonkey P2P system, are found to be *scalable*.'
author:
- |
Joseph S. Kong, Jesse S. A. Bridgewater and Vwani P. Roychowdhury\
Department of Electrical Engineering\
University of California, Los Angeles\
{jskong, jsab, vwani}@ee.ucla.edu
bibliography:
- 'routing\_rcm.bib'
title: A General Framework for Scalability and Performance Analysis of DHT Routing Systems
---
Introduction {#sec:intro}
=============
Developing scalable and fault tolerant systems to leverage and utilize the shared resources of distributed computers has been an important research topic since the dawn of computer networking. In recent years, the popularity and wide deployment of peer-to-peer (P2P) systems has inspired the development of distributed hash tables (DHTs). DHTs typically offer scalable $O(\log n)$ routing latency and efficient lookup interface. According to a recent study [@Parker:cachelogic], the DHT based file-sharing network eDonkey is emerging as one of the largest P2P systems with millions of users and accounting for the largest fraction of P2P traffic, while P2P traffic currently accounts for 60% of the total Internet bandwidth. Given the transient nature of P2P users, analyzing and understanding the robustness of DHT routing algorithms in the asymptotic system size limit under unreliable environments become essential.
In the past few years, there has been a growing number of newly proposed DHT routing algorithms. However, in the DHT routing literature, there have been few papers that provide a general analytical framework to compare across the myriad routing algorithms. In this paper, we develop a method to analyze the performance and scalability of different DHT routing systems under random failures of nodes. We would like to emphasize that we intend to analyze the performance of the *basic* routing geometry and protocol. In a real system implementation, there is no doubt that a system designer have many optional features, such as additional sequential neighbors, to provide improved fault tolerance. Nevertheless, the analysis of the basic routing geometry will give us more insights and good guidelines to compare among systems. In this paper, we investigate the routing performance of five DHT systems with uniform node failure probability $q$. Such a failure model, also known as the *static resilience* model[^1], is assumed in the simulation study done by Gummadi et al. [@Gummadi:impact]. A static failure model is well suited for analyzing performance in the shorter time scale. In a DHT, very fast detection of faults is generally possible through means such as TCP timeouts or keep-alive messages, but establishing new connections to replace the faulty nodes is more time and resource consuming. The applicability of the results derived from this static model to dynamic situations, such as churn, is currently under study. Intuitively, as the node failure probability $q$ increases, the routing performance of the system will worsen. A quantitative metric, called *routability* is needed to characterize the routing performance of a DHT system under random failure:
The *routability* of a DHT routing system is the expected number of routable pairs of nodes divided by the expected number of possible pairs among the surviving nodes. In other words, it is the fraction of survived routing paths in the system. In general, routability is a function of the node failure probability $q$ and system size $N$. \[def:routability\]
As the DHT-based eDonkey is reaching global scale, it is important to study how DHT systems perform as the number of nodes reaches millions or even billions. In fact, we know from site percolation theory[@Stauffer:intro_perc], that if $q>(1-p_c)$, where $p_c$ is called the percolation threshold of the underlying network, then the network will get fragmented into very small-size connected components and for large enough network size. As a result, the routability of the network will approach zero for such failure probability due to the lack of connectivity. However, because of how messages get routed as specified by the underlying routing protocol, all pairs belonging to the same connected component need not be reachable under failure.
In general, the size of the connected components do not directly give us the routability of the subnetworks. Hence, one needs to develop a framework different from the well-known framework of percolation. As a result, this work investigates DHT routability under the random failure model for both finite system sizes and the infinite limit. We will define the *scalability* of a routing system as follows:
A DHT routing system is said to be *scalable* if and only if its routability converges to a nonzero value as the system size goes to infinity for a nonzero failure probability $q$. Mathematically, it is defined as follows: $$\lim_{N \rightarrow \infty} r(N,q) > 0 \ \ for \ 0 < q < 1 - p_c$$ where $r(N,q)$ denotes the routability of the system as a function of system size $N$ and failure probability $q$. Similarly, the system is said to be *unscalable* if and only if its routability converges to zero as the system size goes to infinity for a nonzero failure probability $q$: $$\lim_{N \rightarrow \infty} r(N,q) = 0 \ \ for \ 0 < q < 1 - p_c$$ \[def:scalability\]
We want to emphasize that in a real implementation, there are many system parameters that the system designer can specify, such as the number of near neighbors or sequential neighbors. As a result, the designer can always add enough sequential neighbors to achieve an acceptable routability under reasonable node failure probability for a maximum network size that exceeds the expected number of nodes that will participate in the system. The scalability definition is provided for examining the *theoretical* asymptotic behavior of DHT routing systems, not for claiming a DHT system is unsuitable for any large-scale deployment.
Having specified the definition of the key metrics, we will present the reachable component method (RCM), a simple yet effective method for analyzing DHT routing performance under random failure. We apply the RCM method to analyze the basic routing algorithms used in the following five DHT systems: Symphony [@Manku:symphony], Kademlia [@May:kademlia], Chord [@Stoica:chord], CAN [@Ratnasamy:CAN] and Plaxton routing based systems [@Plaxton:routing]. For all algorithms except Chord routing, we derive the analytical expression for each algorithm’s routability under random failure, while an analytical expression for a tight lower bound is obtained for Chord routing. In fact, our analytical results match the simulation results carried out in [@Gummadi:impact], where different DHT systems were simulated and the percentage of failed paths (i.e., 1-routability) was estimated for $N=2^{16}$, as illustrated in Fig. \[fig:compare\_fig\]. In addition, we also derive the asymptotic performance of the routing algorithm under failure as the system scales.
One interesting finding of this paper is that under random failure, the basic DHT routing systems can be classified into two classes: *scalable* and *unscalable*. For example, the XOR routing scheme of Kademlia is found to be *scalable*, since the routability of the system under nonzero probability of failure converges very fast to a positive limit even as the size of the system tends to infinity. This is consistent with the observation that the Kademlia-based popular P2P network eDonkey is able to scale to millions of nodes. In contrast, as the system scales, the routability of Symphony’s routing scheme is found to quickly converge to zero for any failure probability greater than zero. Thus, the basic routing system for Symphony is found to be *unscalable*. However, as briefly discussed above in this section, a system designer for Symphony can specify enough near neighbors to guarantee an acceptable routability in the system for a maximum network size and a reasonable failure probability $q$.
The rest of this paper is organized as follows. In section \[sec:related\], we discuss previous work on the fault tolerance of P2P routing systems. In section \[sec:overview\], we will give an overview of the DHT routing systems that we intend to analyze. In section \[sec:rcm\], we present the *reachable component method* (RCM) and apply the RCM method on several DHT systems. In section \[sec:scalability\], we examine the scalability of DHT routing systems. In section \[sec:conclusion\], we give our concluding remarks.
Related Work {#sec:related}
============
The study of robustness in routing networks has grown in the past few years with researchers simulating failure conditions in DHT-based systems. Gummadi et al.[@Gummadi:impact] showed through simulation results that the routing geometry of each system has a large effect on the network’s static resilience to random failures. Static resilience represents the performance of the system subject to failures when no measures are taken to compensate for connectivity loss. This is not only a realistic model for many systems, but it also represents a baseline of performance against which different recovery algorithms can be compared. In Section \[sec:intro\] we note that detecting a failed connection can be very fast, thus when a node’s neighbor becomes unavailable the node will not attempt to incorrectly route packets through that neighbor for long. However, while detection of failure is fast, it will not generally be sufficiently fast to create new connections to compensate for the lost connectivity. The existence of these two time-scales makes the static resilience failure model important and relevant. We do not consider recovery algorithms and other dynamics here but will instead focus on the static random failure model. In addition, there have been research work done in the area of analyzing and simulating dynamic failure conditions (i.e. churn) in DHT systems [@Liben:analysis; @Li:; @comparing; @Krish:chord].
Theory work has been done to predict the performance of DHT systems under a static failure model. The two main approaches thus far have been graph theoretic methods[@Angel:routing; @Loguinov:graph; @Lam:evaluation] and Markov processes[@Wang:markov]. Most analytical work to date has dealt with one or two routing algorithms to which their respective methods are well-suited but have not provided comparisons across a large fraction of the DHT algorithms. Angel et al. [@Angel:routing] use percolation theory to place tight bounds on the critical failure probability that can support efficient routing on both hypercube and $d$-dimensional mesh topologies. By efficient they mean that it is possible to route between two nodes with time complexity on the order of the network diameter. While this method predicts the point at which the network becomes virtually unusable, it does not allow the detailed characterization of routability as a function of the failure probability. Wang et al.[@Wang:markov] model CAN with small-world extensions using Markov-chain methods. This method is straightforward and produces detailed performance predictions. However the method’s usefulness is limited somewhat by the fact that the network connectivity structure is represented as a matrix and consequently the level of detail in the model approaches that of a routing simulation. Because of this approach, a fairly-complex numerical computation must be done for each system size and node failure probability to yield the routability. In contrast, the reachable component method (RCM) method exploits the geometries of DHT routing networks and leads to simple analytical results that predict routing performance for arbitrary network sizes and failure probabilities.
Overview of DHT Routing Protocols {#sec:overview}
=================================
We will first review the five DHT routing algorithms that we intend to analyze. An excellent discussion of the geometric interpretation of these routing algorithms (except for Symphony) is provided by Gummadi et al.[@Gummadi:impact] and we use the same terms for the geometric interpretations of DHT routing systems in this paper (e.g. hypercube and ring geometry for CAN and Chord routing systems, respectively). By following the algorithm descriptions in [@Gummadi:impact] as well as the descriptions in this section, one can construct Markov chain models (e.g. Fig. \[fig:two\_mc\]) for the DHT routing algorithms. The application of the Markov chain models will be discussed in section \[ssec:rcm\_des\] and \[ssec:hyper\_example\]. In addition, we will use the notation of *phases* as used in [@Kleinberg:algorithmic]: we say that the routing process has reached phase $j$ if the numeric distance (used in Chord and Symphony) or the XOR distance (used in Kademlia) from the current message holder to the target is between $2^j$ and $2^{j+1}$. In addition, we will use binary strings as identifiers although any other base besides 2 can be used. Finally, for those systems that require resolving node identifier bits *in order*, we use the convention of correcting bits from left to right.
Tree (Plaxton)
--------------
Each node in a tree-based routing geometry has $\log N$ neighbors, with the $i$th neighbor matching the first $i-1$ bits and differ on the $i$th bit. When a source node $S$, wishes to route to a destination, $D$, the routing can only be successful if one of the neighbors of $S$ , denoted $Z$, shares a prefix with $D$ and has the highest-order differing bit. Each successful step in the routing results in the highest-order bit being corrected until no bits differ.
The routing Markov chain (Fig. \[fig:plaxton\_mc\]) for the tree geometry can easily be generated by examining the possible failure conditions during routing. At each step in the routing process, the neighbor that will correct the leftmost bit must be present in order for the message to be routed. Otherwise, the message is dropped and routing fails.
Hypercube (CAN)
---------------
In the hypercube geometry, each node’s identifier is a binary string representing its position in the $d$-dimensional space. The distance between nodes is simply the Hamming distance of the two addresses. The number of possible paths that can correct a bit is reduced by 1 with each successful step in the route. This fact makes the creation of the hypercube routing Markov chain (Fig. \[fig:hypercube\_mc\]) straightforward.
XOR (Kademlia)
--------------
In XOR routing [@May:kademlia], the distance between two nodes is the numeric value of the XOR of their node identifiers. Each node keeps $\log(N)$ connections, with the $i$th neighbor chosen uniformly at random from an XOR distance in the range of $[2^{d-i},2^{d-i+1}]$ away. Messages are delivered by routing greedily in the XOR distance at each hop. Moreover, it is a simple exercise to show that choosing a neighbor at an XOR distance of $[2^{d-i},2^{d-i+1}]$ away is equivalent to choosing a neighbor by matching the first (i-1) bits of one’s identifier, flipping the $i$th bit, and choose random bits for the rest of the bits.
Effectively, this construction is equivalent to the Plaxton-tree routing geometry. As a result, when there is no failures, the XOR routing protocol resolves node identifier bits from left to right as in the Plaxton-tree geometry. However, when the system experiences node failures, nodes have the option to route messages to neighbors that resolve lower order bits when the neighbor that would resolve the highest order bit is not available. Note that resolving lower order bits will also make progress in terms of decreasing the XOR distance to destination. Nonetheless, the progress made by resolving lower order bits is not necessarily preserved in future hops or phases (see Fig. \[fig:xor\_routing\]).
For example, at the start of the routing process, one phase is advanced if the neighbor correcting the leftmost bit exists. Otherwise, the routing process can correct one of the lower order bits. However, if all of the neighbors that would resolve bits have failed, the routing process fails. A Markov chain model for the routing process is illustrated in Fig. \[fig:xor\_mc\].
Ring (Chord)
------------
In Chord [@Stoica:chord], nodes are placed in numerical order around a ring. Each node with identifier $a$ maintains $\log(N)$ connections or fingers, with each finger at a distance $[2^{d-i},2^{d-i+1}]$ away (the randomized version of Chord is discussed here). Routing can be done greedily on the ring. When the system experiences failure, each node will continue to route a message to the neighbor closest to destination (i.e. in a greedy manner). A Markov chain model for the routing process is illustrated in Fig. \[fig:chord\_mc\].
Small-World (Symphony)
----------------------
Small-world routing networks in the $1$-dimensional case have a ring-like address space where each node is connected to a constant number of its nearest neighbors and a constant number of shortcuts that have a $1/d$ distance distribution ($d$ is the ring-distance between the end-points of the shortcut). Each node maintains a constant number of neighbors and uses greedy routing. Due to the distance distribution it will take an average of $O(\log N)$ hops before routing halves the distance to a target node, therefore requiring $O(\log N)$ such phases to reach a target node for a total expected latency of $O(\log^2 N)$.
When the system experiences node failures, some of the shortcuts will be unavailable and the route will have to take “suboptimal” hops. The small-world Markov chain model is fundamentally different from the ones for XOR routing (Fig. \[fig:xor\_mc\]) and ring routing (Fig. \[fig:chord\_mc\]). A routing phase is completed if any of the node’s shortcuts connects to the desired phase. This happens with probability $\frac{k_s}{d}$ where $k_s$ denotes the number of shortcuts that each node maintains. Alternatively, the routing fails if all of the node’s near neighbor and shortcut connections fail, which happens with probability $q^{k_n+k_s}$. If neither of the above happens then the route takes a suboptimal hop, which happens with probability $1-\frac{k_s}{d}-q^{k_n+k_s}$.
Reachable Component Method and its Applications {#sec:rcm}
===============================================
+:----------------------+:----------------------+:----------------------+
| {width="1.5in"} | ing){width="1.5in"} | ing){width="1.5in"} |
| | | |
| | | [c c c c]{} $$h$$ & |
| | | $$n(h)$$ & |
| | | $$\mbox{Pr}(S_h,S_{h+ |
| | | 1})$$\ |
| | | \ |
| | | $$1$$ & |
| | | $$\binom{3}{1}$$ & |
| | | $$1-q^3$$\ |
| | | \ |
| | | $$2$$ & |
| | | $$\binom{3}{2}$$ & |
| | | $$1-q^2$$\ |
| | | \ |
| | | $$3$$ & |
| | | $$\binom{3}{3}$$ & |
| | | $$1-q$$\ |
| | | \ |
+-----------------------+-----------------------+-----------------------+
Method Description {#ssec:rcm_des}
------------------
We now describe the steps of the *reachable component method* (RCM) in calculating the routability of a DHT routing system under random failure. Before we delve into the description, let us first clarify several concepts and notations on DHT routing. First, we allow all DHTs to fully populate their identifier spaces (i.e. node identifier length $d = \log_b N$). Second, when a DHT is not in its perfect topological state, it can be the case that a pair of nodes are in the same connected component but these two nodes cannot route between each other. Thus, the reachable component of node $i$ is the set of nodes that node $i$ can route to under the given routing algorithm. Note that the reachable component of node $i$ is a subset of the connected component containing node $i$. Third, we assume that no “back-tracking” is allowed (i.e. when a node cannot forward a message further, the node is not allowed to return the message back to the node from whom the message was received).
RCM is fairly simple in concept and involves the following five steps:
1. Pick a random node, node $i$, from the system and denote it as the *root node*. Construct the root node’s routing topology from the routing algorithm of the system (i.e. the topology by which the root node routes to all other nodes in the system).
2. Obtain the distribution of the distances (in hops or in phases) between the root node and all other nodes (denoted as $n(h)$); in other words, for each integer $h$, calculate the number of nodes at distance $h$ hops from the root node. Note that the meaning of *hops* or *phases* will be clear from the context.
3. Compute the probability of success, $p(h,q)$, for routing to a node $h$ hops away from the root node under a uniform node failure probability, $q$.
4. Compute the expected size of the *reachable component* from the root node by first calculating the expected number of reachable nodes at distance $h$ hops away (which is simply given by $n(h)*p(h,q)$). Now, we sum over all possible number of hops to obtain the expected size of the reachable component.
5. By inspection, the expected number of routable pairs in the system is given by summing all surviving nodes’ expected reachable component sizes. Then, dividing the expected number of routable pairs by the number of possible node pairs among all surviving nodes produces the routability of the system under uniform node failure probability $q$.
The formula for computing the expected size of the reachable component, $E[S_i]$, described in step 4 is derived as follows: $$\begin{aligned}
E[S_i] &=& E[\displaystyle\sum_{\substack{j=1\\j\neq i}}^{N} Y_j] = \displaystyle\sum_{\substack{j=1\\j\neq i}}^{N} E[Y_j]
= \displaystyle\sum_{h=1}^{d} n(h)p(h,q) \nonumber\end{aligned}$$ where $Y_j$ is Bernoulli random variable for denoting reaching node $j$, and $d$ is the node identifier length. Since nodes in the system are removed with probability $q$, there are $(1-q)N$ or $pN$ nodes that survive on average. In step 5, the formula for calculating the routability, $r$, of the system under uniform failure probability $q$ is given as follows: $$\begin{aligned}
r &=& \frac{M_{rp}}{M_p} = \frac{E\left[\displaystyle\sum_{\small i=1}^{\small pN} S_i\right]}{2\binom{\lfloor pN \rfloor)}{2}} \approx \frac{\displaystyle\sum_{i=1}^{pN} E[S_i]}{pN(pN-1)}\nonumber \\
&=& \frac{ E[S]}{(pN-1)} \label{eq:routability}\end{aligned}$$ where $M_{rp}$ denotes the expected number of *routable pairs* among surviving nodes, and $M_p$ is the expected number of all *possible pairs* among surviving nodes. Note that the last equality follows from the observation that DHTs investigated in this paper have symmetric nodes. Therefore, the routing topology of each node is statistically identical to each other. Thus, all $S_i$’s are identically distributed for all $i$’s: $E[S] = E[S_i] \ \forall i$.
Using the Hypercube Geometry as an Example {#ssec:hyper_example}
------------------------------------------
A simple application of the RCM method is illustrated for the CAN hypercubic routing system in Fig. \[fig:merge\_1\]-\[fig:merge\_3\]. The RCM steps involved are as follows:
**Step 1.** As reviewed in section \[sec:overview\], in a hypercube routing geometry [@Ratnasamy:CAN], the distance (in hops) between two nodes is their Hamming distance. Routing is greedy by correcting bits in any order for each hop.\
**Step 2.** Thus, for any random node $i$ in a hypercube routing system with identifier length of $d$ bits, we have the following distance distribution: $n(h) = \binom{d}{h}$. The justification is immediate: a node at $h$ hops away has a Hamming distance of $h$ bits with node $i$. Since there are $\binom{d}{h}$ ways to place the $h$ differing bits, there are $\binom{d}{h}$ nodes at distance $h$ (see Fig. \[fig:merge\_2\]).\
**Step 3.** The routing process can be modeled as a discrete time Markov chain (Fig. \[fig:merge\_3\] and \[fig:hypercube\_mc\]). The states $S_i's$ of the Markov chain correspond to the number of corrected bits. Note that there are only two absorbing states in the Markov chain: the failure state $F$ and the success state (i.e. $S_h$). Thus, the probability of successfully routing to a target node at distance $h$ hops away is given by the probability of transitioning from $S_0$ to $S_h$ in the Markov chain model: $$\begin{aligned}
p(h,q)&=&\mbox{Pr}(S_0 \rightarrow S_1 \rightarrow ... \rightarrow S_h) \nonumber \\
&=&\mbox{Pr}(S_0 \rightarrow S_1)\mbox{Pr}(S_1 \rightarrow S_2)...\mbox{Pr}(S_{h-1} \rightarrow S_h) \nonumber \\
&=&(1-q^h)(1-q^{h-1})...(1-q) \nonumber \\
&=&\displaystyle\prod^{h}_{m=1}(1-q^m) \label{eq:hypercube_eq}\end{aligned}$$ **Step 4.** Thus, the expected size of the reachable component is given as: $$E[S] = \displaystyle\sum_{h=1}^{d} n(h)p(h,q) = \displaystyle\sum_{h=1}^{d} \binom{d}{h}\displaystyle\prod^{h}_{m=1}(1-q^m)$$ **Step 5.** Using Eq. \[eq:routability\], we obtain the analytical expression for routability: $$\begin{aligned}
r &= \frac{\displaystyle\sum_{h=1}^{d} n(h)p(h,q)}{(1-q)2^d -1} \label{eq:routability_eq}\\
&= \frac{\displaystyle\sum_{h=1}^{d} \binom{d}{h}\displaystyle\prod^{h}_{m=1}(1-q^m)}{(1-q)2^d -1}\end{aligned}$$
[2]{}
[2]{}
. \[fig:compare\_fig\]
Summary of Results for other Routing Geometries {#ssec:summary_other}
-----------------------------------------------
Using the RCM method, the analytical expressions for the other DHT routing geometries can be similarly derived as for the hypercube routing geometry. In all the derivations, the majority of the work involves finding the expression for $p(h,q)$ through Markov chain modeling. Note that the analytical expressions derived in this section are compared with the simulation results obtained by Gummadi et al. [@Gummadi:impact] in Fig. \[fig:analy\_vs\_sim\] and \[fig:chord\_analy\_vs\_sim\].
For ease of exposition, we will use the notation $G(i,j)$, which denotes the probability that, starting at state $i$, the Markov chain ever visits state $j$. By any of the Markov chain models for the routing protocols, we note that $G(S_0,S_1) = 1-Q(h)$, $G(S_1,S_2) = 1-Q(h-1)$, and so forth, where the function $Q(m)$ can be thought of as the probability of failure at the $m$th phase of the routing process. As a result, all of the DHT systems under study have the property that the probability of successfully traveling $h$ hops or phases from the root node, $p(h,q)$, is given by the following common form: $$\begin{aligned}
p(h,q) &=& G(S_0,S_1)G(S_1,S_2)...G(S_{h-1},S_h) \nonumber \\
&=& \prod^{\small h}_{\small m=1}(1-Q(m)) \label{eq:common_eq}\end{aligned}$$ Using Eq. \[eq:routability\_eq\], we see that only the expressions for $n(h)$ and $Q(m)$ are needed to compute the routability of the DHT routing system under investigation. As a result, we will only provide the $n(h)$ and $Q(m)$ expressions for each system for conciseness.
### Tree
For the tree routing geometry, the routing distance distribution, $n(h)$, is $\binom{d}{h}$ by inspection. Furthermore, it is simple to show that $p(h,q) = (1-q)^h$ by examining the Markov chain model (see Fig. \[fig:plaxton\_mc\]). In sum, the expression for routability can be succinctly given as follows: $r = \frac{(2-q)^d-1}{(1-q)2^d-1}$
### XOR
As reviewed in section \[sec:overview\], connecting to a neighbor at an XOR distance of $[2^{d-i},2^{d-i+1}]$ is equivalent to choosing a neighbor by matching the first (i-1) bits of one’s identifier, flipping the $i$th bit, and choose random bits for the rest of the bits. Note that this is equivalent to how neighbors are chosen in the Plaxton-tree routing geometry. As a result, the $n(h)$ expression is given as: $n(h) = \binom{n}{h}$ just as in the tree case.
Now, let’s examine how the Markov chain model (Fig. \[fig:xor\_mc\]) is obtained: in this scenario, a message is to be routed to a destination $h$ phases away; starting at state $S_0$, state $S_1$ is reached if the optimal neighbor correcting the leftmost bit exists, which happens with probability $1-q$ ($S_i$ denotes the state that corresponds to the $i$th advanced phase). However, if all $h$ neighbors have failed (i.e. with probability $q^h$), the failure state $F$ is entered. Otherwise, the routing process can correct one of the lower order bits, which happens with probability $q(1-q^{h-1})$. Note that there is a maximum number of $h-1$ lower order bits that can be corrected in the first phase. All other transition probabilities can be obtained similarly. By inspecting the Markov chain model, we note that $G(S_0,S_1) = 1-Q_{xor}(h)$, $G(S_1,S_2) = 1-Q_{xor}(h-1)$, and so forth, where the function $Q_{xor}(m)$ is defined as follows: $$\begin{aligned}
Q_{xor}(m)&=&q^m + \sum_{k=1}^{m-1}q^m[\displaystyle\prod^{m-1}_{j=m-k} (1-q^j)] \label{eq:xor_q} \\
&\approx&q^{m}(m+\frac{q}{1-q}(q^{m-1}(m-1)-\frac{1-q^{m+1}}{1-q})) \nonumber \end{aligned}$$ The approximation is obtained by invoking the following: $1-x \approx e^{-x}$ for x small.
[2]{}
### Ring
In ring routing as implemented in Chord, when a node takes a suboptimal hop in the routing process, the progress made by taking this suboptimal hop is preserved in later hops. For example, consider the scenario that a message is to be routed to a node at a numeric distance that is $O(N)$ (i.e. the message is to be routed one full circle around the ring), and the fingers are connected to nodes that are half way across the ring, one quarter across the ring, etc. For the message’s first hop, it takes a suboptimal hop which takes the message only one quarter across the ring, because the finger that would have taken the message half way across the ring has failed. Then, for the message’s second hop, none of the finger connections has failed. Thus, the message takes an optimal hop which takes the message half way across the ring. Therefore, after two hops, the message is now three quarters of the way across the ring. Note that the progress made in the first suboptimal hop is this scenario is later preserved by a subsequent hop.
This property that suboptimal hops in ring routing contribute non-trivially to the routing process is not accounted for in the the Markov chain model as illustrated in Fig. \[fig:chord\_mc\]. The reason is that accounting for progress made by suboptimal hops would lead to an exponential blowup in the number of terms that we need to keep track of for computing $p(h,q)$. This simplified Markov chain model essentially makes the assumption that progress made by suboptimal hops do not contribute to the routing process. Therefore, the analytical expression for $p(h,q)$ using this model provides a *lower bound*.
The Markov chain model for ring routing \[fig:chord\_mc\] is very similar to the one for XOR routing (Fig. \[fig:xor\_mc\]). However, fundamental differences exist: first, when a suboptimal hop is taken in Chord, the number of next hop choices does not decrease. For example, in the first phase, there are $h$ choices for the next hop, thus the transition probabilities from the states in the first phase to the failure state are given by $q^h$. In contrast, the corresponding transition probabilities in Fig. \[fig:xor\_mc\] are given by $q^h$, $q^{h-1}$, and so forth. In addition, the maximum number of suboptimal hops in Chord is given by $2^{h-1}$, $2^{h-2}$ and so forth, while the corresponding transition probabilities in Fig. \[fig:xor\_mc\] are given by $h$, $h-1$, and so forth. This difference is due to the fact that in XOR routing, routing fails if all the lower order bits are resolved and the leftmost bit is not yet resolved. However, Chord does not have such restriction.The results for the ring routing geometry is derived by inspecting Fig. \[fig:chord\_mc\]: $$\begin{aligned}
Q_{ring}(m) &= q^m\sum_{k=0}^{2^{m-1}-1}[q(1-q^{m-1})]^k \\
&= q^{m}\frac{1-[q(1-q^{m-1})]^{2^{m-1}}}{1-q(1-q^{m-1})} \end{aligned}$$ In addition, one can easily see by inspection that the $n(h)$ expression for the ring geometry is given by: $n(h) = 2^{h-1}$.
### Symphony
Symphony’s Markov chain model (Fig. \[fig:symphony\_mc\]) is fundamentally different from the ones for XOR routing (Fig. \[fig:xor\_mc\]) and ring routing (Fig. \[fig:chord\_mc\]). Starting at $S_0$, one phase is advanced if any of the node’s shortcuts connects to the desired phase, which happens with probability $\frac{k_s}{d}$ where $k_s$ denotes the number of shortcuts. Alternatively, the routing fails if all of the node’s near neighbor and shortcut connections fail, which happens with probability $q^{k_n+k_s}$. The third possibility is taking a suboptimal hop, which happens with probability $1-\frac{k_s}{d}-q^{k_n+k_s}$. All other transition probabilities in the Markov chain can be similarly derived. Note that we approximate the maximum number of suboptimal hops by $\lceil \frac{d}{1-q} \rceil$.
For the Symphony routing geometry, we note that the expression for the $Q$’s is constant for all phases. The results are similarly derived as the other systems by inspecting Fig. \[fig:symphony\_mc\]: $$\begin{aligned}
Q_{sym} &= q^{k_n+k_s}\sum^{\lceil\frac{d}{1-q}\rceil}_{j=0}(1-\frac{k_s}{d}-q^{k_n+k_s})^j \nonumber \\
&\approx q^{k_n+k_s}( \frac{1-(1-\frac{k_s}{d}-q^{k_n+k_s})^{\frac{d}{1-q}+1}}{1-(1-\frac{k_s}{d}-q^{k_n+k_s})})
\label{eq:symphony_q}\end{aligned}$$ The symbols $k_n$ and $k_s$ denote the number of near neighbors and shortcuts respectively. Similarly to ring routing, the $n(h)$ expression for the Symphony routing algorithm is given by: $n(h) = 2^{h-1}$.
Scalability of DHT Routing Protocols under Random Failure {#sec:scalability}
=========================================================
{width="3.35"}
{width="3.35"}
[2]{}
For a DHT routing system to be scalable, its routability must converge to a non-zero value as the system size goes to infinity (Definition \[def:scalability\]). Alternatively, we examine the asymptotic behavior of $p(h,q)$ with $h$ set to the average routing distance in the system (i.e. $h=O(\log N)$ or $O(\log^2 N)$ for Symphony). Using Eq. \[eq:routability\_eq\], it is simple to show that the equivalent condition for scalability is as follows: $$\lim_{N \rightarrow \infty} p(h,q) = \lim_{h \rightarrow \infty} p(h,q) > 0 \ \mbox{for} \ 0 < q < 1-p_c
\label{scalable_cond}$$ Otherwise, the routing system is unscalable. In other words, the equivalent condition for system scalability states that as the number of routing hops to reach a destination node in the system approaches infinity, the probability of successfully routing to the destination node must not drop to zero for a non-zero node failure probability in the system.
As discussed in section \[ssec:summary\_other\], all of the DHT systems under study have the property that the probability of successfully traveling $h$ hops or phases from the root node is given by the following form: $$p(h,q) = \prod^h_{m=1} (1-Q(m))
\label{scalable_eq}$$ where $Q(m)$ can be thought of as the probability of failure at the $m$th phase of the routing process.
[(*From Knopp* [@Knopp:inf_series])]{} \[series\_thm\] If, for every $n$, $0 \le a_n < 1$, then the product $\prod (1-a_n)$ tends to a limit greater than 0 if, and only if, $\sum a_n$ converges.
Theorem \[series\_thm\] allows us to conveniently convert our problem of determining the convergence of an infinite product to a simpler infinite sum. Thus, $p(h,q)$ is convergent if and only if $\sum Q(m)$ converges.
Tree
----
The case for the tree routing geometry can be trivially shown to be *unscalable*: $$\lim_{h \rightarrow \infty}(1-q)^h = 0 \ \ \mbox{for}\ \mbox{any}\ q > 0$$
Hypercube
---------
For hypercube routing, $p(h,q)$ is given by $p(h,q) = \displaystyle\prod^{h}_{m=1}(1-q^m)$ (Eq. \[eq:hypercube\_eq\]). By invoking Theorem \[series\_thm\], it is trivial to see that $\sum q^m$ converges for $0 < q < 1-p_c$. Thus, the hypercube routing geometry is *scalable*.
XOR
---
In XOR routing, the $Q(m)$ expression given by Eq. \[eq:xor\_q\]. It is simple to show that the $Q(m)$ series involves only $q^m$ and $mq^m$ terms. Thus, $\sum Q(m)$ is convergent and the XOR routing scheme is *scalable*.
Ring
----
We will demonstrate that the ring routing geometry is also scalable by showing that the XOR results derived above is a lower bound for the ring geometry. We compare the Markov chain models for the ring geometry and the XOR geometry (Fig. \[fig:chord\_mc\] and Fig. \[fig:xor\_mc\]). We note that the transition probabilities for the suboptimal hops in ring are strictly greater than the corresponding probabilities for XOR. For example, in Fig. \[fig:chord\_mc\], note that the transition probabilities for $S_0 \rightarrow (0,1)$, $(0,1) \rightarrow (0,2)$ and so forth are given by $q(1-q^{h-1})$. These probabilities are strictly greater than the corresponding transition probabilities in Fig. \[fig:xor\_mc\]. Thus, by comparing these two Markov chain models, it is simple to show that the $p(h,q)$ expression for the ring routing geometry is strictly greater than the $p(h,q)$ expression for XOR routing. Thus, the ring routing geometry is also *scalable*.
Symphony
--------
In Symphony routing, the $Q(m)$ expression given by Eq. \[eq:symphony\_q\]. Note that the $Q(m)$ expression is given by a constant term. Therefore, $\sum Q(m)$ is divergent and the Symphony routing scheme is *unscalable*.\
\
Please refer to Fig. \[fig:asymptotic\] and \[fig:scale\_curves\] for plots of the above scalability results.
Concluding Remarks {#sec:conclusion}
==================
In this work, we present the reachable component method (RCM) which is an analytical framework for characterizing DHT system performance under random failures. The method’s efficacy is demonstrated through an analysis of five important existing DHT systems and the good agreement of the RCM predictions for each system with simulation results from the literature. Researchers involved in P2P system design and implementation can use the method to assess the performance of proposed architectures and to choose robust routing algorithms for application development. In addition, although the analysis presented in this work assumes fully-populated identifier spaces, analytical results for real world DHTs with non-fully-populated identifier spaces can be similarly derived. Detail investigation in this area will be left for future work.
One of the most interesting implications of this analysis is that in the large-network limit, some DHT routing systems are incapable of routing to a constant fraction of the network if there is any non-zero probability of random node failure. These DHT algorithms are therefore considered to be *unscalable*. Other algorithms are more robust to random node failures, allowing each node to route to a constant fraction of the network even as the system size goes to infinity. These systems are considered to be *scalable*. Now that real DHT implementations have on the order of millions of highly transient nodes, it is increasingly important to characterize how the size and failure conditions of a DHT will affect its routing performance.
Acknowledgments
===============
We would like to thank Krishna Gummadi for furnishing the simulation results for DHT systems. We wish to thank Nikolaos Kontorinis for his feedback and suggestions. This work was in part supported by the NSF grants ITR:ECF0300635 and BIC:EMT0524843.
[^1]: The term *static* refers to the assumption that a node’s routing table remains unchanged after accounting for neighbor failures.
|
---
author:
- 'Elise E. Cawley'
bibliography:
- 'ref.bib'
date: 'July 17, 1991'
title: 'The [Teichmüller]{} space of an Anosov diffeomorphism of $T^{2}$'
---
Introduction
============
In this paper we consider the space of smooth conjugacy classes of an Anosov diffeomorphism of the two-torus. A diffeomorphism $f$ of a manifold $M$ is Anosov if there is a continuous invariant splitting of the tangent bundle $TM = E^{s} \bigoplus E^{u}$, where the subbundle $E^{s}$ is contracted by $f$, and $E^{u}$ is expanded. More precisely, if $\| \cdot \|$ is a Riemannian metric on $M$, there are constants $c > 0$ and $\lambda < 1$ such that $$\begin{aligned}
\| Df^{n} \cdot v \| & \leq & c \lambda^{n} \| v \|
\ \mbox{for v in}
\ E^{s} \\
\| Df^{-n} \cdot v \| & \leq & c \lambda^{n} \| v \|
\ \mbox{for v in}
\ E^{u} \end{aligned}$$
for all positive integers $n$. If $f$ is $C^{1 + \alpha}$ for $0 < \alpha < 1$, it can be shown that the splitting is in fact Hölder continuous.
The only 2-manifold that supports an Anosov diffeomorphism is the 2-torus [@F]. Franks and Manning showed that every Anosov diffeomorphism of ${{\bf T}}^{2}$ is topologically conjugate to a linear example; that is, to an automorphism defined by a hyperbolic element of $GL(2,{\bf Z})$ whose determinant has absolute value one [@F], [@Ma]. Consider $f$ and $g$ which are topologically conjugate, so there is a homeomorphism taking the orbits of $f$ to the orbits of $g$. If the conjugacy is in fact smooth, then $f$ and $g$ must have the same expanding and contracting eigenvalues at corresponding periodic points. De la Llave, Marco, and Moriyon have shown that the eigenvalues at periodic points are a [*complete*]{} smooth invariant: if the eigenvalues of $f$ are the same as the eigenvalues of $g$ at periodic points that correspond under a topological conjugacy, then the conjugacy is smooth [@L],[@MM1], [@MM2].
The question arises: what sets of eigenvalues occur as the Anosov diffeomorphism ranges over a topological conjugacy class? The information in the set of expanding eigenvalues of $f$ is recorded by a Hölder cyclic cohomology class associated to $f$. The real-valued function $$x \mapsto \phi_{u}(x) = -{\rm log}
\parallel Df(x) \parallel _{u}\;,$$ where $\parallel Df \parallel _{u}$ is the Jacobian of $f$ along the unstable bundle $E^{u}$, can be described as a “cocycle” over $f$. The cohomology class of this cocycle, that is, its residue class modulo the space of coboundaries $x \mapsto
u(f(x)) - u(x)$, is independent of the choice of Riemannian metric on $M$. Moreover, the cohomology class of a Hölder cocycle defined over an Anosov diffeomorphism is determined by the sums of values of the cocycle over the various periodic orbits, which for $\phi _{u}$ is simply minus the logarithm of the expanding eigenvalue at the periodic point. Similarly, the information in the set of contracting eigenvalues is recorded by the cohomology class of the cocycle defined by $\phi_{s} = {\rm log} \parallel
Df \parallel _{s}$ where $\parallel Df \parallel _{s}$ is the Jacobian of $f$ along the stable bundle $E^{s}$. The sign convention that makes these cocycles negative is chosen for consistency with the notation of Bowen, Ruelle, and Sinai in the theory of Gibbs and equilibrium measures for Anosov systems.
The question asked above can be reformulated: what pairs of cohomology classes (one determined by the expanding eigenvalues, and one by the contracting eigenvalues) occur as the diffeomorphism ranges over a topological conjugacy class? The cohomology is defined over the entire conjugacy class by pulling the Jacobian cocycles back to a fixed representative of the conjugacy class. The purpose of this paper is to answer this question: [*all*]{} pairs of Hölder reduced cohomology classes occur. (The reduced cohomology is the cyclic cohomology divided out by the constant cocycles. The pair of reduced cohomology classes is still sufficient information to determine the smooth conjugacy class.)
The Teichmüller space $T(f)$ of an Anosov diffeomorphism $f$ is defined to be the set of smooth structures preserved by the topological dynamics determined by $f$. This is the smooth category version of the Teichmüller space of a rational map, which was studied by McMullen and Sullivan [@MS]. We show that for an Anosov diffeomorphism of ${{\bf T}}^{2}$, there is a natural bijection from the Teichmüller space to the product $G(f) \times G(f^{-1})$, where $G(f)$ denotes the real-valued [Hölder]{} reduced cyclic cohomology over $f$.
The main theorem is:
Let $f:{\bf T}^{2} \rightarrow {\bf T}^{2}$ be an Anosov diffeomorphism. Then there is a natural bijection $T^{1 + H}(f) \leftrightarrow
G(f) \times G(f^{-1})$.
An easy corollary will be
Let $f:{\bf T}^{2} \rightarrow {\bf T}^{2}$ be a volume preserving Anosov diffeomorphism. Then there is a natural bijection $T_{vol}^{1 + H}(f) \leftrightarrow G(f)$.
Here $G(f)$ is the Hölder reduced cyclic cohomology over $f$, where the Hölder exponent is allowed to vary between $0$ and $1$. More precisely: let $C^{\alpha}$ denote the space of $\alpha$-[Hölder]{} functions on $T^2$. Let $C^{H} = \cup_{\alpha \in (0,1)} C^{\alpha}$. Then $G(f)$ is the quotient of $C^{H}$ by the subspace of “almost coboundaries” $$x \mapsto u(f(x)) - u(x) + K\;,$$ where $u$ is a function on $T^2$ and $K \in {\bf R}$. (It follows in this setting that $u \in C^{H}$). $T^{1 + H}(f)$ is the Teichmüller space of $C^{1+H}$ invariant smooth structures. An Anosov diffeomorphism is called [*volume preserving*]{} if it admits an invariant measure that is absolutely continuous with repect to Lebesgue. $T_{vol}^{1 + H}
(f)$ is the restriction of this Teichmüller space to the volume preserving elements. See the appropriate sections for precise definitions.
The main theorem can be restated as follows:
[**Theorem $1'$**]{} [*Let $L$ be a hyperbolic automorphism of the torus ${{\bf T}}^{2}$. Given two [Hölder]{}functions $\phi_{u}$ and $\phi_{s}$ from ${{\bf T}}^{2}$ to $\bf R$, there exist uniquely defined constants $P_{u}$ and $P_{s}$, and a unique $C^{1 + H}$ smooth structure on ${{\bf T}}^2$ which is preserved by $L$, and determines the cohomology classes $< \phi_{u} - P_{u}>$ and $<\phi_{s} - P_{s}>$. Moreover, $L$ is Anosov in this smooth structure.*]{}
[**Remark.**]{} The [Hölder]{} exponent of the new smooth structure depends on the [Hölder]{} norms and exponents of the pair of functions, and on a dynamically defined norm (the [*Bowen*]{}, or [*variation*]{} norm) of the cohomology classes. [@B3]
[**Remark.**]{} The topological conjugacy between $C^{1 + H}$ Anosov diffeomorphisms is in fact [Hölder]{} continuous [@Mn]. Therefore this description is independent of the choice of “base point” as the linear mapping.
Let $\lambda_{u}$ and $\lambda_{s}$ denote the unstable and stable eigenvalues at a periodic point of period $n$. The numbers $|\lambda_{u}|^{1/n}$ and $|\lambda_{s}|^{1/n}$ can be prescribed arbitrarily on any finite set of periodic points, up to (non-unique) constant factors ${\rm exp}(-P_{u})$ and ${\rm exp}(P_{s})$, respectively.
[**Remark.**]{} The correction factors in the corollary are asymptotically unique along any sequence of periodic point sets $P_i$ with the property that the normalized dirac mass on $P_{i}$ converges to Haar measure (which is also the measure of maximal entropy) on ${\bf
T}^{2}$.
We give a sketch of the proof of Theorem 1. The main step is to show that a cohomology class over an Anosov diffeomorphism $f$ determines a canonical invariant transverse measure class to the stable foliation. This is constructed using the Gibbs measure class defined by the cohomology class. When the stable foliation is co-dimension 1, the transverse measure class can be interpreted as a transverse smooth structure. In dimension 2, when both foliations are co-dimension 1, the two transverse structures (determined by the two given cohomology classes) define a product smooth structure, which is invariant by $f$.
Ruelle and Sullivan [@RS], and Sinai [@Si], gave a transverse interpretation of a particular Gibbs measure, namely that associated to the constant cocycle. In this case, and only this case, one has a transverse measure, as opposed to a transverse measure class. Under the isomorphism of Theorem 1, this corresponds to the linear member of the topological conjugacy class. The present paper shows how to extend the decomposition of a Gibbs measure into transverse stable and transverse unstable part, as described in [@RS] for the constant cocycle measure, to the general case.
The organization of the paper is as follows. In section 2 we define the Teichmüller space $T(f)$. In section 3 we recall facts about cyclic cohomology over a ${\bf Z}$ action, over a foliation or an equivalence relation, and describe the Jacobian and Radon-Nykodym cocycles. In Section 4, the map which gives the isomorphism in Theorem 1 is described. Section 5 describes the cocycle properties of Gibbs measures, and collects some necessary results. Section 6 gives the statements of the transverse structure realization results, and proves Theorem 1 assuming these. The main body of the proof is in Section 7, where the tranverse measure class is constructed. Section 8 gives a simple description of the smooth structure defined by a pair of cohomology classes in terms of explicit coordinates on rectangles in a Markov partition.
[**Acknowledgements.**]{} It is a pleasure to thank Dennis Sullivan for many hours of conversation, for inspiration and encouragement. Jack Milnor made many helpful comments on the manuscript. I also thank IHES and the Institute for Mathematical Sciences at Stony Brook for their hospitality while this work was being completed.
The Teichmüller space of an Anosov diffeomorphism
=================================================
Let $f:M \rightarrow M$ be a $C^{r}$ Anosov diffeomorphism, $0 < r \leq
\omega$. The $C^{r}$ [Teichmüller]{} space $T^{r}(f)$ is defined as follows. Consider triples $(h, N, g)$ where $g:N \rightarrow N$ is a $C^{r}$ Anosov diffeomorphism, and $h:M \rightarrow N$ is a homeomorphism satisfying $g \circ h = h \circ
f$. We call such a triple a [*marked Anosov diffeomorphism modeled on $f:M \rightarrow M$*]{}. Two such triples $(h_{1},N_{1},g_{1})$ and $(h_{2},N_{2},g_{2})$ are [*equivalent*]{} if the homeomorphism $s:N_{1} \rightarrow N_{2}$ defined by $s \circ h_{1} = h_{2}$ is in fact a $C^{1}$ diffeomorphism. The [Teichmüller]{} space $T^{r}(f)$ is the space of equivalence classes of triples.
(195,165)
(13,90)[$f$]{} (30,90)[$M$]{} (150,90)[$s$]{} (75,135)[$h_{1}$]{} (75,45)[$h_{2}$]{} (135,150)[$N_{1}$]{} (135,45)[$N_{2}$]{} (163,150)[$g_{1}$]{} (163,45)[$g_{2}$]{}
(60,97.5)[(3,2)[68]{}]{}
(60,82.5)[(2,-1)[68]{}]{} (142.5,135)[(0,-1)[75]{}]{}
(24,93)[(6,6)\[l\]]{} (24,93)[(6,6)\[t\]]{} (29,90)[(1,0)[0]{}]{}
(154,153)[(6,6)\[r\]]{} (154,153)[(6,6)\[t\]]{} (149,150)[(-1,0)[0]{}]{}
(154,48)[(6,6)\[r\]]{} (154,48)[(6,6)\[t\]]{} (149,45)[(-1,0)[0]{}]{}
We also define the $C^{1 + H}$ [Teichmüller]{} space $T^{1 + H}(f)$. Let $f:M \rightarrow M$ be a $C^{1+\alpha}$ diffeomorphism for some $0 <
\alpha < 1$. Consider all marked Anosov diffeomorphisms $g:N \rightarrow
N$ modeled on $f$ where $g$ is $C^{1 + \alpha^{\prime}}$ for some $0 < \alpha^{\prime} < 1$ (that is, the [Hölder]{} exponent of $g$ is not necessarily the same as that of $f$). Then $T^{1 + H}$ is the space of equivalence classes, where equivalence is defined just as it was in the $C^{r}$ category.
Cyclic cohomology.
==================
Group actions and the Jacobian and Radon-Nykodym cocycles.
----------------------------------------------------------
We describe here the notion of cocycle over a group action, and the associated notions of coboundary and cohomology (see [@Z] and [@K].) We work in the topological category since all cocycles we are interested in have at least this degree of regularity. We consider $\Gamma$, a locally compact, second countable group, and a continuous right action of $\Gamma$ on a topological space $M$. We will be especially interested in the case of a ${\bf Z}$ action defined by a diffeomorphism of a manifold $M$.
A real-valued (additive) cocycle over the action of $\Gamma$ is a continuous map $$\Phi:M \times \Gamma \rightarrow {\bf R}$$ satisfying: $$\Phi(x,\gamma_{1}\cdot\gamma_{2}) = \Phi(x,\gamma_{1}) +
\Phi(x \cdot \gamma_{1}, \gamma_{2})$$
Here $x \rightarrow x \cdot \gamma$ denotes the action of $\gamma$ on the point $x$. A [*coboundary*]{} is a cocycle of the form $\Phi(x,\gamma) =
u(x\cdot \gamma) - u(x)$ where $u:M \rightarrow {\bf R}$ is a continous function. The function $u$ is called the [*transfer function*]{} of the coboundary $\Phi$. Two cocycles are [*equivalent*]{} or [*cohomologous*]{} if their difference is a coboundary. The cohomology over the action of $\Gamma$ is the space of equivalence classes of cocycles. A cocycle of the form $\Phi(x,\gamma) = u(x\cdot \gamma) - u(x) +
K(\gamma)$, where $K:\Gamma \rightarrow {\bf R}$ is a homomorphism, is called an [*almost coboundary*]{}. The space of cocycles modulo almost coboundaries is the [*reduced cohomology*]{} over the action. If $\Phi$ is a cocycle, we denote its cohomology class and reduced cohomology class $<\Phi>$ and $<\Phi>_{*}$, respectively. When the space $M$ has additional structure, we can consider e.g. [Hölder]{}, Lipshitz, or smooth cocycles (and coboundaries and cohomology).
The cohomology equivalence relation on cocycles has the following meaning. Let $F(M,{\bf R})$ be the space of continuous maps from $M$ to ${\bf R}$. Then a cocycle $\Phi$ defines an action of $\Gamma$ on $F(M,{\bf R})$ as follows. If $T:M \rightarrow {\bf R}$, then $(\gamma \cdot T)(x) =
\Phi(x,\gamma) + T(x \cdot \gamma)$. Note that if $\Phi$ is the identically $0$ cocycle, then this action is just the usual pull-back of functions by the group action. When the two cocycles $\Phi$ and $\Psi$ are cohomologous, the actions they define are equivalent. If $u$ is the transfer function of the coboundary that relates $\Phi$ to $\Psi$, then the map $U:F(M,{\bf R}) \rightarrow F(M,{\bf R})$ defined by $T \rightarrow T + u$ is an isomorphism which conjugates the action defined by $\Phi$ to the action defined by $\Psi$.
[**Example 1: The Jacobian cocycle.**]{} Suppose that $\Gamma$ acts by diffeomorphisms on a Riemannian manifold $M$. We define the (additive) Jacobian cocycle $J:M \times \Gamma \rightarrow {\bf R}$ by $J(x,\gamma) = {\rm log} \parallel D\gamma (x) \parallel$ where $\parallel
\cdot \parallel$ is the Riemannian metric. The chain rule for differentiation is precisely the cocycle condition. If we choose a new Riemannian metric $\parallel \cdot \parallel _{1}$ on $M$, then the new Jacobian cocycle is cohomologous to the original one. The transfer function is simply the logarithm of the ratio of volume elements with repect to the two metrics. Hence there is a cohomology class, the Jacobian class $<J>$, naturally associated to a smooth group action on a smooth manifold. The apriori coarser invariant, the reduced Jacobian class $<J>_{*}$, is also defined.
[**Example 2: The Radon-Nykodym cocycle.**]{} Suppose $\Gamma$ acts on the measure space $(M,\mu)$, quasi-preserving the measure $\mu$. The (additive) Radon-Nykodym cocycle $R:M \times \Gamma \rightarrow
{\bf R}$ is defined by $R(x,\gamma) = {\rm log}
\frac{d\mu(\gamma(x))}{d\mu(x)}$. (Since we have retricted to the topological category, we assume that the Radon-Nykodym is continuous.) Again the cocycle condition is the chain rule. If $\nu$ is a measure that is equivalent to $\mu$, with Radon-Nykodym derivative $r = \frac{d\nu}{d\mu}$, then the corresponding cocycles are cohomologous, via the transfer function $u = {\rm log (r)}$. So there is a cohomology class, the Radon-Nykodym class $<R>$, and a reduced cohomology class $<R>_{*}$, naturally associated to an action that preserves a measure class.
We now focus on the case $\Gamma = {\bf Z}$. A cocycle over a ${\bf Z}$ action is determined by its values on the generator: $\phi(x) =: \Phi(x,1)$. The cocycle condition implies that $\Phi(x,n) = \sum_{k=0}^{n-1} \phi(x\cdot
k)$ where $x \cdot k$ denotes the action of $k$ on the point $x$. If $f:M \rightarrow M$ is the action of the generator, then we write $\Phi(x,n) = \sum_{k=0}^{n-1} \phi \circ f^{k}(x)$. Hence the cocycles over a ${\bf Z}$ action can be identifed with continuous functions on $M$. A coboundary is a function of the form $u \circ f - u$. An almost coboundary is a function of the form $u \circ f - u + K$, where $K$ is a constant.
[**Example 3: the BRS classes of an Anosov diffeomorphism.**]{} Suppose $f:M \rightarrow M$ is an Anosov diffeomorphism, with $TM = E^{s}
\bigoplus E^{u}$. Let $\parallel \cdot \parallel$ be a Riemannian metric on $M$. Since the subbundle $E^{u}$ is preserved by $f$, we can define the [*unstable Jacobian cocycle*]{} to be the cocycle over the action of $f$ determined by the function $\phi_{u}(x) = -{\rm log} \parallel Df(x)
\parallel _{u}$. Here $\parallel Df(x) \parallel _{u}$ denotes the Jacobian in the unstable direction. (The minus sign is a convention in the theory of Bowen, Ruelle, and Sinai.) Similarly we can define a stable Jacobian cocycle $\phi_{s}(x) = {\rm log} \parallel
Df(x)
\parallel_{s}$. (With this sign convention, the stable Jacobian cocycle of $f$ is the same as the unstable Jacobian cocycle of $f^{-1}$.) If $g$ is an Anosov diffeomorphism which is smoothly conjugate to $f$, say $g \circ h = h \circ f$, then the corresponding Jacobian cocycles are cohomologous. Namely, $\phi_{u}^{g} \circ h = \phi_{u}^{f} + u \circ f -
u$ where $u = -{\rm log} \parallel Dh \parallel_{u}$, and $\phi_{s}^{g} \circ h =
\phi_{s}^{f} +
v \circ f - v$ where $v = {\rm log} \parallel Dh \parallel_{s}$. In other words, the smooth conjugacy class of $f$ naturally determines a pair of cohomology classes, $(<\phi_{u}>,<\phi_{s}>)$, which we will refer to as the unstable and stable [*BRS classes*]{} (for Bowen, Ruelle, and Sinai) of $f$. The corresponding reduced classes will be referred to as the reduced BRS classes of $f$.
Cocycles over a foliation.
--------------------------
It will be useful in what follows to have the notion of a cocycle over a foliation $\cal F$ of a manifold $M$. The definition depends on the [*graph*]{}, or holonomy groupoid, of the foliation, which was constructed by Winkelnkemper [@Co],[@W]. An element $\gamma$ of the graph $GR({\cal F})$ is a pair of points of $M$, $x = s(\gamma)$ and $y = r(\gamma)$, together with an equivalence class of smooth paths $\gamma(t)$ from $x$ to $y$, tangent to the foliation. Two paths $\gamma_{1}(t)$, $\gamma_{2}(t)$ from $x$ to $y$ are equivalent if the holonomy of the path $\gamma_{2} \circ \gamma_{1}^{-1}$ from $x$ to $x$ is the identity. There is a natural composition law on $GR$, defined when the endpoint of one pair is the first point of another pair. A cocycle over the foliation $\cal F$ is a function $$\Phi:GR({\cal F}) \rightarrow {\bf R}$$ such that $$\Phi(\gamma_{1} \circ \gamma_{2}) = \Phi(\gamma_{1}) + \Phi(\gamma_{2})$$ whenever the composition $\gamma_{1} \circ \gamma_{2}$ is defined. A coboundary is a cocycle of the form $\Phi(\gamma) = u(r(\gamma)) -
u(s(\gamma))$ where $u$ is a function on $M$. If the leaves of $\cal F$ have trivial holonomy, then the point $\gamma \in
GR({\cal F})$ depends only on the pair $x = r(\gamma)$ and $y = s(\gamma)$. In this case we will use the notation $\Phi(x \rightarrow y)$ for the cocycle.
Cocycles over an equivalence relation
-------------------------------------
Let ${\cal F} \subset M \times M$ be defined by an equivalence relation $\sim$ on $M$. Then a cocycle over ${\cal F}$ is a function $$\Phi : {\cal F} \rightarrow {\bf R}$$ satisfying $$\Phi(x,z) = \Phi(x,y) + \Phi(y,z)$$ whenever $y$ and $z$ belong to the equivalence class of $x$. See [@HK] A coboundary is a cocycle of the form $\Phi(x,y) = u(y) - u(x)$ where $u: M \rightarrow {\bf R}$.
The Bowen-Ruelle-Sinai map
==========================
Let $f:{{\bf T}}^{2} \rightarrow {{\bf T}}^{2}$ be a $C^{1 + \alpha}$ Anosov diffeomorphism. Let $G(f)$ denote the [Hölder]{} reduced cohomology over $f$, where the [Hölder]{} exponent is allowed to vary. The Bowen-Ruelle-Sinai map, $$BRS:T^{1 + H}(f) \rightarrow G(f) \times G(f^{-1})$$ is defined as follows. Let $(h,N,g)$ be a representative of a point in the [Teichmüller]{} space $T^{1 + H}(f)$. Let $\phi_{u}(g)$ and $\phi_{s}(g) = \phi_{u}(g^{-1})$ be the unstable and stable Jacobian cocycles of $g$. We map the [Teichmüller]{} point to the pair of reduced cohomology classes $(<\phi_{u}(g) \circ h >_{*}, < \phi_{u}(g^{-1}) \circ
h>_{*})$. Because the topological conjugacy between two $C^{1 + \alpha}$ Anosov diffeomorphisms is always [Hölder]{}, these are [Hölder]{} cohomology classes. The image point is independent of the choice of representative, because a smooth conjugacy changes the Jacobian cocycles by a coboundary.
The map $BRS$ is injective.
[**Remark.**]{} Two points need to be added to the theorem proved by de la Llave, Marco, and Moriyon ([@L],[@MM1],[@MM2]) to give the stated result. They consider $C^{r}$ diffeomorphisms where $2 \leq r \leq \omega$. They prove that the pair of BRS cohomology classes is a complete $C^{r}$ conjugacy invariant. The first point is that if $\phi$ and $\psi$ are BRS cocycles over an Anosov diffeomorphism whose difference is an almost coboundary, then in fact the difference is a coboundary. (See the remarks about the [*pressure*]{} in the next section.) The second point is that the $C^{1 + H}$ smoothness case is simpler than the higher smoothness case, because the foliations have $C^{1 + H}$ transverse smoothness. An easy modification of the ideas of de la Llave, Marco, and Moriyon proves that the BRS cohomology classes are a complete $C^{1 + H}$ conjugacy invariant.
To prove Theorem 1, we need to show that the map $BRS$ is [*surjective*]{}. That is, given a pair of reduced cohomology classes $(<\phi_{u}>_{*}, <\phi_{s}>_{*})$ over $f$, we need to construct a marked Anosov diffeomorphism $(h,N,g)$ modeled on $f$, whose pair of unstable and stable reduced BRS cohomology classes, pulled back to $f:M \rightarrow M$ is this pair.
This problem can be restated as follows. We consider the smooth torus $M$ to be defined by a $C^{1 + \alpha}$ system of charts $(U_{\beta},
\eta_{\beta})$ on a topological torus ${{\bf T}}^{2}$. The mapping $f$ is defined on the topological torus, and is assumed to define a $C^{1 +
\alpha}$ Anosov diffeomorphism when viewed in the smooth charts $(U_{\beta},\eta_{\beta})$. Given a pair of reduced cohomology classes $(<
\phi_{u}>_{*},<\phi_{s}>_{*})$ over $f$, the problem is to construct a new smooth system of charts $(U_{\beta}^{\prime},
\eta_{\beta}^{\prime})$ on the topological torus ${{\bf T}}^{2}$, with the property that in these charts the mapping $f$ is $C^{1 + \alpha^{\prime}}$, for some $0 < \alpha^{\prime} < 1$, and the reduced BRS cohomology classes in this smooth structure are $<\phi_{u}>_{*}$ and $<\phi_{s}>_{*}$. In other words, we identify the underlying point sets of the smooth tori $M$ and $N$ via the homeomorphism $h$, and vary the point in [Teichmüller]{} space by varying the smooth structure on this point set.
[**Proof of Theorem 2 from Theorem 1.**]{} An Anosov diffeomorphism is volume preserving if and only if the forward and backward BRS cohomology classes coincide (under the canonical identification of the forward and backward cohomology) [@B]. Hence the BRS map restricted to the volume preserving diffeomorphisms maps onto the “diagonal” in $G(f) \times G(f^{-1})$, which is naturally identified with $G(f)$.
Gibbs measures and associated cocycles
======================================
In this section we recall some properties of Gibbs measures. We emphasize the dynamically defined [*equivalence relations*]{} and associated cocycles that define a Gibbs measure. The transverse measure class constructed in this paper can be viewed as a “transverse Gibbs measure.” It is obtained by focusing on a different dynamically defined equivalence relation and its associated cocycle.
Gibbs measures are defined for dynamical systems with a local product structure. See Baladi’s thesis [@Ba] for a nice exposition and proofs of some basic results for the general case. Here we only need consider the two classes: Anosov diffeomorphisms and subshifts of finite type. The latter arise because an Anosov diffeomorphism has a presentation as a quotient of a subshift of finite type.
Let $A$ be an $r \times r$ matrix of $0$’s and $1$’s. Consider the setof bi-infinite sequences $\Sigma_{A} = \lbrace {\bf x} \rbrace$, where ${\bf x} = \ldots x_{-k}x_{-k+1}\ldots x_{-1}x_{0}x_{1}
\ldots x_{l}x_{l+1}\ldots$ is in $\Sigma_{A}$ if and only if $x_{i} \in \lbrace 1,\ldots,r \rbrace$ and $A_{x_{i}x_{i+l}} = 1$ for all $i$. The shift map $\sigma:\Sigma_{A}
\rightarrow
\Sigma_{A}$ is defined by $\sigma({\bf x}) = {\bf y}$ where $y_{i} =
x_{i+1}$. We give $\Sigma_{A}$ the topology defined by the metric $d({\bf x},{\bf y}) = \Sigma_{x_{i} \neq y_{i}}
2^{-| i |}$. The metric space $\Sigma_{A}$ together with the shift map is called the [*subshift of finite type*]{} defined by $A$.
If ${\bf x} \in \Sigma_{A}$, we define the [*stable set*]{} $W^{s}({\bf x})$ to be the set of all ${\bf y} \in \Sigma_{A}$ such that $d(\sigma^{n}
{\bf x},\sigma^{n}{\bf y}) \rightarrow 0$ as $n \rightarrow \infty$. Similarly we define the [*unstable set*]{} $W^{u}({\bf x})$ to be the set of ${\bf y} \in
\Sigma_{A}$ such that $d(\sigma^{-n}{\bf x},\sigma^{-n}{\bf y}) \rightarrow
0$ as $n \rightarrow \infty$. A [*local stable set*]{} $W^{s}_{\epsilon}({\bf x})$ is the set of points ${\bf y}$ such that $d(\sigma^{n}({\bf x}),\sigma^{n}({\bf y})) < \epsilon$ for $n \geq 0$. Local unstable sets are defined similarly. A stable set has an intrinsic topology defined by the metric $d^{-}({\bf x},{\bf y}) =
\sum_{x_{i} \neq y_{i}} 2^{i}$. Similarly, an unstable set has an intrinsic topology defined by the metric $d^{+}({\bf x},{\bf y}) = \sum_{x_{i} \neq y_{i}} 2^{-i}$. We will refer to the collection of stable (unstable) sets as the stable (unstable) foliation of $\Sigma_{A}$.
If $f: M \rightarrow M$ is an Anosov diffeomorphism, the stable foliation ${\cal W}^{s}$ and unstable foliation ${\cal W}^{u}$ are defined similarly, and have tangent distributions $E^s$ and $E^u$, respectively.
Anosov diffeomorphisms and subshifts of finite type have a [*local product structure*]{}. For every point $x$ in $M$ (or $\Sigma_{A}$) there is a neighborhood $U$ of $x$, and a homeomorphism $$u: W^{s}_{\epsilon}(x) \times W^{u}_{\epsilon}(x) \rightarrow U$$ that takes verticals $\lbrace w \rbrace \times W^{u}_{\epsilon}(x)$ onto local unstable sets, and horizontals $W^{s}_{\epsilon}(x) \times
\lbrace z \rbrace$ onto local stable sets.
There is an equivalence relation on $M$, and $\Sigma_{A}$, defined by the pair of foliations ${\cal W}^{s}$ and ${\cal W}^{u}$. Namely, $$x \sim y \Leftrightarrow x \in W^{s}(y) \cap W^{u}(y).$$ If $x \sim y$, then there are neighborhoods $U_{x}$ of $x$ and $U_{y}$ of $y$, and a homeomorphism $$\theta : U_{x} \rightarrow U_{y}$$ such that $z \sim \theta (z)$ for all $z \in U_{x}$. These are called [*conjugating homeomorphims*]{}. The pseudogroup of conjugating homeomorphisms generates this equivalence relation (referred to in the sequel as the Gibbs equivalence relation).
The following definition is due to Capocaccia [@Ca].
Let $f: M \rightarrow M$ be Anosov. Let $\phi : M \rightarrow {\bf R}$ be continuous. A measure $\mu$ on $M$ is a [*Gibbs measure for $\phi$*]{} if $${\rm log}(\frac{d\mu(\theta(x))}{d\mu(x)}) = \sum_{k = - \infty}^{\infty}
(\phi \circ f^{k} ( \theta(x))
- \phi \circ f^{k}(x))$$ for every conjugating homeomorphism $\theta$.
It is implicit in the definition that both sides of the equation are well-defined. The left hand side of the equation is the logarithmic Radon-Nykodym cocycle associated to the Gibbs equivalence relation and the measure $\mu$. The function $\phi$ should be regarded as a cocycle over $f$. If $\phi$ is changed by an almost coboundary $u \circ f - u + K$ the expression on the right hand side does not change. Hence a Gibbs measure is associated to a reduced cohomology class over $f$, and is defined by an associated cocycle (referred to as the Gibbs cocycle) over the Gibbs equivalence relation. The definition of Gibbs measure in the subshift of finite type case is exactly analagous.
Now we describe the cocycle properties of the transverse measure class we are going to construct. Instead of the Gibbs equivalence relation, we consider only the stable foliation, and the corresponding holonomy pseudogroup. If $\phi : M \rightarrow {\bf R}$ is continuous, and if $$\Phi(x \rightarrow y) =: \sum_{k = 0}^{\infty}
(\phi \circ f^{k} (y)
- \phi \circ f^{k}(x))$$ is finite whenever $x \in W^{s}(y)$, and this expression defines a continuous function of $x$ on a small transversal to ${\cal W}^{s}$, then $\Phi$ is a cocycle over $W^{s}$, which we will refer to as the transverse Gibbs cocycle. Moreover, if $\phi$ is changed by an almost coboundary $u \circ f - u + K$, then $\Phi$ changes by the coboundary $U(x \rightarrow y) = -u(y) + u(x)$. The transverse Gibbs measure class to be constructed will have the following properties. We associate to a reduced cohomology class $< \phi >_{*}$ over the mapping the cohomology class $< \Phi >$ over the stable foliation defined by equation 5.1. If $\mu$ is a representative measure on a small transversal $\tau$, then the logarithmic Radon-Nykodym cocycle of $\mu$ over the holonomy ${\rm hol}: \tau \rightarrow \tau$ will be of the form $\Phi(x \rightarrow y) + u(y) - u(x)$, where $u: \tau \rightarrow {\bf R}$ is a continuous function. (In fact $u$ will be [Hölder]{} but we postpone the discussion of this until later.)
The following result of Bowen leads us to consider [Hölder]{} cocycles over $f$.
[@B] If $\phi : M \rightarrow {\bf R}$ is [Hölder]{}, then the associated Gibbs cocycle and transverse Gibbs cocycle exist, and are [Hölder]{}.
That is, the Gibbs and transverse Gibbs cocycles are [Hölder]{} on the domains of the homeomorphisms in the associated pseudogroups. (In fact they are [Hölder]{} cocycles over the relevant [*metric*]{} equivalence relations, but we will not need this.) The proposition is true for [Hölder]{} cocycles over a subshift of finite type, as well.
[**Remark.**]{} There is a larger class of cocycles for which the Gibbs cocycles are finite. In fact there is a natural norm identified by Bowen (the [*variation norm*]{}) which defines a Banach space of cocycles with well-defined associated Gibbs cocycles. [@B3] One easily carries out the construction of a transverse [*continuous*]{} measure class for these cocycles. An open problem is to determine a regularity description of these transverse measure classes.
The [Hölder]{}cohomology over an Anosov diffemorphism is naturally associated to the topological conjugacy class of the map. This is because the conjugacy between two $C^{1 + H}$ Anosov diffeomorphisms is always [Hölder]{} continuous. The [Hölder]{} coboundaries over an Anosov diffemorphism form a [*closed*]{} subspace of the [Hölder]{} cocycles. This is a consequence of the Livshitz theorem, which states that the cohomology class of a [Hölder]{} cocycle $\phi$ over an Anosov diffeomorphism is determined by the values of the cocycle over periodic orbits, i.e. by the sums $\sum_{k=0}^{n-1} \phi
\circ
f^{k}(p)$ where $f^{n}(p) = p$ [@Li]. Since these sums are $0$ for a coboundary, and this is a closed condition, the coboundaries are a closed subspace.
We now collect various results which will be needed in the sequel. The [*one-sided subshift of finite type*]{} $\Sigma_{A}^{+}$ associated to a $0-1$ matrix $A$ is defined just as in the subshift of finite type case, but one considers only one-sided sequences. The shift map $\sigma : \Sigma_{A}^{+} \rightarrow \Sigma_{A}^{+}$ is then an expanding endomorphism.
The following lemma is due to Sinai and Bowen [@B]:
Let $\phi: \Sigma_{A} \rightarrow {\bf R}$ be a [Hölder]{} function. Then $\phi$ is cohomologous (via a [Hölder]{} transfer function) to a [Hölder]{} function $\phi^{+}$ with the property that $\phi^{+}(\bf x)$ depends only on the forward part of the sequence, i.e. on $x_{0},x_{1},x_{2},\ldots$.
So the [Hölder]{} cyclic cohomologies over $(\Sigma_{A}, \sigma)$ and over $(\Sigma_{A}^{+}, \sigma)$ are isomorphic.
We give a brief description of Bowen’s proof, as we will need to know the form of $\phi^{+}$. Let $[i]$ denote the “rectangle” consisting of those sequences ${\bf x} \in \Sigma_{A}$ with $x_{0} = i$. Let $W^{u}({\bf x},i)$ be the local unstable set through ${\bf x}$ intersected with $[i]$. For each $i$, choose ${\bf x}^{i} \in [i]$. Define $r:\Sigma_{A}
\rightarrow
\Sigma_{A}$ by projecting along local stable sets in the rectangle $[i]$, onto $W^{u}({\bf x}^{i},i)$. Let $u:\Sigma_{A} \rightarrow {\bf R}$ be defined by $$u({\bf y}) = \sum_{k=0}^{\infty}(\phi \circ \sigma^{k}({\bf y})
- \phi \circ \sigma^{k}
(r({\bf y})))$$ The function $u$ is [Hölder]{} along local unstable sets, by Proposition 1 above in the subshift of finite type setting. It can be checked that $$\phi^{+} = \phi + u \circ \sigma - u$$ depends only on the forward part of a sequence.
$\Box$
In the general setting of a homeomorphism $f:M \rightarrow M$ of a compact metric space, the [*pressure*]{} function $P:C(M) \rightarrow {\bf R}$ is defined on the space $C(M)$ of continuous functions on $M$. See [@Wa3] or [@B]. In fact, the pressure is defined on the set of cohomology classes: if $\phi$ is cohomologous to $\psi$, then $P(\phi) = P(\psi)$. In addition, $P(\phi + K) = P(\phi) + K$. For the purposes of the present paper, the pressure can be viewed as defining an imbedding of the reduced cohomology into the (unreduced) cohomology. Namely, the reduced class $<\phi>_{*}$ is mapped to the unreduced class $<\phi - P(\phi)>$. The image is precisely the set of cohomology classes with pressure zero. We note that if $f$ is an Anosov diffeomorphism, and $<\phi>$ is the unstable BRS class, then $P(<\phi>) = 0$ [@B].
Let $\Sigma_{A}^{+}$ be a transitive one-sided subshift of finite type (transitive means that there is a dense orbit). Let $\phi : \Sigma_{A}^{+} \rightarrow {\bf R}$ be a [Hölder]{} function with $P(\phi) = P$. Then there is a unique probability measure $\mu$ on $\Sigma_{A}^{+}$ such that ${\rm log}\frac
{d\mu(\sigma({\bf x}))}{d \mu({\bf x})} = -\phi({\bf x}) + P$.
See [@B],[@Wa2]. The measure $\mu$ is positive on open sets.
The measure $\mu$ has the following “one-sided Gibbs” property. Suppose that for some $n > 0$, we have $\sigma^{n}({\bf x})
= \sigma^{n}({\bf y})$. Then there is a homeomorphism $T_{{\bf x}{\bf y}}$ from a neighborhood of ${\bf x}$ to a neighborhood of ${\bf y}$, defined by the property $\sigma^{n}(T_{{\bf x}{\bf y}}({\bf z}))
= \sigma^{n}({\bf z})$. For $\mu$ as in the BRS Theorem, we have $$\log(\frac{d\mu(T_{{\bf x}{\bf y}}({\bf z}))}{d\mu({\bf z})}) =
\sum_{k = 0}^{n-1}(\phi (\sigma^{k}(T_{{\bf x}{\bf y}}({\bf z})))
- \phi (\sigma^{k} ({\bf z}))).$$
We will need the following
Let $\Sigma_{A}^{+}$ be a transitive one-sided subshift of finite type. Let $\phi : \Sigma_{A}^{+} \rightarrow {\bf R}$ be a [Hölder]{} function. Let $P = P(\phi)$. Let $\nu$ be a finite measure, supported on an open set $V \subset \Sigma_{A}^{+}$. Suppose that $\nu$ satisfies the property: ${\rm log}(\frac{d\nu(\sigma^{n}({\bf
x})}
{d\nu({\bf x})}) = -\sum_{k=0}^{n-1}\phi \circ \sigma^{k}({\bf x}) + nP$, whenever ${\bf x }\in V$ and $\sigma^{n}({\bf x}) \in V$. Then $\nu$ coincides, up to a constant factor, with the measure $\mu$ associated to $\phi$ by the Bowen-Ruelle-Sinai Theorem, restricted to $V$.
[**Proof of Corollary 1.**]{} There is an open subset $U \subset V$ and an $n$ such that $\sigma^{n}$ is injective on $U$ and $\sigma^{n}(U) = \Sigma_{A}^{+}$. Define a measure $\tilde{\nu}$ on all of $\Sigma_{A}^{+}$ by $$\frac{d\tilde{\nu}(\sigma^{n}({\bf x}))}{d\nu({\bf x})} =
{\rm exp}(-\sum_{k=0}^{n-1}\phi \circ \sigma^{k}({\bf x}) + nP)$$ where ${\bf x} \in U$. Then $\tilde{\nu}$ agrees with $\nu$ on $V$, by the derivative hypothesis on $\nu$, and ${\rm log}\frac{d\tilde{\nu}(\sigma({\bf x}))}{d\tilde {\nu}({\bf x}}
= - \phi({\bf x}) + P$. Now apply the uniqueness part of the Bowen-Ruelle-Sinai Theorem.
$\Box$
Let $\Sigma_{A}^{+}$ be a transitive subshift of finite type, and let $\phi:\Sigma_{A}^{+} \rightarrow {\bf R}$ be [Hölder]{}. Then there exists a [Hölder]{} function $h$ with the following property. Let $\phi^{\prime} =
\phi + h - h \circ \sigma - P(\phi)$. Then $\phi^{\prime} < 0$ and $\mu_{\phi^{\prime}}$ (the measure associated to $ \phi^{\prime}$ by the BRS theorem) is invariant under $\sigma$.
See [@Le],[@Wa]. Let $\mu_{\phi}$ and $\mu_{\phi^{\prime}}$ be the measures associated to $\phi$ and $\phi^{\prime}$, respectively, by the BRS theorem. Then the construction yields $$\frac{d \mu_{\phi^{\prime}}}{d \mu_{\phi}} = \log ( h).$$ Since $P(\phi^{\prime}) = 0$, the Radon-Nykodym derivative of $\mu_{\phi^{\prime}}$ under the shift map is $\exp(-\phi^{\prime}) \geq c > 1$. Hence the invariant measure is expanded by the shift map.
We now address an important subtlety of the smooth structure construction. Ultimately we obtain a smooth structure by integrating a representative measure on a transversal. A different representative measure differs by a [Hölder]{} Radon-Nykodym derivative, that is [*[Hölder]{} with respect to the underlying metric of, say, the linear toral diffeomorphism.*]{} We need to know that the Radon-Nykodym derivative is [Hölder]{} with respect to the [*new*]{} smooth coordinate, namely the measure itself. This will follow from the following proposition.
Suppose we have a mixing one-sided subshift of finite type $\Sigma_{A}^{+}$. A cylinder set $C_{n}$ of length $n$ associated to a finite word $x_{0}x_{1}
\ldots x_{n-1}$ is the set of all sequences in $\Sigma_{A}^{+}$ that begin with this word. Suppose we have a [Hölder]{} map $\pi:\Sigma_{A}^{+}
\rightarrow I$ onto an interval $I \subset {\bf R}$ satisfying the following properties. First, $\pi$ is such that $\pi({\bf x}) = \pi({\bf y})$ implies that $\pi(\sigma({\bf x})) = \pi(\sigma({\bf y}))$. Second, we assume that the image by $\pi$ of a cylinder set is an interval. Let $\phi^{\prime} $ be a [Hölder]{} function on I. Let $\mu$ be the measure on $\Sigma_{A}^{+}$, associated to the pull-back of $\phi$ by $\pi$, constructed in the Bowen-Ruelle-Sinai theorem. Let $\mu_{0}$ be the measure corresponding to the constant function. Assume that $\pi$ is injective on a set of full measure with respect to both $\mu_{0}$ and $\mu$. There are two metrics $d_{0}$ and $d_{\phi}$ defined on $I$ by the push-forward by $\pi$ of $\mu_{0}$ and $\mu$, respectively.
The identity map $\iota: (I,d_{0}) \rightarrow (I,d_{\phi})$ is quasisymmetric.
See [@Ja] and [@Ji] for the proof of quasisymmetry. We give the outline of the proof in the appendix.
A function on $I$ is [Hölder]{} in the $d_{0}$ metric if and only if it is [Hölder]{} in the $d_{\phi}$ metric.
[**Proof.**]{} A quasisymmetric homeomorphism is [Hölder]{} [@A].
Realizing cohomology classes as transverse structures
=====================================================
Transverse measure class to a foliation
---------------------------------------
Let $M$ be a smooth $n$-dimensional manifold, and let ${\cal F}$ be a $k$-dimensional foliation of $M$. That is, $M = \cup_{F \in {\cal F}} F$ where each $F \in {\cal F}$ is a smooth submanifold of $M$. $M$ is covered by [*flow-boxes*]{} $D^{k} \times
D^{n-k}$ with the property that each leaf $F \in {\cal F}$ meets a flow-box in a collection of disks of the form $D^{k} \times \lbrace y \rbrace$. A [*transversal*]{} $\tau$ to the foliation is a smooth $(n-k)$-dimensional submanifold that meets each leaf $F$ transversely.
A [*transverse measure*]{} $\mu_{\cal F}$ assigns to each small transversal a measure, with finite total mass, with the property that the measure is invariant under the holonomy pseudogroup [@Co],[@RS].
By relaxing the condition on invariance under holonomy, we arrive at the notion of a [*transverse measure class*]{}, ${\mbox{\boldmath $\mu$}}_{\cal F}$. This object assigns to each small transversal a measure [*class*]{}, with finite total mass, which is invariant under the holonomy pseudogroup. If the foliation has sufficient transverse regularity so that it preserves a smoothness class $\Lambda$, e.g. where $\Lambda$ denotes [Hölder]{}, Lipshitz, or $C^{r}$ regularity, then we can define a [*transverse $\Lambda$ measure class*]{} by requiring that the representative measures on a transversal are equivalent with Radon-Nykodym derivatives in the class $\Lambda$.
[**Example.**]{} Suppose $\cal F$ is a foliation of $M$ with transverse regularity $C^{ 1 + H}$, i.e. the holonomy maps on transversals are $C^{1 + H}$. Then Lebesgue measure on transversals defines a transverse [Hölder]{} measure class.
Let $f:M \rightarrow M$ be a diffeomorphism, preserving the foliation $\cal
F$. Then we say that $f$ preserves the transverse $\Lambda$ measure class ${\mbox{\boldmath $\mu$}}$ if for every small transversal $\tau$, and measurable subset $E \subset \tau$, $E$ has positive ${\mbox{\boldmath $\mu$}}$-measure if and only if $f(E) \subset f(\tau)$ has positive ${\mbox{\boldmath $\mu$}}$-measure, and moreover the Radon-Nykodyn derivative has regularity $\Lambda$.
There is a $\Lambda$ cohomology class over $f$ naturally associated to an $f$-invariant transverse $\Lambda$ measure class ${\mbox{\boldmath $\mu$}}_{\cal F}$. It can be defined as follows. Pick a finite covering of $M$ by flow-boxes $B_{i} = D^{k}_{i} \times D^{n-k}_{i}$. Choose representative measures $\mu_{i}$ on transversals $\tau_{i} = \lbrace x_{i}
\rbrace
\times D^{n-k}_{i}$ in each flow-box. Let $\lbrace \alpha_{i} \rbrace$ be a smooth partition of unity subordinate to the covering by flow-boxes. Define $$\phi(x) = -{\rm log} \frac{d\mu(f(x))}{d\mu(x)}$$ where $\mu = \sum_{i} \alpha_{i}(x) \mu_{i}(x)$, regarded as a measure on the local quotient space obtained by projecting along the leaf factors $D_{i}^{k} \times \lbrace y \rbrace$ in the flow boxes containing $x$.
The $\Lambda$ cohomology class of $\phi$ is independent of the choice of covering by flow-boxes, the representative measures, and the partition of unity. We will call this the Radon-Nykodym class of $f$ acting on the transverse measure class ${\mbox{\boldmath $\mu$}}_{\cal F}$.
In a similar way, we can define the notion of a [*transverse smooth structure*]{}. This is an assignment of a smooth structure to each small transversal, with the property that the holonomy pseudogroup acts smoothly with respect to this smooth structure. Note that the assigned smooth structure on a transversal in general will have nothing to do with that induced on a transversal by the ambient smooth structure of the manifold $M$. If, as above, $f$ is a diffeomorphism preserving the foliation $\cal F$, then an [*f-invariant transverse smooth structure*]{} is one in which the action of $f$ on transversals is smooth (with respect to the assigned structure). There is a natural cohomology class associated to the action of $f$, defined as in the transverse measure class case, but with Radon-Nykodym derivative replaced by the Jacobian. We will refer to this as the transverse Jacobian class of the action of $f$ on the transverse smooth structure.
Radon-Nykodym realization
-------------------------
We are now ready to describe the main step in the construction of an invariant smooth structure from a pair of cohomology classes over an Anosov diffeomorphism $f$.
Let $f:M \rightarrow M$ be an Anosov diffeomorphism. Let $<\phi>_{*}$ be a [Hölder]{}reduced cohomology class over $f$. Then there is an f-invariant transverse [Hölder]{} measure class ${\mbox{\boldmath $\mu$}}$ to the stable foliation $W^{s}$, with the property that the reduced Radon-Nykodym class of $f$ acting on ${\mbox{\boldmath $\mu$}}$ is $<\phi>_{*}$.
The transverse measure class ${\mbox{\boldmath $\mu$}}$ in the theorem has the additional property that the measure class on a transversal is positive on open subsets of the transversal. When $M = T^{2}$, transversals to $W^{s}$ are one-dimensional. In this case, an $\alpha-$ [Hölder]{} transverse measure class that is positive on open sets is equivalent to a $C^{1 + \alpha}$ transverse smooth structure.
Let $f:T^{2} \rightarrow T^{2}$ be Anosov. Let $<\phi>_{*}$ be a [Hölder]{}reduced cohomology class over $f$. Then there is an f-invariant transverse $C^{1 + \alpha}$ smooth structure to the stable foliation $W^{s}$, with the property that the reduced Jacobian class of $f$ acting on this transverse smooth structure is $<\phi>_{*}$.
The Radon-Nykodym realization theorem is proved in section 7.
Complementary transverse smooth structures
------------------------------------------
Let $\cal F$ and $\cal G$ be foliations of complementary dimension. We say that the foliations [*intersect transversely*]{} if the leaves of $\cal F$ and $\cal G$ meet transversely. We assume that there is a system of simultaneous flow-boxes of the form $$D^{k} \times D^{l}$$ where each leaf of $\cal F$ meets a flow-box in a collection of disks of the form $D^{k} \times \lbrace y \rbrace$, and each leaf of $\cal G$ meets a flow-box in a collection of disks of the form $\lbrace x \rbrace \times
D^{l}$.
Let $\cal F$ and $\cal G$ be foliations of complementary dimension, intersecting transversely. Then a pair of transverse smooth structures, one for each foliation, determines a canonical smooth structure on $M$ as follows. Consider a simultaneous flow-box $D^{k} \times D^{l}$. Pick a point $(x,y)$ in the flow-box. Then the disk $\lbrace x \rbrace \times D^{l}$ is a transversal to the foliation $\cal G$. Similarly the disk $D^{k} \times \lbrace y
\rbrace$ is a transversal to the foliation $\cal F$. So there is a product smooth structure on the flow-box determined by the transverse smooth structures on these disks. The overlap maps for these charts are block diagonal, with the blocks being the derivative of the holonomy for each foliation. So the product structure has the same degree of smoothness as the transverse structures.
[**Proof of Theorem 1**]{} We show that the BRS map is surjective, assuming the Jacobian realization theorem. Let $<\phi_{u}>_{*}$ and $<\phi_{s}>_{*}$ be the reduced cohomology classes which we want to realize. We have complementary, transverse foliations $W^{s}$ and $W^{u}$, which by the Jacobian realization lemma can be equipped with a transverse smooth structures, with associated reduced Jacobian classes equal to $<\phi_{u}>_{*}$ and $<\phi_{s}>_{*}$. We define a product structure as just described. In this smooth structure, the unstable reduced Jacobian class of $f$ is simply the tranverse reduced Jacobian class of $f$ acting on the transverse smooth structure, i.e. $<\phi_{u}>_{*}$. Similarly, the reduced stable Jacobian class is $<\phi_{s}>_{*}$. It remains to see that $f$ is Anosov in the new smooth structure. This follows from Proposition 2 of Section 5, and the construction of the transverse measure from the Gibbs measure associated to $\phi$. We postpone this simple argument to Section 8, where explicit charts in the smooth structure are decribed.
$\Box$
Proof of Radon-Nykodym realization
==================================
An Anosov diffeomorphism has a presentation as the quotient of the shift map on a subshift of finite type [@Si]. The subshift of finite type is defined by the transition properties of rectangles in a Markov partition under the action of the diffeomorphism. We will show that a [Hölder]{} cocycle over the shift map defines a “transverse [Hölder]{} measure class” to the stable sets in the subshift of finite type. The transverse measure class pushes down to a transverse measure class to the stable foliation on $M$ provided the cocycle passes down to a cocycle on $M$, i.e. when it is generated by a function on the subshift which is constant on fibers of the quotient to $M$.
This section is organized as follows. In subsection 1, we recall the definition of a Markov partition for an Anosov diffeomorphism, and construct the quotient from the shift to the diffeomorphism. We define the notion of transverse measure class for subshifts of finite type in subsection 2, and in subsection 3 show that a [Hölder]{} cocycle determines a [Hölder]{} transverse measure class. In subsection 4 we show that the transverse measure class pushes forward to the quotient space when the cocycle does.
Markov partitions
-----------------
The simultaneous flow-boxes for the stable and unstable foliations of an Anosov diffeomorphism $f:M \rightarrow M$ define a [*local product structure*]{} $D^{s} \times D^{u}$ on the manifold $M$. A [*rectangle*]{} is a closed set of small diameter which is a product in the local product structure: $R = A^{s} \times A^{u}$. A rectangle is [*proper*]{} if it is the closure of its interior. We define the [*stable*]{} and [*unstable boundary of R*]{} respectively: $\partial^{s}R = A^{s} \times \partial A^{u}$; $\partial^{u}R = \partial
A^{s} \times
A^{u}$.
A [*Markov partition*]{} for $f$ is a set ${\cal C} = \lbrace R_{1}, \ldots,
R_{r}
\rbrace$ of small proper rectangles whose union is $M$, and satisfying:
i.
: each $R_{i}$ is connected
ii.
: $\mbox{int}(R_{i}) \cap \mbox{int}(R_{j}) = \emptyset$ for $i
\neq j$
iii.
: $f(\partial^{s}{\cal C}) \subset \partial^{s}{\cal C}$ where $\partial^{s}{\cal C} = \cup_{i=1}^{i=r}
\partial^{s}R$
iv.
: $f^{-1}(\partial^{u}{\cal C}) \subset \partial^{u}{\cal C}$ where $\partial^{u}{\cal C} =
\cup_{i=1}^{i=r}\partial^{u}R$
Sinai proved that Anosov diffeomorphisms have Markov partitions of arbitrarily small diameter, where the diameter of the partition is defined to be the largest diameter of a rectangle in the partition [@Si].
A Markov partition defines a 0-1 matrix $A$ where $A_{ij} = 1$ if $f(\mbox{int}R_{i}) \cap \mbox{int}R_{j} \neq \emptyset$ and is $0$ otherwise. The properties of the Markov partition guarantee that if ${\bf x} \in
\Sigma_{A}$, then the intersection $\cap_{i=-\infty}^{\infty}f^{-i}R_{x_{i}}$ consists of a single point. This defines a map $\pi: \Sigma_{A} \rightarrow M$ which semi-conjugates the shift map to the mapping $f$.
Transverse structures on a subshift of finite type.
---------------------------------------------------
We recall that a subshift of finite type $\Sigma_{A}$ has a [*local product structure*]{} defined by the stable and unstable foliations. That is, there is a homeomorphism defined on a neighborhood $U$ of a point $\bf x$, $$u:U \rightarrow W^{s}_{\epsilon}({\bf x}) \times W^{u}_{\epsilon}({\bf x})$$ with the property that local stable sets map to sets of the form $W^{s}_{\epsilon}
({\bf x}) \times \lbrace {\bf z} \rbrace$, and local unstable sets map to sets of the form $\lbrace {\bf w} \rbrace \times W^{u}_{\epsilon}({\bf x})$.
A small [*transversal*]{} to the stable foliation is a set which in the local product structure is represented as the graph of a continuous function $\tau: W^{u}_{\epsilon}({\bf x}) \rightarrow
W^{s}_{\epsilon}({\bf x})$.
We assume that $\Sigma_{A}$ is a transitive subshift of finite type, i.e. there is a dense orbit. Suppose ${\bf x} \in W^{s}({\bf y})$. Then there is $\epsilon > 0$ and a canonical homeomorphism $h: W^{u}_{\epsilon}({\bf
x})
\rightarrow W^{u}_{\epsilon}({\bf y})$, such that for every ${\bf z} \in
W^{u}_{\epsilon}({\bf x})$, $h({\bf z}) \in W^{s}({\bf z})$. These homeomorphisms are called (unstable) conjugating homeomorphisms. The [*holonomy pseudogroup of the stable foliation*]{} is defined to be the pseudogroup of homeomorphisms between transversals generated by projections onto the unstable factor in local product charts, and unstable conjugating homeomorphisms between local unstable sets.
A [*transverse measure class*]{} to the stable foliation is an assignment of a measure class to each small transversal, with the property that the holonomy transformations preserve the measure class. A [Hölder]{} tranverse measure class is one in which the representative measures on a transversal are required to be equivalent with [Hölder]{} Radon-Nykodym derivative.
A shift-invariant transverse measure class is defined as for the diffeomorphism case. Note that the shift map preserves the class of transversals. A shift-invariant transverse [Hölder]{} measure class defines a [Hölder]{} Radon-Nykodym cohomology class over the action of the shift, just as in the foliation case.
Radon-Nykodym realization for subshifts of finite type.
-------------------------------------------------------
The following theorem follows easily from the standard Gibbs theory described in Section 5.
Let $\Sigma_{A}$ be a transitive subshift of finite type. Let $<\phi>_{*}$ be a reduced [Hölder]{} cohomology class over the shift map. Then there is a unique shift-invariant transverse measure class ${\mbox{\boldmath $\mu$}}$ to the stable foliation such that the associated reduced Radon-Nykodym class is $<\phi>_{*}$.
[**Proof.**]{} We are given a [Hölder]{} cohomology class $<\phi>$ on $\Sigma_{A}$. We apply Lemma 1 of Section 5 to obtain a [Hölder]{} function $\phi^{+}$ in the cohomology class $<\phi>$, which we can view as a function on the one-sided shift $\Sigma_{A}^{+}$. We can also view $\Sigma_{A}^{+}$ as a subset of the two-sided shift, in fact as a transversal to the stable foliation, as follows. For each symbol $i
\in \lbrace
1,\ldots, r \rbrace$, pick a point ${\bf y}^{i}$ with $y^{i}_{0} = i$. Then $\tau = \cup _{i=1}^{r}W^{u}({\bf y^{i}},i)$ is canonically isomorphic to $\Sigma_{A}^{+}$, and is a union of small transversals to the stable foliation. Moreover, $\tau$ meets every stable set. We define the measure class on $\tau$ to be the [Hölder]{} measure class containing the measure $\mu_{{\phi}^{+}}$ determined by the Bowen-Ruelle-Sinai theorem applied to $\phi^{+}$. We define the measure class on any small transversal to be the pull-back of $\mu_{{\phi}^{+}}$ by a holonomy transformation to the transversal $\tau$ (which exists since $\tau$ meets every stable set). To see that this defines a transverse [Hölder]{} measure class, it suffices to check that the holonomy transformations between local stable sets in $\tau$ preserve the measure class of $\mu_{{\phi}^{+}}$, with [Hölder]{} Radon-Nykodym derivative. But this is precisely the “one-sided Gibbs” property of $\mu_{{\phi}^{+}}$. Finally we note that the Radon-Nykodym class associated to this transverse measure class is the reduced cohomology class of $\phi^{+} - P$, as desired.
$\Box$
Pushing the transverse measure class forward.
---------------------------------------------
Let $W^{u}_{\epsilon}(x)$ be the $\epsilon$-ball about $x$ in $W^{u}(x)$. If $i$ is a rectangle in the Markov partition, let $W^{u}(x,i) =
W^{u}_{\epsilon}(x) \cap R$, where $\epsilon$ is chosen so that this is a single horizontal slice of $R$. Let $[i]^{+} \subset \Sigma_{A}^{+}$ be the set of sequences $y_{0}y_{1}y_{2}\ldots$ with $y_{0} = i$. If $x \in M$ and $x = \pi({\bf x})$ where ${\bf x} = \ldots x_{-1}x_{0}x_{1}
\ldots$ with $x_{0} = R$, then we define a quotient map: $$\pi_{{\bf x},R}:[R]^{+} \rightarrow W^{u}(x,R)$$ by $\pi_{{\bf x},R}({\bf y}) = \pi(\ldots
x_{-2}x_{-1}y_{0}y_{1}y_{2}\ldots)$.
For each rectangle $R$ and $x \in R$ with $x = \pi({\bf y})$, the measure class on the transversal $W^{u}(x,R)$ is defined to be the image by $\pi_{{\bf y},R}$ of the measure class ${\mbox{\boldmath $\mu$}}_{\phi}$ on $[R]^{+}$.
We need to check that if parts of $W^{u}(x,R)$ and $W^{u}(x^{\prime},R^{\prime})$ correspond under local projection along the stable foliation, then the measure classes defined by $\pi_{{\bf x},R}$ and $\pi_{{\bf x}^{\prime},R^{\prime}}$ also correspond. If $R = R^{\prime}$, this follows from the fact that, if $p_{x,x^{\prime}}$ is the projection along local stable leaves from $W^{u}_{\epsilon}(x)$ to $W^{u}_{\epsilon}(x^{\prime})$, and if $\pi({\bf x}) = x$ and $\pi({\bf x}^{\prime}) = x^{\prime}$, then $$\pi_{{\bf x},R} = p_{x,x^{\prime}} \circ \pi_{{\bf x}^{\prime},R}$$ If $R \neq R^{\prime}$ we consider representative measures on the $W^{u}(x,R)$ and $W^{u}(x^{\prime}, R^{\prime})$. Let $\mu_{x,R}$ denote the image of $\mu_{{\phi}^{+}}$ restricted to $[R]^{+}$ by $\pi_{{\bf x},R}$.
Let $U \subset W^{u}(x,R)$ and $V \subset W^{u}(x^{\prime},R^{\prime})$ be such that $p_{x,x^{\prime}}:
U \rightarrow V$ is a homeomorphism. There are two main points:
1.
: Let $y \in U$, and $y^{\prime} = p_{x,x^{\prime}}(y)$. We can make sense of the expression $\Phi^{+}(y \rightarrow y^{\prime})$ [*on $U$*]{} as follows. Recall $$\Phi(y \rightarrow y^{\prime}) = \Sigma_{k=0}^{\infty}(\phi \circ
f^{k}(y^{\prime})
- \phi \circ f^{k}(y)).$$ is defined and [Hölder]{} on $U$. Note that the transfer function $u:\Sigma_{A} \rightarrow {\bf R}$ which makes $\phi$ cohomologous to $\phi^{+}$ pushes forward to a well-defined and [Hölder]{}function $u_{R}$ on each of the individual quotients $W^{u}(x,R)$ Therefore we [*define*]{} a [Hölder]{} function: $$\Phi^{+}(y \rightarrow y^{\prime}) = \Phi(y \rightarrow y^{\prime}) +
u_{R^{\prime}}(y^{\prime})
- u_{R}(y).$$
2.
: Let $\mu_{U}$ be $\mu_{{\bf x},R}$ restricted to $U$. Define $\mu_{V}$ similarly. Then $$\frac{d((p_{x,x^{\prime}})_{*}(\mu_{V}))}{d\mu_{U}} = {\rm exp}(\Phi^{+}(y
\rightarrow y^{\prime})).$$
We need to prove the second statement. We define a measure $\nu$ supported on $V$ by $\nu = p_{x,x^{\prime}}^{*}({\rm exp}(\Phi^{+})\mu_{U})$. We want to pull back to $\Sigma_{A}^{+}$ both $\nu$ and $\mu_{V}$, where we will show them to be equal. For this to make sense, and imply the second statement, we need the following lemma. Let $\tilde{V} = \pi_{x,R^{\prime}}^{-1}(V)$.
$\pi_{{\bf x},R^{\prime}}:\tilde{V} \rightarrow V$ is one-to-one over a set of full measure with respect to both $(p_{x,x^{\prime}})^{*}\mu_{U}$ and $\mu_{V}$.
[**Proof.**]{} Let $Y \subset V$ be the set of points with more than one preimage by $\pi_{{\bf x},R^{\prime}}$. We need to show that $\mu_{U}(p_{x,x^{\prime}}^{-1}(Y)) = 0$ and $\mu_{V}(Y) = 0$
Let $\mu$ be the invariant Gibbs measure on $M$ associated to the cohomology class $<\phi>$. We need the following two facts. The technical proofs are included in the appendix.
1.
: Let $R = A^{s} \times A^{u}$ in the local product structure. Let $x \in R$, and $Y \subset W^{u}(x,R)$. Then $\mu(A^{s} \times Y) = 0$ implies $\mu_{x,R}(Y) = 0$.
2.
: $\mu(\partial {\cal C}) = 0$.
Now we prove the lemma. If $y \in Y$, then $d(f^{n}(y),\partial
{\cal C})
\rightarrow 0$ as $n \rightarrow \infty$. The same is therefore true of points in $Y^{\prime} := p_{x,x^{\prime}}^{-1}(Y)$. Let $R = A^{s}(R) \times A^{u}(R)$ and $R^{\prime}
= A^{s}(R^{\prime}) \times A^{u}(R^{\prime})$. Let $Z = A^{s}(R) \times Y$ and $Z^{\prime} = A^{s}(R^{\prime}) \times Y^{\prime}$. Then $d(f^{n}(Z),\partial {\cal C}) \rightarrow 0$ and $d(f^{n}(Z^{\prime},\partial
{\cal C}) \rightarrow 0$ as $n \rightarrow \infty$. Since $\mu(\partial
{\cal C}) = 0$ , and $\mu$ is invariant, $\mu(Z) = 0$ and $\mu(Z^{\prime}) = 0$, and we conclude that $\mu_{V}(Y) = 0$ and $\mu_{U}(Y^{\prime}) = 0$.
$\Box$
We return to the proof of statement 2 relating the measures on $U$ and $V$. We want to show that the measure $\tilde{\nu} = (\pi_{x,R^{\prime}})_{*}\nu$ coincides with $(\pi_{x,R^{\prime}})_{*}\mu_{V}$. The latter is simply $\mu_{\phi^{+}}$ restricted to $\tilde{V}$. By the Local Uniqueness corollary to the Bowen-Ruelle-Sinai theorem, it suffices to show that $\tilde{\nu}$ has Radon-Nykodym derivative ${\rm exp}(-\sum_{k=0}^{n-1}\phi^{+}\circ
\sigma^{k} + nP)$ under ${\bf x} \rightarrow \sigma^{n}({\bf x})$ whenever ${\bf x}$ and $\sigma^{n}({\bf x})$ are both in $\tilde{V}$.
Recall $\nu = p_{x,x^{\prime}}^{*}({\rm exp}(\Phi^{+})\mu_{U})$. So $$\frac{d({\tilde{\nu}(\sigma({\bf y})}))}{d\nu({\bf y})} =
\pi^{*}(p_{x,x^{\prime}})_{*}
\frac{d({\rm exp} (\Phi^{+})\mu_{U})(f(y))}{d({\rm
exp}(\Phi^{+})\mu_{U})(y)}$$ The main point in the calculation is the following. $\phi^{+}$ is well-defined on the individual quotients $W^{u}(x,R)$, namely $\phi^{+}_{R} = \phi +
u_{R} \circ
f - u_{R}$. Then $$\Phi^{+}(f(y)\rightarrow f(y^{\prime}))
- \Phi^{+}(y \rightarrow y^{\prime}) = -\phi^{+}_{R^{\prime}}(y^{\prime}) +
\phi^{+}_{R}(y)$$ where $y^{\prime} = p_{x,x^{\prime}}(y)$ and $f(y^{\prime}) = f(p_{x,x^{\prime}})
= p_{x,x^{\prime}}(f(y))$.
Thus $$\begin{aligned}
\frac{d({\rm exp} (\Phi^{+})\mu_{U})(f(y))}{d({\rm
exp}(\Phi^{+})\mu_{U})(y)}
& = & {\rm exp}(-\phi^{+}_{R^{\prime}}(y^{\prime}) +
\phi^{+}_{R}(y)) \cdot {\rm exp} (-\phi^{+})\\
& = & {\rm exp}(-\phi^{+}_{R^{\prime}}(p_{x,x^\prime}(y)) +
P)\end{aligned}$$ which is the desired result.
Gibbs charts
============
A Markov partition for a hyperbolic automorphism $L$ of $T^{2}$ can be constructed as follows [@AW]. Let $E^{s}$ and $E^{u}$ be the stable and unstable eigenspaces respectively. Project into the torus a segment in $E^{s}$ through the origin, and a segment in $E^{u}$ through the origin. Extend these segments until they cut the torus into parallelograms. The segment in the stable direction should map into itself under $L$, and the segment in the unstable direction should map into itself under $L^{-1}$. This decomposition of the torus is a Markov partition. Let $A$ be the transition matrix of the partition, and let $\pi:\Sigma_{A} \rightarrow T^{2}$ be the quotient map from the subshift of finite type defined by $A$. The unstable segment $\tau_{u}$ (which is also the unstable boundary of the partition) is the image by $\pi$ of a copy of the [*one-sided shift*]{} $\Sigma_{A}^{+}$ specified by fixing the backward part of a sequence to be the backward part of some fixed pre-image of the origin. Similarly, the stable segment $\tau_{s}$ is the image of a copy of the backward one-sided shift defined by $A$, or equivalently of the one-sided shift defined by the $A^{t}$, the transpose of $A$.
The smooth structure determined by a pair of reduced cohomology classes $<\phi_{u}>_{*}$ and $<\phi_{s}>_{*}$ has the following explicit description. We can assume that the functions $\phi_{u}$ and $\phi_{s}$ have pressure 0. If not, we can just subtract the pressure, which will not change the reduced cohomology class. Let $\tilde{\phi_{u}}$ be the pull-back of $\phi_{u}$ to $\Sigma_{A}$, by $\pi$. $\tilde{\phi_{s}}$ is defined similarly. Now change $\tilde{\phi_{u}}$ by a coboundary to get a function $\phi_{u}^{+}$ that depends only on the forward part of a sequence. Similarly, one gets a function $\phi_{s}^{-}$ which is cohomologous to $\tilde{\phi_{s}}$ and depends only on the backward part of a sequence. We can regard these as functions on $\Sigma_{A}^{+}$ and $\Sigma_{A^{t}}^{+}$, respectively. Now let $\mu_{+}$ be the unique probability measure on $\Sigma_{A}^{+}$ satisfying $$\frac{d\mu_{+}(\sigma({\bf x}))}{d\mu_{+}({\bf x})} = {\rm
exp}(\phi_{u}^{+}({\bf x}))$$ Let $\mu_{-}$ be the unique probability measure on $\Sigma_{A^{t}}^{+}$ with $$\frac{d\mu_{-}(\sigma({\bf y}))}{d\mu_{-}({\bf y})} = {\rm
exp}(\phi_{s}^{-}({\bf y}))$$ These measures push-forward to measures $\nu^{+}$ and $\nu^{-}$ on $\tau_{u}$ and $\tau_{s}$ respectively.
Smooth coordinate charts (in the new structure) are obtained by integrating the measures $\nu^{+}$ and $\nu^{-}$ along two side of a “rectangle” in the partition, and taking the product structure. The main point in the proof of the theorem is that if the segment $\tau_{u}$ is presented [*differently*]{} as the image of $\Sigma_{A}^{+}$, i.e. by choosing a different pre-image of the origin, with different backward part, then the smooth coordinate along $\tau_{u}$ determined by pushing forward the measure $\nu^{+}$ by this different presentation is smoothly equivalent to the original one.
It is now clear from Proposition 2 of Section 5 that $f$ is Anosov in the new smooth structure. We simply use the expanding measure in the [Hölder]{} measure class associated to $<\phi_{u}>_{*}$ to define the coordinate along $\tau_{u}$. Similarly, use the expanding measure associated to $<\phi_{s}>_{*}$ to define the coordinate along $\tau_{s}$.
[**Remark.**]{} The two maps $\pi_{1}:\Sigma_{A}^{+} \rightarrow
\tau_{u}$ and $\pi_{2}:\Sigma_{A}^{+} \rightarrow \tau_{u}$, determined by viewing $\tau_{u}$ from the “clockwise” side or the “counterclockwise” side, determine a comparison map $\alpha:\Sigma_{A}^{+} \rightarrow \Sigma_{A}^{+}$ on a set of full measure, defined by $\pi_{2}(\alpha({\bf x})) = \pi_{1}({\bf x})$. The measure $\mu^{+}$ has [Hölder]{} Radon-Nykodym derivative under $\alpha$.
[**Appendix**]{}
Quasisymmetric equivalence of Gibbs structures
==============================================
We show that the conjugacy between the linear map in a topological conjugacy class, and the map constructed from a [Hölder]{} cocycle, is quasisymmetric along the leaves of the stable and unstable foliations. The proof is adapted from [@Ji].
A homeomorphism $h: I \rightarrow I$ is [*quasisymmetric*]{} if there exists a $K > 0$ such that for every pair of adjacent intervals $I$ and $J$ of equal length, $$1/K \leq \frac{|h(I)|}{|h(J)|} \leq K.$$ The number $K$ is the [*quasisymmetry constant*]{} of the map, and $h$ is called $K-quasisymmetric$. The composition of a quasisymmetric map with a $C^{1}$ map is again quasisymmetric.
We recall Proposition 2 from Section 5. Let $\Sigma_{A}^{+}$ be a mixing one-sided subshift of finite type. Let $\pi: \Sigma_{A}^{+} \rightarrow
I$ be a [Hölder]{} map onto an interval $I \subset {\bf R}$, with the property that $\pi(\sigma({\bf x})) = \pi(\sigma({\bf y}))$ whenever $\pi({\bf x})
= \pi({\bf y})$. In addition we assume that the image by $\pi$ of a cylinder set $C_{n}$ is an interval. Let $\phi^{\prime}$ be a [Hölder]{} function on $I$, and let $\phi$ be the pull-back of $\phi^{\prime}$ to $\Sigma_{A}^{+}$ by $\pi$. Let $\mu_{0}$ be the measure of maximal entropy on $\Sigma_{A}^{+}$, and let $\mu_{\phi}$ be the measure associated to $\phi$ as constructed in the Bowen-Ruelle-Sinai theorem, that is, the unique probability measure with Radon-Nykodym derivative $\phi - P(\phi)$ where $P(\phi)$ is the pressure of $\phi$. Assume that $\pi$ is injective on a set of full measure, with respect to both $\mu_{0}$ and $\mu_{\phi}$. There are two metrics $d_{0}$ and $d_{\phi}$ on $I$, defined by the push-forward by $\pi$ of $\mu_{0}$ and $\mu_{\phi}$, respectively.
The identity map $\iota:(I,d_{0}) \rightarrow (I,d)$ is quasisymmetric.
We note that the hypotheses of the proposition are satisfied by the map $\pi:\Sigma_{A}^{+} \rightarrow \tau_{u}$ defined in section 8.
The proof given in [@Ji] applies in a more general context. The basic idea is the following. The image of the partitions into cylinder sets of the subshift of finite type defines a nested sequence of partitions of the interval $I$. The identity map of course preserves these partitions. The quasisymmetry estimate follows from bounds on the geometry of this sequence of partitions, in both the $d_{0}$ and the $d_{\phi}$ metrics.
We can assume that $\phi$ is in fact the Radon-Nykodym derivative of the expanding (equilibrium) measure on $\Sigma_{A}^{+}$ associated to the reduced cohomology class $<\phi>_{*}$. This is because the expanding measure, and the measure $\mu_{\phi}$ are equivalent with a [Hölder]{} Radon-Nykodym derivative, and therefore the corresponding metrics on $I$ are $C^{1 + \alpha}$ equivalent.
Let ${\cal C}_{n}$ denote the partition of $\Sigma_{A}^{+}$ by cylinder sets of size $n$. Let ${\cal D}_{n}$ be the corresponding sequence of partitions of $I$.
In the following lemma, we consider $I$ with the $d_{0}$ metric, and show how to approximate the intervals defined by an equally spaced triple of points by elements of the partitions ${\cal D}_{n}$.
There exists a positive integer $N = N(\Sigma_{A}^{+})$ with the following property. Let $x,y \in I$, and let $z$ be the midpoint of the interval $[x,y]$. We will write $R = [x,z]$ and $S = [z,y]$. Let $n$ be the smallest integer such that there exists $D \in {\cal D}_{n}$ with $R \cup S \subset D$. Then there are $D_{R}, D_{S} \in {\cal D}_{n + N}$ with $D_{R} \subset R$ and $D_{S} \subset S$.
[**Outline of the proof.**]{} Let $h$ be the topological entropy of $\Sigma_{A}^{+}$, and let $M$ be the mixing time, that is, $A^{M}$ has all positive entries. Then $N \geq 4M + 1 + \frac{log 2}{log h}$ has this property. This follows easily from the following three geometric properties of the partitions ${\cal D}_{n}$ in the $d_{0}$ metric. Let $\lambda = \exp
h$. We denote the length of an interval $D$ in the $d_{0}$ metric by $|D|_{0}$.
1. Exponentially decreasing geometry.
: Let $D_{n + m} \in {\cal D}_{n + m}$, $D_{n} \ in {\cal D}_{np}$, with $D_{n + m} \subset D_{n}$. Then $$\frac{|D_{n + m}|_{0}}{|D_{n}|_{0}} \leq \lambda^{m - M}.$$
2. Bounded ratio geometry.
: Let $D_{n} \in {\cal D}_{n}$, $D_{n + 1} \in {\cal D}_{n + 1}$, with$D_{n + 1} \subset D_{n}$. Then $$\frac{|D_{n + 1}|_{0}}{|D_{n}|_{0}} \geq \lambda^{-(M + 1)}.$$
3. Bounded nearby geometry.
: Let $D, E \in {\cal D}_{n}$ be adjacent intervals. Then $$\lambda^{-M} \leq \frac{|D|_{0}}{|E|_{0}} \leq \lambda^{M}.$$
$\Box$
[**Outline of proof of the proposition.**]{} Analagous geometric properties hold for the partitions ${\cal D}_{n}$ in the $d_{\phi}$ metric. The quasisymmetry estimate follows from these. We define some preliminary quantities.
Let ${\rm S}_{n}\phi ( {\bf x}) = \sum_{k = 0}^{n - 1}
(\phi(\sigma^{k}({\bf x}))$. Let $${\rm var}_{n} \phi = {\rm sup} \lbrace | {\rm S}_{n}\phi({\bf x}) -
{\rm S}_{n}\phi({\bf y}) |
\ {\rm where} \ {\bf x},{\bf y} \in C_{n} \ {\rm for \ some} \
C_{n} \in {\cal C}_{n} \rbrace.$$ Since $\phi$ is [Hölder]{}, there exist $c > 0$ and $\beta < 1$ such that ${\rm var}_{n} \phi < c \beta^{n}$. Therefore $${\rm var} \phi =: \sum_{k= 0}^{\infty} {\rm var}_{k} \phi < \infty.$$ Let $\| \phi \| = {\rm sup} \lbrace | \phi({\bf x}) | : {\bf x} \in
\Sigma_{A}^{+} \rbrace$. Define $L = \exp (2 {\rm var} \phi) \cdot \exp ( M \| \phi \| )$.
In what follows, all lengths are with respect to the $d_{\phi}$ metric, where $\phi$ is the Radon-Nykodym derivative of the expanding measure on $\Sigma_{A}^{+}$ associated to the reduced cohomology class $<\phi>_{*}$. The $d_{\phi}$ length of an interval $D$ is denoted $|D|_{\phi}$.
1. Bounded ratio geometry.
: Let $D_{n + 1} \in {\cal D}_{n + 1}$, $D^{n} \in {\cal D}_{n}$, with$D_{n + 1} \subset D_{n}$. Then $$\frac{|D_{n + 1}|_{\phi}}{|D_{n}|_{\phi}} \geq L^{-1} \cdot \exp (- \| \phi
\|) .$$
2. Bounded nearby geometry.
: Let $D, E \in {\cal D}_{n}$ be adjacent intervals. Then $$\frac{1}{L} \leq \frac{|D|_{\phi}}{|E|_{\phi}} \leq L.$$
These properties follow from Bowen’s estimate for the $\mu_{\phi} - $measure of a cylinder set $C_{n}$, when $\phi$ is the Radon-Nykodym derivative of the invariant measure.[@B] For any ${\bf x} \in C_{n}$, $$c_{1} \leq \frac{\mu_{\phi}(C_{n})}{exp (S_{n}\phi({\bf x}))} \leq c_{2}$$ where $c_{1} = \exp (-M \| \phi \| ) \cdot \exp (-{\rm var} \phi)$, and $c_{2} = \exp ( {\rm var}\phi)$.
Using these properties, and the Lemma, one obtains an estimate for the quasisymmetry constant $K$: $$K \leq L^{(2N + 2)} \cdot \exp ((2N + 1) \| \phi \|).$$
[**Remark.**]{} This estimate is not sharp, as can be seen by considering $\phi$ to be the constant function with pressure zero. The estimate can be improved by a more careful comparison of the partition elements lying in the pair of adjacent intervals.
Gibbs measures and Markov partition boundaries
==============================================
We prove here the technical facts needed in Section 7.4 Let $f:M
\rightarrow M$ be an Anosov diffeomorphism, $\cal C$ be a Markov partition for $f$, and $\mu$ be the Gibbs measure associated to the cohomology class $< \phi >$.
Let $R \in {\cal C}$ be a rectangle, with $R = A_s \times A_u$ in the local product structure. Let $x \in R$, and $Y \subset W^{u}(x,R)$. Let $\mu_{x,R}$ be the push-forward of the one-sided measure $\mu_{\phi^{+}}$, as defined in Section 7.4. Then implies $\mu_{x,R}(Y) = 0$.
[**Proof.**]{} We recall how $\mu$ on $T^{2}$ can be constructed from $\mu_{\phi^{+}}$ on $\Sigma_{A}^{+}$. See [@B] for details. If $\pi: \Sigma_{A} \rightarrow T^{2}$ is the quotient map determined by the Markov partition, then $\mu$ is the push-forward by $\pi$ of the measure $\tilde{\mu}$ on $\Sigma_{A}$, defined as follows. Let $\nu$ be the [*shift-invariant*]{} probability measure on $\Sigma_{A}^{+}$ equivalent to $\mu_{\phi^{+}}$. (The Radon-Nykodym derivative of $\nu$ with respect to $\mu_{\phi^{+}}$ is given explicitly up to a constant factor as the unique positive eigenvector of the Perron-Fröbenius operator associated to $\phi^{+}$.) The measure $\tilde{\mu}$ on $\Sigma_{A}$ is obtained from the measure $\nu$ on $\Sigma_{A}^{+}$ as follows. If $g$ is a continuous function on $\Sigma_{A}$, define $g^{*}$ on $\Sigma_{A}^{+}$ by $$g^{*}({\bf x}) = {\rm min} \lbrace g({\bf y})\ {\rm where} \ y_{i} = x_{i} \
{\rm for \ all} \ i \geq 0 \rbrace$$ Then $\lim_{n \rightarrow \infty} \nu ( (g \circ \sigma^{n})^{*})$ exists, and we define $\tilde{\mu}(g)$ to be this limit. This linear functional defines the measure $\tilde{\mu}$. The proposition now follows easily.
$\Box$
$\mu(\partial{\cal C}) = 0$.
[**Proof.**]{} The following proof is adapted from Bowen’s in the case $\mu$ is the measure associated to the constant cocycle. [@B1] The proof relies on the [*variational principle*]{}, and the fact that the topological pressure always decreases when the dynamics is restricted to an invariant subset.
[**Variational Principle.**]{} If a cohomology class over $f$, say $< \phi >$ has been fixed, we define the [*measure theoretic pressure*]{} of an invariant probability measure $\nu$ to be $h_{\nu} + \int \phi d\nu$, where $h_{\nu}$ is the measure theoretic entropy of $f$. We denote this $P_{\nu}(\phi,f)$. The topological pressure of $\phi$ is denoted $P(\phi,f)$. Note that these depend only on the cohomology class of $\phi$. The variational principle states: $$P_{\nu}(\phi,f) \leq P(\phi,f) \ {\rm and } \ \sup_{\nu} P_{\nu}(\phi,f) =
P(\phi,f)$$ where the supremum is over all invariant probability measures. The variational principle is true for any homeomorphism of a compact metric space. If the mapping is Anosov, and the cocycle is [Hölder]{} then supremum is achieved precisely at the Gibbs measure $\mu$ associated to $<\phi>$. Recall that if $W$ is a compact f-invariant subset, then $P(\phi_{\mid W}, f_{\mid W}) < P(\phi,f)$.
Let $\mu$ be the Gibbs measure associated to the cohomology class $<\phi>$. We will show that $\mu(\partial^{s}{\cal C}) = 0$. A similar argument shows that $\mu(\partial^{u}{\cal C}) = 0$. Let $W = \cap_{n \geq 0}
f^{n}(\partial^{s}{\cal C})$. Suppose that $\mu(\partial^{s}{\cal C}) = a > 0.$ Then $\mu(W) = a$, and $\mu(\cup f^{n}(\partial {\cal C}) \setminus W) = 0$. Define $\nu_{1}$ on $W$ by $\nu_{1} = \frac{1}{a} \mu$, and $\nu_{2}$ on $M$ by $\nu_{2}(E) =
\frac{1}{1-a}
\mu(E \setminus W)$. Then $\nu_{1}$ and $\nu_{2}$ are f-invariant, have disjoint support, and $\mu = a \nu_{1} + (1 - a) \nu_{2}$. Therefore $$P_{\mu}(\phi,f) = a P_{\nu_{1}}(\phi,f) + (1 - a) P_{\nu_{2}}(\phi,f).$$ The variational principle implies that $P_{\nu_{2}}(\phi,f) \leq P(\phi,f)$ and $$\begin{aligned}
P_{\nu_{1}}(\phi,f) & = & P_{\nu_{1}}(\phi_{\mid W}, f_{\mid W}) \\
& \leq & P(\phi_{\mid W},f_{\mid W}) \\
& < & P(\phi,f). \end{aligned}$$ But then $P_{\mu}(\phi, f) < P(\phi,f)$, a contradiction since $\mu$ achieves the supremum.
$\Box$
|
---
abstract: 'We collected optical and near IR linear polarization data obtained over 20–30 years for a sample of 51 blazars. For each object, we calculated the probability that the distribution of position angles was isotropic. The distribution of these probabilities was sharply peaked, with 27 blazars showing a probability $<$ 15% of an isotropic distribution of position angles. For these 27 objects we defined a preferred position angle. For those 17 out of 27 blazars showing a well-defined radio structure angle (jet position angle) on VLBI scales (1–3mas), we looked at the distribution of angle differences – the optical polarization relative to the radio position angles. This distribution is sharply peaked, especially for the BL Lac objects, with alignment better than 15 for half the sample. Those blazars with preferred optical position angles were much less likely to have bent jets on 1–20mas scales. These results support a shock-in-jet hypothesis for the jet optical emission regions.'
author:
- 'Michael J. Yuan$^1$, Hien Tran$^2$, Beverley Wills$^1$, D. Wills$^1$'
title: 'The Physics of Blazar Optical Emission Regions I: Alignment of Optical Polarization and the VLBI Jet'
---
\#1[[*\#1*]{}]{} \#1[[*\#1*]{}]{} =
\#1 1.25in .125in .25in
Introduction
============
Polarization observations have long been a very important probe of the internal structure of blazar jets. Bright spatially resolved knots often show radio polarization ([**E**]{} vector) aligned with the projected jet direction, indicating a perpendicular magnetic field. This suggests that shocks are responsible for compressing the jet magnetic field and accelerating the synchrotron-emitting electrons (e.g. Aller, Aller & Hughes 1985).
The relation between optical polarization and VLBI structure provides a unique tool for investigating the regions of jet formation on $\la$ parsec scales. While previous statistical investigations have shown a tendency for optical polarization to be aligned with the jet (Impey et al. 1991, Rusk & Seaquist 1985), the interpretation of optical polarization is less clear because the emitting regions in blazars are not resolved, blazars often show violent short-term optical polarization variability, and the old radio observations did not have sufficient angular resolution to probe the region near the optically emitting core. A few quasi-simultaneous optical-VLBI observations indicate that the optical polarization is aligned with the direction of newly-ejected blobs at the highest VLBI resolutions (Gabuzda & Sitko 1996, Lister & Smith 2000). The optical polarization may originate in shocks at the base of the jet.
We re-address the question of optical alignment, taking advantage of a more extensive optical polarization database, and more and improved VLBI maps.
Data and Derived Parameters
===========================
Our sample consists of 31 BL Lac type objects (BLLs) and 20 high polarization QSOs (simply called QSOs) with extensive optical linear polarization data and high quality VLBI maps taken from the literature. Optical polarization data from observations spanning 20–30 years were collected from the literature and McDonald Observatory archives.
We determined the following parameters for each blazar:
1. The probability that the measured optical polarization vectors are drawn from an isotropic distribution.
2. The preferred optical polarization position angle. This is the angle of the vector average of the unit vectors corresponding to each polarization measurement. We calculate this for objects with an isotropic distribution probability $<$ 15%. In these cases, our data are consistent with a single preferred angle (for an exception with two preferred angles, see the paper by Cross & Wills, these Proceedings).
3. The position angle for the VLBI inner structure. While some blazar jets are straight, many are curved even on very small scales (Gomez et al. 1999, Kellermann et al. 1998). Therefore, we measure position angles on both 1–3mas and 5–20mas scales, and determine a jet bending angle (the difference between them).
Results
=======
1. Most BLLs show long-term, preferred optical-polarization angles despite their violent short-term variability (Figure 1–left). The probability of this distribution arising by chance is $<< 10^{-4}$ for BLLs alone, and for BLLs and QSOs combined. The QSOs’ distribution is significantly different from the BLLs’ (0.5% chance for the two to arise from the same underlying distribution) and consistent with an isotropic angle distribution.
2. When we look only at the objects with preferred optical polarization position angles most BLLs have preferred optical polarization aligned with the VLBI 1–3mas jet. For BLLs, or BLLs and QSOs combined, the probability that Figure 1–right represents an isotropic distribution of angles is $<$0.1%.
3. The objects with preferred optical polarization angles show a strong tendency to have straight VLBI jets (bending angle $< 15^{\circ}$) compared with objects with no preferred optical polarization angles (Figure 2). The probability for the objects with preferred optical angles to have the same VLBI bending angle distribution as the ones with no preferred angles, is less than 1%. Objects with preferred optical polarization angle and small VLBI bending are mostly BLLs.
Discussion
==========
A natural explanation for the result that optical polarization tends to align with the jet, is that the optical synchrotron emission arises from a shock front in which the jet magnetic field has been compressed, on average, perpendicular to the jet. The large scatter in the optical polarization angles for a given blazar suggests that, in the inner jet region, the compressed magnetic field changes direction with time. Possible explanations are that the inner jet is internally unstable, or shocks may form via interaction with gas surrounding the central engine. The jets of QSOs may be affected by gas in the NLR and BLR, present in QSOs but absent in BLLs. This may explain why QSOs show preferred optical polarization angles less frequently. The variations may be enhanced by the effects of foreshortening and relativistic beaming.
The tendency that objects with well-defined preferred optical polarization directions also have very small VLBI scale jet bending indicates that a well-behaved straight jet on parsec to Kpc scales corresponds to a well-behaved jet on sub-parsec (optical) scales. Large curvature is likely to be the effect of projection of small jet curvature at very small viewing angles (Gower et al. 1982). Possible causes of jet curvature are \[1\] an interaction with the environment, or \[2\] an apparent curvature. In the first case, how does the base of the jet know about the environment on much larger scales? The angular resolution of optical observations is at best a factor of 100 worse than VLBI, often $> 100$mas. So we do not have direct evidence to test our assumption that the more energetic optical photons arise near the base of the jet. The optical emission could arise in the same shocks giving rise to cm-wavelength emission. The observation of rapid polarization variation at cm wavelengths, outside the core, gives credence to this idea (Gabuzda et al. 2000). In the second case, the direction of particle ejection may vary with time, for example, via a precession jet (e.g. Hummel et al. 1997). Present data are inadequate to address changes in optical polarization position angles on precession time scales.
Aller, H. D., Aller, M. F. & Hughes, P. A. 1985, , 298, 296 Gabuzda, D. C., Kochenov, P. et al. 2000, , 313, 627 Gabuzda, D. C., Sitko, M. L. & Smith, P. S. 1996, , 112, 1877 Gomez, J., Marscher, A. P. et al. 1999, , 519, 642 Gower, A. C., Gregory, P. C., Unruh, W. G. & Hutchings, J. B. 1982, , 262, 478 Hummel, C. A., Krichbaum, T. P. et al. 1997, , 324 857 Impey, C. D., Lawrence, C. R. & Tapia, S. 1991, , 375, 46 Lister, M. L. & Smith, P. S. 2000, , 541, 66 Kellermann, K. I., Vermeulen, R. C. et al., 1998, , 115, 1295 Rusk, R. & Seaquist, E. R. 1985, , 90, 30
|
---
abstract: 'We aim to give a pedagogic presentation of the open system dynamics of a periodically driven qubit in contact with a temperature bath. We are specifically interested in the thermodynamics of the qubit. It is well known that by combining the Markovian approximation with Floquet theory it is possible to derive a stochastic Schrödinger equation in $\mathbb{C}^2$ for the state of the qubit. We follow here a different approach. We use Floquet theory to embed the time-non autonomous qubit dynamics into time-autonomous yet infinite dimensional dynamics. We refer to the resulting infinite dimensional system as the dressed-qubit. Using the Markovian approximation we derive the stochastic Schrödinger equation for the dressed-qubit. The advantage of our approach is that the jump operators are ladder operators of the Hamiltonian. This simplifies the formulation of the thermodynamics. We use the thermodynamics of the infinite dimensional system to recover the thermodynamical description for the driven qubit. We compare our results with the existing literature and recover the known results.'
author:
- Brecht Donvil
bibliography:
- 'lit.bib'
date: 'June 16, 2017'
title: Thermodynamics of a Periodically Driven Qubit
---
Introduction
============
In a recent paper [@PekCal] an experimental setup was proposed to perform a calorimetric measurement of work performed on a quantum system. The setup consists of a driven qubit in contact with a finite sized electron bath at a certain temperature. The electron bath acts as a calorimeter, its temperature changes due to interactions between the qubit and electrons. By monitoring the temperature one can indirectly track the evolution of the qubit.
The experimental setup of [@PekCal] was theoretically studied by [@Paolo] in the case of an adiabatic or weak drive. The authors of [@Paolo] derived a fluctuation relation and verified the first law of thermodynamics: the change of the internal energy of the qubit is equal to the work performed on the system minus the heat dissipated to the environment. Finally [@Paolo] showed how the thermodynamic relations recover the known expressions in the infinite calorimeter limit [@Breuer2].
We want to extend the results of [@Paolo] beyond a weak or adiabatic treatment of the drive to a more general periodic drive. To do this it is essential to have a clear understanding of thermodynamics of a periodically driven qubit in contact with an infinite sized environment. This is what we aim for in the present paper. To develop a formalism well attuned to extend the work of [@Paolo] to a general periodic drive [@DonvilCal].
We do this by lifting the problem to an infinite dimensional Hilbert space. The reason for doing so is that we can use Floquet theory to embed the periodically driven qubit into a time-autonomous dynamics in this infinite dimensional Hilbert space. We refer to the infinite dimensional time-autonomous system as the dressed-qubit. For the dressed-qubit we can derive a stochastic Schrödinger equation, see e.g. [@openpaper]. The advantage of the infinite dimensional Hilbert space is that the jump operators are ladder operators of the Hamiltonian of the dressed-qubit. This fact simplifies the formulation of the thermodynamics. We show how to recover from the infinite dimensional level of description the thermodynamics for the driven qubit. We compare our results with earlier results in [@Cuetara; @AlickiFloquet; @Gasparinetti1; @Gelbwaser] and we show that our approach is in agreement. We get the same heat currents and fluctuation relation. From the fluctuation relation we obtain an expression for the path-wise entropy production. The entropy production is an important indicator for the design and control of efficient engines at the micro and nano-scales see e.g. the discussion in [@DechantHeat; @CampisiJukka] in the quantum case and [@PaoloKay] and references therein for the classical counterpart.
The study of periodically driven systems in contact with environments is an active field with a rich literature. Some recent works in relation to quantum nanodevices include [@Cuetara; @AlickiFloquet; @Szczygielski; @Gasparinetti1; @SzAl2015; @LeAlKo2012; @KoDiHa1997; @HoKeKo2009; @Gelbwaser] and in relation to quantum-computers [@AlickiMaster]. It is therefore worth to briefly discuss how to put the contribution of the present work in the context of the existing literature.
Reference [@BPfloquet] combined Floquet theory [@Shirley; @Zeldovich1] and the Markov approximation to derive a stochastic Schrödinger equation in $\mathbb{C}^2$. The authors of [@BPfloquet] also noticed the correspondence between transitions occurring in driven qubit dynamics and those between dressed-atom states [@tan]. By dressed-atom we mean an atom interacting with a fully second quantised electromagnetic field. The infinite dimensional dressed-qubit is known to be equivalent with the dressed-atom picture in a certain limit [@Shirley; @Swain1; @Guerin], hence the name dressed-qubit. We embed the stochastic dynamics of the driven qubit in the stochastic dynamics of the dressed-qubit. This confirms the interpretation made by [@BPfloquet] of the dynamics of the driven qubit in terms of atom-dressed states.
To the best of our knowledge the study of the thermodynamics of the periodically driven qubit in the existing literature is based on the analysis of the Lindblad-Gorini-Kossakowski-Sudarshan equation, see e.g. [@open; @Rivas], for the state operator acting on the $\mathbb{C}^2$ Hilbert space. See in particular [@Cuetara; @AlickiFloquet; @Gasparinetti1; @LeAlKo2012; @Gelbwaser]. References [@Cuetara; @Gasparinetti1] extend this approach to the counting statistics formalism. We show that the thermodynamic description of the driven qubit which we derive from the dressed-qubit gives the same fluctuation relation and heat currents as [@Cuetara; @AlickiFloquet; @Gelbwaser; @Gasparinetti1].
Reference [@LevyKosloff] thoroughly discusses local and global master equations for interacting subsystems separately interacting with a heat bath. It is instructive to characterize the Floquet approach in the language of [@LevyKosloff]. In the local approach one derives a master equation by neglecting the interaction between both subsystems. In the global approach one derives a master equation by solving the two subsystems without neglecting the interaction. For this reason Floquet theory is a form of a global approach. One first solves the dynamics of the qubit and the drive. The weak drive limit is a local approach. For strong driving this approach can result in violations of the second law [@GeKoSk95].
The paper is structured as follows: Section \[sec:Floquet\] focusses on Floquet theory for a closed periodically driven qubit [@Shirley; @Zeldovich1]. In particular, we recall how the time-non autonomous periodic quantum dynamics on a finite dimensional Hilbert space can be mapped on a time-autonomous infinite dimensional system [@Shirley; @Swain1; @Guerin].
In Section \[sec:model\] we introduce the model of the periodically driven qubit interacting with a infinite sized electron bath.
Section \[sec:SSE\] consists of two parts. In Subsection \[subsec:sseq\] we recall the results of [@BPfloquet] on the stochastic Schrödinger equation for the periodically driven qubit. Subsection \[subsec:qp\] contains the core result of this paper. First we formulate a master equation for the dressed-qubit following the method of [@openpaper]. Then we show that the stochastic evolution of the periodically driven qubit can be embedded in the stochastic evolution of the dressed-qubit. The periodically driven qubit corresponds to the dressed-qubit with a specific set of initial conditions. This ensures that the heat dissipated to the environment in the case of the dressed-qubit and in the case of the driven qubit are equal.
The last part of this paper leisurely discusses the recovery of the thermodynamics of the periodically driven qubit from the thermodynamic relations of the infinite dimensional dressed-qubit. In Sections \[sec:mast\] and \[sec:ther\] we recover fluctuation relation and heat currents by [@Gasparinetti1; @Cuetara; @AlickiFloquet]. In Section \[sec:mast\] we derive a Pauli master equation for the populations in the Floquet states. Using the definition of Lebowitz and Spohn [@Lebowitz] we obtain an expression for the average entropy production. We also derive a fluctuation relation which holds for both the driven qubit and the dressed-qubit. In Section \[sec:ther\] we study the thermodynamics of the dressed-qubit: we verify the first and second law of thermodynamics for the dressed-qubit and the driven qubit.
The example of a constant drive is discussed in Section \[sec:Const\]. We also consider the weak drive limit for the dressed-qubit. We recover the thermodynamic relations derived in the weak drive regime. In Section \[sec:Mon\] we look at the monochromatic drive. We derive the stochastic Schrödinger equation, discuss the thermodynamics of this example and compare to earlier results by [@AlickiFloquet].
Floquet theory {#sec:Floquet}
==============
In this Section we aim to give a brief overview on the Floquet approach for a closed periodically driven qubit, with an emphasis of what is needed in the rest of the paper. The qubit has a Hamiltonian $H_q(t)=H+H_d(t)$. The periodic part of the Hamiltonian $H_d(t)$ is called the drive. The time evolution of the two level system is found by solving the set of equations $$\label{eq:set}
\begin{cases}
(H_q(t)-i\hbar\partial_t)\psi(t)=0\\
\psi(0)=\psi_0,
\end{cases}$$ were the Hamiltonian $H_q(t+T)=H_q(t)$ is periodic.
By Floquet’s theorem there exists a periodic matrix $P_{t+T,0}=P_{t,0}$ and a matrix $D$ such that the solution of the set of equations can be written into the form $$\psi(t)=P_{t,0}e^{-iD t}\psi_0.$$ References [@Shirley] and [@Zeldovich1] showed that the matrix $D$ is Hermitian. It can be diagonalised in an orthonormal basis $\nu_\pm$ with real eigenvalues $\epsilon_\pm$, which are also called quasi-energies. The eigenvectors evolve as $$\begin{aligned}
\varphi_\pm(t) &=P_{t,0}e^{-iD t}\nu_\pm\\
&=e^{-i\epsilon_\pm t}\phi_\pm(t)\label{eq:ineed}.\end{aligned}$$ We have defined the Floquet states $\phi_{\pm}(t)\equiv P_{t,0}\nu_\pm$, which are periodic. It is clear that we can add $n\hbar 2\pi/T$ to the quasi-energy in the last line and replace $\phi_\pm(t)$ by $\phi_{\pm,n}(t)\equiv e^{in (2\pi/T) t}\phi_\pm(t)$ and still get the same solution $\varphi_\pm(t)$. In this sense we are free to choose what we call $\epsilon_\pm$. In the rest of this paper we choose $\epsilon_\pm$ to be in the zeroth Brillouin zone $]-\pi/T,\pi/T]$. The operator $P_{t',t}$ is then the operator that evolves the Floquet states $\phi_\pm(t)$ to $\phi_\pm(t')$.
Note that the above is only an existence result. To find the Floquet states in practice one solves the problem $$\label{eq:floquet}
\begin{cases}
(H_q(t)-i\hbar\partial_t)\phi_{\pm,n}(t)=\left(\epsilon_\pm+n\frac{2\pi}{T}\right)\phi_{\pm,n}(t)\\
\phi_{\pm,n}(t+T)=\phi_{\pm,n}(t).
\end{cases}$$ Then $\nu_\pm=\phi_{\pm}(0)$. This problem is an eigenvalue equation in a larger space: our original Hilbert space $\mathbb{C}^2$ extended with the space of $T$-periodic functions $L^2[0,T]$. The scalar product is extended by $$\label{eq:scalar}
\langle f,g\rangle_{ L^2}=\frac{1}{T}\int_0^T {\mathop{}\!\mathrm{d}}\tau \langle f(\tau),g(\tau)\rangle,$$ for periodic functions $f$, $g\in \mathbb{C}^2\otimes L^2[0,T]$. In the enlarged Hilbert space the Floquet states form an orthonormal basis. We define the Floquet Hamiltonian $H_F(\tau)$ as $$\label{eq:floquetHam}
H_F(\tau)\equiv H_q(\tau)-i\hbar\partial_\tau,$$ it is Hermitian for the scalar product $\langle\,.\, ,\,.\, \rangle_{L^2}$. The Floquet Hamiltonian is the energy operator of the infinite dimensional system we call the dressed-qubit.
The action of the periodic matrix $P_{t,0}$ on a vector $\psi\in\mathbb{C}^2$ can be expressed in terms of the Floquet states: $P_{t,0}\psi=\sum_{r=\pm}\phi_r(t)\langle \nu_r,\psi\rangle$. The solution of the set of equations (\[eq:set\]) is $$\begin{aligned}
\psi(t) &=\sum_{r=\pm} e^{-i\epsilon_r t}\phi_r(t)\langle\nu_r,\psi_0\rangle\label{eq:soleq}\\
&=\sum_{r=\pm}\sum_{n\in\mathbb{Z}} e^{-i(\epsilon_r +n\frac{2\pi}{T})t}\phi_{r,n}(t)\frac{1}{T}\int_0^T {\mathop{}\!\mathrm{d}}s e^{in \frac{2\pi}{T}s} \langle\nu_r,\psi_0\rangle\\
&=\sum_{r=\pm}\sum_{n\in\mathbb{Z}} e^{-i(\epsilon_r +n\frac{2\pi}{T})t}\phi_{r,n}(t)\frac{1}{T}\int_0^T {\mathop{}\!\mathrm{d}}s \, e^{in \frac{2\pi}{T}s} \langle P_{s,0}\nu_r,P_{s,0}\psi_0\rangle\label{eq:soleq2}.\end{aligned}$$ Going to the last line, we used the unitarity of $P_{s,0}$.
The dynamics in of the dressed-qubit in the space $\mathbb{C}^2\otimes L^2[0,T]$ are given by the set of equations $$\label{eq:set2}
\begin{cases}
(H_F(\tau)-i\hbar\partial_t)\Psi(t,\tau)=0\\
\Psi(0,\tau)=P_{\tau,0}\psi_0,
\end{cases}$$ it is a one dimensional vector valued Schrödinger equation with periodic boundary conditions in the variable $\tau$ and $t$ is the time coordinate. The second equation lifts the initial state of the qubit $\psi_0$ to the larger space. In principle one could take any initial state, but this one reproduces the evolution of the driven qubit. $\Psi(t,\tau)$ is the coordinate representation of a time dependent vector $\Psi(t)\in \mathcal{H}\otimes L^2[0,T]$. The advantage of this larger space is that the Hamiltonian $H_F(\tau)$ is time-autonomous. The solution of the set of equations (\[eq:set2\]) can be expressed in terms of the Floquet states, the energy eigenbasis of $H_F(\tau)$, in a straightforward way: $$\begin{aligned}
\label{eq:soleq3}
\Psi(t,\tau)=\sum_r\sum_{n\in\mathbb{Z}} e^{-i(\epsilon_r +n\frac{2\pi}{T})t}\phi_{r,n}(\tau)\frac{1}{T}\int_0^T {\mathop{}\!\mathrm{d}}s\, e^{in \frac{2\pi}{T}s} \langle \phi_r(s),P_{s,0}\psi_0\rangle.\end{aligned}$$ Comparing the solutions for the driven qubit and the dressed-qubit given by (\[eq:soleq\]) and (\[eq:soleq3\]) we see that both solutions are related by $$\label{eq:closedeq}
\Psi(t,\tau)=P_{\tau,t}\psi(t).$$ The above relation holds between the driven qubit and the dressed-qubit initialised in the zeroth Brillouin zone. By multiplying the lift of the initial conditions and the left hand side of by $e^{in (2\pi/T) \tau}$, the relation holds for the dressed-qubit initialised in any single Brillouin zone. Equation implies the relation $\psi(t)=\Psi(t,\tau)\big|_{\tau=t}$ [@Peskin1; @Pfeifer1].
![On the left side is the dressed-qubit with Hamiltonian $H_F(\tau)=\frac{\hbar\omega_q}{2}\sigma_z+H_d(\tau)-i\hbar\partial_\tau$, where $H_d(\tau)$ has period $2\pi/\omega_L$. In the appropriate limit [@Shirley; @Zeldovich1; @Guerin] it is equivalent to the system portrayed on the right: the atom interacting with the single mode of an electromagnetic field. $H_{SP}$ is the interaction between the system and photon corresponding to the drive $H_d(t)$.[]{data-label="fig:equiv"}](equiv.eps)
(0,0) (-67,12.5)[$H_d(t)$]{} (-68,23.3)[$\hbar\omega_q$]{} (-31,15.7)[$\hbar\omega_q$]{} (-26,12)[$H_{SP}$]{} (-12,14.7)[$\hbar\omega_L$]{} (-90,-3.5)[$H_F(\tau)=\frac{\hbar\omega_q}{2}\sigma_z+H_d(\tau)-i\hbar\partial_\tau$]{} (-38,-3.5)[$\frac{\hbar\omega_q}{2}\sigma_+\sigma_-+H_{SP}+\hbar\omega_La^\dagger a$]{}
Moreover there is an established between the dressed-qubit system and the two level system interacting with a single mode of a fully second quantised electromagnetic field with frequency $\omega_L= 2\pi/T$. In the limit of an intense laser field [@Shirley; @Zeldovich1; @Guerin] the Floquet states $\phi_{\pm,n}$ correspond to dressed-atom states with energy $\epsilon_\pm+n\hbar\omega_L$. The periodic phase factors $e^{in\omega_L \tau}$ can be interpreted as increasing the amount of photons by $n$ and $-\frac{i}{\omega_L}\partial_\tau$ is the number operator for the photons. The periodic drive $H_D(\tau)$ is the interaction between the photons and the atom. The periodically driven qubit is a semi-classical approximation of this interaction, where we completely forget about the interaction with the photons. The equivalence between the two systems is illustrated in Figure \[fig:equiv\].
The model {#sec:model}
=========
From now on we will focus on the actual system of interest: a periodically driven qubit in contact with a thermal electron bath. The Hamiltonian of the full system consists of three terms $$\label{eq:hamiltonian1}
H(t)=H_q(t)+H_e+H_I.$$ The first term on the right hand side is the Hamiltonian of the periodically driven qubit. It is time dependent and periodic with period $T$ $$\label{eq:hamQ}
H_q(t)=\frac{\hbar\omega_q}{2}\sigma_z+H_d(t),$$ where $\sigma_z$ is the canonical Pauli matrix and the time dependent term $H_d(t)$ represents the drive. Let us define the angular frequency $\omega_L=\frac{2\pi}{T}$. The energy of the electrons is given by $$\label{eq:hamE}
H_e=\sum_k \eta_k a^\dagger_k a_k,$$ $a_k$ and $a^\dagger_k$ are ladder operators satisfying the fermionic anticommutation relations. The sum over $k$ is over all degrees of freedom for the electrons. The last term on the right hand side of equation (\[eq:hamiltonian1\]) represents the interaction between the two systems $$\label{eq:hamI}
H_I=\sum_{kl}g_{kl}(\sigma_++\sigma_-)a^\dagger_ka_l.$$ The diagonal elements of the interaction are assumed to be zero, i.e. $g_{kk}=0$ for all $k$ and $\sigma_\pm$ are ladder operators in the qubit space. Note that one could also be interested in coupling with a different thermal bath, e.g. a photon bath. The results derived in the following Sections do not depend on the specific thermal bath. It is only important that the jump rates satisfy detailed balance.
Stochastic Schrödinger equation {#sec:SSE}
===============================
In this Section we discuss the evolution of a periodically driven qubit in contact with an electron bath. In Subsection \[subsec:sseq\] we recall a result by [@BPfloquet] to obtain a stochastic Schrödinger equation in $\mathbb{C}^2$ for the state of the driven qubit. In second Subsection \[subsec:qp\] we derive a stochastic Schrödinger equation for the dressed-qubit system. We show how the stochastic evolution of the driven qubit can be embedded in the stochastic evolution of the dressed-qubit.
Periodically driven qubit {#subsec:sseq}
-------------------------
In [@BPfloquet] a stochastic Schrödinger equation is derived for a periodically driven system in contact with a thermal photon bath. The approach can be extended in a straightforward way to systems in contact with a thermal electron bath with inverse temperature $\beta$. The only significant difference is in the jump rates. The exact form of the jump rates is given in Appendix \[sec:appSSE1\]. For the results derived in the next Sections it is only important that the rates satisfy detailed balance.
The derivation by [@BPfloquet] assumes the existence of four timescales. The first timescale is $\tau_B$ over which the bath correlation functions decay. Secondly $\tau_R$ is the relaxation time of the open system, the qubit. $\tau_m$ is set by inverse of the dressed-qubit frequencies, i.e. the inverse of the difference in the quasi-energies $\epsilon_\pm/\hbar$ and $\omega_L$. The last timescale $\tau_T$ is the one over which the transitions are evaluated. It is assumed that $\tau_T$ relates to the others as $\tau_B,\tau_m\ll\tau_T\ll\tau_R$.
We formulate the stochastic Schrödinger equation in the interaction picture. A vector $\bar{\psi}$ is transformed to the interaction picture by setting $\psi=U^\dagger(t,0)\bar{\psi}$, where ${\mathop{}\!\mathrm{d}}U^\dagger(t,0)/{\mathop{}\!\mathrm{d}}t=iH_q(t)U^\dagger(t,0)/\hbar$. The stochastic Schrödinger equation describing the evolution of the qubit consists of two parts: a continuous evolution, proportional to ${\mathop{}\!\mathrm{d}}t$, which is interrupted by sudden jumps $$\label{eq:SSE1}
{\mathop{}\!\mathrm{d}}\psi(t)=-\frac{i}{\hbar}G(\psi(t)) {\mathop{}\!\mathrm{d}}t +\sum_\omega \left(\frac{A(\omega)\psi(t)}{\|A(\omega)\psi(t)\|}-\psi(t)\right){\mathop{}\!\mathrm{d}}N(\omega).$$ The operators $A(\omega)$ are called effect or jump operators, they determine which jumps the qubit can make. The operators are labelled by the amount of energy $\hbar \omega$ that is transferred from the qubit to the environment with the jump. They are expressed in terms of the Floquet states as [@BPfloquet] $$\begin{aligned}
\label{eq:effect1}
A(\omega)&=\sum_{r,s=\pm}\sum_n \alpha_{r,s,n} |\phi_{r}(0)\rangle\langle\phi_s(0)|\end{aligned}$$ where $\phi_r$’s are the Floquet states, with the constraint on the sums for the energies $$\label{eq:energyconstraint}
\hbar\omega=\epsilon_s-\epsilon_r+n\hbar\omega_L$$ and the matrix element $\alpha_{r,s,n}$ is defined by $$\label{eq:alpha}
\alpha_{r,s,n}=\frac{1}{T}\int_0^T {\mathop{}\!\mathrm{d}}\tau\langle\phi_{r}(\tau),(\sigma_++\sigma_-)\phi_{s,n}(\tau)\rangle.$$
The first term on the right hand side is proportional to the short time increment ${\mathop{}\!\mathrm{d}}t$. It gives the continuous evolution of the wave function when no jumps occur $$\label{eq:contev3}
G(\psi)=-\frac{i\hbar}{2}\sum_\omega \gamma(\omega)[A^\dagger(\omega)A(\omega)-\|A(\omega)\psi\|^2]\psi.$$
The rest of the terms on the right hand side describe the jump behaviour. They are proportional to increments of Poisson processes ${\mathop{}\!\mathrm{d}}N(\omega)$. These increments are either 1, meaning the qubit makes a jump, or 0, no jump occurs. The expectation value of the increments, conditioned that at a time $t$ the qubit is in a state $\psi$, is $$\label{eq:condexpfloq}
\mathbb{E}({\mathop{}\!\mathrm{d}}N(\omega)|\psi)=\gamma(\omega)\|A(\omega)\psi\|^2 {\mathop{}\!\mathrm{d}}t,$$ where $\gamma(\omega)$ is called the jump rate. The rates satisfy detailed balance $$\label{eq:detbal}
\frac{\gamma(\omega)}{\gamma(-\omega)}=e^{\beta\hbar \omega}.$$
From the definition of the effect operators given by equation (\[eq:effect1\]) one can see that it is possible for the qubit to exchange different amounts of energy while doing the same jump. For example: the qubit can jump from $\phi_{+,n}$ to $\phi_{-,0}$ exchanging the difference in quasi-energies $\epsilon_+-\epsilon_-$ with the environment plus $n\hbar\omega_L$ on top of that. Interpreting the Floquet states as dressed-atom states, as suggested by [@BPfloquet], would explain this. The change in Brillouin zone means the creation or annihilation of photons with energy $\hbar\omega_L$. In the next section we derive a stochastic Schrödinger equation for the dressed-qubit. We embed the dynamics of the driven qubit in the dynamics of the dressed-qubit. In the end of Section \[sec:Floquet\] we discussed the correspondence between dressed-qubit and dressed-atom states. This correspondence in combination with the embedding of the driven qubit in the dressed-qubit implies the interpretation proposed by [@BPfloquet].
To show the above statements we make an extra assumption on the difference of the quasi-energies. We want to exclude the case in which the difference of the quasi-energies is a multiple of the photon energy. The physical consequence is that one can distinguish between a jump in the qubit state and a change in Brillouin zone, the creation of photons, by monitoring the energy exchange with the environment. We assume that $$\label{eq:assump}
\epsilon_+-\epsilon_-\neq k\hbar\omega_L, \quad \textrm{with} \quad k\in \mathbb{Z}.$$ An equality in the above equation would correspond to a very specific choice of physical parameters. The difference in quasi-energies for the monochromatic drive, see Section \[sec:Mon\], is given by $\sqrt{\hbar^2(\omega_q-\omega_L)^2+4\lambda^2}$. For this to be equal to an multiple of $\hbar\omega_L$ a very non-generic choice of parameters is required.
Under assumption there is a unique integer $n_\omega\in \mathbb{Z}$ such that the energy balance in equation (\[eq:energyconstraint\]) holds. For each jump there is a unique amount of photons annihilated in the drive. The effect operators simplify to $$\begin{aligned}
\label{eq:effect2}
A(\omega)&= \sum_{r,s=\pm}\alpha_{r,s,n_\omega} |\phi_{r}(0)\rangle\langle\phi_s(0)|.\end{aligned}$$
Dressed-qubit {#subsec:qp}
-------------
Let us start by formulating a stochastic Scrödinger equation for the dressed-qubit. Remember that by dressed-qubit we mean the infinite dimensional system for which the closed dynamics are governed by the Floquet Hamiltonian $H_F$ . The Hamiltonian for the dressed-qubit in contact with the electron bath is given by $$H=H_{F}+H_I+H_E,$$ where the different terms have been defined in equations (\[eq:floquetHam\]), (\[eq:hamE\]) and (\[eq:hamI\]). The dynamics are completely time-autonomous, the Hamiltonian does not explicitly depend on time anymore. Therefore a stochastic Schrödinger equation can be formulated by following the method described by [@openpaper], this derivation requires the same timescales as discussed in the beginning of Subsection \[subsec:sseq\]. We rewrite the interaction term in terms of energy eigenoperators, the jump operators, $$H_I=\sum_{kl}\sum_\omega g_{kl}B(\omega)a_k^\dagger a_l.$$ The energy eigenoperators lower the energy of the system with $\hbar\omega$ and are defined by projecting different energy eigenstates onto $\sigma_++\sigma_-$. They act on a state $\Psi(t)$ as $$\begin{aligned}
\label{eq:effectph}
B(\omega)\Psi(t)=\sum_{r,s=\pm}\sum_{k,l}\phi_{r,k}\langle \phi_{r,k},(\sigma_++\sigma_-) \phi_{s,l}\rangle_{L^2}\langle\phi_{s,l},\Psi(t)\rangle_{L^2}\end{aligned}$$ where the constraint (\[eq:energyconstraint\]) on the energies holds with $n=l-k$. Comparing the above expression with expression (\[eq:effect1\]) for the matrix elements of the effect operators $A(\omega)$, we see that they are equal in the Floquet basis. The matrix element $\langle \phi_{r,k},(\sigma_++\sigma_-) \phi_{s,l}\rangle_{L^2}$ is equal to $\alpha_{r,s,l-k}$ as defined in equation . The effect operators determine which jumps a system can make. The qubit and the dressed-qubit can perform the same jumps exchanging the same amount of energy with the environment. Under assumption the change in Brillouin zone $l-k=n_\omega$. The action of the effect operator on a state $\Psi(t)$ becomes
\[eq:effectph2\] $$\begin{aligned}
\begin{split}
B(\omega)\Psi(t)=\sum_{r,s=\pm}\sum_{k} \alpha_{r,s,n_\omega}\phi_{r,k} \langle\phi_{s,k+n_\omega},\Psi(t)\rangle_{L^2}.
\end{split}\end{aligned}$$ A more explicit version of this equality reads $$\begin{aligned}
\begin{split}
(B(\omega)\Psi(t))(\tau)=\sum_{r,s=\pm}\sum_{k} \alpha_{r,s,n_\omega}\phi_{r,k}(\tau) \langle\phi_{s,k+n_\omega},\Psi(t)\rangle_{L^2}.
\end{split}\end{aligned}$$
The stochastic Schrödinger equation for the dressed-qubit in contact with the electron bath is given by $$\begin{aligned}
\label{eq:sseph}
{\mathop{}\!\mathrm{d}}\Psi(t)=-\frac{i}{\hbar}K(\Psi(t)) {\mathop{}\!\mathrm{d}}t +\sum_\omega \left(\frac{B(\omega) \Psi(t)}{\|B(\omega) \Psi(t)\|_{L_2}}-\Psi(t)\right){\mathop{}\!\mathrm{d}}M(\omega).\end{aligned}$$ The continuous evolution is defined analogous to last subsection $$\begin{aligned}
\label{eq:ssecontph}
K(\Psi)&=-\frac{i\hbar}{2}\sum_\omega \gamma(\omega)[B^\dagger(\omega) B(\omega)-\|B(\omega) \Psi\|_{ L_2}^2]\Psi\end{aligned}$$ and the conditional average of the Poisson processes are $$\label{eq:condexpph}
\mathbb{E}({\mathop{}\!\mathrm{d}}M(\omega)|\Psi)=\gamma(\omega)\|B(\omega)\Psi\|_{ L_2}^2 {\mathop{}\!\mathrm{d}}t.$$ The jump rates $\gamma(\omega)$ are equal to those in equation .
Let us now discuss the relation between the stochastic evolution for the periodically driven qubit and for the dressed-qubit, given by equations and respectively. We have already shown that both systems can make similar jumps exchanging the same amount of energy and with the same jump rates. A mapping between the driven-qubit and dressed-qubit can be found if the dressed qubit is initialised as in $$\label{eq:init}
\psi_0 \longleftrightarrow\Psi_0=P_{\tau,0}\psi_0= \phi_{+,0}(\tau) \langle \phi_+(0),\psi_0\rangle +\phi_{-,0}(\tau) \langle \phi_{-}(0),\psi_0\rangle$$ In fact it does not matter in which Brillouin zone the dressed-qubit is initialised, as we show below, as long as the initial state is in only one of them.
Under assumption we know at any time $t$ the amount $\mu(t)$ the quanta of $\hbar \omega_L$ gained by the environment, the total decrease (or increase when $\mu(t)<0$) in Brillouin zone during the process. During the continuous evolution there is no change in Brillouin zone, while a jump $A(\omega)$ lowers it by $n_\omega$. We can write the evolution of the qubit state $\psi(t)$ and the decrease in Brillouin zone $\mu(t)$ by a couple of stochastic differential equations $$\label{eq:couple}
\begin{cases}
{\mathop{}\!\mathrm{d}}\psi(t)=-\frac{i}{\hbar}G(\psi)+\sum\limits_{\omega} \left(\frac{A(\omega)\psi}{\|A(\omega)\psi\|}-\psi\right){\mathop{}\!\mathrm{d}}N(\omega)\\
{\mathop{}\!\mathrm{d}}\mu(t)=\sum\limits_\omega n_\omega {\mathop{}\!\mathrm{d}}N(\omega).
\end{cases}$$ In the Appendix \[sec:appSSE2\] we show that the dressed-qubit and the driven qubit state are connected by $$\label{eq:equivvv}
\Psi(t,\tau)=e^{-i\mu(t)\omega_L\tau}P_{\tau,0}\psi(t),$$ the short time increment of the state $\Psi(t,\tau)$ satisfies the stochastic Schrödinger equation for the dressed-qubit with initial condition . Under this initial condition the Poisson processes $N(\omega)$ in and $M(\omega)$ in are the same. We can initialise the dressed-qubit in an arbitrary Brillouin zone $m$ by replacing $\mu(t)$ by $\mu(t)-m$ in the above equation.
In summary relation shows how the stochastic evolution of the driven qubit can be embedded in the stochastic evolution of the dressed-qubit by introducing the photon counting process $\mu(t)$. In general the dressed-qubit shows a more complex behaviour than described by , it can be initialised in multiple Brillouin zones. The equivalence discussed at the end of Section \[sec:Floquet\] implies that the jumps between Floquet states and the change Brillouin zones for the periodically driven qubit can be interpreted as jumps between dressed-atom states and the creation or annihilation of photons.
Finally, let us define the projector $P_n$ which projects $\Psi(t)$ onto the n-th Brillouin zone: $$\label{eq:projector}
P_n \Psi(t)=\phi_{+,n}\langle\phi_{+,n}, \Psi(t)\rangle_{L^2}+\phi_{-,n}\langle\phi_{-,n}, \Psi(t)\rangle_{L^2}.$$ The state $\Psi_n(t)= P_n \Psi(t)$ is living in the $n$-the Brillouin zone. From equation we see that when the dressed-qubit is initialised as , all Brillouin zones are empty, i.e. $\Psi_n(t)=0$, except when $n=\mu(t)$. Figure \[fig:fol\] shows a visualisation of the dressed-qubit process.
![A visualisation of the stochastic process given by the couple of equations and the process with jumps at times $t_1$, $t_2$, ... The qubit state is initialised in the lower Floquet state $\phi_-$ and initially no photons have been created or annihilated in the drive, so $\mu(t_0)=0$. At the jump times, the qubit can jump between different Floquet states and the photon number changes. []{data-label="fig:fol"}](jumpsss.eps)
(0,0) (-10,-4)[$t$]{} (-92,67)[$\Psi(t_1)$]{} (-88,-2)[$t_1$]{} (-90.5,38.5)[$\psi(t_1)$]{} (-95,52)[$2$]{} (-95,42)[$1$]{} (-109,33)[$\mu(t_1)=$]{} (-95,33)[$0$]{} (-98,23)[$-1$]{} (-98,13)[$-2$]{} (-58,67)[$\Psi(t_2)$]{} (-54,-2)[$t_2$]{} (-56.5,48)[$\psi(t_2)$]{} (-61,52)[$2$]{} (-61,42)[$1$]{} (-75,42)[$\mu(t_2)=$]{} (-61,33)[$0$]{} (-64,23)[$-1$]{} (-64,13)[$-2$]{} (-24,67)[$\Psi(t_3)$]{} (-20,-2)[$t_3$]{} (-22.5,38.5)[$\psi(t_3)$]{} (-27,52)[$2$]{} (-27,42)[$1$]{} (-41,33)[$\mu(t_3)=$]{} (-27,33)[$0$]{} (-30,23)[$-1$]{} (-30,13)[$-2$]{}
Master equation and entropy production {#sec:mast}
======================================
We calculate the average entropy production of the periodically driven qubit in two ways. In the next subsection we calculate it path-wise by comparing the probability measure of a realisation of the process and its backwards process. But first we start out by deriving a (classical) master equation for the Floquet states, a Pauli master equation, from the set of differential equations With an equation like this we can use the classical definition for the entropy production rate by Lebowitz and Spohn [@Lebowitz].
Master Equation
---------------
Following [@openpaper] we define the state operator for the dressed-qubit system as $\rho(t)=\mathbb{E}(\Psi(t) \Psi^*(t))$. From the stochastic Schrödinger equation it is straightforward to check that the state operator satisfies the Lindblad-Gorini-Kossakowski-Sudarshan equation in the interaction picture $$\label{eq:lind}
\frac{{\mathop{}\!\mathrm{d}}}{{\mathop{}\!\mathrm{d}}t}\rho(t)=\sum_\omega \left( B(\omega)\rho(t)B^\dagger(\omega)-\frac{1}{2}\{B^\dagger(\omega)B(\omega),\rho(t)\}\right).$$ Remember that $P_n$ as defined in equation is the projector on the $n$-th Brillouin zone. The diagonal blocks of the state operator $\rho_n(t)=P_n\rho(t)P_n$ corresponding to the $n$-th Brillouin zone satisfy a closed set of coupled (Lindblad) equations. Indeed projecting both sides of the above equation on the left and right with the projector $P_n$ we find, with de definition of the effect operators with , that $$\label{eq:lindd}
\frac{{\mathop{}\!\mathrm{d}}}{{\mathop{}\!\mathrm{d}}t}\rho_n(t)=\sum_\omega \left( B(\omega)\rho_{n+n_\omega}(t)B^\dagger(\omega)-\frac{1}{2}\{B^\dagger(\omega)B(\omega),\rho_n(t)\}\right).$$
We define $P(r,n,t)=\langle\phi_{r,n},\rho_n(t)\phi_{r,n}\rangle_{L_2}$ as the population in the state $\phi_{r,n}$. From equation we get a closed equation for the probabilities $$\label{eq:master}
\frac{{\mathop{}\!\mathrm{d}}}{{\mathop{}\!\mathrm{d}}t}P(r,n,t)=\sum_s\sum_m\left[ W_{m-n}(r|s)P(s,m,t)-W_{n-m}(s|r)P(r,n,t)\right].$$ This equation is of the form of the master equation derived for dressed-atom states by Cohen-Tannoudji and Reynaud [@tan]. The rates only depend on the difference between Brillouin zones and the Floquet states $$W_{m-n}(r|s)=\gamma(\epsilon_s/\hbar-\epsilon_r/\hbar+(m-n)\omega_L)|\alpha_{r,s, m-n}|^2,$$ where the matrix element $\alpha_{r,s, m-n}$ was defined in equation . From the definition of Lebowitz and Spohn [@Lebowitz] we get the average entropy production rate corresponding to the master equation $$\label{eq:entprod}
\sigma(t) = \frac{1}{2}\sum_{r,s}\sum_{k,m}\left[\bigg( W_{n-m}(r|s)P(s,n,t)-W_{m-n}(s|r)P(r,m,t)\bigg)\log\left(\frac{W_{n-m}(r|s)P(s,n,t)}{W_{m-n}(s|r)P(r,m,t)}\right)\right].$$ This is the average entropy production obtained by [@Cuetara].
Summing over $n$ on both sides of equations we get a Lindblad equation for the state operator of the qubit $$\label{eq:linddd}
\frac{{\mathop{}\!\mathrm{d}}}{{\mathop{}\!\mathrm{d}}t}\bar{\rho}(t)=\sum_\omega \left( A(\omega)\bar{\rho}(t)(t)A^\dagger(\omega)-\frac{1}{2}\{A^\dagger(\omega)A(\omega),\bar{\rho}(t)\}\right),$$ where $\bar{\rho}(t)=\sum_{n=-\infty}^{+\infty}\rho_n(t)$. It is straightforward to check that this is the Lindblad equation stemming from the stochastic Schrödinger equation for the driven qubit . The master equation for the populations $P(r)=\langle\phi_{r}(0),\bar{\rho}(t)\phi_{r}(0)\rangle=\sum_n P(r,n)$ can be obtained by taking the diagonal elements in the Floquet basis $\phi_\pm(0)$. The result is $$\label{eq:master2}
\frac{{\mathop{}\!\mathrm{d}}}{{\mathop{}\!\mathrm{d}}t}P(r,t)=\sum_s \left( W(r|s)P(s,t)-W(s|r)P(r,t)\right),$$ where the rates $W(r|s)=\sum_n W_n(r|s)$ do not satisfy detailed balance any longer. The above equation can also be obtained by summing over $n$ on both sides of equation .
The entropy equation of Lebowitz and Spohn [@Lebowitz] still also applies to and gives a positive definite entropy production $\bar{\sigma}(t)$. Using the log sum inequality one can show that $\sigma(t)\geq \bar{\sigma}(t)$ [@Cuetara].
Pathwise entropy production
---------------------------
We aim to calculate the path-wise entropy production for a realization of the stochastic process given that the dressed-qubit is initialised in a single Brillouin zone. Note that one could do the same derivation from the set of equations , both approaches give the same result. We compare the probability density of a realization of the process, with the density of its time reversed process. We follow the same procedure as [@Paolo].
Let us specify the measurement process. We wait until the dressed-qubit makes a jump at a time $t_i$ such that it is certainly in one of the Floquet states $\phi_{\pm,n}$, to start the measurement. In an experiment one can take $t_i$ to be very large compared to the time scale of the evolution of the qubit such that it has almost certainly made a jump to one of the Floquet states. The Floquet states are stationary states for the continuous evolution of the stochastic Schrödinger equation . We are dealing with a pure jump process between the Floquet states.
The dressed-qubit has an initial probability distribution $$P_i=P(i,n,t_i),$$ where $i=\pm$ and $n\in\mathbb{Z}$ are the quantum specifying the initial state $\phi_{i,n}$.
At the end of the measurement $t_f$ the dressed-qubit is in the state $\phi_{f,m}$. We get a final distribution $$P_f=P(f,m,t_f).$$ In between the initial and final time a realisation is characterised by a set of times $t_i=t_1<...<t_n<t_{n+1}=t_f$ and jumps $B(\omega_i)$, which we denote as the set $\{t_j, B(\omega_j)\}_{j=2}^n$. In between the jumps the qubit stays in its current Floquet state. The probability density of this realisation is given by [@open] $$\label{eq:probfl}
\mathbb{P}(\{t_j, B(\omega_j)\}_j,\phi_{f,m}|\phi_{i,n})=\prod_{j=2}^n \gamma(\omega_j) |\langle\phi_{f,m}, \prod_{k=2}^n B(\omega_k) \phi_{i,n}\rangle_{L^2}|^2.$$ The reverse process is characterised by the initial state $\phi_{f,m}$, the final state $\phi_{i,n}$ and the set of jump times and jumps $\{t_f-t_{n-j}, B^\dagger (\omega_{n-j})\}_j$. The reversed process thus has probability density $\mathbb{P}(\{t_f-t_{n-j}, B^\dagger (\omega_{n-j})\}_j,\phi_{i,n}|\phi_{f,m})$.
Comparing the probability of a realisation and its time reversed equivalent we get a fluctuation relation $$\label{eq:fluct}
\mathbb{P}(\{t_j, B(\omega_j)\},\phi_{f,m}|\phi_{i,n})=e^{ J}\mathbb{P}(\{t_f-t_j, B^\dagger (\omega_{n-j})\},\phi_{i,n}|\phi_{f,m}),$$ where $J$ is the inverse temperature $\beta$ times the total heat dissipated to the environment $$J=\beta\sum_k \hbar \omega_k.$$
Combining the all contributions to the entropy production, we arrive at the total pathwise entropy production $$\label{eq:entpath}
S=S_f-S_i + \beta \sum_k \hbar \omega_k.$$ The first two terms on the right hand side are the entropy of the initial preparation and the final measurement $S_{i/f}=-\log P_{i/f}$. The third term is the inverse temperature $\beta$ times the heat dissipated from the dressed-qubit to the electron bath, due to jumps. The above equation recovers the result by [@Cuetara].
The instantaneous average entropy production is given by [@MaesEnt] $$\begin{aligned}
\sigma&=\lim_{t\downarrow 0}\frac{1}{t}\langle S\rangle,\end{aligned}$$ where $t$ is the length of the time interval for which we consider the entropy production. This corresponds to equation .
Thermodynamics {#sec:ther}
==============
In this section we discuss the thermodynamics of the dressed-qubit and the driven qubit. More precisely, we will use the thermodynamics relations obtained for the dressed-qubit to formulate those for the driven qubit.
Let us examine the thermodynamics of the dressed-qubit. From now on we will be working in the interaction picture. This means the $H_F$ is added to the continuous evolution of the stochastic process . We suppose that the system is in a state $\Psi(t)$ at time $t$. By construction the Hamiltonian $H_F$ is time-independent. Thus we expect the work performed on the infinite dimensional system to be zero. If we apply the definition of work by [@Pusz; @AlickiHeatEngine] we find $$\label{eq:work}
\mathcal{W}(t)=\langle \Psi(t) ,\left(\frac{{\mathop{}\!\mathrm{d}}}{{\mathop{}\!\mathrm{d}}t}H_{F}\right) \Psi(t)\rangle_{L^2}=0.$$ We emphasise that in the above equation $t$ is the time parameter. In this sense $H_F$ is time-independent, but it does have a periodic dependence on the variable $\tau$. The $L^2$ scalar product appearing in is the integral over $\tau$. The heat $\mathcal{Q}$ dissipated from the dressed-qubit system to the electron bath at time $t$ is $$\label{eq:heat}
\mathcal{Q}= \sum_\omega \hbar \omega {\mathop{}\!\mathrm{d}}M_\omega,$$ where ${\mathop{}\!\mathrm{d}}M_\omega$ are the Poisson processes appearing in the stochastic Schrödinger equation . The embedding of the driven qubit in the dressed-qubit tells us that the above expression is also the heat dissipated by the driven qubit. The internal energy of the system is given by $$\label{eq:internalE}
\mathcal{E}(t)=\langle \Psi(t) ,H_{F}\Psi(t)\rangle_{L^2}.$$ We retrieve the first law of thermodynamics in the form $$\label{eq:firstlaw1}
\mathbb{E}({\mathop{}\!\mathrm{d}}\mathcal{E}(t) |\Psi(t))=-\mathbb{E}( \mathcal{Q}|\Psi(t)),$$ where $\mathbb{E}(.|\Psi(t))$ is the average over all realisations of the stochastic process conditioned on the state $\Psi(t)$. The change in energy of dressed-qubit is equal to minus the heat dissipated to the environment. The second law of thermodynamics follows from equation . We have $$S_f-S_i=S-\beta \mathcal{Q}.$$ The average of the entropy production S is positive by definition . Taking the average over all realisations of the stochastic process , we get $$\label{eq:secondlaw}
\mathbb{E}(S_f-S_i) \geq -\beta\, \mathbb{E}(\mathcal{Q}).$$ This finishes the description of the thermodynamics for the dressed-qubit. Our aim is now to reinterpret these results in terms of quantities defined in the finite dimensional space of the driven qubit.
The identity $H_F(\tau)=H_Q(\tau)-i\hbar\partial_\tau$ implies that we can write the internal energy of the qubit as the sum of two contributions. We interpret $$E(t)=\langle \Psi(t) ,H_Q\Psi(t)\rangle_{L^2}$$ as the internal energy of the qubit. The differential of the second contribution $$\label{eq:workwork}
W(t)=-{\mathop{}\!\mathrm{d}}\langle \Psi(t) ,-i\hbar\partial_\tau\Psi(t)\rangle_{L^2}.$$ we interpret as the work performed by the drive on the qubit. It follows immediately from that $$\label{eq:firstlaw2}
\mathbb{E}({\mathop{}\!\mathrm{d}}E(t) |\Psi(t))=\mathbb{E}(W(t)|\Psi(t)){\mathop{}\!\mathrm{d}}t-\mathbb{E}( \mathcal{Q}|\Psi(t)).$$ This is the energy balance relation from the qubit, the first law of thermodynamics for the qubit. The second law of thermodynamics derived for the dressed-qubit also holds for the driven qubit.
Let us now examine the above equation when the qubit reaches its steady state. By this we mean the stationary solution of equation . The stationary solution is $$\rho_s=\frac{1}{\Gamma_++\Gamma_-}\begin{pmatrix}
\Gamma_+ & 0\\0&\Gamma_-
\end{pmatrix},$$ where $\Gamma_\pm=\sum_{n}\gamma(\epsilon_\mp-\epsilon_\pm +n\hbar\omega_L)$. A straightforward calculation shows that equation becomes $$\begin{aligned}
\label{eq:statn}
\mathbb{E}({\mathop{}\!\mathrm{d}}W(t)|\rho_s) &=-\hbar\omega_L\mathbb{E}\left(\sum_\omega n_\omega{\mathop{}\!\mathrm{d}}N(\omega)\bigg|\rho_s\right)\\
&=-\hbar\omega_L\mathbb{E}\left({\mathop{}\!\mathrm{d}}\mu(t)|\rho_s\right).\end{aligned}$$ Remember that $\mu(t)$ is the process introduced in Subsection \[subsec:qp\] to keep track of the amount of photons annihilated in the drive. With equations and we recover the heat current by [@Gasparinetti1; @Cuetara] and the work at steady state by [@Cuetara]. Figure \[fig:energy\] shows a schematic representation of the energy flows on the two different levels of description: the dressed-qubit and the driven qubit.
![A schematic representation of the energy flows between the different subsystems for the two different levels of description: the dressed-qubit and the driven qubit. The curly (black) arrow shows the energy dissipated to the environment. The (black) dotted arrow is the work performed on the dressed-qubit, which is zero. For the the driven qubit we identify two contributions to the heat. The normal (blue) arrow shows qubit component of the heat flow to the environment, i.e. the change in internal energy of the qubit ${\mathop{}\!\mathrm{d}}E$. The (green) dashed line is the drive contribution to the heat, the work performed by the drive.[]{data-label="fig:energy"}](Energy.eps)
(0,0) (-32,24.5)[$\mathcal{Q}{\mathop{}\!\mathrm{d}}t$]{} (-32,18) (-34,11.5) (-47,35)[$\mathcal{W}{\mathop{}\!\mathrm{d}}t=0$]{}
The above equations show us how heat is not only dissipated to the environment by the qubit but also by the drive, due to the indirect coupling of the drive to the environment. In the limit that the strength of the drive goes to zero, the heat dissipated to the environment is equal to the heat dissipated by the qubit. Moreover they are equal to the heat dissipated to the environment when the stochastic Schrödinger equation is derived in the weak drive limit [@Breuer2]. In this limit the thermodynamics of both approaches are equivalent, it is sufficient to work in $\mathbb{C}^2$. This will be shown in Section \[sec:Const\].
Constant drive {#sec:Const}
==============
It is instructive to look at the case of a constant drive for two reasons. The first is that it allows a complete analytical treatment. The second is because we need it to recover from the infinite dimensional space the normal weak drive case as discussed at the end of Section \[sec:ther\]. The time independent Hamiltonian of the qubit with constant drive modulo a unitary transformation is given by $$\label{eq:timeind}
H_Q=\frac{\hbar\omega}{2}\sigma_z.$$ The explicit solution of the spectral problem is given by the set $\{\pm\hbar\omega/2+n\hbar\omega_L\}_{n\in\mathbb{Z}}$ of eigenvalues and eigenvectors $\phi_{\pm,n}(\tau)=e^{in\omega_L\tau}|\pm\rangle$. The vectors $|\pm\rangle$ are the energy eigenstates of the time independent Hamiltonian $H_Q$.
The only possible jumps for the dressed-qubit system are those exchanging no photons. The matrix elements of the jump operators are zero when photons are being created or annihilated. The jump operators are $|\pm\rangle\langle\mp|\otimes \mathbb{I}$, where $\mathbb{I}$ is the identity operator in $L^2$. We observe that the jumps are purely between the qubit energy states $|\pm\rangle$.
The results from Sections \[sec:mast\] and \[sec:ther\] hold, with all transitions creating or annihilating photons set to zero. The work performed by the drive $W$ as defined in equation equals zero. While the heat dissipated to the environment is given by $$\label{eq:thweak}
\mathcal{Q}=\hbar\omega[{\mathop{}\!\mathrm{d}}N(\omega) -{\mathop{}\!\mathrm{d}}N(\omega)].$$
Let us now consider the weak drive limit. The time independent Hamiltonian $H_Q$ is perturbed by the periodic term $\lambda H_D(t)$. To zeroth order in the strength of the drive $\lambda$ we obtain the above expressions for the jump operators and heat. The dressed-qubit only jumps between qubit energy eigenstates exchanging $\pm\hbar \omega$ with the environment. The heat dissipated to the environment is given by is exactly the heat dissipated when the stochastic Schrödinger equation is derived in the weak drive limit [@Breuer2]. This proves our claim made in the end of Section \[sec:ther\].
Monochromatic drive {#sec:Mon}
===================
Let us now consider the specific example of a monochromatic drive, which is of the form $$H_d(t)=\lambda(e^{-i\omega_L t}\sigma_++e^{i\omega_L t}\sigma_-).$$ We would like to find an expression for the Floquet states to get the effect operators as in equation . We can avoid solving the eigenvalue problem directly by finding a gauge transformation $U_t=e^{iF(t)}$, where $F(t)$ is a Hermitian matrix with the same period as $H_Q(t)$, such that the operator $$\label{eq:gunit}
G=U_t H_Q(t) U^\dagger_t +i\hbar (\partial_t U_t)U^\dagger_t$$ is time independent. The Floquet states can now be given in terms of the eigenvectors $|r\rangle$, with eigenvalues $\epsilon_r$ of $G$. It can checked that the states defined as $$\label{eq:ffloquett}
|\phi_r(t)\rangle= U^\dagger_t |r\rangle,$$ are indeed eigenstates of the Floquet Hamiltonian with quasi-energies $\epsilon_r$, i.e. they solve the eigenvalue problem giving in equation . For the monochromatic drive we can take [@GeKoSk95] $$\label{eq:hunit}
F(t)=\omega_L \sigma_+\sigma_-t.$$ The Floquet states and quasi-energies can thus be found by diagonalising the matrix $$\label{eq:ggunit}
G=\frac{\hbar}{2}(\omega_q-\omega_L)(\sigma_+ \sigma_- -\sigma_- \sigma_+)+\lambda\left(\sigma_++\sigma_-\right)-\frac{\omega_L}{2}\mathbb{I}.$$ The Floquet states and effect operators are explicitly calculated in the appendix. The difference of the quasi-energies is $\epsilon_+-\epsilon_-=\hbar\nu=\sqrt{\hbar^2(\omega_q-\omega_L)^2+4\lambda^2}$. To formulate the thermodynamic quantities for this model it is instructive to have stochastic Schrödinger equation for the dressed-qubit $$\begin{aligned}
\label{eq:seemon}
{\mathop{}\!\mathrm{d}}\Psi(t) &=-\frac{i}{\hbar}K(\Psi(t))+\sum_{s=-1,0,1}\bigg[\left(\frac{ B(\omega_L+s\nu)\Psi(t)}{\|B(\omega_L+s\nu)\Psi(t)\|}-\Psi(t)\right){\mathop{}\!\mathrm{d}}M(\omega_L+s\nu)\\
&+\left(\frac{B(-\omega_L+s\nu)\Psi(t)}{\|B(-\omega_L+s\nu)\Psi(t)\|}-\Psi(t)\right){\mathop{}\!\mathrm{d}}M(-\omega_L+s\nu)\bigg].\end{aligned}$$ The operator $K$ has been defined in equation and the jump operators $B$ are explicitly defined in Appendix \[sec:appMon\]. The dressed-qubit can make six different jumps. They come in three pairs that have the same effect on the qubit but a different amount of energy is exchanged. For example the jumps governed by the Poisson processes $M(\omega_L-\nu)$ and $M(-\omega_L-\nu)$ both bring the qubit to the excited Floquet state. According to our earlier discussions we can interpret the difference in energy as creation or annihilation of photons. In the first jump one extra photon gets annihilated in the drive, in the second jump one photon gets created. Figure \[fig:realisationn\] shows a possible realisation of the stochastic process for the driven qubit. The rates satisfy the detailed balance relation .
![Possible realisation of the qubit stochastic process. When the photon energy $\omega_L$ is larger than $\nu$, the rate $\gamma(\omega_L-\nu)$ is larger than $\gamma(\nu-\omega_L)$. In this way the qubit gets driven out of the lower Floquet state.[]{data-label="fig:realisationn"}](sim.eps)
(0,0) (-35,37.5)[$\hbar\omega_L-\nu$]{} (-21,37.5)[$\hbar\omega_L+\nu$]{} (-5,37.5)[$\hbar\omega_L$]{} (-43,37.5)[$\hbar\omega_L$]{} (-77,23)[$|\langle\phi_+(0),\psi(t)\rangle|$]{} (-10,1)[$t$]{}
We can now apply the definitions of Section \[sec:ther\] to get the thermodynamic quantities for the monochromaticly driven system. The dissipated heat is give by $$\label{eq:heatMon}
\mathcal{Q}=\sum_{s=-1,0,1}(\hbar\omega_L+s\nu)[{\mathop{}\!\mathrm{d}}M(\hbar\omega_L+s\nu)-{\mathop{}\!\mathrm{d}}M(-\hbar\omega_L-s\nu)]$$ The thermodynamics of this model have also been studied by [@AlickiFloquet]. Their analysis is based on a Markovian master equation of the state operator of a qubit, see [@AlickiMaster] and [@Szczygielski]. The heat current dissipated to the environment they obtain is the same as the average over all realisations $\mathbb{E}[.]$ of the expression in equation . We identify the contribution to the heat current as follows. The packets $\hbar \omega_L$ come from the drive. Since the Floquet states are qubit-dressed states, the exchange of $\hbar\nu$ comes from a change in internal energy of the qubit and the drive.
Conclusion
==========
In this manuscript we considered a periodically driven qubit in contact with a thermal environment. Using Floquet theory we embedded the periodically driven qubit into an infinite dimensional time-autonomous system. This system we called the dressed-qubit.
We studied the thermodynamics of a periodically driven qubit in contact with a thermal environment by reformulating the thermodynamics of the dressed-qubit. This is where our contribution differs from the existing literature, where the analysis is based on the Lindblad-Gorini-Kossakowski-Sudarshan equation for the state operator in $\mathbb{C}^2$. The advantage of the dressed-qubit is that it is time-autonomous.
Our approach can be used for the analysis of microscopic or nano-scale devices like [@PekCal]. It can be extended to systems with more that two energy levels or multiple heat baths in a straightforward way. It also offers a very clear interpretation in terms of atom-dressed states.
I would like to thank Jukka Pekola, Paolo Muratore-Ginanneschi, Kay Schwieger and Christian Maes for fruitful discussions and comments.
The research has been supported by the Academy of Finland via the Centre of Excellence in Analysis and Dynamics Research (project No. 307333)
Stochastic Schrödinger equation {#sec:appSSE}
===============================
Jump rates {#sec:appSSE1}
----------
The jump rates (for both the driven qubit and dressed-qubit) are given by [@openpaper; @BPfloquet] $$\gamma(\omega)=\int_{-\infty}^{+\infty}{\mathop{}\!\mathrm{d}}t e^{i(\hbar\omega +\eta_k-\eta_l) t} \langle\sum_{k,l}g^*_{kl}a^\dagger_k a_l\sum_{p,q}g_{pq}a^\dagger_p a_q)\rangle_\beta$$ where the average is over the thermal state of the electron bath with inverse temperature $\beta$. Calculating the average and peforming the standard approximation of the sums over different wave numbers as integrals, which is for example explained in Chapter 2 of [@solid], we arrive at $$\begin{aligned}
\gamma(\omega) &=\frac{9 N^2\hbar}{4E_F^3}\int_0^\infty {\mathop{}\!\mathrm{d}}E_1\int_0^\infty {\mathop{}\!\mathrm{d}}E_2\int_{-\infty}^{+\infty}{\mathop{}\!\mathrm{d}}t e^{i(\omega +E_1-E_2) t} (1-f(E_1))f(E_2)g^2\sqrt{E_1E_2}\end{aligned}$$ where $N$ is the average amount of electrons in the bath, $E_F$ is the Fermi energy and $f(x)=\frac{1}{1+e^{\beta(E-\mu)}}$ is the Fermi distribution and we assumed that the coupling $g_{kl}=k$ is constant. Evaluating the time integral gives us a delta function over the energies divided by $2\pi$, evaluating one of the integrals over the energies gives us $$=\frac{9 N^2\hbar}{4E_F^3(2\pi)}\int_0^\infty {\mathop{}\!\mathrm{d}}E (1-f(E))f(E+\hbar\omega)\sqrt{E(E+\hbar\omega)}.$$ From this one can see that the rates indeed satisfy detailed balance.
Embedding {#sec:appSSE2}
---------
To show that the relation indeed embeds the driven qubit into the dressed-qubit we want to show that the short time increment of the dressed-qubit state ${\mathop{}\!\mathrm{d}}\Psi(t,\tau)=\Psi(t+{\mathop{}\!\mathrm{d}},\tau)-\Psi(t,\tau)$ satisfies the stochastic Schrödinger equation for the dressed-qubit .
First we show that the form implies that $$\label{eq:probs}
\|A(\omega)\psi(t)\|=\|B(\omega)\Psi(t)\|_{L^2}$$ Writing $\psi(t)=\phi_+(0)\langle \phi_+(0),\psi(t)\rangle+\phi_-(0)\langle \phi_-(0),\psi(t)\rangle$ in the Floquet basis, then $$\label{eq:lift}
\Psi(t,\tau)=e^{i\mu(t)\omega_L \tau}P_{\tau,0}\psi(t)=e^{-i\mu(t)\omega_L\tau}\phi_+(\tau)\langle \phi_+(0),\psi(t)\rangle+e^{-i\mu(t)\omega_L\tau}\phi_-(\tau)\langle \phi_-(0),\psi(t)\rangle.$$ Let us calculate $A(\omega)\psi(t)$ and $B(\omega)\Psi(t)$ using definitions and . $$A(\omega)\psi(t)=\sum_{r=\pm} \alpha_{r,+,n_\omega}\phi_r(0)\langle \phi_+(0),\psi(t)\rangle +\alpha_{r,-,n_\omega}\phi_r(0)\langle \phi_-(0),\psi(t)\rangle$$ $$B(\omega)\Psi(t)=\sum_{r=\pm} \alpha_{r,+,n_\omega}\phi_{r,-\mu(t)-n_\omega}\langle \phi_+(0),\psi(t)\rangle +\alpha_{r,-,n_\omega}\phi_{r,-\mu(t)-n_\omega}\langle \phi_-(0),\psi(t)\rangle$$ From the above equations a straightforward calculation shows that indeed holds. Note that $$\label{eq:proof1}
(B(\omega)\Psi(t))(\tau)=e^{-i(\mu(t)+n_\omega)\omega_L\tau}P_{\tau,0}A(\omega)\psi(t),$$ and from this it also follows that $$\label{eq:proof2}
(B^\dagger(\omega)B(\omega)\Psi(t))(\tau)=e^{-i\mu(t)\omega_L\tau}P_{\tau,0}A^\dagger(\omega) A(\omega)\psi(t).$$
Let us now show that the state $\Psi(t,\tau)=e^{i\mu(t)\omega_L \tau}P_{\tau,0}\psi(t)$ defined as in equation evolves by the stochastic Schrödinger equation for the dressed-qubit. We take the time increment on both sides of equation , following the rules of stochastic calculus see e.g. [@Kurt], we find $$\begin{aligned}
{\mathop{}\!\mathrm{d}}\Psi(t,\tau) =&e^{-i\mu(t+{\mathop{}\!\mathrm{d}}t)\omega_L\tau}P_{\tau,t_0}\psi(t+{\mathop{}\!\mathrm{d}}t)-e^{-i\mu(t)\omega_L\tau}P_{\tau,t_0}\psi(t)\\
=&\left( e^{-i\mu(t+{\mathop{}\!\mathrm{d}}t)\omega_L\tau}-e^{-i\mu(t)\omega_L\tau}\right) P_{\tau,t_0}\psi(t)+ e^{-in(t)\omega_L\tau}P_{\tau,t_0}{\mathop{}\!\mathrm{d}}\psi(t)\nonumber\\
&+\left( e^{-i\mu(t+{\mathop{}\!\mathrm{d}}t)\omega_L\tau}-e^{-i\mu(t)\omega_L\tau}\right) P_{\tau,t_0}{\mathop{}\!\mathrm{d}}\psi(t)\end{aligned}$$ With the definition of ${\mathop{}\!\mathrm{d}}\mu(t)$ and the fact that ${\mathop{}\!\mathrm{d}}N(\omega){\mathop{}\!\mathrm{d}}N(\omega')={\mathop{}\!\mathrm{d}}N(\omega)\delta_{\omega,\omega'}$ we get $$\begin{aligned}
=&e^{-i\mu(t)\omega_L\tau} \sum_\omega {\mathop{}\!\mathrm{d}}N(\omega)\left( e^{i\mu_\omega\omega_L\tau}-1\right) P_{\tau,t_0}\psi(t)+ e^{in(t)\omega_L\tau}P_{\tau,t_0}{\mathop{}\!\mathrm{d}}\psi(t)\\
&+e^{i\mu(t)\omega_L\tau}\sum_\omega{\mathop{}\!\mathrm{d}}N(\omega)\left( e^{i\mu_\omega\omega_L\tau}-1\right) P_{\tau,t_0}{\mathop{}\!\mathrm{d}}\psi(t)\end{aligned}$$ Using the explicit expression of the stochastic Schrödinger equation and cancelling out terms we arrive to $$\begin{aligned}
= -\frac{i}{\hbar} e^{i\mu(t)\omega_L\tau} P_{\tau,0}G(\psi(t)){\mathop{}\!\mathrm{d}}t+\sum_\omega \left( e^{i\mu(t)\omega_L\tau} e^{in_\omega\omega_L\tau}\frac{P_{\tau,t_0}A(\omega)\psi}{\|A(\omega)\psi(t)\|}-\Psi(t,\tau)\right){\mathop{}\!\mathrm{d}}N(\omega)\end{aligned}$$
From equations and it follows that $e^{i\mu(t)\omega_L\tau} P_{\tau,0}G(\psi(t))=(K(\Psi(t)))(\tau)$. Using and we find that $${\mathop{}\!\mathrm{d}}\Psi(t,\tau)= -\frac{i}{\hbar} (K(\Psi(t)))(\tau){\mathop{}\!\mathrm{d}}t+\sum_\omega \left( \frac{(B(\omega)\Psi(t))(\tau)}{\|B(\omega)\Psi(t)\|_{L_2}}-\Psi(t,\tau)\right){\mathop{}\!\mathrm{d}}N(\omega)$$ This expression is almost the evolution equation for the dressed-qubit . The only difference is that the jumps are governed by different Poisson processes. However since $\|A(\omega)\psi(t)\|=\|B(\omega)\Psi(t)\|_{L^2}$ we see that the conditional expectation values, as defined in equations and , are equal $\mathbb{E}({\mathop{}\!\mathrm{d}}N(\omega)|\psi(t))=\mathbb{E}({\mathop{}\!\mathrm{d}}M(\omega)|\Psi(t))$ at all times. This proves that the above stochastic process is equal to the stochastic Schrödinger equation for the dressed-qubit.
Monochromatic drive {#sec:appMon}
===================
The Floquet can be found by diagonalising and using equation . We get $$\begin{aligned}
\phi_+(\tau)&=(\cos(\theta/2)e^{-i\omega_L \tau}, \sin(\theta/2))\\
\phi_+(\tau)&=(-\sin(\theta/2)e^{-i\omega_L \tau}, \cos(\theta/2)),\end{aligned}$$ where $\cos\theta=\frac{\omega_q-\omega_L}{\sqrt{\hbar^2(\omega_q-\omega_L)^2+4\lambda^2}}$. The difference in the quasi-energies is $\epsilon_+-\epsilon_-=\hbar\nu=\sqrt{(\omega_q-\omega_L)^2+4\lambda^2}$. The matrix elements of $(\sigma_++\sigma_-)$ in the Floquet basis are $$\begin{aligned}
\langle \phi_+(\tau),(\sigma_++\sigma_-)\phi_+(\tau)\rangle&=e^{-i\omega_L \tau
}\frac{\sin\theta}{2}+ e^{i\omega_L \tau}\frac{\sin\theta}{2}\\
\langle \phi_-(\tau),(\sigma_++\sigma_-)\phi_-(\tau)\rangle&=-e^{-i\omega_L \tau}\frac{\sin\theta}{2}-e^{i\omega_L \tau}\frac{\sin\theta}{2}\\
\langle \phi_+(\tau),(\sigma_++\sigma_-)\phi_-(\tau)\rangle&=-e^{-i\omega_L \tau}\sin^2(\theta/2)+e^{i\omega_L \tau}\cos^2(\theta/2)\\
\langle \phi_-(\tau),(\sigma_++\sigma_-)\phi_+(\tau)\rangle&=e^{-i\omega_L \tau}\cos^2(\theta/2)-e^{i\omega_L \tau}\sin^2(\theta/2).\end{aligned}$$ From the above matrix elements we can see that there are six jump operators, acting on a state $\Psi$ as $$\begin{aligned}
(B(\omega_L)\Psi)(\tau)&=\frac{\sin\theta}{2}\sum_n(\phi_{+,n-1}(\tau)\langle\phi_{+,n},\Psi\rangle_{L^2}-\phi_{-,n-1}(\tau)\langle\phi_{-,n},\Psi\rangle_{L^2})\\
(B(\nu+\omega_L)\Psi)(\tau)&=\sin^2(\theta/2)\sum_n\phi_{-,n-1}(\tau)\langle\phi_{+,n},\Psi\rangle_{L^2}\\
(B(\nu-\omega_L)\Psi)(\tau)&=\cos^2(\theta/2)\sum_n\phi_{-,n-1}(\tau)\langle\phi_{+,n},\Psi\rangle_{L^2}\end{aligned}$$ and their complex conjugates. From equation we can see that it is possible to get rid of all the sines and consines in the definition of the jump operators by redefining the jump rates as the original rates times the prefactor of the corresponding jump operator squared (for the jumps themselves the prefactors do not matter, the jumps are normalised). For example we define $\frac{\sin^2\theta}{4}\gamma(\omega_L)\rightarrow \gamma(\omega_L)$. We do this for notational simplicity.
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harvmac.tex
\#1[0= 0=0 1= 1=1 0>1 \#1 / ]{} \#1\#2[[=14.4pt plus 0.3pt[^1][\#2]{}]{}]{}
Kenneth Lane
Department of Physics, Boston University
590 Commonwealth Avenue, Boston, MA 02215
.3in
**Abstract**
I present a model of topcolor-assisted technicolor that can have topcolor breaking of the desired pattern, hard masses for all quarks and leptons, mixing among the heavy and light generations, and explicit breaking of all technifermion chiral symmetries except electroweak $SU(2) \otimes $U(1). These positive features depend on the outcome of vacuum alignment. The main flaw in this model is tau-lepton condensation.
It is not difficult to construct a dynamical model of electroweak symmetry breaking. We have known how at least since 1973 \[\]. The difficulties lie in extending this dynamics to flavor: accounting for the masses of all known fermions, including the top quark’s; breaking technifermion chiral symmetries to prevent light technipions with axion-strength couplings to quarks and leptons; and evading the many phenomenological pitfalls—flavor-changing neutral currents to name the most famous and ubiquitous example—that plague any theory of flavor \[\]. This paper develops further the topcolor-assisted technicolor approach to accomplishing all this.
Topcolor-assisted technicolor (TC2) is the only scheme known in which there is an explicit dynamical and natural mechanism for breaking electroweak symmetry and generating the fermion masses including $m_t \simeq 175\,\gev$. In TC2, there are no elementary scalar fields and no unnatural or excessive fine-tuning of parameters \[\]. In Hill’s simplest TC2 model, the third generation of quarks and leptons transforms under strongly-coupled color and hypercharge groups, $\suone
\otimes \uone$, with the usual charges, while the light generations transform under weakly-coupled $\sutwo \otimes \utwo$. Near $1\,\tev$, these four groups are broken, somehow, to the diagonal subgroup of ordinary color and hypercharge, $\suc\otimes \uy$. The desired pattern of heavy quark condensation occurs because $\uone$ couplings are such that the spontaneously broken $\suone \otimes \uone$ interactions are supercritical only for the top quark.
Hill did not address the issues of topcolor breaking, generational mixing and chiral symmetry breaking. In addition to these concerns, there are stringent constraints on model-building from the conflict between custodial isospin conservation and the large topcolor $\uone$ coupling \[\], and from limits on $B_d-\ol B_d$ mixing \[\]. These constraints, the cancellation of $U(1)$ anomalies, and the dynamics of generational mixing and topcolor breaking to $\suc \otimes \uy$ were considered in Refs. \[\] and \[\]. The main features of the models developed in these studies are:
[(1)]{} The $\uone$ charges of technifermions are custodial-isospin symmetric.
[(2)]{} Above the electroweak scale, third-generation quarks transform under strongly-coupled $\suone$ while the two light-generation quarks transform under the weaker $\sutwo$. However, [*all*]{} quarks and leptons transform under the strongly-coupled $\uone$.
[(3)]{} In order that $Z^0$ couplings be nearly standard, the breakdown $\uone \otimes \utwo \ra \uy$ necessarily occurs at a somewhat higher scale than $SU(2) \otimes \uy \ra U(1)_{EM}$. This is effected by a higher dimensional technifermion $\psi$ whose condensate is $SU(2) \otimes \uy$ invariant. The $\psi$-condensate gives rise to a 2–3 TeV $Z'$ boson with much interesting phenomenology , , \[\], \[\], \[\].
[(4)]{} The breaking of the color and electroweak symmetries to $\suc
\otimes U(1)_{EM}$ is due to technifermions in the fundamental representation of the TC gauge group, assumed to be $SU(N)$. In particular, the $\suone \otimes \sutwo$ breaking condensate $\condab$, where $T^i$ is a triplet of $SU(3)_i$, is driven by an attractive strong $\uone$ interaction.
[(5)]{} Generational mixing is produced by an extended technicolor (ETC) operator which induces the transition $d_L,s_L \leftra b_R$, but [*not*]{} $d_R,s_R \leftra b_L$. In this way, the excessive $B_d-\ol B_d$ mixing discussed in Ref. is avoided.
[(6)]{} Nontrivial solutions exist to all the $U(1)$ anomaly-cancellation equations.
These constraints led to a proliferation of technifermions and a large chiral $SU(N_T)_L \otimes SU(N_T)_R$ symmetry. In the models considered, it was not possible to break explicitly all unwanted chiral symmetries, so that massless or very light technipions occurred.
Explicit chiral symmetry breaking and generational mixing, in the form of quark mass $\ol q T \ol T q$ and technipion mass $\ol T T \ol T T$ operators, are induced mainly by ETC interactions. Here, $T = (U,D)$ are technifermion isodoublets. Let us define a “complete set” of $SU(2)
\otimes U(1)$-invariant 4T operators $\ol T^i_L \gamma^\mu T^j_L \ts\ts \ol
T^k_R \gamma_\mu (a + b \sigma_3) T^l_R$ as one for which no technifermion global symmetry generator commutes with every member of the set. In a complete set, every left-handed and right-handed technifermion field appears in at least one of the operators. (This excludes operators in which the left or right-handed currents involve the same technifermion twice, e.g., operators generated by diagonal ETC or $\uone$ interactions.) Since I have not specified an ETC group and its breaking pattern, it is necessary to assume that the required operators exist, [*provided*]{} they respect all known gauge interactions, including $\uone \otimes \utwo$. For the type of model considered in Ref. , I was unable to find a complete set of 4T operators.
Even a complete set of operators is not sufficient to guarantee that all technipion masses are large. It is also necessary that condensates form so that all 4T operators have nonzero vacuum expectation values, i.e., that they contribute to the vacuum energy Here, the hamiltonian $\CH'$ is the sum over all allowed 4T operators and $\CW$ is an $SU(N_T)_L \otimes SU(N_T)_R$ transformation. Finding the transformation $\CW^0$ which minimizes $E(\CW)$ is known as vacuum alignment \[\]. In the correct vacuum, $\langle \ol T^i_L T^j_R \rangle \propto
W^0_{ij}$, where $W^0$ is the corresponding $SU(N_T)$ matrix. The models under consideration have a large number of technifermions and 4T operators, and minimization is a complicated numerical task, now under study.
I present here a type of TC2 model which does allow a complete set of 4T operators. For the models of Ref. , the difficulty of constructing such a set was due at least in part to the fact that light and heavy quarks transform under different color $SU(3)$ groups. Then their hypercharges were tightly constrained by cancellation of $U(1) \ts [SU(3)]^2$ anomalies and there was no complete set invariant under $\uone \otimes \utwo$. In the model presented here, I adopt the “flavor-universal topcolor” of Chivukula, Cohen and Simmons \[\]; also see Ref. . Their model was motivated by the apparent excess of high-$E_T$ events in the CDF jet data \[\]. They used two $SU(3)$ groups, but assumed all quarks transform under only the stronger $\suone$ color group. I find that this allows simpler quark hypercharges than in Ref. and, so, the $U(1)$ constraints for a complete set of 4T operators can be met. A dynamical mechanism for breaking $\suone
\otimes \sutwo \ra \suc$ was not provided in Refs. and . I shall use the condensation of technifermions transforming under the two color groups to effect this breaking. The model I present has one obvious bad feature: the tau-lepton has very strong, attractive $\uone$ interactions and, therefore, it has a large condensate and mass.
The fermions in this new model, their color representations and $U(1)$ charges are listed in Table 1. Technifermions $T^i_{L,R}$ transform under $SU(N)$ as fundamentals, while $\psi_{L,R}$ are antisymmetric tensors. As noted above, the condensate $\langle \ol \psi_L \psi_R \rangle$ breaks $\uone \otimes \utwo \ra \uy$ and $\condab \neq 0$ breaks $\suone \otimes
\sutwo \ra \suc$. The condition on the $\uone$ charges of $T^1$ and $T^2$ required to form this condensate is, in the walking technicolor and large-$N$ limits , Here, $\alpha_3$ and $\alpha_1$ are the $\suone$ and $\uone$ couplings near 1 TeV. Note that this requires and $(u_1 - u'_1)^2 > (u_1 - v_1) (v_1
- u'_1) > 0$. To preserve $U(1)_{EM}$, $T^1$ and $T^2$ must have equal electric charges, i.e., $u_1 + u_2 = u'_1 + u'_2 = v_1 + v_2$
To give mass to quarks and leptons, I assume the following ETC operators: To generate $d_L,s_L \leftra b_R$, I require the operator Of course, the technifermions in these operators must condense in the correctly aligned ground state. To forbid the transition $d_R,s_R \leftra
b_L$ and unacceptably large $B_d - \ol B_d$ mixing, ETC interactions must not generate any of the operators $\ol q^h_L \gamma^\mu T^i_L \ts\ts \ol
D^j_R \gamma_\mu d^l_{R}$ for any $i,j$. This gives the constraints $b \neq
0, u_1 - u'_1, y_1 - y'_1, z_1 - z'_1$, etc.
A complete set of allowed $SU(2) \otimes U(1)$-invariant 4T operators is Note that the equal-charge conditions $x_1 + x_2 = y_1 + y_2 = z_1 + z_2$ are implied by these operators. In addition to this set, diagonal 4T operators from broken ETC and $\uone$ interactions contribute to the chiral-breaking hamiltonian, $\CH'$.
The requirement that gauge anomalies cancel further constrains $U(1)$ charge assignments. Taking account of the equal-charge conditions, there are four independent conditions which are linear in the hypercharges (the $U(1)_{1,2}[\sutwo]^2$ condition is automatically satisfied): Taken together with the hypercharge conditions, Eqs. and , there follow the relations: Note that $bb' > 0$ and $bb'' < 0$ which favors top, but not bottom, condensation.
There are four anomaly conditions that are cubic in the hypercharges. However, the $\uy[SU(2)]^2$ anomaly cancellation guarantees that the $[\uy]^3$ anomaly also cancels, leaving three independent conditions. They are conveniently given for $[\uone]^3$, $[\uone]^2 \uy$, and $[\uone]^3 + [\utwo]^3 - 3[\uone]^2 \uy$: These conditions have an infinite number of solutions. Following Ref. , I found one with $\vert u_1 - u'_1\vert = \CO(1)$, as required for naturally large couplings in Eq. , as follows: I assumed $u_1 = - u'_1$ and $\xi = - \xi'$. Then, for $N=4$, I chose $z_1
=8$ and $z_1 + z_2 = 2$. This input has the nontrivial solution The other hypercharges are to be chosen in accord with Eqs.–.
The large values $a \simeq a' \simeq -2 z_1$ found in the solutions to Eqs. are unavoidable: The $[\uone]^3$ condition has no real solution for $u_1 - u'_1 \neq 0$ and $\vert a \vert \simle \vert u_1 - u'_1
\vert$. The large positive value of $aa'$ then suggests that the $\uone$ interactions generate a tau-condensate $\langle \ol \tau_L \tau_R \rangle
\sim \langle \ol t_L t_R \rangle$. Such a hypercharge also raises the question of the triviality of the $\uone$ interaction: does the Landau pole occur at an energy significantly lower than the one at which we can envisage $\uone$ being unified into an asymptotically free ETC group ? I know of no choice of chiral symmetry breaking ETC operators and associated hypercharge assignments within the present simple model of flavor-universal topcolor which evades $aa'/(u_1 - u'_1)^2 \gg 1$. It may be possible to find an acceptable model, including a complete set of 4T operators, by enlarging the technifermion sector and/or complicating the light generation hypercharge assignments.
In conclusion, I have constructed a TC2 model with flavor-universal topcolor that seems capable of satisfying all major phenomenological constraints except those involving the tau-lepton. To my mind, the more important task ahead is to show that a nontrivial vacuum-alignment solution exists that results in nonzero masses and mixings for all the fundamental fermions and composite technipions.
I am grateful to E. Simmons for a careful reading of the manuscript and valuable comments. This research was supported in part by the Department of Energy under Grant No. DE–FG02–91ER40676.
TABLE 1: Lepton, quark and technifermion colors and hypercharges.
[^1]: \#1
|
---
abstract: 'Optical spectroscopic data are presented here for quasars from the Molonglo Quasar Sample (MQS), which forms part of a complete survey of 1-Jy radio sources from the Molonglo Reference Catalogue. The combination of low-frequency selection and complete identifications means that the MQS is relatively free from the orientation biases which affect most other quasar samples. To date, the sample includes 105 quasars and 6 BL Lac objects, 106 of which have now been confirmed spectroscopically. This paper presents a homogenous set of low-resolution optical spectra for 79 MQS quasars, the majority of which have been obtained at the Anglo-Australian Telescope. Full observational details are given and redshifts, continuum and emission-line data tabulated for all confirmed quasars.'
author:
- 'Joanne C. Baker'
- 'Richard W. Hunstead'
- 'Vijay K. Kapahi and C.R. Subrahmanya'
title: |
THE MOLONGLO REFERENCE CATALOG 1-JY RADIO SOURCE SURVEY\
IV: OPTICAL SPECTROSCOPY OF A COMPLETE QUASAR SAMPLE
---
\#1
INTRODUCTION {#sec:intro}
============
It has been widely claimed that orientation plays a crucial role in the classification of active galactic nuclei (AGN), acting to increase the observed diversity. This idea has been formulated into the ‘unified schemes’ for AGN (reviewed by Antonucci 1993) which attempt to reduce the diversity by finding evidence that some classes of AGN are identical except for viewing direction. One of the most successful applications has been the description of core- and lobe-dominated quasars as being identical except for radio-jet orientation. In this picture, core-dominated quasars are simply foreshortened lobe-dominated quasars viewed with Doppler-boosted cores (Orr & Browne 1982; Kapahi & Saikia 1982).
Implicit in the unified schemes for AGN is the presence of anisotropic emission in many wavebands. As well as predicting orientation dependencies, anisotropic emission also implies that selection biases must affect all samples of AGN to some degree; brighter and therefore preferentially-oriented AGN will tend to be favoured. To overcome or compensate for such selection effects, samples should be arguably well defined (eg.Hewett & Foltz 1994) and if possible selected by some [*isotropic*]{} property, such as extended radio emission. However, additional biases can be introduced by imposing inappropriate flux limits at other wavelengths, most commonly the optical. For example, the paucity of lobe-dominated quasars in the B2 sample compared with the 3CR (de Ruiter et al. 1986) has been attributed to missing faint optical counterparts below the Palomar Sky Survey plate limit (Kapahi & Shastri 1987). This follows if the optical continuum in quasars is aspect dependent.
Many observations have pointed to the optical continuum emission in quasars being highly anisotropic (Browne & Wright 1985; Jackson & Browne 1989). However, none of these early studies had been able to disentangle the aspect dependence from the effects of sample selection. Indeed, with the exception of the 3CR (Laing, Riley & Longair 1983), all previous studies have used data from samples with unknown biases and selection effects. To rectify this, and also to quantify the effects of imposing optical constraints on other samples, we have defined a new sample of southern, radio-loud quasars, the Molonglo Quasar Sample (MQS), defined below. The MQS has been defined using minimal selection criteria; it includes all radio-loud quasars in a region of sky down to a radio flux density limit of 0.95 Jy at 408 MHz. Sources were drawn initially from the Molonglo Reference Catalogue (MRC; Large et al. 1981). The quasar identifications have proceeded largely in parallel with identifications of all the other sources in the same sky strip, which has ensured that no quasars have remained unidentified because they are optically faint, for example, or because they have large radio-optical positional offsets as a result of asymmetric radio structure.
In this paper, optical spectroscopic data are presented for the MQS, including tabulated redshifts, continuum and emission-line data. Optical spectra are shown here for 79 MQS quasars; spectra for the remaining quasars will be published elsewhere. This paper is the fourth in a series giving basic radio and optical data for quasars and radio galaxies from the MRC/1-Jy sample. The radio galaxy identifications are listed in Paper I (McCarthy et al. 1996) and radio images are presented in Paper II (Kapahi et al. 1998a). Radio data for the quasars are presented in Paper III (Kapahi et al. 1998b), together with a detailed description of the identification of the quasar sample. A full set of optical finding charts is included in Paper III. Other papers based on an earlier listing of the MQS have been published, including investigations into the aspect dependence of the optical continuum in the MQS by Baker & Hunstead (1995) and Baker (1997), and the X-ray properties by Baker, Hunstead & Brinkmann (1995). Many follow-up projects are underway, including infrared spectroscopy and intermediate-resolution optical spectroscopy of high-redshift MQS quasars (Baker & Hunstead 1996; Baker 1998).
THE MOLONGLO QUASAR SAMPLE
==========================
As described in a companion paper (Paper III), the selection criteria for the MQS were designed to minimise the orientation-dependent biases present in most other radio-selected samples. To achieve this, initial selection was made at low frequency where the radio emission is dominated by the steep-spectrum extended components, and complete optical identifications were then sought.
The complete flux-limited radio sample from which the quasar sample was selected consists of 557 sources with flux densities exceeding 0.95 Jy at 408 MHz in the MRC in a $10^{\circ}$ declination strip ($-20^{\circ}>\delta>-30^{\circ}$), excluding those with low Galactic latitude $|b|<20^{\circ}$ and those in the RA range 14[$^{\rm h}$]{}03–20[$^{\rm h}$]{}20 (due to constraints on observing time). Within this strip, sources were imaged first at 843 MHz with the Molonglo Observatory Synthesis Telescope (MOST) (Subrahmanya & Hunstead 1986) and then at higher-resolution (1) with the VLA, mostly at 5 GHz (see Paper III). The VLA snapshot images have allowed the separation of compact and extended radio components from which the core-to-lobe flux density ratios, $R$, have been estimated. The values of $R$ at an emitted frequency of 10 GHz, $R_{10}$, are listed in Paper III for all sources except compact, steep-spectrum (CSS) sources which we define as having radio linear sizes $l<20$ kpc ($H_0=50$ Mpc$^{-1}$ and $q_0=0.5$ assumed throughout) and spectral indices steeper than $\alpha = 0.5$ ($S \propto \nu^{-\alpha}$) between 408 MHz and 5 GHz; see, for example, Fanti & Fanti (1994). In common with other studies, $R$ is assumed to be an indicator of jet-axis orientation (e.g., Orr & Browne 1982). The distribution of $R$ values for the MQS (see Baker 1997) is consistent with the sample being randomly oriented apart from an excess of core-dominated quasars whose Doppler-enhanced emission pushes them above the radio flux limit. We note that no such bias arises in the optical because complete identifications have been made.
Optical counterparts near the radio core and/or radio centroid positions were identified first by eye (as described in Paper III) by their ‘stellar’ appearance on the UK Schmidt IIIaJ plates, down to the limiting magnitude of $b_{\rm J}\approx22.5$ (where $b_{\rm J} = B - 0.23(B-V)$, Bahcall & Soneira 1980). Deep CCD imaging (to $r\sim 24$) of the fields of most of the 557 sources at Las Campanas, as described in Paper I, also detected a number of quasar candidates, including some close to the plate limit. Spectroscopy was then sought to confirm the identifications. The MQS comprises a total of 111 quasar candidates, of which 106 have been confirmed spectroscopically to date (including 6 BL-Lacs, only one with a measured redshift). The acquisition and reduction of the optical spectra is described in Section \[sec:aatspec\].
OPTICAL SPECTROSCOPY {#sec:aatspec}
====================
Low-resolution optical spectra have been sought for all MQS quasar candidates with mostly the 3.9-m Anglo-Australian Telescope (AAT) and also the 4-m telescope at the Cerro Tololo Inter-American Observatory (CTIO). To date, AAT spectra have been obtained for 77 quasar candidates (plus 2 BL Lac objects). Another two quasar spectra were obtained with the ESO 3.6m telescope (Wall & Shaver, 1993, private communication). Low signal-to-noise spectra, confirming only redshifts and classifications, have been obtained at CTIO for fourteen more quasars but were inadequate for accurate emission-line measurements and have not been included in this compilation. Further spectroscopy is being sought. Another five candidates await spectroscopic confirmation, including two lobe-dominated and three CSS targets. Seven more quasars have spectral data published in the literature and have not been reobserved. Data for these remaining objects will be published separately. The observations and reduction of the AAT spectra are described below.
*Journal of Observations*
-------------------------
The majority of spectra were obtained in three observing runs at the AAT: 1989 August 2–3, 1991 March 15–16 and 1993 June 22–23. In addition, a small number of spectra were obtained through AAT service observations and by collaborators as part of their back-up programs. Table 1 summarises the instrumental setup and observing conditions for the AAT observations.
On the AAT, we used the RGO spectrograph at the f/8 Cassegrain focus. A dichroic beam splitter directed the red light to FORS (Faint Object Red Spectrograph) and the blue light to either the IPCS (Image Photon Counting System), or the Blue Thomson (BT) or Tektronix (Tek) CCD detectors. The spectral resolution of FORS is about 25Å FWHM (10.1Å per CCD pixel) over the range 5200–10500Å. The blue detectors were used with the 250B grating, yielding considerably higher resolution, about 3Å per pixel or 7–8Å FWHM, over a range extending from the atmospheric cutoff to the dichroic cutoff at 5400Å.
*Observing Strategy*
--------------------
Observations were made on dark nights with a slit typically $2$ wide and $2^{\prime}$ long on the sky, oriented at the parallactic angle. Differential atmospheric refraction along the slit from 3500–9000Å ranged from $<0$2 for zenith distances ${\rm ZD}<20$[$^{\circ}$]{} to $\sim2$ at ${\rm ZD}\sim50$[$^{\circ}$]{}. A narrow slit was favoured to optimise signal-to-noise ratios, but in cases of poor seeing ($>2$) the slit width was increased to reduce slit losses (Table 1). Each quasar was observed in consecutive exposures with the object in one of two fixed positions on the slit. This strategy was used to optimise sky subtraction beneath the object spectrum and reduce the effect of local pixel variations. To minimise atmospheric extinction, observations were made as close to the zenith as possible. Standard stars were observed on each night at appropriate airmasses to enable flux calibration and to remove atmospheric absorption features. Observations of a Cu-Ar comparison lamp were used for wavelength calibration.
*Spectral Data Reduction Method* {#sec:figarored}
--------------------------------
The AAT spectra were reduced using the FIGARO package at the University of Sydney. First, the raw CCD images were bias-subtracted and divided by a normalised flat-field; this was not necessary for the IPCS data. Sky flats were used in preference to dome flats as they showed greater uniformity across the image. Cosmic ray hits were removed interactively from regions of the CCD images close to the quasar spectrum, and then automatically over the whole image; in both cases we used interpolation from neighbouring pixels. No cosmic ray rejection was required for the IPCS data. To maximise the signal-to-noise ratio, spectra were extracted from the central few pixels of the seeing profile, consistent with the image spread due to atmospheric dispersion. Wavelength scales for both the object and sky spectra were set from the Cu-Ar lamp spectrum, achieving a typical wavelength accuracy of $<$1Å rms.
To remove strong telluric absorption bands, the red spectra were divided by that of a smooth-spectrum star observed at a similar airmass to the quasar. The strong absorption bands were generally well-subtracted, except above 9000Å where the residuals are magnified by the flux calibration which is poorly constrained in this region. Corrections were also applied for atmospheric extinction in the blue in order to recover accurate continuum slopes; these corrections were mostly negligible except below about 4500Å in a few objects observed at large air masses.
Flux calibration was applied using spectrophotometric standard stars observed on the same night as the quasars and at similar air masses. Estimated errors in flux density are typically in the range 20–50%. The main limitations were from slit losses in poor and variable seeing, a consequence of the narrow slit used to attain high signal-to-noise ratios. The loss of narrow-line flux from extended emission regions is expected to be negligible for the majority of quasar targets (e.g. Baum et al. 1988 and Tadhunter et al. 1993).
After individually extracting and flux-calibrating the FORS and IPCS/Tek spectra, they were then joined together at wavelengths around 5400Å. The blue spectra were re-binned first to match the resolution of FORS, improving the signal-to-noise ratio for emission-line measurements, particularly in the early IPCS data. In general, the red and blue spectra matched well in the region of the join. In cases where the match was poor the IPCS/Tek data were rescaled vertically to the FORS data. All the spectra have been shifted to a heliocentric frame of reference and the wavelength scale corrected from air to vacuum. The final wavelength coverage of the joined spectra is typically 3400–10000Å, with an overall spectral resolution of $\sim$25Å FWHM and average signal-to-noise ratio exceeding ten.
*Confirmation of Quasar IDs*
----------------------------
After spectroscopy, the original MQS candidate list yielded 100 confirmed quasars and 6 BL Lacs, leaving five still requiring confirmation. These five quasar candidates are retained in the list for completeness.
During the ongoing definition of the MQS, a handful of IDs were excluded at the telescope on the basis of their spectral properties. These included two targets mis-identified with Galactic stars and seven objects classified after spectroscopy as narrow-line radio galaxies. The main spectral requirement for an ID to be classified as a quasar was the presence of broad emission lines. Stars were readily eliminated, and galaxies were recognised by their narrow emission lines or the 4000Å break and other characteristic absorption features.
For the quasars comprising the MQS, the distribution of IIIaJ Schmidt-plate magnitudes $b_{\rm J}$ (drawn from the COSMOS Southern Sky Catalogue) is shown in Figure \[bjdist\]. The distribution peaks at magnitudes significantly brighter than the plate limit, giving confidence that very few faint quasar IDs have been missed and the sample is relatively unbiased. Consistent with this, most very faint new optical counterparts from the deep CCD imaging program have turned out spectroscopically to be galaxies.
THE SPECTRA {#sec:spectra}
===========
Seventy-nine observed (AAT plus two ESO) spectra for the MQS are presented in Figure \[spectra\]. Quasars with published optical data and not shown here have references included in Table 2. Also, spectra of BL Lacs (MRCB0118$-$272, MRCB1309$-$216 and MRCB2240$-$260) have been left out because data with much higher signal-to-noise ratios exist in the literature. Spectra for two borderline quasars/radio galaxies (MRCB0032$-$203, MRCB0201$-$214) are included separately in Figure \[bord\] — neither shows obvious broad lines although the line ratios indicate higher excitation than is typical for radio galaxies. Until further investigation, it was decided that these objects should remain in the galaxy subsample.
*Comments on Individual Spectra* {#sec:indiv}
--------------------------------
: A CSS object with an unusual optical spectrum. The peak of the strong line at about 8800Å would indicate that this line is 6583 rather than although the broad base may contain . This interpretation is supported by the presence of strong 6717,6734 and weak or absent . The low-excitation spectrum is reminiscent of a starburst galaxy, or possibly a LINER, although is still relatively strong.
: The and lines appear to be asymmetric with a tail to the blue.
: This quasar has an unusual spectrum showing copious emission and weak . is very weak.
: The $\lambda4959$/5007 line ratio appears to be less than $1/3$, perhaps indicating a problem with sky subtraction or possible blending with 4924. The 3000Åbump is prominent. The redshift was first reported by Hunstead et al. (1978); an early spectrum was published by Wilkes et al. (1983).
: This spectrum is very similar to that of MRCB0123$-$226. The redshift was originally determined by Jauncey et al. (1978) and spectra have been published by Wilkes et al. (1983) and Wilkes (1986). Again, $\lambda4959$ appears relatively weak, possibly as a result of blending with 4924.
: and show absorption features just blueward of their peaks. This $z_{\rm abs}\approx z_{\rm em}$ absorption is also seen in the noisier spectrum published by Wilkes (1986). $\lambda1640$ is prominent.
: A marked upturn is present in the continuum blueward of due to the 3000Å bump.
: A quasar with $z_{\rm abs}\approx z_{\rm em}$ absorption in .
: A well-known Gigahertz Peaked Spectrum (GPS) quasar noted for an apparent overdensity of intervening metal absorption lines. Previous studies have been made by Wilkes et al. (1983), Wilkes (1986), Heisler, Hogan & White (1989) and Foltz et al.(1993). Correction for atmospheric absorption distorts the AAT spectrum at wavelengths $>9000$Å, and makes the relative flux measurement of unreliable.
: This is the highest redshift quasar in the MQS ($z=2.914$). The continuum is very faint but and were detected clearly.
: The spectrum is very noisy at $\lambda>9000$Åmaking a measurement of the equivalent width impossible.
: ESO/3.6m spectrum, observed by Wall and Shaver (1993, private communication).
: ESO/3.6m spectrum, observed by Wall and Shaver (1993, private communication).
: A bright quasar showing strong broad lines. An intervening absorption system corresponding to at $z_{\rm
abs}=0.477$ (Wright et al. 1979) is clearly visible to the blue of the emission peak; this absorption system has been well studied in the optical and UV (e.g., Bergeron & Kunth 1984; Kinney et al. 1985).
: The blue wing of may be cut off by absorption, similar to that seen in MRCB1244$-$255.
: This bright low-redshift quasar has been imaged extensively both in the radio and optical by Hutchings, Crampton & Campbell (1984), Gower & Hutchings (1984) and Stockton & MacKenty (1987). It shows a nebulous optical extension to the NW, which appears to align with the stronger radio lobe. The extended nebulosity has also been studied spectroscopically by Boisson et al. (1994).
: The 3000Å bump is very prominent in this quasar and is notably weak.
: The and emission lines show absorption just blueward of the line peaks.
: The line appears to be heavily absorbed, and appears to show absorption close to the line peak. Uncertainties in flux calibration and sky subtraction longward of 9000Å affect the relative strength of , which may also show absorption.
: The emission line has a rounded profile with no evidence for a narrow component, while shows both a narrow and a very broad component. The possible emission feature at about 5100Å may be accentuated by the join between the FORS and IPCS spectra.
: The region around the spectral join at 5400Å is noisy and so the scaling between the FORS and IPCS spectra is uncertain.
: A faint quasar with poor signal-to-noise ratio above 9000Å. and are discernible but the other lines are weak.
: Only a FORS spectrum is available — the broad emission line is identified with . The measurement of is uncertain due to the limited wavelength coverage and presence of the 3000Å bump.
: shows a prominent narrow component.
: The prominence of the 3000Å bump makes uncertain.
: The spectrum is noisy above 9000Å, making and fluxes uncertain.
: A BL-Lac object studied by Blades et al. (1980); no spectrum is given here. No emission lines were found by Blades et al. but a redshift limit of $z\geq 1.49$ was set by the detection of absorption lines.
: This quasar was observed on two occasions at the AAT, showing little change apart from a possible fall in the strength of between 1991 March and 1993 June. The most recent observing date only is listed in Table 2, and the corresponding spectrum shown in Figure \[spectra\].
: is very weak compared with , and appears to be very broad with no narrow component. The continuum is also very red.
: 5007 is affected by atmospheric A-band absorption.
: The emission lines are very weak in this spectrum; the strongest line is assumed to be at $z=0.602$. An emission feature just beyond 8000Å is consistent with weak 5007 at the same redshift.
: A number of strong absorption lines are present in the spectrum; these can be identified as complexes of 2382,2586,2600 and 2796,2803 at $z_{\rm abs}
\sim 1.70$ and 1.75.
: A quasar with a steep optical spectrum and weak emission lines. may be heavily absorbed.
: Clearly at high-redshift ($z=2.272$), this quasar was noted in the sample of Dunlop et al. (1989) with an incorrect redshift.
: has a broad, flat-topped profile and has an asymmetric profile, possibly due to absorption. $\lambda1640$ is prominent.
: Absorption is evident at the central emission wavelength of and possibly also .
: This quasar was observed on two occasions (1989 Sept 26 and 1993 June 22). and were better recorded in the later spectrum shown here in Figure \[spectra\]. The most recent observing date only is listed in Table 2.
: The continuum shape appears to be dominated by the 3000Å bump and prominent emission-line blends; , therefore, is uncertain.
SPECTROSCOPIC DATA {#sec:obsdata}
==================
Table 2 lists the observational details and optical properties for the MQS, including observing dates, redshifts, optical spectral indices and IIIaJ blue magnitudes, $b_{\rm J}$, measured by COSMOS. References are supplied for quasars with published redshifts.
*Redshifts* {#sec:redshifts}
-----------
Redshifts, listed in Table 2, were measured from the peaks of strong emission lines. Figure \[zdis\] shows the distribution of measured redshifts for the MQS, which span the range 0.0–3.0 (median $z\approx1$). Redshifts have not been established for five BL-Lac objects. Narrow lines were used in preference to broad lines for redshift measurements to avoid possible bias arising from systematic differences (velocity shifts $\sim1000$ kms[$^{-1}$]{}) which have been reported between narrow and broad lines (Gaskell 1982; Tytler & Fan 1992). Such velocity shifts were seen in some MQS spectra, and will be discussed in a separate paper.
*Optical Spectral Indices* {#sec:aopt}
--------------------------
Optical spectral indices, ($f_{\nu}\propto \nu^{-\alpha}$), have been measured over the observed wavelength range 3400–10000Å. In practice, the observed quasar continuum rarely follows a simple power law over this range and so large uncertainties in fitted spectral index result, of the order $\pm 0.2$. One of the most obvious spectral features which produces a deviation from the power-law continuum is the so-called ‘3000Å bump’, believed to arise from blended and Balmer continuum emission over the restframe wavelength range 2000–4000Å (Oke, Shields & Korykansky 1984; Wills, Netzer & Wills 1985).
The MQS includes quasars with a wide range of optical continuum slope, $-0.3<$$<3$ with median $\approx 1$ (Figure \[aopt\]). Furthermore, a tail of red quasars appears in the distribution of in Figure \[aopt\].
In Figure \[aoptbj\] — a plot of against blue magnitude $b_{\rm J}$ — a trend is evident, despite the large scatter, for red quasars (steep ) to be optically faint. For example, all MQS quasars with $> 1.5$ are fainter than $b_{\rm J}=18$. The correlation in Figure \[aoptbj\] has a (Kendall’s tau) probability of $P=0.001$ of occurring by chance (note the correlation remains significant for core-dominated quasars alone, $P=0.02$). The correlation may be viewed alternatively as a paucity of optically-bright, red quasars in the MQS, which is surprising since such objects are not excluded directly by our selection criteria. The fact that red MQS quasars are also optically faint may argue in favour of reddening of quasar light in these objects. Interestingly, all the quasars with $> 1.5$ are lobe-dominated and CSS quasars, pointing to an intrinsic (possibly orientation-dependent) origin. The trend for lobe-dominated and CSS quasars to be reddened more than core-dominated quasars is confirmed by the average spectral properties of the MQS (Baker & Hunstead 1995), and is consistent with other indicators such as Balmer decrements and narrow-line equivalent widths. This issue is addressed in a paper by Baker (1997), to which the reader is referred.
*Emission-Line Fluxes* {#sec:emlines}
----------------------
Restframe emission-line properties have been measured for prominent lines (see Table 3) using [starlink]{} DIPSO. The local continuum level was fitted in most cases by eye because of the difficulty in obtaining an acceptable fit to a power law or polynomial over the full wavelength range covered and to take account of broad emission features, such as the 3000Å bump. Where possible, a polynomial continuum was fitted first as a guide. Errors in line fluxes were estimated empirically with repeated measurements using extreme continuum placements.
Broad and the narrow $\lambda\lambda4959,5007$ doublet were separated by integrating the emission above the red wing of and subtracting it from the total + flux. The doublet itself was not separated for flux measurements because both lines were often blended at the resolution of FORS. As the doublet ratio should be constant, combining the lines should not introduce bias. Due to the low spectral resolution no attempt has been made to separate contributions to prominent broad lines from blended species (e.g. and have not been separated, nor have , and ). It is recommended that individual spectra be viewed to judge the uncertainty of line measurements.
Rest-frame fluxes for strong lines are given in Table 3 relative to for low-redshift spectra and relative to at higher redshifts where was not visible. Neither nor commonly suffers absorption in quasar spectra although at low resolution these lines may be blended with and , respectively (Wills, Netzer & Wills 1985; Steidel and Sargent 1991). Because of the low spectral resolution, we have not attempted to separate the complex blends from the other broad-line emission in the 4500–5400Å region. Line flux ratios are not affected by errors in the absolute flux calibration; however, uncertainties will be introduced by differences in the relative scaling of the red and blue spectra, wavelength-dependent errors in flux calibration and extinction corrections. Errors in relative fluxes are large, estimated to be $\sim30$%, possibly greater for weak lines or in regions of poor signal-to-noise. Neighbouring lines, such as and will have more accurately determined flux ratios.
Generally, the broad lines show a small dispersion in relative fluxes. On the other hand, Figure \[rfluxo\] shows that the ratios of and to flux span nearly four and two decades, respectively. The MQS spectra show a range of / Balmer decrements from 3 to 12, suggesting that reddening is important in the MQS (see Baker 1997). Line blends may affect the fluxes of both and , although not significantly for strong lines. Other diagnostic line ratios in the MQS include broad / which ranges between 0.8–10, and narrow / which spans 1–200.
*Equivalent Widths* {#sec:ews}
-------------------
Equivalent widths, , were measured in the manner described above. Uncertainties in equivalent widths are typically 5–15% (see Table 3), and are independent of flux calibration. The errors are slightly larger for the and lines due to uncertainty in setting the continuum level in the (rest frame) 2000–4000Å region. Where possible, a polynomial was fitted through line-free regions of the spectrum avoiding the 3000Å bump, and this level was used to calculate . The tabulated uncertainties in include any differences between interpolated (polynomial fit) and local continuum flux.
Figure \[ewall\] shows the distributions of equivalent widths for the eight most prominent lines in the quasar spectrum: narrow 4959,5007 and 3727, broad 6563, 4861, 2798, 1909, 1549,1551 and 1216. As expected from Figure \[rfluxo\], the narrow and lines show the greatest spread: 4 and 2 decades respectively. The small range of equivalent widths is notable, a consequence of the tight correlation between optical continuum and line luminosity observed in virtually all types of AGN (Yee 1980). The distributions of in Figure \[ewall\] for the broad lines show no prominent asymmetry, except perhaps for . In general, the ranges measured for the MQS are similar to those found in other samples of both radio-loud and -quiet quasars (e.g. Baldwin, Wampler & Gaskell 1989; Boroson & Green 1992; Jackson & Browne 1991).
*Emission-Line Widths* {#sec:linewidths}
----------------------
Line widths (FWHM) have been measured for the prominent broad lines and are listed in Table 3 in . No deconvolution of broad and narrow components was attempted in general; most line profiles were smooth and dominated by the broad component. In a few cases, however, with a clearly separated unresolved component sitting on top of a broad line, the width was measured from the broad component alone using eyeball deconvolution. Again, the measurement uncertainties are large, about $20$%. The widths may be unreliable for emission lines with absorption occurring near the line peak. Amongst MQS quasars with $z>1.4$, where and/or is visible, absorption systems within a few thousand of the line peak are seen in approximately 50% of the spectra, comparable with other studies of steep-spectrum quasars (e.g. Anderson et al. 1987) (the occurrence of associated absorption in the MQS will be addressed elsewhere, see e.g. Baker & Hunstead 1996). Because a typical emission line profile is rarely Gaussian, as well as the possibility of absorption, observed FWHMs provide only an approximate measure of line width, particularly for strong lines with very broad wings. The line profiles and their aspect dependence will be studied in more detail in a later paper.
Figure \[fwhmdis\] shows the distribution of velocity widths (kms[$^{-1}$]{}) for the four strongest broad lines. All four lines show a similar spread in line widths, from 1000–20000 kms[$^{-1}$]{}. seems to have an asymmetric distribution, with a tail extending to narrower lines. Again, these values are comparable with velocity widths for quasars in other samples (e.g. Baldwin, Wampler & Gaskell 1989).
NOTES TO TABLES
===============
The spectroscopic data are presented in Tables 2 and 3. In Table 2, for each MRC quasar named (B1950 convention) the observing date is listed in column 2; the source classification (Q=quasar; Q?=likely quasar; B=BL Lac) in column 3; the redshift in column 4; the optical spectral index (3000–10000Å observed; $S_{\nu}
\propto \nu^{-\alpha}$) in column 5; and the COSMOS IIIaJ magnitude (r=$R$-band CCD magnitude) in column 6. References are supplied for individual quasars in column 7: \[1\] Blades (1980); \[2\] Boisson (1994); \[3\] Chu (1986); \[4\] Dekker & D’Odorico (1984); \[5\] Dunlop (1989); \[6\] Gower & Hutchings (1984); \[7\] Hunstead (1991 priv. comm.); \[8\] Hunstead (1978); \[9\] Jauncey (1978); \[10\] Murdoch (1984); \[11\] McCarthy (1994 priv. comm.); \[12\] O’Dea (1991); \[13\] Rawlings (1993 priv. comm.); \[14\] Savage (1976); \[15\] Stickel (1993a); \[16\] Stickel (1993b); \[17\] Wall & Shaver (1993 priv. comm.); \[18\] White (1988); \[19\] Wilkes et al. (1983), Wilkes (1986); \[20\] Wright (1979); \[21\] Wright (1983). An ‘N’ in column 8 denotes additional notes in Section 4, an ‘S’ is given if the spectrum is shown in Figure 4.
Emission line data are presented in Table 3 for the lines 6563 ($+$), \[O[iii]{}\]4959,5007, 4861, 4340 (blended with 4363), 3727, 3426, 2798, 1909, 1549,1551, Si [iv]{}/O [iv]{}\]1400 (blend) and 1216 (blended 1240). The observed wavelength of the line peak is given in column 3 (only 5007 is given for the \[O[iii]{}\] doublet), or references given for data taken from the literature: \[1\] Hunstead (priv. comm.); \[2\] White et al. (1988); \[3\] Wilkes (1986). Uncertain values are indicated with a colon throughout the Table. In column 4 the ratio of the integrated line flux is given relative to broad or (when no ); letters show if the line measurement was affected by bad sky subtraction (s), absorption (a), noise (n), being near the edge of the spectrum (e) or a join (j), or the line is weak (w). Restframe equivalent widths (Å) are listed (with their measurement uncertainties) in column 5; velocity widths (FWHM) are given in column 7 ($\times 10^{3}$) for mostly unblended permitted lines (forbidden lines are indicated by ‘u’).
CONCLUSIONS
===========
Low-resolution optical spectroscopy has been completed for 106/111 quasars (including 6 BL Lacs) which comprise the MQS; 79 quasar spectra are published here (plus two borderline radio galaxy spectra). The high signal-to-noise ratios achieved, even for faint ($\sim22$mag) targets, have allowed measurements of redshifts, spectral slopes and a wide range of emission-line properties. Redshifts for the MQS range from $z\approx 0.1$ to 3, reaching significantly higher $z$ than the only other completely-identified, low-frequency-selected sample, the 3CRR (Laing et al. 1983), which contains only one $z>2$ quasar.
The MQS quasars show a wide range of spectral properties including a large fraction of red quasars, which are mainly lobe-dominated and CSS quasars. The emission-line properties of the MQS are in broad agreement with other studies, showing similar ranges in line widths and equivalent widths for the broad lines. The narrow oxygen lines, notably , span a particularly wide range in equivalent width (see Baker 1997). Further analysis of the MQS optical data and follow-up observations are ongoing and will be presented in forthcoming papers.
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
We thank the referee, Chris Impey, for his helpful comments. JCB was supported for part of this work by a postgraduate scholarship from the Special Research Centre for Theoretical Astrophysics, University of Sydney. RWH acknowledges funding from the Australian Research Council. Staff at the Anglo-Australian Telescope are thanked greatly for their support.
[99]{}
å[A&A]{}
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[lcccl]{} \[obs\] 89-08-02 & [fors+ipcs]{} &1.0 &1.0 &clear 89-08-03 & “ &2.0 &1.5 &cloud 89-09-26 & ” &3.0 &1.5 &poor 91-03-15 & “ &2.0 &1.6 &clear 91-03-16 & ” &1.0 &1.3 &clear 92-06-04 & [fors+bt]{} &2-4 &1.5 &clear 92-11-29 & [fors+tek]{} &2.0 &2.0 &clear 92-11-30 & “ &2.0 &2.0 &clear 93-06-22 & ” &1.5 &1.7 &cloud 93-06-23 & “ &4.5 &2.0 &haze 93-11-15 & ” &1.5 &2.5 &mostly clear 94-04-16 & [fors]{} &1.7 &1.8 &clear
\[table\_opt\]
\[ewtab\]
|
---
abstract: 'We give a new proof of a polynomial recurrence result due to Bergelson, Furstenberg, and McCutcheon, using idempotent ultrafilters instead of IP-limits.'
address: |
The Ohio State University\
231 West 18th Avenue\
Columbus, OH 43210
author:
- Christian Schnell
title: Idempotent ultrafilters and polynomial recurrence
---
Introduction {#sec:0 .unnumbered}
============
In the thirty or so years since H. Furstenberg reproved Szemerédi’s theorem using methods from ergodic theory, many striking discoveries have been made in the area now known as *Ergodic Ramsey theory*. Perhaps the most surprising of these is the discovery that recurrence results can be obtained for polynomial sets, meaning sets of values of polynomials. The following pretty theorem, a special case of a more general theorem proved by V. Bergelson, H. Furstenberg, and R. McCutcheon in [@BFM], is a typical result in this direction.
Let $\mathcal{F}$ be the collection of all non-empty finite subsets of $\NN$. For any polynomial $p \in {{\mathbb{Z}}\lbrack x_1, \dotsc, x_k \rbrack}$ satisfying $p(0, \dotsc,
0) = 0$, and for any IP-sets $\{n_{\alpha}^{(1)}\}_{\alpha \in \mathcal{F}},
\dotsc, \{n_{\alpha}^{(k)}\}_{\alpha \in \mathcal{F}}$, the set $$R = {\bigl\{p(n_{\alpha}^{(1)}, \dotsc, n_{\alpha}^{(k)}) \thinspace\big\vert\thinspace \alpha \in \mathcal{F} \bigr\}}$$ is a set of nice recurrence.
To say that $R \subseteq \ZZ$ is a set of *nice recurrence* means that for any probability space ${(X, \mathcal{B}, \mu)}$, and any invertible measure-preserving transformation $T$ on $X$, one has $$\limsup_{n \in R} \mu(A \cap T^n A) \geq \mu(A)^2$$ for all $A \in \mathcal{B}$. Moreover, an *IP-set* is any set of the form $${\Bigl\{n_{\alpha} = \sum_{i \in \alpha} n_i \thinspace\Big\vert\thinspace \text{$\alpha \in \mathcal{F}$} \Bigr\}},$$ for positive integers $n_0, n_1, n_2, \dotsc$.
As in Furstenberg’s result, this inequality has immediate combinatorial applications. It also turned out that the above theorem was only a first step; much stronger results—combining IP-convergence, multiple recurrence as in Szemerédi’s theorem, and polynomial sets—have since been established, for instance in [@BM].
The purpose of the present paper is to give a different proof for the central result of [@BFM], using *idempotent ultrafilters* instead of IP-limits. While this approach is less constructive, it has the advantage of “making the statements and proofs cleaner and more algebraic,” in the words of the survey paper [@BSur]. It also follows the general philosophy that for each result about IP-sets, there should be an analogous result about idempotent ultrafilters.
The main theorem and its proof are presented in Section \[sec:V\]; however, a better point to begin reading is probably Section \[sec:II\], which treats a special but typical case, and explains the method of proof in some detail. Section \[sec:VI\] contains a small number of applications, of the type mentioned above.
Since ultrafilters on groups and semigroups are used throughout the paper, their basic properties are reviewed in Section \[sec:I\]; readers who are already familiar with $\beta{\mathbb{N}}$, for instance from [@BSur], will recognize all the material, despite the more general context. To keep the paper self-contained, several generally known results about operators and integer-valued polynomials have also been included; these make up Sections \[sec:III\] and \[sec:IV\].
Note {#note .unnumbered}
----
Vitaly Bergelson, who advised me during my first two years in graduate school, suggested the problem of reproving the results in [@BFM] using idempotent ultrafilters. I am very grateful to him for his help, as well as for countless pleasant conversations. Unlike wine, the paper has failed to mature during the several years that it has been stored on the hard drive of my computer; nevertheless, I have decided to make it available, since it is in my opinion a nice application of idempotent ultrafilters to recurrence results.
The Stone-Čech compactification of a discrete semigroup {#sec:I}
=======================================================
Ultrafilters {#ultrafilters .unnumbered}
------------
We begin by reviewing the definition and several basic properties of the space of ultrafilters. Let $(S, \circ)$ be a commutative semigroup. An on $S$ is a collection $p$ of subsets of $S$ with the following four properties:
1. $S \in p$ and $\emptyset \not\in p$.
2. If $A \in p$, and $B \supseteq A$, then $B \in p$.
3. If $A,B \in p$, then $A \cap B \in p$.
4. For every $A \subseteq S$, either $A \in p$, or $S \setminus A \in p$.
For every $s \in S$, there is a or *trivial* ultrafilter consisting of all subsets containing $s$; the construction of other ultrafilters requires the Axiom of Choice.
The space $\beta S$ of all ultrafilters on $S$, suitably topologized, is the Stone-Čech compactification of the discrete space $S$. After briefly stating the basic properties of $\beta S$, we will consider two examples: one where $S$ is the group ${{\mathbb{Z}}^{n}}$, and a second one where $S$ equals $\mathcal{F}$, the set of nonempty finite subsets of ${\mathbb{N}}$. A good and very comprehensive reference for this topic is the book by Hindman and Strauss [@HiS].
Terminology {#terminology .unnumbered}
-----------
Since ultrafilters are collections of sets, the following terminology is convenient when dealing with their members. If $p$ is an ultrafilter on $S$, we call a set if it is contained in $p$; we shall also use the phrase ‘’ to mean ‘for all $s$ in some $p$-big set.’ In the case of several variables, we shall say that $\langle \text{statement} \rangle$ holds ‘’ if $${\bigl\{s_1 \in S \thinspace\big\vert\thinspace {\bigl\{s_2 \in S \thinspace\big\vert\thinspace \cdots {\bigl\{s_n \in S \thinspace\big\vert\thinspace \langle \text{statement}
\rangle \bigr\}} \in p \cdots \bigr\}} \in p \bigr\}} \in p.$$ In other words, there should be $p$-many $s_1$, for which there are $p$-many $s_2$, for which …, for which there are $p$-many $s_n$, for which $\langle \text{statement}
\rangle$ is true. Nested sets of exactly this form will play a role during the proof of the main theorem in Section \[sec:V\].
Basic properties {#basic-properties .unnumbered}
----------------
As was said above, we let $\beta S$ be the set of ultrafilters on $S$, and consider $S$ as a subset of $\beta S$, by identifying an element of $S$ with the principal ultrafilter it generates. One can put a topology on $\beta S$, in which the sets $$\begin{aligned}
{\overbar{A}{3pt}{3pt}}= {\bigl\{p \in \beta S \thinspace\big\vert\thinspace A \in p \bigr\}} && \text{(for $A \subseteq S$)}\end{aligned}$$ give a basis for the closed sets; each ${\overbar{A}{3pt}{3pt}}$ is both closed and open. The result is a compact space (this includes the Hausdorff property) that has $S$ as a discrete and dense subspace.
The semigroup operation $\circ$ extends to $\beta S$; given $p$ and $q$ in $\beta S$, their product $p \circ q$ may be defined by the property that for any $A \subseteq S$, $$\label{eq:circ}
A \in p \circ q \quad \Longleftrightarrow \quad {\bigl\{s \in S \thinspace\big\vert\thinspace {\bigl\{t \in S \thinspace\big\vert\thinspace s \circ t
\in A \bigr\}} \in q \bigr\}} \in p.$$ The new operation is associative and continuous from the left (meaning that for any $q$, the map $p \mapsto p \circ q$ is continuous), and makes $\beta S$ into a compact left-topological semigroup.
Idempotent ultrafilters {#idempotent-ultrafilters .unnumbered}
-----------------------
An ultrafilter $p \in \beta S$ is called if it satisfies the relation $p \circ p = p$. Idempotent ultrafilters are closely related to , which are sets of the form $$\begin{aligned}
{\Bigl\{\prod_{i \in \alpha} s_i \thinspace\Big\vert\thinspace \text{$\alpha \subset {\mathbb{N}}$ finite, nonempty} \Bigr\}},\end{aligned}$$ for a given sequence ${( s_i )_{i \in {\mathbb{N}}}}$. Any member of an idempotent ultrafilter contains an IP-set, and conversely, every IP-set is contained in some idempotent ultrafilter. This fact is sometimes called Hindman’s theorem (see [@BSur]\*[Theorem 3.4]{} for details); it implies that one can find many idempotent ultrafilters (provided, as usual, that the Axiom of Choice is assumed).
When a finite sequence $s_1,\dotsc, s_n$ is used in place of an infinite one, we shall denote the resulting by ${\operatorname{IP}\bigl( s_1, \dotsc, s_n \bigr)}$. The proof that any member of an idempotent has to contain an IP-set allows a much stronger conclusion if we are only looking for finite IP-sets.
\[lem:FPsets\] Let $p \in \beta S$ be an idempotent ultrafilter. If $A$ is a $p$-big set, then for any $n \in {\mathbb{N}}$ one has $${\bigl\{s_1 \in A \thinspace\big\vert\thinspace \cdots {\bigl\{s_n \in A \thinspace\big\vert\thinspace {\operatorname{IP}\bigl( s_1, \dotsc, s_n \bigr)} \subseteq A \bigr\}} \in p
\cdots \bigr\}} \in p.$$ In the terminology introduced above, one can say that there are $p$-many $s_1,
\dotsc, s_n$ in $A$ such that ${\operatorname{IP}\bigl( s_1, \dotsc, s_n \bigr)} \subseteq A$.
Limits along ultrafilters {#limits-along-ultrafilters .unnumbered}
-------------------------
Another useful notion is that of a , or a limit along some ultrafilter. Let $p \in \beta S$ be an ultrafilter. Given a map $f \colon S \to Y$ into some topological space $Y$, we say that a point $y$ is a limit of $f$ along $p$, written $$y = {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_s f(s),$$ if for every neighborhood $U$ of $y$, the set $f^{-1}(U)$ is $p$-big. When the target space $Y$ is compact, all $p$-limits exist and are unique.
This notion of limit is related to the Stone-Čech compactification in the following manner. A of a Hausdorff space $X$ is a compact space containing $X$ as a dense subspace. The $\beta X$ is the universal compactification, in the sense that for any compact space $Y$ and any continuous map $f \colon X \to Y$, there is one and only one continuous extension ${f_{\ast}}$ from $\beta X$ to $Y$, as illustrated in the diagram.
$$\begindc{\commdiag}[25]
\obj(1,3){$X$}
\obj(3,3)[$bX$]{$\smash[b]{\beta}X$}
\obj(1,1){$Y$}
\mor(1,3)(1,1){$\smash{f}$}[\atright, \solidarrow]
\mor(1,3)(3,3){}[\atright, \injectionarrow]
\mor(3,3)(1,1){$\smash[b]{{f_{\ast}}}$}[\atleft, \dasharrow]
\enddc$$
Every other compactification is a quotient of $\beta X$; furthermore, if $g \colon Y
\to Z$ is a second continuous map of compact spaces, one has ${(fg)_{\ast}} = {f_{\ast}}
{g_{\ast}}$ because of the uniqueness statement.
Now the space $\beta S$, as defined above, is the Stone-Čech compactification of the discrete topological space $S$; given any map $f \colon S \to Y$ into a compact space $Y$, the required extension ${f_{\ast}} \colon \beta S \to Y$ is given by $$\begin{aligned}
{f_{\ast}}(p) = {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_s f(s) &&\text{(for $p \in \beta S$),}\end{aligned}$$ which is continous as a map from $\beta S$ to $Y$.
The following lemma is an immediate consequence of the universal property.
\[lem:phistar\] Any map $\phi \colon S \to T$ between two semigroups $S$ and $T$ induces a continuous map ${\phi_{\ast}}\colon \beta S \to \beta T$, given by ${\phi_{\ast}}(p) = {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_s \phi(s)$. A set $B$ is ${\phi_{\ast}}(p)$-big if, and only if, its preimage $\phi^{-1}(B)$ is $p$-big. For any map $f : T \to Y$ into a compact space $Y$, one has $${\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_s f(\phi(s)) = {\mathop{{\phi_{\ast}}(p)\text{-}}\negthinspace\mathop{\mathrm{lim}}}_t f(t).$$ If $\phi$ is multiplicative, so is ${\phi_{\ast}}$; in particular, ${\phi_{\ast}}(p)$ is then always idempotent for idempotent $p \in \beta S$.
There is another important property of $p$-limits, especially useful for our purposes.
\[lem:plimits\] Let $p$ and $q$ be two elements of $\beta S$. The equality $${\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_s {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_t f(s \circ t) = {\mathop{(p \circ q)\text{-}}\negthinspace\mathop{\mathrm{lim}}}_s f(s)$$ holds for any map $f \colon S \to Y$ into a compact space $Y$.
Let $y = {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_s {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_t f(s \circ t)$; for any neighborhood $U$ of $y$, the set $${\bigl\{s \thinspace\big\vert\thinspace {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_t f(s \circ t) \in U \bigr\}}$$ is $p$-big. Equivalently, $${\bigl\{s \thinspace\big\vert\thinspace {\bigl\{t \thinspace\big\vert\thinspace f(s \circ t) \in U \bigr\}} \in q \bigr\}} \in p,$$ and this is nothing but the condition $f^{-1}(U) \in p \circ q$. It follows that the right-hand limit ${\mathop{(p \circ q)\text{-}}\negthinspace\mathop{\mathrm{lim}}}_s f(s)$ also equals $y$.
The lemma explains one useful aspect of idempotent ultrafilters—if $p$ is an idempotent, one has $$\label{eq:doubleplimit}
{\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_s {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_t f(s \circ t) = {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_s f(s),$$ and this relation is at the base of all applications of ultrafilters to recurrence results.
As an application, let us prove a lemma known as , for $p$-limits. It provides a useful sufficient condition for a weak $p$-limit in a Hilbert space to be zero.
\[lem:VanDerCorput\] Let $Y$ be a closed ball in a Hilbert space $\Hil$, endowed with the weak topology (and thus compact). Given a map $f \colon S \to Y$ and an idempotent $p \in \beta S$, let $y = {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_s f(s)$. If $${\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_s {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_t {\bigl\langle f(s \circ t), f(t) \bigr\rangle} = 0,$$ then $y = 0$.
One uses in a clever way. Notice that for any $N \in
{\mathbb{N}}$, $$y = {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{s_1} \cdots {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{s_N} \frac{1}{N} \sum_{n = 1}^N f(s_n \circ
\dotsb \circ s_N).$$ Using weak lower semi-continuity of the norm, we obtain $$\begin{aligned}
{\bigl\| y \bigr\|}^2 &\le
{\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{s_1} \cdots {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{s_N} \frac{1}{N^2} {\Bigl\| \sum_{n = 1}^N f(s_n
\circ \dotsb \circ s_N) \Bigr\|}^2 \\
&= {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{s_1} \cdots {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{s_N} \frac{1}{N^2} {\Bigl\langle \sum_{m = 1}^N f(s_m
\circ \dotsb \circ s_N), \sum_{n = 1}^N f(s_n \circ \dotsb \circ s_N) \Bigr\rangle} \\
&= \frac{1}{N^2} \sum_{m, n} {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{s_1} \cdots {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{s_N} {\bigl\langle f(s_m
\circ \dotsb \circ s_N), f(s_n \circ \dotsb \circ s_N) \bigr\rangle}, \\
\intertext{and after collapsing the multiple $p$-limits with the help of
\eqref{eq:doubleplimit}, this becomes}
&= \frac{1}{N^2} \sum_n {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_s {\bigl\| f(s) \bigr\|}^2 + \frac{2}{N^2} \Re
\sum_{m < n} {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_s {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_t {\bigl\langle f(s \circ t), f(t) \bigr\rangle} \\
&= \frac{1}{N} {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_s {\bigl\| f(s) \bigr\|}^2.\end{aligned}$$ Since $N$ was arbitrary, we see that $y = 0$.
We shall now discuss two concrete examples of semigroups and their Stone-Čech compactifications, namely $\beta {{\mathbb{Z}}^{n}}$ and $\beta \mathcal{F}$.
Abelian groups and ultrafilters {#abelian-groups-and-ultrafilters .unnumbered}
-------------------------------
We are going to use vector notation for elements of ${{\mathbb{Z}}^{n}}$, such as ${\vec{a}_{}}= (a_1,
\dotsc, a_n)$. Even though ${{\mathbb{Z}}^{n}}$ is a group, the space of ultrafilters $\beta {{\mathbb{Z}}^{n}}$ is only a semigroup, because there are in general no inverses for elements. Still, we can get information about ultrafilters in $\beta {{\mathbb{Z}}^{n}}$ from the group structure of ${{\mathbb{Z}}^{n}}$; in particular, we shall investigate the relationship between subgroups and idempotent ultrafilters.
Every subgroup of ${{\mathbb{Z}}^{n}}$ is itself free, of rank between $0$ and $n$. The first observation is that subgroups of rank $n$ are contained in every idempotent ultrafilter.
\[lem:Lattice\] For every idempotent $p \in \beta {{\mathbb{Z}}^{n}}$, all rank $n$ subgroups are $p$-big.
A subgroup $L$ of rank $n$ necessarily has finite index. As an ultrafilter, $p$ thus has to contain one of the cosets, say ${\vec{z}_{\thinspace}}+ L$, and as an idempotent, it then has to contain the set $${\bigl\{{\vec{a}_{}}\in {\vec{z}_{\thinspace}}+ L \thinspace\big\vert\thinspace {\bigl\{{\vec{b}_{}}\in {\vec{z}_{\thinspace}}+ L \thinspace\big\vert\thinspace {\vec{a}_{}}+ {\vec{b}_{}}\in {\vec{z}_{\thinspace}}+ L \bigr\}} \in p \bigr\}}$$ as well. In particular, that set is nonempty. The resulting equation ${\vec{z}_{\thinspace}}+ {\vec{z}_{\thinspace}}\equiv
{\vec{z}_{\thinspace}}\mod L$ gives ${\vec{z}_{\thinspace}}+ L = L$, and we can conclude that $L$ itself is $p$-big.
We now define the of an ultrafilter $p$, denoted $\dim p$, to be the smallest possible rank of a $p$-big subgroup of ${{\mathbb{Z}}^{n}}$. Since we expect $p$-big sets to be large (especially when $p$ is an idempotent), it would be nice if the dimension of an ultrafilter in $\beta {{\mathbb{Z}}^{n}}$ was always $n$. This is not true; for instance, the principal ultrafilter generated by $0$ is idempotent, and has dimension zero. But as the following lemma shows, in all such examples, the ultrafilter in question really lives on a smaller group.
\[lem:Dimension\] Let $p \in \beta{{\mathbb{Z}}^{n}}$ be an ultrafilter, of dimension $s \in \{0, \dotsc, n\}$. If $G \subseteq {{\mathbb{Z}}^{n}}$ is an arbitrary $p$-big subgroup of rank $s$, then there is an injective group homomorphism $\phi \colon {{\mathbb{Z}}^{s}} \to {{\mathbb{Z}}^{n}}$ with image $G$, and an $s$-dimensional ultrafilter $q \in \beta {{\mathbb{Z}}^{s}}$, such that $p = {\phi_{\ast}}(q)$. If $p$ is idempotent, then any such $q$ is idempotent as well.
Let $G$ be a $p$-big subgroup of rank $s$ in ${{\mathbb{Z}}^{n}}$. Since it is free, it is isomorphic to ${{\mathbb{Z}}^{s}}$, and so there is an injective group homomorphism $\phi \colon
{{\mathbb{Z}}^{s}} \to {{\mathbb{Z}}^{n}}$ whose image is exactly $G$. Define $$q = {\bigl\{A \subseteq {{\mathbb{Z}}^{s}} \thinspace\big\vert\thinspace \phi(A) \in p \bigr\}};$$ since $G$ is $p$-big, it is easily verified that $q \in \beta {{\mathbb{Z}}^{s}}$, and that ${\phi_{\ast}}(q) = p$. Now $q$ has to have dimension $s$, for otherwise ${{\mathbb{Z}}^{s}}$, and therefore also $G$, would contain a $p$-big subgroup of smaller rank, contradicting the choice of $s$.
Now assume that $p$ is an idempotent ultrafilter. Since ${\phi_{\ast}}$ is a homomorphism, we get $${\phi_{\ast}}(q \circ q) = {\phi_{\ast}}(q) \circ {\phi_{\ast}}(q) = p \circ p = p;$$ but as $q$ is obviously uniquely determined by the condition that ${\phi_{\ast}}(q) = p$, it follows that $q \circ q = q$, and so $q$ is idempotent as well.
IP-sets and ultrafilters {#ip-sets-and-ultrafilters .unnumbered}
------------------------
Our second example is the Stone-Čech compactification of $\mathcal{F}$, the set of finite nonempty subsets of ${\mathbb{N}}$. For any two such finite sets $\alpha$ and $\beta$, we may form their union $\alpha \cup \beta$; this operation makes $\mathcal{F}$ into a commutative semigroup. An ultrafilter in this setting is now a set of sets of finite subsets of ${\mathbb{N}}$; to avoid confusion, we shall reserve the letters $\alpha,
\beta, \gamma$ for points of $\mathcal{F}$. We also continue to write $\circ$ for the semigroup operation on $\beta \mathcal{F}$. The character of this operation is utterly different from addition on $\beta {\mathbb{Z}}$; for instance, any principal ultrafilter is now idempotent.
We are mostly going to look at IP-sets in ${\mathbb{N}}$ and $\mathcal{F}$ from the point of view of $\beta \mathcal{F}$. Since the semigroup operation on $\NN$ is addition, an IP-set is now a set of the form $${\Bigl\{\sum_{i \in \alpha} n_i \thinspace\Big\vert\thinspace \text{$\alpha \in \mathcal{F}$} \Bigr\}},$$ where ${( n_i )_{i \in \NN}}$ is a sequence of positive integers. This can also be considered as a map $${n_{\bullet}}\colon \mathcal{F} \to {\mathbb{N}}, \qquad \alpha \mapsto n_{\alpha}
= \sum_{i \in \alpha} n_i$$ that satisfies $$\label{eq:nadditive}
n_{\alpha \cup \beta} = n_{\alpha} + n_{\beta}$$ for disjoint $\alpha, \beta \in \mathcal{F}$. It induces a map ${n_{\ast}}$ from $\beta
\mathcal{F}$ to $\beta {\mathbb{N}}$ that we would like to be structure-preserving, in particular with regard to idempotents, but this cannot be true. The problem is that the map ${n_{\bullet}}$ fails to be additive because holds for disjoint sets only. One answer is to look at a subclass of ultrafilters in $\beta
\mathcal{F}$, excluding—among other things—the principal ones.
One quickly sees that in order to make use of , it has to be possible, when choosing $\beta$ inside a member of some ultrafilter, to make it disjoint from a given $\alpha$. To accomplish this, we introduce the following notion. We let $$\begin{aligned}
C_n = {\bigl\{\alpha \in \mathcal{F} \thinspace\big\vert\thinspace n \in \alpha \bigr\}} && \text{(for $n \in {\mathbb{N}}$),}\end{aligned}$$ and call an ultrafilter *congested* if it contains one of the $C_n$, or *uncongested* if it contains none. Certainly, every principal ultrafilter is congested. We also introduce the notation $\alpha < \beta$ to express that the maximum of the finite set $\alpha$ is less than the minimum of $\beta$.
Whenever $p$ is an uncongested ultrafilter and $\alpha \in \mathcal{F}$, the set $${\bigl\{\beta \in \mathcal{F} \thinspace\big\vert\thinspace \alpha < \beta \bigr\}}$$ is obviously $p$-big, being an intersection of complements of certain $C_n$. This means that for any set $A \in p$, the set of $\beta \in A$ with $\alpha < \beta$ is still $p$-big, and so we can impose the even stronger condition $\alpha < \beta$ when choosing an element $\beta$ from $A$.
Now let us see what the set ${\beta\mathcal{F}^{\mathit{un}}}$ of all uncongested ultrafilters looks like. An *IP-ring* is a special type of IP-set in $\mathcal{F}$; it consists of an infinite sequence ${( \alpha_i )_{i \in {\mathbb{N}}}}$ of elements of $\mathcal{F}$ satisfying $\alpha_0 < \alpha_1 < \alpha_2 < \dotsb$, together with all possible finite unions of these. The notation $${\operatorname{IP}( \alpha_i )_{i \in \NN}}$$ will be used for such IP-rings. We then have the following result about ${\beta\mathcal{F}^{\mathit{un}}}$ and its connection with IP-rings.
\[lem:Uncongested\] ${\beta\mathcal{F}^{\mathit{un}}}$ is a closed (hence compact) sub-semigroup of $\beta \mathcal{F}$. Every idempotent $p \in {\beta\mathcal{F}^{\mathit{un}}}$ has the property that if a set is $p$-big, it contains an IP-ring. Conversely, every IP-ring is a member of some uncongested idempotent.
By definition, $${\beta\mathcal{F}^{\mathit{un}}}= \bigcap_{n \in {\mathbb{N}}} \cl{\mathcal{F} \setminus C_n}$$ is an intersection of closed sets, hence closed. Let us show that it is also a semigroup. If a product $p \circ q$ is congested, it has to contain $C_n$ for some $n$, and so $${\bigl\{\alpha \thinspace\big\vert\thinspace {\bigl\{\beta \thinspace\big\vert\thinspace n \in \alpha \cup \beta \bigr\}} \in q \bigr\}} \in p.$$ If $C_n$ is not in $p$, one of the $\alpha$ in the outer set does not contain $n$, in which case the inner $q$-big set equals $C_n$. Either way, one of the two factors is congested; products of uncongested ultrafilters are therefore uncongested. It follows that ${\beta\mathcal{F}^{\mathit{un}}}$ is a compact semigroup.
To verify the second statement—existence of IP-rings in $p$-big sets for idempotent $p$—the same proof as for Hindman’s theorem will work; when choosing elements, one simply follows the recipe mentioned above.
Third, let us show that every IP-ring is contained in some uncongested idempotent (this also proves the existence of uncongested idempotents). Let ${\operatorname{IP}( \alpha_i )_{i \in \NN}}$ be an IP-ring and set $A_n = {\operatorname{IP}( \alpha_i )_{i \geq n}}$, with obvious meaning. Following the usual procedure, we will show that the intersection $$\mathcal{A} = \bigcap_{n \in {\mathbb{N}}} \cl{A_n}$$ is a closed, nonempty subsemigroup of ${\beta\mathcal{F}^{\mathit{un}}}$; by Ellis’ theorem it then has idempotents, and any such idempotent is uncongested and contains our IP-ring.
$\mathcal{A}$ is certainly closed and nonempty (use the finite intersection property of the compact space $\beta \mathcal{F}$). To verify that it is a semigroup, we need to check that for $p, q \in \mathcal{A}$ and any $n \in {\mathbb{N}}$, the set $A_n$ is a member of $p \circ q$. Given $\alpha \in A_n$, there is some $k > n$ for which $\alpha \cup \beta \in A_n$ for every $\beta \in A_k$. Hence $${\bigl\{\beta \thinspace\big\vert\thinspace \alpha \cup \beta \in A_n \bigr\}}$$ is a $q$-big set for every $\alpha \in A_n$; as a consequence, we have $$\begin{aligned}
{\bigl\{\alpha \thinspace\big\vert\thinspace {\bigl\{\beta \thinspace\big\vert\thinspace \alpha \cup \beta \in A_n \bigr\}} \in q \bigr\}} \in p
&& \text{(for $n \in {\mathbb{N}}$),}\end{aligned}$$ and thus $p \circ q \in \mathcal{A}$. Finally, every $p \in \mathcal{A}$ has to be uncongested, for $A_{n+1}$ and $C_n$ are always disjoint.
One conclusion is that uncongested idempotents do exist; more importantly, they naturally arise when one is looking at IP-rings in terms of ultrafilters. Indeed, the lemma is the exact analogon of Hindman’s theorem for the case of IP-rings.
Another useful property of uncongested ultrafilters is stated in the last lemma of this section; it closely follows our thoughts after .
\[lem:uncongestedidp\] For any IP-set ${n_{\bullet}}\colon \mathcal{F} \to \NN$, the induced map ${n_{\ast}} \colon {\beta\mathcal{F}^{\mathit{un}}}\to \beta\NN$ is a homomorphism of semigroups. In particular, ${n_{\ast}}(p)$ is idempotent for each uncongested idempotent $p \in
{\beta\mathcal{F}^{\mathit{un}}}$.
Let $p, q \in {\beta\mathcal{F}^{\mathit{un}}}$ be arbitrary uncongested ultrafilters; we need to show that $${n_{\ast}}(p \circ q) = {n_{\ast}}(p) \circ {n_{\ast}}(q).$$ For any subset $A \subseteq \NN$, we have $A \in {n_{\ast}}(p)$ exactly when ${\bigl\{\alpha \thinspace\big\vert\thinspace n_{\alpha} \in A \bigr\}} \in p$; referring back to the definition of the operation $\circ$ on page , we then find that $$\begin{aligned}
A \in {n_{\ast}}(p \circ q) \quad & \Longleftrightarrow \quad
{\bigl\{\alpha \thinspace\big\vert\thinspace {\bigl\{\beta \thinspace\big\vert\thinspace n_{\alpha \cup \beta} \in A \bigr\}} \in q \bigr\}} \in p,
\intertext{while}
A \in {n_{\ast}}(p) \circ {n_{\ast}}(q) \quad & \Longleftrightarrow \quad
{\bigl\{\alpha \thinspace\big\vert\thinspace {\bigl\{\beta \thinspace\big\vert\thinspace n_{\alpha} + n_{\beta} \in A \bigr\}} \in q \bigr\}} \in p.\end{aligned}$$ But since $q$ is uncongested, these two conditions are actually equivalent. Indeed, given $\alpha \in \mathcal{F}$, we have $n_{\alpha \cup \beta} = n_{\alpha} +
n_{\beta}$ whenever $\beta > \alpha$, and so $$\begin{aligned}
{\bigl\{\beta \thinspace\big\vert\thinspace n_{\alpha \cup \beta} \in A \bigr\}} \cap {\bigl\{\beta \thinspace\big\vert\thinspace \beta > \alpha \bigr\}}
&= {\bigl\{\beta \thinspace\big\vert\thinspace \text{$n_{\alpha \cup \beta} \in A$ and $\beta > \alpha$} \bigr\}} \\
&= {\bigl\{\beta \thinspace\big\vert\thinspace \text{$n_{\alpha} + n_{\beta} \in A$ and $\beta > \alpha$} \bigr\}} \\
&= {\bigl\{\beta \thinspace\big\vert\thinspace n_{\alpha} + n_{\beta} \in A \bigr\}} \cap {\bigl\{\beta \thinspace\big\vert\thinspace \beta > \alpha \bigr\}}.\end{aligned}$$ Now $q$ always contains the set ${\bigl\{\beta \thinspace\big\vert\thinspace \beta > \alpha \bigr\}}$, and so $${\bigl\{\beta \thinspace\big\vert\thinspace n_{\alpha \cup \beta} \in A \bigr\}} \in q \quad \Longleftrightarrow \quad
{\bigl\{\beta \thinspace\big\vert\thinspace n_{\alpha} + n_{\beta} \in A \bigr\}} \in q.$$ This shows that $A \in {n_{\ast}}(p \circ q)$ if, and only if, $A \in {n_{\ast}}(p) \circ
{n_{\ast}}(q)$, and thus proves the lemma.
This result will later allow us to transfer results from $\beta{\mathbb{N}}$ or $\beta{{\mathbb{Z}}^{n}}$ to the space $\beta\mathcal{F}$. We shall see applications of this idea in Section \[sec:V\], after we have proved the main theorem.
An extended example {#sec:II}
===================
In this section, we want to give an in-depth discussion of a special case of the main results, Theorem \[thm:MainA\] and Theorem \[thm:MainB\]. We hope that this will help the reader understand the character of the argument—in particular, how the induction used in the proof works. We are going to consider the following theorem.
\[thm:A\] Let $U$ be an arbitrary unitary operator on a Hilbert space $\Hil$, and let ${m_{\bullet}}, {n_{\bullet}}\colon \mathcal{F} \to {\mathbb{N}}$ be any two IP-sets. If $p \in \beta
\mathcal{F}$ is any uncongested idempotent, the operator $P$ defined by the weak operator limit $$P = {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{\alpha} U^{m_{\alpha} n_{\alpha}}$$ is an orthogonal projection.
The given (weak operator) limit abbreviates the equality $${\bigl\langle P x, y \bigr\rangle} = {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{\alpha} {\bigl\langle U^{m_{\alpha} n_{\alpha}} x, y \bigr\rangle}$$ for all $x, y \in \Hil$.
We are not going to prove this directly, because the presence of the two IP-sets is inconvenient, in that it obscures part of the underlying structure. Suppose, for example, that the two IP-sets were (more or less) equal; then the essentially two-dimensional situation of the theorem would collapse down to a one-dimensional one, and surely something in the proof will have to change, too. The problem, in other words, is that there appears to be a notion of *dimension* behind the theorem—but it is cumbersome to deal with dimension for IP-sets.
On the other hand, as shown by Lemma \[lem:Dimension\], there is a good definition of dimension for idempotent ultrafilters in $\beta{{\mathbb{Z}}^{2}}$. Instead of trying to prove Theorem \[thm:A\] in its present form, we should pass instead to the group ${{\mathbb{Z}}^{2}}$, where we can talk about the rank of subgroups and the dimension of ultrafilters.
To this end, define a map $\phi \colon \mathcal{F} \to {{\mathbb{Z}}^{2}}$ by $$\begin{aligned}
\phi(\alpha) = \bigl( m_{\alpha}, n_{\alpha} \bigr)
&& \text{(for $\alpha \in \mathcal{F}$).}\end{aligned}$$ Because $p$ is uncongested, the new ultrafilter $q = {\phi_{\ast}}(p)$ is an idempotent in $\beta {{\mathbb{Z}}^{2}}$ by Lemma \[lem:uncongestedidp\]; moreover, Lemma \[lem:phistar\] changes the limit defining $P$ into $$P = {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} U^{z_1 z_2},$$ where the notation ${\vec{z}_{\thinspace}}$ is again used for elements of ${{\mathbb{Z}}^{2}}$. The following more general statement now suggests itself.
\[thm:B\] Let $U$ be a unitary operator on a Hilbert space $\Hil$. If $q \in \beta {{\mathbb{Z}}^{2}}$ is any idempotent, the operator $P$ defined by the weak operator limit $$P = {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} U^{z_1 z_2}$$ is an orthogonal projection.
Even in this special case, a proof seems to require two separate steps. We begin by introducing an auxiliary operator $$Q = {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} U^{a_1 z_2 + a_2 z_1},$$ the polynomial in the exponent arising from the original $z_1 z_2$ as $$(a_1 + z_1)(a_2 + z_2) - z_1 z_2 - a_1 a_2.$$
Step 1 {#step-1 .unnumbered}
------
For the time being, we are going to assume that $Q$ is a projection operator, and use the splitting of the Hilbert space $\Hil = \ker Q \oplus \im Q$ it induces to prove Theorem \[thm:B\]. To show that $P$ is an orthogonal projection, we appeal to Lemma \[lem:Projections\]: $P$ is clearly normal, being a limit of unitary operators, and so all we need to do is prove the relation $P^2 = P$. To help with that, let us also create, for each ${\vec{a}_{}}\in {{\mathbb{Z}}^{2}}$, the operator $$Q_{{\vec{a}_{}}} = {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} U^{a_1 z_2 + a_2 z_1}.$$ Since the polynomials involved are linear in ${\vec{z}_{\thinspace}}$, the reader will prove without much effort that each $Q_{{\vec{a}_{}}}$ is an orthogonal projection; for example, one can use the identity in to show that $Q_{{\vec{a}_{}}}^2 = Q_{{\vec{a}_{}}}$, and then apply Lemma \[lem:Projections\]. Of course, any two of those operators commute, since they are all limits of powers of $U$.
The point is that under our assumption on $Q$, the weak operator limits $$Q = {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} Q_{{\vec{a}_{}}} = {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} U^{a_1 z_2 + a_2 z_1}$$ are actually strong ones, as we shall see. To prove the identity $P^2 = P$, let us first consider the situation on the space $\ker Q$. If $x$ satisfies $Q x = 0$, we get $${\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} {\bigl\| Q_{{\vec{a}_{}}} x \bigr\|}^2 = {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} {\bigl\langle Q_{{\vec{a}_{}}} x, x \bigr\rangle} =
{\bigl\langle Q x, x \bigr\rangle} = 0,$$ and because we have convergence in the norm, we can apply van der Corput’s trick to show $P x = 0$. The condition in Lemma \[lem:VanDerCorput\], $$\begin{aligned}
{\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} {\Bigl\langle U^{(a_1 + z_1)(a_2 + z_2)} x, U^{z_1 z_2} x \Bigr\rangle}
&= {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} {\Bigl\langle U^{a_1 z_2 + a_2 z_1} x, U^{-a_1 a_2} x \Bigr\rangle} \\
&= {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} {\Bigl\langle Q_{{\vec{a}_{}}} x, U^{-a_1 a_2} x \Bigr\rangle} = 0,\end{aligned}$$ is satisfied, and we conclude that $P x = 0$, hence $P^2 x = P x$.
Next, let us see what happens if $x \in \im Q$. In this case, $Q x = x$, and we can write $$\begin{aligned}
{\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} &{\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} {\bigl\| U^{a_2 z_2 + a_2 z_1} x - x \bigr\|}^2 \\
&= 2 {\bigl\| x \bigr\|}^2 - 2 \cdot \Re {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} {\bigl\langle U^{a_1 z_2 + a_2
z_1} x, x \bigr\rangle}
= 2 {\bigl\| x \bigr\|}^2 - 2 {\bigl\langle Qx, x \bigr\rangle} = 0, \end{aligned}$$ from which it follows that $$\label{eq:no1}
{\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} {\bigl\| U^{(a_1 + z_1)(a_2 + z_2)} x - U^{a_1 a_2} U^{z_1
z_2} x \bigr\|} = 0.$$ To obtain $P^2 x = P x$, we make use of the identity in for double $q$-limits; together with , we obtain $$\begin{aligned}
P x &= {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} U^{z_1 z_2} x = {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} U^{(a_1 + z_1)(a_2
+ z_2)} x \\
&= {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} U^{a_1 a_2} U^{z_1 z_2} x = P^2 x. \end{aligned}$$
We are therefore able to show that $P$ is an orthogonal projection, provided that $Q$ is one. The device of getting strong from weak convergence is frequently useful, by the way; it is formalized in Lemma \[lem:StrongConvergence\] below.
Step 2 {#step-2 .unnumbered}
------
So far, we have been able to reduce Theorem \[thm:B\] to the proof of the following, simpler result.
\[thm:C\] Let $U$ be a unitary operator on a Hilbert space $\Hil$. For any idempotent $q
\in \beta {{\mathbb{Z}}^{2}}$, the operator $$Q = {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} U^{a_1 z_2 + a_2 z_1}$$ is an orthogonal projection.
Just as in Step 1, everything hinges on having a good splitting of the underlying Hilbert space $\Hil$. But which splitting one should use depends on the ultrafilter $q$, more precisely on its dimension—which could be 0, 1, or 2. We will treat these as separate cases here; in the proof of the main theorem, we shall of course want a unified approach.
### Dimension 0 {#dimension-0 .unnumbered}
If $q$ is 0-dimensional, it contains the set $\bigl\{(0,0)\bigr\}$, and since we can restrict to a $q$-big set when taking limits, $Q$ is simply the identity operator. So this case is trivial.
### Dimension 1 {#dimension-1 .unnumbered}
In case $\dim q = 1$, we can find a subgroup ${\mathbb{Z}}{\vec{c}_{}}$ (with ${\vec{c}_{}}\neq 0$) of rank one in $q$. Accordingly, we will use the splitting $\Hil
= \Hil_1 \oplus \Hil_1^{\bot}$, where $$\Hil_1 = \bigcap_{n \neq 0} \ker Q_{n {\vec{c}_{}}}
\qquad \text{and} \qquad
\Hil_1^{\bot} = \overline{\sum_{n \neq 0} \im Q_{n {\vec{c}_{}}}}.$$ It is then straightforward to show that $Q$ is orthogonal projection onto $\Hil_1^{\bot}$.
Indeed, if $x$ is an element of $\Hil_1$, then $Q_{n {\vec{c}_{}}} x = 0$ holds for all nonzero $n$, and since ${\vec{c}_{}}$ generates a $q$-big subgroup and $\bigl\{(0,0)\bigr\}$ is not $q$-big, we get $$Q x = {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} Q_{{\vec{a}_{}}} x = 0$$ On the other hand, to show that $Q$ restricted to $\Hil_1^{\bot}$ is the identity, we need only consider $x \in \im Q_{n {\vec{c}_{}}}$, as the span of these vectors is dense. For any such $x$, we have $$Q_{n {\vec{c}_{}}} x = {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} U^{n(c_1 z_2 + c_2 z_1)} x = x,$$ which, as before, can be strengthened to $$\label{eq:no2}
{\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} {\bigl\| U^{n(c_1 z_2 + c_2 z_1)} x - x \bigr\|} = 0.$$ Now we need to extend this equality, true for only one vector $n {\vec{c}_{}}$, to some $q$-big set of vectors. We leave it for the reader to check that actually gives $${\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} {\bigl\| U^{N n(c_1 z_2 + c_2 z_1)} x - x \bigr\|} = 0.$$ for any $N \in {\mathbb{Z}}$. (Hint: Use a telescoping sum.) But the set ${\mathbb{Z}}\cdot n {\vec{c}_{}}$ is again a $q$-big subgroup (since $q$ is idempotent), and so $${\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} {\bigl\| U^{a_1 z_2 + a_2 z_1} x - x \bigr\|} = 0,$$ which implies $Q x = x$. So $Q$ is indeed an orthogonal projection, with image $\Hil_1^{\bot}$.
### Dimension 2 {#dimension-2 .unnumbered}
Finally, let us treat the really interesting case of a two-dimensional $q$. We use the same argument as before, only the splitting has to be adjusted a bit; instead of focusing on one specific subgroup (like ${\mathbb{Z}}{\vec{c}_{}}$), we shall consider all of them. So let $$\Hil_2 = \bigcap_{{\vec{a}_{}}, {\vec{b}_{}}} \ker Q_{{\vec{a}_{}}} Q_{{\vec{b}_{}}}
\qquad \text{and} \qquad
\Hil_2^{\bot} = \overline{\sum_{{\vec{a}_{}}, {\vec{b}_{}}} \im Q_{{\vec{a}_{}}} Q_{{\vec{b}_{}}}}$$ be the two complementary subspaces, where both the intersection and the sum are taken over those ${\vec{a}_{}}, {\vec{b}_{}}\in {{\mathbb{Z}}^{2}}$ for which the subgroup ${\mathbb{Z}}{\vec{a}_{}}+ {\mathbb{Z}}{\vec{b}_{}}$ has rank two. Again, it will turn out that $Q$ is orthogonal projection onto $\Hil_2^{\bot}$.
To prove that $Q$ fixes every vector in $\Hil_2^{\bot}$, we may again limit our attention to elements $x \in \im Q_{{\vec{a}_{}}} Q_{{\vec{b}_{}}}$ for two vectors ${\vec{a}_{}}$ and ${\vec{b}_{}}$ that span a rank two subgroup. The same argument as before shows that $${\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} {\bigl\| U^{M (a_1 z_2 + a_2 z_1) + N (b_1 z_2 + b_2 z_1)} x - x \bigr\|} = 0$$ for any $M, N \in {\mathbb{Z}}$; the group generated by ${\vec{a}_{}}$ and ${\vec{b}_{}}$ is $q$-big (remember that it contains some lattice), and so we have $${\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} {\bigl\| U^{c_1 z_2 + c_2 z_1} x - x \bigr\|} = 0$$ for $q$-many ${\vec{c}_{}}\in {{\mathbb{Z}}^{2}}$. Taking the $q$-limit over ${\vec{c}_{}}$ then gives the result, namely that $Q x = x$.
To finish the proof, we have to deal with an arbitrary $x \in \Hil_2$ and show that $Q x = 0$. What we know is that $Q_{{\vec{a}_{}}} Q_{{\vec{b}_{}}} x = 0$ for any two vectors ${\vec{a}_{}}$ and ${\vec{b}_{}}$ with a two-dimensional span. This is a lot of information, since there are many such pairs—in fact, for any nonzero ${\vec{a}_{}}\in {{\mathbb{Z}}^{2}}$, a $q$-big set of ${\vec{b}_{}}$ has the required property. For suppose, to the contrary, that $q$-many vectors ${\vec{b}_{}}$ could span only a subgroup of rank one together with ${\vec{a}_{}}$. As ${\mathbb{Z}}{\vec{a}_{}}+ {\mathbb{Z}}{\vec{b}_{}}$ is of rank one if and only if ${\vec{b}_{}}$ is a multiple of ${\vec{a}_{}}/g$ (here $g$ is the greatest common divisor of the components of ${\vec{a}_{}}$), it would follow that ${\mathbb{Z}}{\vec{a}_{}}/g$ was a $q$-big subgroup of ${{\mathbb{Z}}^{2}}$, contradicting our assumption on the dimension of $q$.
In particular, we know $Q_{{\vec{a}_{}}} Q_{{\vec{b}_{}}} x = 0$ for sufficiently many ${\vec{a}_{}}$ and ${\vec{b}_{}}$ to conclude that $$Q^2 x = {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{b}_{}}} Q_{{\vec{a}_{}}} Q_{{\vec{b}_{}}} x = 0;$$ but now the operator $Q$ is very evidently self-adjoint and so $Q x = 0$ as well. This shows that $Q$ is a projection and ends the proof of Theorem \[thm:C\].
Conclusions {#conclusions .unnumbered}
-----------
Let us end this section with several remarks concerning the nature of the proof. Firstly, the reader will have observed the balance—crude in the case of one-dimensional $p$, slightly more subtle for two dimensions—between the two spaces of the splitting. In the first space, $\Hil_1$ or $\Hil_2$, where we use the null spaces of projections, we need to intersect a large number of them to make up for the weakness of each individual piece; for each operator, we only know that one particular $p$-limit is zero, and that amounts to nothing by itself. On the other hand, the orthogonal complements, $\Hil_1^{\bot}$ or $\Hil_2^{\bot}$, involve image spaces of projections; the knowledge that we gain from each piece is far stronger here, and so we can afford to have this knowledge in only one case.
Secondly, it is clear that the dimension of the ultrafilter is important. It was pointed out before that, although the same concept is lurking around in Theorem \[thm:A\], it is less easily quantified and dealt with there. The passage from $\beta\mathcal{F}$ to $\beta{{\mathbb{Z}}^{2}}$ helps to make it visible, by removing the IP-sets. Moreover, it is of course unnecessary to handle the various dimensions by different arguments; the proof is really the same in all cases. Indeed, in Section \[sec:V\], when proving the main theorem, the first step will be to adjust the dimension of the surrounding group to make it match that of the ultrafilter $p$. This is where Lemma \[lem:Dimension\] will play its part.
Finally, the more general result in the main theorem requires more effort to prove; although the proof is, in essence, the same as the one given here, there are several technical points that need to be dealt with. In particular, the presence of polynomials of higher degree needs special care. The following two sections contain a few tools that will be helpful; all necessary results about polynomials are collected in Section \[sec:IV\].
Orthogonal projections and limits {#sec:III}
=================================
In this section, we prove two simple but useful results about orthogonal projections and limits; these are well-known, of course. The first, which has already been used, gives a condition for an operator to be a projection.
\[lem:Projections\] A normal operator $P$ on a Hilbert space $\Hil$ is an orthogonal projection if, and only if, it satisfies $P^2 = P$.
Necessity is clear. If $P$ meets the condition, the product $Q = P^{\ast} P$ of $P$ and its adjoint also does. The latter is self-adjoint in addition, hence satisfies ${\bigl\langle Q x, x - Q x \bigr\rangle} = {\bigl\langle x, Q (x - Q x) \bigr\rangle} = 0$ and is therefore an orthogonal projection onto the image space of $Q$. For $x \in \ker Q$, one has ${\bigl\| P x \bigr\|}^2 = {\bigl\langle x, Q x \bigr\rangle} = 0$; for $x \in \im Q$, one has $P x = P Q x =
P P^{\ast} P x = P^{\ast} P x = x$. Consequently, $P = Q$ is an orthogonal projection.
Our second lemma deals with the question of when certain ‘weak’ limits in a Hilbert space $\Hil$ are ‘strong’ limits and is meant to collect the pieces of reasoning used in the previous section. The whole discussion is somewhat vague but the result is useful, though nearly self-evident. Let ‘$\lim$’ be an abbreviation for some unspecified $p$-limit, maybe even a multiple one, and let $I$ be the corresponding index set. So for example, $\lim$ might equal ${\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{b}_{}}}$, with both ${\vec{a}_{}}$ and ${\vec{b}_{}}$ ranging over ${{\mathbb{Z}}^{2}}$, in which case the index set $I$ would be ${{\mathbb{Z}}^{2}} \times {{\mathbb{Z}}^{2}}$.
By what we said in Section \[sec:I\], the limit $\lim x_i$ is defined, in the weak topology, for every bounded family ${( x_i )_{i \in I}}$ of points, as any closed ball in $\Hil$ is weakly compact. $\lim x_i = x$ thus means that for any $y \in \Hil$, $$\lim {\bigl\langle x_i, y \bigr\rangle} = {\bigl\langle x, y \bigr\rangle}.$$ On the other hand, the convergence is called *strong* if $$\lim {\bigl\| x_i - x \bigr\|} = 0.$$ The norm is weakly lower semi-continuous—if $x = \lim x_i$, then $${\bigl\| x \bigr\|} \le \lim {\bigl\| x_i \bigr\|}.$$
One can also define the notions of weak and strong *operator limits*; in fact, we have already been using these. We say, for example, that $T$ is the weak operator limit of a family ${( T_i )_{i \in I}}$ of operators—and write $T = \lim T_i$—if $$\begin{aligned}
\lim {\bigl\langle T_i x, y \bigr\rangle} = {\bigl\langle T x, y \bigr\rangle} && \text{(for $x, y \in \Hil$)}.\end{aligned}$$ A few simple calculations then give the following result.
\[lem:StrongConvergence\] Let $\lim$ and $I$ be defined as above.
1. If $T = \lim U_i$ is the weak operator limit of a family ${( U_i )_{i \in I}}$ of unitary operators, then $T$ is normal. For $x \in \Hil$, one has $\lim
{\bigl\| T x - U_i x \bigr\|} = 0$ if, and only if, ${\bigl\| T x \bigr\|} = {\bigl\| x \bigr\|}$. In case $T$ is an orthogonal projection, this happens exactly when $T x = x$.
2. If $T = \lim P_i$ is the weak operator limit of a family ${( P_i )_{i \in I}}$ of orthogonal projections, then $T$ is self-adjoint. For $x \in \Hil$, one has $\lim {\bigl\| T x - P_i x \bigr\|} = 0$ if, and only if, ${\bigl\langle T x, x - T x \bigr\rangle} = 0$. In case $T$ is itself an orthogonal projection, this condition is always satisfied.
The result looks innocent enough, but it will be used frequently.
Polynomials {#sec:IV}
===========
We shall be using polynomials in several variables for which the following notation seems appropriate. Lower-case Roman letters with arrows will usually denote $n$-dimensional vectors, e.g. ${\vec{z}_{\thinspace}}= (z_1, \dotsc, z_n)$. We shall be speaking of polynomials *in the variable ${\vec{z}_{\thinspace}}$*, meaning really polynomials in the $n$ variables $z_1, \dotsc, z_n$. The *degree* of such a polynomial will be its total degree. We shall also consider polynomials in several multi-dimensional variables: $f({\vec{a}_{}}, {\vec{c}_{}})$, say, would be a polynomial in both sets of variables; the *degree in ${\vec{a}_{}}$* is the total degree of $f$ as a polynomial in $a_1, \dotsc, a_n$, and so on.
If $G$ is any Abelian group, we shall let ${G \lbrack {\vec{z}_{1}}, \dotsc, {\vec{z}_{s}} \rbrack}$ stand for the additive group of polynomials in ${\vec{z}_{1}}, \dotsc, {\vec{z}_{s}}$ with coefficients in $G$; we shall ignore the multiplicative structure. For the subgroup of those polynomials in ${{\mathbb{Q}}\lbrack {\vec{z}_{1}}, \dotsc, {\vec{z}_{r}} \rbrack}$ that produce integer values for integer arguments, we shall write ${{\mathrm{Int} \lbrack {\vec{z}_{1}}, \dotsc, {\vec{z}_{s}} \rbrack}}$.
In the one-dimensional case, ${{\mathrm{Int} \lbrack x \rbrack}}$ consists of all polynomials $f \in
{{\mathbb{Q}}\lbrack x \rbrack}$ with $f({\mathbb{Z}}) \subseteq {\mathbb{Z}}$. It is a free group with basis consisting of the polynomials $$\begin{aligned}
\binom{x}{i} = \frac{x (x-1) \dotsm (x - i + 1)}{i!} && \text{(for $i \ge 0$).} \end{aligned}$$ Indeed, if $f$ is any polynomial in ${{\mathrm{Int} \lbrack x \rbrack}}$ and $m$ its degree, one may find $m + 1$ integers $a_0, \dotsc, a_m$ such that $$f(x) = \sum_{i = 0}^m a_i \binom{x}{i},$$ by evaluating successively at $x = 0, 1, \dotsc, m$, and solving the resulting system of equations. For any number $d \ge 0$, the polynomials of degree at most $d$ form a free subgroup of rank $d + 1$.
The same argument, applied inductively, proves the following.
\[lem:Int\] ${{\mathrm{Int} \lbrack {\vec{z}_{1}}, \dotsc, {\vec{z}_{s}} \rbrack}}$ is always a free group; for any integer $d \ge 0$, the polynomials of total degree at most $d$ constitute a free subgroup of finite rank, and so do the polynomials of degree at most $d$ in each variable.
We now introduce one more useful notion. In the example in Section \[sec:II\], when dealing with the polynomial $f(x, y) = xy$, we found it useful to form the new polynomial $$f(a + x, b + y) - f(a, b) - f(x, y),$$ essentially because its degree in $(x, y)$ was lower. An appropriate generalization is as follows. Given a polynomial $f({\vec{z}_{\thinspace}}) \in {G \lbrack {\vec{z}_{\thinspace}}\rbrack}$ and an integer $s \ge 1$, we recursively define a new polynomial ${\Delta \negthinspace ^{s} f({\vec{z}_{1}}, \dotsc,
{\vec{z}_{s}})}$, by letting ${\Delta \negthinspace ^{1} f({\vec{z}_{1}})} = f({\vec{z}_{1}})$, and $$\begin{gathered}
\label{eq:DeltaRecursively}
{\Delta \negthinspace ^{s+1} f({\vec{z}_{1}}, \dotsc, {\vec{z}_{s+1}})} =\\
{\Delta \negthinspace ^{s} f({\vec{z}_{1}}, \dotsc, {\vec{z}_{s}} + {\vec{z}_{s+1}})} - {\Delta \negthinspace ^{s} f({\vec{z}_{1}}, \dotsc, {\vec{z}_{s}})}
- {\Delta \negthinspace ^{s} f({\vec{z}_{1}}, \dotsc, {\vec{z}_{s+1}})}.\end{gathered}$$ Of course, ${\Delta \negthinspace ^{s} }$ can be described explicitly as $${\Delta \negthinspace ^{s} f({\vec{z}_{1}}, \dotsc, {\vec{z}_{s}})} = \sum_{\emptyset \neq \alpha \subseteq
\{1, \dotsc, s \}} (-1)^{s - {\left| \alpha \right|}} \cdot f \bigl( \sum_{i \in \alpha} {\vec{z}_{i}}
\bigr),$$ and the symmetry in all arguments is more apparent from this description.
Let us investigate some properties of ${\Delta \negthinspace ^{s} }$. First, we have the following easy lemma.
\[lem:Degree\] The polynomial ${\Delta \negthinspace ^{2} f({\vec{a}_{}}, {\vec{z}_{\thinspace}})} = f({\vec{a}_{}}+ {\vec{z}_{\thinspace}}) - f({\vec{a}_{}}) - f({\vec{z}_{\thinspace}})$ is of lower degree in each variable than $f({\vec{z}_{\thinspace}})$ itself, whenever $f({\vec{z}_{\thinspace}}) \in {G \lbrack {\vec{z}_{\thinspace}}\rbrack}$ is nonzero.
Now let $f({\vec{z}_{\thinspace}})$ be of degree $d \ge 1$. Since ${\Delta \negthinspace ^{s} f({\vec{z}_{1}}, \dotsc, {\vec{z}_{s}})}$ is symmetric in its $s$ arguments, the lemma—together with the relations —immediately shows that its degree in any variable can be at most $(d + 1 - s)$. It follows that ${\Delta \negthinspace ^{d+1} f({\vec{z}_{1}}, \dotsc, {\vec{z}_{d+1}})}$ is a constant, with value $${\Delta \negthinspace ^{d+1} f(0, \dotsc, 0)} = \sum_{k=1}^{d+1} (-1)^{d+1-k} \binom{d+1}{k}
f(0) = (-1)^d f(0).$$
If $f$ happens to satisfy $f(0) = 0$, one has ${\Delta \negthinspace ^{d+1} f({\vec{z}_{1}}, \dotsc, {\vec{z}_{d+1}})} = 0$. For reasons of symmetry, ${\Delta \negthinspace ^{d} f({\vec{z}_{1}}, \dotsc, {\vec{z}_{d}})}$ is then linear in each of its $d$ arguments.
We have shown the following.
\[lem:Delta\] For any polynomial $f({\vec{z}_{\thinspace}}) \in {G \lbrack {\vec{z}_{\thinspace}}\rbrack}$ of degree $d \ge 1$, one has the relation $${\Delta \negthinspace ^{d+1} f({\vec{z}_{1}}, \dotsc, {\vec{z}_{d+1}})} = (-1)^d f(0).$$ If $f(0) = 0$, then ${\Delta \negthinspace ^{d} f({\vec{z}_{1}}, \dotsc, {\vec{z}_{d}})}$ is a linear function of each argument.
A third lemma deals with the case of homogeneous $f$.
\[lem:Homogeneous\] Let $f({\vec{z}_{\thinspace}}) \in {G \lbrack {\vec{z}_{\thinspace}}\rbrack}$ be a homogeneous polynomial of degree $d \ge 1$. Then $${\Delta \negthinspace ^{d} f({\vec{a}_{}}, \dotsc, {\vec{a}_{}})} = d! f({\vec{a}_{}}).$$
Using homogeneity, we have $${\Delta \negthinspace ^{s} f({\vec{a}_{}}, \dotsc, {\vec{a}_{}})} = \sum_{k=1}^s (-1)^{s-k} \binom{s}{k}
f(k {\vec{a}_{}}) = \sum_{k=1}^s (-1)^{s-k} \binom{s}{k} k^d \cdot f({\vec{a}_{}}).$$ We obviously have to evaluate sums of the form $$\begin{aligned}
C(s,m) = \sum_{k=1}^s (-1)^{s-k} \binom{s}{k} k^m
&& \text{(for $s \ge 1, m \ge 0$);}\end{aligned}$$ in particular, $C(d,d) = d!$ is what we need to show. From the previous lemma, we already know that $C(s,m) = 0$ whenever $s > m$. Now we compute $$\begin{aligned}
C(m+1,m+1) &= \sum_{k=1}^{m+1} (-1)^{m+1-k} \binom{m+1}{k} k^{m+1} \\
&= \sum_{k=1}^{m+1} (-1)^{m-(k-1)} (m+1) \binom{m}{k-1} k^m \\
&= (m+1) \sum_{l=0}^{m} (-1)^{m-l} \binom{m}{l} (l+1)^m \\
&= (m+1) \sum_{l=0}^{m} (-1)^{m-l} \binom{m}{l} \Bigl( 1 + \sum_{i=1}^m
\binom{m}{i} l^i \Bigr) \\
&= (m+1) \sum_{l=0}^{m} (-1)^{m-l} \binom{m}{l} + (m+1) \sum_{i=1}^m
\binom{m}{i} C(m,i) \\
&= (1-1)^m + (m+1) C(m,m) = (m+1) C(m,m),\end{aligned}$$ and together with $C(1,1) = 1$ this proves the lemma by induction.
We will now use the previous results to establish an important technical lemma; it is essential for the proof of the main theorem in Section \[sec:V\]. Note that it introduces a feature not present in the example of Section \[sec:II\], where we had to deal with polynomials of no more than first degree. It does, however, fit in with the general philosophy behind the argument—there is one situation in the proof where one has to make a lot from apparently nothing, meaning where one has to create useful $p$-big sets from useless ones, and the following lemma does just that.
\[lem:Key\] Suppose that $p \in \beta{{\mathbb{Z}}^{n}}$ is an $n$-dimensional idempotent. Let $G$ be an Abelian group and let $v({\vec{z}_{\thinspace}}) \in {G \lbrack {\vec{z}_{\thinspace}}\, \rbrack}$ be a polynomial in ${\vec{z}_{\thinspace}}= (z_1, \dotsc, z_n)$. Fix a subgroup $V \subseteq G$. If the set $$B = {\bigl\{{\vec{b}_{}}\in {{\mathbb{Z}}^{n}}\thinspace\big\vert\thinspace \text{$N \cdot v({\vec{b}_{}}) \in V$ for some $N \ne 0$} \bigr\}}$$ is $p$-big, then so is the set $$A = {\bigl\{{\vec{a}_{}}\in {{\mathbb{Z}}^{n}}\thinspace\big\vert\thinspace v({\vec{a}_{}})-v(0) \in V \bigr\}}.$$
The idea of the proof is simple: Whenever $B$ contains ${\operatorname{IP}\bigl( {\vec{b}_{1}}, \dotsc,
{\vec{b}_{s}} \bigr)}$, there is some integer $N \neq 0$ such that $N \cdot {\Delta \negthinspace ^{s} v({\vec{b}_{1}}, \dotsc,
{\vec{b}_{s}})} \in V$. Using this and the previous results, we can extract from $v$ its homogeneous parts of different degrees, and show that they are each contained in $V$ for $p$-many ${\vec{z}_{\thinspace}}$. Proceeding stepwise, we shall prove two things:
1. Without loss of generality, it may be assumed that $v(0) = 0$.
2. The set $A = {\bigl\{{\vec{a}_{}}\thinspace\big\vert\thinspace v({\vec{a}_{}}) \in V \bigr\}}$ is $p$-big.
The details are as follows.
Step I {#step-i .unnumbered}
------
Let $d$ be the degree of $v({\vec{z}_{\thinspace}})$. The set $B$ is a member of the idempotent $p$, and by Lemma \[lem:FPsets\], we may select $d + 1$ elements ${\vec{b}_{1}}, \dotsc,
{\vec{b}_{d+1}}$ in $B$ with $$F = {\operatorname{IP}\bigl( {\vec{b}_{1}}, \dotsc, {\vec{b}_{d+1}} \bigr)} \subseteq B.$$ For each ${\vec{b}_{}}$ in this finite $\mathrm{IP}$-set, there exists some $N \neq 0$ such that $N \cdot v({\vec{b}_{}}) \in V$; if we let $N_1$ be the greatest common divisor of these numbers, we guarantee that $N_1 \cdot v({\vec{b}_{}}) \in V$ for every ${\vec{b}_{}}\in F$. By Lemma \[lem:Delta\], we now have $$N_1 \cdot v(0) = N_1 \cdot (-1)^d {\Delta \negthinspace ^{d+1} v({\vec{b}_{1}}, \dotsc, {\vec{b}_{d+1}})} \in V.$$ If ${\vec{b}_{}}$ is any element of $B$ and $N \cdot v({\vec{b}_{}}) \in V$, the expression $N_1 N \cdot \bigl( v({\vec{b}_{}}) - v(0) \bigr)$ is also an element of $V$; this means that the set $${\bigl\{{\vec{b}_{}}\thinspace\big\vert\thinspace \text{$N \cdot \bigl( v({\vec{b}_{}}) - v(0) \bigr)\in V$ for some $N \ne 0$} \bigr\}}$$ is equally $p$-big. We may therefore replace $v({\vec{z}_{\thinspace}})$ by $v({\vec{z}_{\thinspace}}) - v(0)$ and assume $v(0) = 0$.
Step II {#step-ii .unnumbered}
-------
We decompose $v$ into homogenous polynomials, $$v({\vec{z}_{\thinspace}}) = \sum_{i=1}^d h_i({\vec{z}_{\thinspace}}),$$ say, with $h_i({\vec{z}_{\thinspace}})$ homogeneous of degree $i$. For $1 \le i \le d$, let $$A_i = {\bigl\{{\vec{a}_{}}\thinspace\big\vert\thinspace h_i({\vec{a}_{}}) \in V \bigr\}}.$$ We shall argue that $A_d \in p$; once this is known, the same reasoning applies to the polynomial $v({\vec{z}_{\thinspace}}) - h_d({\vec{z}_{\thinspace}})$ where it gives $A_{d-1} \in p$, and so on, until one has $A_i \in p$ for each $i$. The result follows because $A$ contains the intersection of all $A_i$.
To show that $A_d$ is $p$-big, we use a similar—but more careful—approach as before. By Lemma \[lem:FPsets\], there are actually $p$-many ${\vec{b}_{1}}, \dotsc, {\vec{b}_{d}}
\in B$ with ${\operatorname{IP}\bigl( {\vec{b}_{1}}, \dotsc, {\vec{b}_{d}} \bigr)} \subseteq B$, and again, to each choice we may find some $N_1 \ne 0$ such that $N_1 \cdot {\Delta \negthinspace ^{d} v({\vec{b}_{1}}, \dotsc, {\vec{b}_{d}})}$ is an element of $V$. When selecting only ${\vec{b}_{1}}, \dotsc, {\vec{b}_{d-1}}$ from these, the set $$B_d = {\bigl\{{\vec{b}_{}}\thinspace\big\vert\thinspace \text{$N \cdot {\Delta \negthinspace ^{d} v({\vec{b}_{1}}, \dotsc, {\vec{b}_{d-1}}, {\vec{b}_{}})} \in V$ for
some $N \ne 0$} \bigr\}}$$ is then always $p$-big by construction.
$B_d$ is also a subgroup of ${{\mathbb{Z}}^{n}}$, the polynomial ${\Delta \negthinspace ^{d} v({\vec{z}_{1}}, \dotsc, {\vec{z}_{d}})}$ being linear in each variable (see Lemma \[lem:Delta\]). Since $p$ is $n$-dimensional, this subgroup has to have rank $n$ and has to contain a set of the form $L \cdot {{\mathbb{Z}}^{n}}$ for some nonzero $L$. For each ${\vec{b}_{}}\in {{\mathbb{Z}}^{n}}$, one gets $$N L \cdot {\Delta \negthinspace ^{d} v({\vec{b}_{1}}, \dotsc, {\vec{b}_{d-1}}, {\vec{b}_{}})} = N \cdot {\Delta \negthinspace ^{d} v({\vec{b}_{1}},
\dotsc, {\vec{b}_{d-1}}, L \cdot {\vec{b}_{}})} \in V$$ for some $N \neq 0$, and so we conclude that $B_d = {{\mathbb{Z}}^{n}}$.
Next, consider the set $$B_{d-1} = {\bigl\{{\vec{b}_{}}\thinspace\big\vert\thinspace \text{for some $N \neq 0$, $N \cdot {\Delta \negthinspace ^{d} v({\vec{b}_{1}}, \dotsc,
{\vec{b}_{d-2}}, {\vec{b}_{}}, {\vec{a}_{d}})} \in V$ for all ${\vec{a}_{d}}$} \bigr\}};$$ by the above, it is $p$-big, and a repetition of the argument shows that $B_{d-1} =
{{\mathbb{Z}}^{n}}$, too. Continuing in this way, we eventually find a nonzero integer $N_2$ such that $$N_2 \cdot {\Delta \negthinspace ^{d} v({\vec{a}_{1}}, \dotsc, {\vec{a}_{d}})} \in V$$ for any choice of ${\vec{a}_{1}}, \dotsc, {\vec{a}_{d}} \in {{\mathbb{Z}}^{n}}$.
By Lemma \[lem:Degree\], we have $${\Delta \negthinspace ^{d} h_d({\vec{a}_{1}}, \dotsc, {\vec{a}_{d}})} = {\Delta \negthinspace ^{d} v({\vec{a}_{1}}, \dotsc, {\vec{a}_{d}})},$$ because all terms of degree less than $d$ disappear. Finally, using Lemma \[lem:Homogeneous\] we get $$N_2 d! \cdot h_d({\vec{a}_{}}) = N_2 \cdot {\Delta \negthinspace ^{d} v({\vec{a}_{}}, \dotsc, {\vec{a}_{}})} \in V$$ for all ${\vec{a}_{}}$, and hence $h_d({\vec{a}_{}}) \in V$ whenever ${\vec{a}_{}}\in N_2 d! \cdot {{\mathbb{Z}}^{n}}$. The latter set is $p$-big and so $A_d \in p$. This ends the proof of the second part, and establishes the lemma.
Statement and proof of the main results {#sec:V}
=======================================
After all the preliminary work in the previous two sections, we are now ready to state and prove the main result. The notation is somewhat heavy, but this generality is needed because of the inductive nature of the proof.
\[thm:MainA\] For $j=1, \dotsc, s$, let $p_j \in \beta{{\mathbb{Z}}^{n_j}}$ be an idempotent, and let $U_1, \dotsc, U_m$ be commuting unitary operators on a Hilbert space $\Hil$. Given any $m$ polynomials $f_1, \dotsc, f_m \in {{\mathrm{Int} \lbrack {\vec{z}_{1}},
\dots,{\vec{z}_{s}} \rbrack}}$—with ${\vec{z}_{j}}$ of dimension $n_j$—satisfying $f_i(0) = 0$ for all $i = 1, \dotsc, m$, define an operator $P$ on $\Hil$ by $$P = {\mathop{p_1\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{1}}} \cdots {\mathop{p_s\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{s}}} {\prod_{i=1}^{m} U_i^{f_i({\vec{z}_{1}}, \dotsc, {\vec{z}_{s}})}}.$$ Then $P$ is always an orthogonal projection. Any two operators defined in this way commute.
We will suppose that all $n_j$ are equal to some $n$ and all $p_j$ are equal to some $p$, in order to simplify notation. The argument need not be changed in any way to accommodate the more general situation—because of the inductive character of the proof, we find ourselves working on no more than one $p$-limit at a time anyway.
Let us first observe that the last part of the statement—commutativity of different projections—is obviously true, for all operators generated for various selections of polynomials are certain weak limits of commuting unitary operators. We may therefore assume commutativity wherever needed.
The remainder of the proof is essentially by induction on the number $s$ of $p$-limits taken, but there are some complications involving the case $s=1$. In fact, different arguments are needed for $s = 1$ and for $s \ge 2$, since the outmost $p$-limit is one of unitary operators in the former situation, but one of projections in the latter. To make the induction be more transparent, we shall use the abbreviation $(s,d)$ when referring to the statement of the theorem for a certain value of $s$ and all possible choices of polynomials $f_i$ of degree at most $d$ in any of their variables ${\vec{z}_{j}}$.
The proof will be divided into several steps, the second and third inductive in nature:
1. We argue that the ultrafilter $p$ may be assumed to be $n$-dimensional, without loss of generality.
2. \[item:II\] We establish the case $(1,d)$, that is, we show that the operator $$P = {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} {\prod_{i=1}^{m} U_i^{f_i({\vec{z}_{\thinspace}})}},$$ with $f_i$ of degree at most $d$, is an orthogonal projection, assuming the statement of the theorem in the two cases $(1,d-1)$ and $(2,d-1)$. Specifically, we need to assume that the operators $$Q_{{\vec{a}_{}}} = {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} {\prod_{i=1}^{m} U_i^{f_i({\vec{a}_{}}+ {\vec{z}_{\thinspace}}) - f_i({\vec{a}_{}}) - f_i({\vec{z}_{\thinspace}})}}$$ and $$Q = {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} Q_{{\vec{a}_{}}}$$ are orthogonal projections.
3. \[item:III\] For $s \ge 2$, we derive $(s,d)$ from $(s-1,d)$ and $(1,d)$. Essentially, we introduce the new polynomials $$f_i'({\vec{a}_{}}, {\vec{z}_{2}}, \dotsc, {\vec{z}_{s}}) = f_i({\vec{a}_{}}, {\vec{z}_{2}}, \dotsc, {\vec{z}_{s}}) - f_i({\vec{a}_{}}, 0, \dotsc, 0)$$ and assume that for each ${\vec{a}_{}}\in \ZZ^n$, the operator $$P_{{\vec{a}_{}}} = {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{2}}} \cdots {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{s}}} {\prod_{i=1}^{m} U_i^{f_i'({\vec{a}_{}}, {\vec{z}_{2}},
\dotsc, {\vec{z}_{s}})}}$$ is an orthogonal projection. We then construct a suitable splitting of the underlying Hilbert space $\Hil$.
4. Using the splitting introduced in the previous part, we show that $P$ is an orthogonal projection.
Once we have established all of the previous, our work will be done. For the statement of the theorem is definitely true in the case $(1,0)$—if all $f_i$ equal zero, $P$ is just the identity—and then \[item:II\] and \[item:III\] suffice to prove the entire theorem by induction. Let us now take a detailed look at the four steps.
Step I {#step-i-1 .unnumbered}
------
Using Lemma \[lem:Dimension\], we begin by adjusting the situation to make sure that the dimension of $p$ is equal to the rank of the group. Of course, this will change the polynomials under consideration; but since their degrees are not increased, it does not affect the proof. To write this down precisely is somewhat cumbersome; so let us look at the case of just one operator $U$ and one polynomial $f({\vec{z}_{\thinspace}})$ to see what happens. We shall use ${\vec{z}_{\thinspace}}$ for an $n$-dimensional and ${\vec{w}_{}}$ for an $s$-dimensional variable. If $s = \dim p$ and $\phi$ are as in Lemma \[lem:Dimension\], then $q = {\phi_{\ast}}(p)$ is idempotent and $s$-dimensional. By Lemma \[lem:plimits\], $$\label{eq:Transformation}
{\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} U^{f({\vec{z}_{\thinspace}})} = {\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{w}_{}}} U^{f(\phi^{-1}({\vec{w}_{}}))}.$$ But $g({\vec{w}_{}}) = f(\phi^{-1}({\vec{w}_{}})) \in {{\mathrm{Int} \lbrack {\vec{w}_{}}\rbrack}}$ satisfies $g(0) =
0$, and is of degree no larger than that of $f$; thus a proof that the right-hand operator in is a projection gives the result for the left-hand one, too. The same is true in the general setting of the theorem, though somewhat unpleasant to write down in detail.
In any case, we shall assume from now on that $\dim p = n$. Lemma \[lem:Key\] is then applicable; it will make its entry in the third step of the proof.
Step II {#step-ii-1 .unnumbered}
-------
As stated above, we shall now assume that both $Q$ and all the $Q_{{\vec{a}_{}}}$ are orthogonal projections; this is permissible because each polynomial $$f_i({\vec{a}_{}}+ {\vec{z}_{\thinspace}}) - f_i({\vec{a}_{}}) - f_i({\vec{z}_{\thinspace}})$$ has degree at most $(d - 1)$ in ${\vec{a}_{}}$ and ${\vec{z}_{\thinspace}}$ (see Lemma \[lem:Degree\]). The projection $Q$ induces a splitting $\Hil = \ker Q \oplus \im Q$ of the underlying Hilbert space; we shall use it to conclude that $P^2 x = P x$ for all $x \in
\Hil$.
First, consider $x \in \ker Q$. Since $Q x = {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} Q_{{\vec{a}_{}}} x$, Lemma \[lem:StrongConvergence\] implies that the convergence is strong, $${\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} {\bigl\| Q_{{\vec{a}_{}}} x \bigr\|} = 0.$$ We now use van der Corput’s trick to get $P x = 0$, the condition $$\begin{aligned}
& {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} {\Bigl\langle {\prod_{i=1}^{m} U_i^{f_i({\vec{a}_{}}+ {\vec{z}_{\thinspace}})}} x, {\prod_{i=1}^{m} U_i^{f_i({\vec{z}_{\thinspace}})}} x \Bigr\rangle} \\
=& {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} {\Bigl\langle {\prod_{i=1}^{m} U_i^{f_i({\vec{a}_{}}+ {\vec{z}_{\thinspace}}) - f_i({\vec{a}_{}}) - f_i({\vec{z}_{\thinspace}})}} x, {\prod_{i=1}^{m} U_i^{-f_i({\vec{a}_{}})}} x \Bigr\rangle} \\
=& {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} {\Bigl\langle Q_{{\vec{a}_{}}} x, {\prod_{i=1}^{m} U_i^{-f_i({\vec{a}_{}})}} x \Bigr\rangle} = 0 \end{aligned}$$ in Lemma \[lem:VanDerCorput\] being fulfilled. A fortiori, $P^2 x = P x$.
Second, consider an arbitrary $x \in \im Q$, which then satisfies $Q x = x$. We again get strong convergence from Lemma \[lem:StrongConvergence\], so that $${\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} {\Bigl\| {\prod_{i=1}^{m} U_i^{f_i({\vec{a}_{}}+ {\vec{z}_{\thinspace}})}} x - {\prod_{i=1}^{m} U_i^{f_i({\vec{a}_{}}) + f_i({\vec{z}_{\thinspace}})}} x \Bigr\|} = 0.$$ But then $$\begin{aligned}
P^2 x &= {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} {\prod_{i=1}^{m} U_i^{f_i({\vec{a}_{}}) + f_i({\vec{z}_{\thinspace}})}} x
= {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} {\prod_{i=1}^{m} U_i^{f_i({\vec{a}_{}}+ {\vec{z}_{\thinspace}})}} x \\
&= {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} {\prod_{i=1}^{m} U_i^{f_i({\vec{z}_{\thinspace}})}} x = P x. \end{aligned}$$ We now have $P^2 = P$; obviously, $P$ is normal, and the result—that $P$ is an orthogonal projection—follows from Lemma \[lem:Projections\].
Step III {#step-iii .unnumbered}
--------
This is the most interesting part of the argument. We start from the inductive assumption that each $P_{{\vec{a}_{}}}$ is an orthogonal projection, and aim for a useful splitting of the space $\Hil$, depending on the projections $P_{{\vec{a}_{}}}$ and the polynomials $$f_i'({\vec{a}_{}}, {\vec{z}_{2}}, \dotsc, {\vec{z}_{s}}) = f_i({\vec{a}_{}}, {\vec{z}_{2}}, \dotsc, {\vec{z}_{s}})
- f_i({\vec{a}_{}}, 0, \dotsc, 0).$$ Note that $f_i'({\vec{a}_{}}, 0, \dotsc, 0) = 0$ for any ${\vec{a}_{}}$, which means that the new polynomials $f_i'$ still satisfy the conditions of the theorem, while having fewer variables.
After the introduction of $g_i({\vec{a}_{}}) = f_i({\vec{a}_{}}, 0, \dotsc, 0)$, the operator $P$ is then given by the limit $$P = {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{}}} \Bigl( {\prod_{i=1}^{m} U_i^{g_i({\vec{a}_{}})}} \Bigr) P_{{\vec{a}_{}}}.$$
We shall let $F \subseteq {\mathrm{Int} \lbrack {\vec{z}_{2}}, \dotsc, {\vec{z}_{s}} \rbrack}$ denote the set of polynomials of degree at most $d$ in each variable; $F$ is a free group of finite rank by Lemma \[lem:Int\]. The product $F^m$ is also free, as are all of its subgroups, and for any ${\vec{a}_{}}\in {{\mathbb{Z}}^{n}}$, the vector $$v({\vec{a}_{}}) = \bigl( \tilde{f}_1({\vec{a}_{}}, {\vec{z}_{2}}, \dotsc, {\vec{z}_{s}}), \dotsc, \tilde{f}_m({\vec{a}_{}}, {\vec{z}_{2}},
\dotsc, {\vec{z}_{s}}) \bigr)$$ is an element of $F^m$. We introduce the notation $V({\vec{a}_{1}}, \dotsc, {\vec{a}_{r}})$ for the subgroup of $F^m$ generated by the vectors $v({\vec{a}_{1}}), \dotsc, v({\vec{a}_{r}})$.
The *crucial idea* is to let $r \ge 0$ be the maximal integer for which $${\bigl\{{\vec{a}_{1}} \thinspace\big\vert\thinspace \cdots {\bigl\{{\vec{a}_{r}} \thinspace\big\vert\thinspace \text{$V({\vec{a}_{1}}, \dotsc, {\vec{a}_{r}})$ has rank $r$} \bigr\}}
\in p \cdots \bigr\}} \in p.$$ Such an $r$ has to exist, because we are working inside a fixed group of finite rank; if not even ${\bigl\{{\vec{a}_{1}} \thinspace\big\vert\thinspace \text{$V({\vec{a}_{1}})$ has rank one} \bigr\}}$ is in $p$, we set $r=0$ to keep the notation consistent. Whenever ${\vec{a}_{1}}, \dotsc, {\vec{a}_{r}}$ are taken, in the correct order, from these nested sets, the rank of the group $V({\vec{a}_{1}}, \dotsc,
{\vec{a}_{r}})$ is $r$.
With $r$ being defined in that manner, one also has $${\bigl\{{\vec{a}_{1}} \thinspace\big\vert\thinspace \cdots {\bigl\{{\vec{a}_{r+1}} \thinspace\big\vert\thinspace \text{$V({\vec{a}_{1}}, \dotsc, {\vec{a}_{r+1}})$ has rank less
than $r$} \bigr\}} \in p \cdots \bigr\}} \in p.$$ Intersecting with the previous set and using that $p$ is an ultrafilter, we obtain $$\begin{aligned}
{\bigl\{{\vec{a}_{1}} \thinspace\big\vert\thinspace \cdots {\bigl\{{\vec{a}_{r}} \thinspace\big\vert\thinspace &\text{$V({\vec{a}_{1}}, \dotsc, {\vec{a}_{r}})$ has rank $r$
and} \\
&{\bigl\{{\vec{b}_{}}\thinspace\big\vert\thinspace \text{$V({\vec{a}_{1}}, \dotsc, {\vec{a}_{r}}, {\vec{b}_{}})$ also has rank $r$} \bigr\}} \in p \bigr\}} \in p
\cdots \bigr\}} \in p.\end{aligned}$$ But if $V({\vec{a}_{1}}, \dotsc, {\vec{a}_{r}})$ and $V({\vec{a}_{1}}, \dotsc, {\vec{a}_{r}}, {\vec{b}_{}})$ both have rank $r$, it means that some nonzero multiple of ${\vec{b}_{}}$ has to lie in the first group. We can therefore conclude from the previous line that $$\label{eq:BigSet} \begin{split}
{\bigl\{{\vec{a}_{1}} \thinspace\big\vert\thinspace \cdots {\bigl\{{\vec{a}_{r}} \thinspace\big\vert\thinspace &\text{$V = V({\vec{a}_{1}}, \dotsc, {\vec{a}_{r}})$ has rank $r$
and} \\
&{\bigl\{{\vec{b}_{}}\thinspace\big\vert\thinspace \text{$N \cdot v({\vec{b}_{}}) \in V$ for some $N \neq 0$} \bigr\}} \in p \bigr\}} \in p
\cdots \bigr\}} \in p.
\end{split}$$
Finally, let $\mathcal{A}$ denote the set of $r$-tuples $\bigl( {\vec{a}_{1}}, \dotsc, {\vec{a}_{r}} \bigr)$, taken in the right order from the nested sets in ; for any one of them, the group $V = V({\vec{a}_{1}}, \dotsc, {\vec{a}_{r}})$ has rank $r$ and the set ${\bigl\{{\vec{b}_{}}\thinspace\big\vert\thinspace \text{$N
\cdot v({\vec{b}_{}}) \in V$ for some $N \neq 0$} \bigr\}}$ is $p$-big.
We have now arrived at our destination—we shall use the splitting $\Hil = \Hil_1 \oplus
\Hil_1^{\bot}$, where $$\Hil_1 = \bigcap_{\mathcal{A}} \ker P_{{\vec{a}_{1}}}\dotsm P_{{\vec{a}_{r}}}
\qquad \text{and} \qquad
\Hil_1^{\bot} = \overline{\sum_{\mathcal{A}} \im P_{{\vec{a}_{1}}}\dotsm P_{{\vec{a}_{r}}}}.$$ This ends the third step; the proof, based on this splitting, that $P$ is an orthogonal projection is contained in the remaining part of the proof.
Step IV {#step-iv .unnumbered}
-------
It remains to prove that the operator $P$ really is a projection. Because of the splitting from \[item:III\], we have two subspaces to consider. Let us begin with the one that is easier to handle, and show that $P$ is zero on $\Hil_1$. If $x \in \Hil_1$, we have $P_{{\vec{a}_{1}}} \dotsm P_{{\vec{a}_{r}}} x = 0$ for $p$-many ${\vec{a}_{1}}, \dotsc, {\vec{a}_{r}}$, thus $$P^r x = {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{1}}} \cdots {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{a}_{r}}} \Bigl( {\prod_{i=1}^{m} U_i^{g_i({\vec{a}_{1}}) +
\dotsb + g_i({\vec{a}_{r}})}} \Bigr) P_{{\vec{a}_{1}}} \dotsm P_{{\vec{a}_{r}}} x = 0.$$ As $P$ is self-adjoint, $P^r x = 0$ quickly leads to $P x = 0$.
The complementary subspace $\Hil_1^{\bot}$, on the other hand, requires more attention. Here, we shall show that $P$ is equal to another projection $P'$, to be defined below, and constructed with the help of the inductive assumptions. So suppose that $x \in \im P_{{\vec{a}_{1}}} \dotsm P_{{\vec{a}_{r}}}$ for a certain tuple $\bigl(
{\vec{a}_{1}}, \dotsc, {\vec{a}_{r}} \bigr) \in \mathcal{A}$; we shall reason that $P x = P' x$, which, by the usual density argument, is sufficient for equality on all of $\Hil_1^{\bot}$.
Since $x$ lies in the image of the product $P_{{\vec{a}_{1}}} \dotsm P_{{\vec{a}_{r}}}$, we get $P_{{\vec{a}_{k}}} x = x$ for each $k = 1, \dotsc, r$. Apply Lemma \[lem:StrongConvergence\] to get strong convergence, in the form $${\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{2}}} \cdots {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{s}}} {\Bigl\| {\prod_{i=1}^{m} U_i^{f_i'({\vec{a}_{k}}, {\vec{z}_{2}},
\dotsc, {\vec{z}_{s}})}} x - x \Bigr\|} = 0.$$ One easily derives that for any $r$ integers $N_1, \dotsc, N_r$, $${\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{2}}} \cdots {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{s}}} {\Bigl\| {\prod_{i=1}^{m} U_i^{N_1 f_i'({\vec{a}_{1}}, {\vec{z}_{2}},
\dotsc, {\vec{z}_{s}}) + \dotsm + N_r f_i'({\vec{a}_{r}}, {\vec{z}_{2}}, \dotsc, {\vec{z}_{s}})}} x - x \Bigr\|} = 0.$$ The vectors $v({\vec{a}_{k}})$ span the group $V = V({\vec{a}_{1}}, \dotsc, {\vec{a}_{r}})$, and so we can conclude that the equality $$\label{eq:StrongLimit}
{\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{2}}} \cdots {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{s}}} {\Bigl\| {\prod_{i=1}^{m} U_i^{h_i({\vec{z}_{2}}, \dotsc, {\vec{z}_{s}})}}
x - x \Bigr\|} = 0$$ is true for any element $(h_1, \dotsc, h_m) \in V$.
We already saw, after , that the set $${\bigl\{{\vec{b}_{}}\thinspace\big\vert\thinspace \text{$N \cdot v({\vec{b}_{}}) \in V$ for some $N \neq 0$} \bigr\}};$$ is $p$-big, because of how the integer $r$ was chosen. As a consequence of Lemma \[lem:Key\], which applies because $\dim p = n$, the set $${\bigl\{{\vec{c}_{}}\thinspace\big\vert\thinspace v({\vec{c}_{}}) - v(0) \in V \bigr\}}$$ is now also $p$-big. Together with , this gives $${\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{2}}} \cdots {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{s}}} {\Bigl\| {\prod_{i=1}^{m} U_i^{f_i'({\vec{c}_{}}, {\vec{z}_{2}}, \dotsc,
{\vec{z}_{s}}) - f_i'(0, {\vec{z}_{2}}, \dotsc, {\vec{z}_{s}})}} x - x \Bigr\|} = 0$$ for $p$-many ${\vec{c}_{}}\in {{\mathbb{Z}}^{n}}$, hence $${\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{c}_{}}} {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{2}}} \cdots {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{s}}} {\Bigl\| {\prod_{i=1}^{m} U_i^{f_i'({\vec{c}_{}},
{\vec{z}_{2}}, \dotsc, {\vec{z}_{s}})}} x - {\prod_{i=1}^{m} U_i^{f_i(0, {\vec{z}_{2}}, \dotsc, {\vec{z}_{s}})}} x \Bigr\|} = 0.$$
The following computation now ends the proof of the fourth and last step: $$\begin{aligned}
P x &= {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{c}_{}}} \Bigl( {\prod_{i=1}^{m} U_i^{g_i({\vec{c}_{}})}} \Bigr) P_{{\vec{c}_{}}} x = \\
&= {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{c}_{}}} \Bigl( {\prod_{i=1}^{m} U_i^{g_i({\vec{c}_{}})}} \Bigr) {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{2}}} \cdots
{\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{s}}} {\prod_{i=1}^{m} U_i^{f_i'({\vec{c}_{}}, {\vec{z}_{2}}, \dotsc, {\vec{z}_{s}})}} x \\
&= \Bigl( {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{c}_{}}} {\prod_{i=1}^{m} U_i^{g_i({\vec{c}_{}})}} \Bigr) \Bigl( {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{2}}}
\cdots {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{s}}} {\prod_{i=1}^{m} U_i^{f_i(0, {\vec{z}_{2}}, \dotsc, {\vec{z}_{s}})}} \Bigr) x\end{aligned}$$ By the hypotheses $(1,d)$ and $(s-1,d)$, both bracketed expressions define orthogonal projections, which moreover commute with each other. Let $P'$ be the projection operator defined as their product; then we have shown that $P x = P' x$, for all $x \in \im P_{{\vec{a}_{1}}} \dotsm P_{{\vec{a}_{r}}}$. This proves that $P$ is also an orthogonal projection when restricted to $\Hil_1^{\bot}$, and thus completes the proof.
Comparison with the original {#comparison-with-the-original .unnumbered}
----------------------------
Since the main purpose of this paper is to reprove the result of [@BFM] using ultrafilters, it may be worthwhile to compare the proof there with the one just given. The overall argument is the same—a proof by induction, relying on a splitting defined in terms of certain groups, which depend on the data (the given IP-sets in [@BFM], the idempotent ultrafilter here). To make the inductive step work out, we have to allow for multiple $p$-limits, necessitating a more intricate argument; our proof is consequently slightly longer than the original one.
We should point out that the third step of the proof uses the same splitting as that in [@BFM]. The notion of dimension, mentioned before, is also apparent in the original paper, and has to be dealt with in much the same way. While we use Lemma \[lem:Dimension\] for this purpose, Bergelson, Furstenberg, and McCutcheon rely on the Milliken-Taylor theorem to handle the different possible dimensions in a unified manner. Several other auxiliary results, proved or quoted in the other paper, also occur at some point in our proof.
Lastly, IP-limits have been replaced by limits along ultrafilters, which means that no subsequences (or more strictly sub-IP-rings) have to be chosen to get convergence. This adds much convenience to the argument.
An IP-version {#an-ip-version .unnumbered}
-------------
As in the example in Section \[sec:II\], we can derive from the previous theorem a version with IP-sets; because of the many subscripts and superscripts, it is more complicated to write down.
\[thm:MainB\] For $j=1, \dotsc, s$, let $q_j \in \beta\mathcal{F}$ be an uncongested idempotent, and let $U_1, \dotsc, U_m$ be commuting unitary operators on a Hilbert space $\Hil$. Given $m$ polynomials $f_1, \dotsc, f_m \in {{\mathrm{Int} \lbrack {\vec{z}_{1}}, \dots,{\vec{z}_{s}} \rbrack}}$—with ${\vec{z}_{j}}$ of dimension $n_j$—satisfying $f_i(0) = 0$ for all $i = 1, \dotsc, m$, and given additionally IP-sets $w_{\bullet}^{k,j}$, indexed by $j = 1, \dotsc, s$ and $k=1, \dotsc, n_j$, define an operator $P$ on $\Hil$ by $$P = {\mathop{q_1\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{\alpha}_{1}}} \cdots {\mathop{q_s\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{\alpha}_{s}}}
{\prod_{i=1}^{m} U_i^{f_i(w_{{\vec{\alpha}_{1}}}^{1, 1}, \dotsc, w_{{\vec{\alpha}_{1}}}^{n_1,1}, \dotsc,
w_{{\vec{\alpha}_{s}}}^{1,s}, \dotsc, n_{{\vec{\alpha}_{s}}}^{n_s,s})}}$$ Then $P$ is always an orthogonal projection. Any two of the operators defined in this way commute.
We use Lemma \[lem:phistar\] and define the following maps. For each $j=1, \dotsc,
s$, let $$\phi_j \colon \mathcal{F} \to \ZZ^{n_j}, \qquad
\phi_j(\alpha) = \bigl( w_{\alpha}^{1,j}, \dotsc, w_{\alpha}^{n_j,j} \bigr),$$ and introduce new ultrafilters $$p_j = \phi_{j\ast}(q_j) \in \beta{\mathbb{Z}}^{n_j}.$$ Since the original ultrafilters were uncongested, all $p_j$ are idempotents by virtue of Lemma \[lem:uncongestedidp\], and we obtain $$P = {\mathop{p_1\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{1}}} \cdots {\mathop{p_s\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{s}}} {\prod_{i=1}^{m} U_i^{f_i({\vec{z}_{1}}, \dotsc, {\vec{z}_{s}})}}$$ from Lemma \[lem:phistar\]. The result now follows from the previous theorem.
Now Lemma \[lem:Uncongested\] states that any IP-ring is contained in an uncongested idempotent of $\beta\mathcal{F}$; it follows that the conclusion of Theorem \[thm:MainB\] holds equally well after replacing each ultrafilter limit by a limit over some IP-ring. We thus recover the main theorem of the original paper [@BFM], as we had set out to do.
Consequences {#sec:VI}
============
From the two theorems in the previous section, we can now derive several other results. In order to simplify the statements, we shall only consider single $p$-limits. Let us begin by showing why it is useful that the weak operator limits we considered are orthogonal projections.
\[thm:Consequence\] Let ${(X, \mathcal{B}, \mu)}$ be a probability measure space, and let $A \subseteq X$ be a measurable set. Let $T_1, \dotsc, T_m$ be commuting invertible measure-preserving transformations on $X$. Furthermore, assume that polynomials $f_1, \dotsc, f_m \in {{\mathrm{Int} \lbrack {\vec{z}_{\thinspace}}\rbrack}}$ are given, where ${\vec{z}_{\thinspace}}= (z_1, \dotsc, z_n)$, such that $f_i(0) = 0$ for all $i = 1, \dotsc, m$.
1. For any idempotent $p \in \beta{{\mathbb{Z}}^{n}}$, one has $${\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} {\mu\biggl(A \cap {\Bigl(\prod_{i=1}^{m} T_i^{f_i({\vec{z}_{\thinspace}})}\Bigr)^{-1}} A\biggr)} \geq {\mu(A)}^2.$$
2. For any uncongested idempotent $q \in \beta\mathcal{F}$ and for any collection of IP-sets $w_{\bullet}^j$, with $j=1, \dotsc, n$, one has $${\mathop{q\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{\alpha} {\mu\biggl(A \cap {\Bigl(\prod_{i=1}^{m} T_i^{f_i(w_{\alpha}^1, \dotsc,
w_{\alpha}^n)}\Bigr)^{-1}} A\biggr)} \geq {\mu(A)}^2.$$
We shall only prove the first statement; the argument will likely be familiar to the reader anyway. On the Hilbert space $\Hil = {\mathit{L}^2(X, \mu)}$, introduce $m$ commuting unitary operators $U_1, \dotsc, U_m$, defining $U_i$ by the rule $U_i g = g \circ T_i$ for $g
\in \Hil$. By virtue of Theorem \[thm:MainA\], the operator $$P = {\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} {\prod_{i=1}^{m} U_i^{f_i({\vec{z}_{\thinspace}})}}.$$ is an orthogonal projection onto some closed subspace of $\Hil$. Write $g$ for the characteristic function of the set $A$, and introduce the abbreviation $$T = \prod_{i=1}^m T_i^{f_i({\vec{z}_{\thinspace}})}.$$ Then we have $${\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} \mu\bigl(A \cap T^{-1} A\bigr) =
{\mathop{p\text{-}}\negthinspace\mathop{\mathrm{lim}}}_{{\vec{z}_{\thinspace}}} {\Bigl\langle g, {\prod_{i=1}^{m} U_i^{f_i({\vec{z}_{\thinspace}})}}g \Bigr\rangle} = {\bigl\langle g, Pg \bigr\rangle}.$$ Now $P$ is a projection; therefore, if $e \equiv 1$ denotes the function identically equal to 1, of norm ${\bigl\| e \bigr\|} = 1$, $${\bigl\langle g, Pg \bigr\rangle} = {\bigl\| Pg \bigr\|}^2 = {\bigl\| Pg \bigr\|}^2 {\bigl\| e \bigr\|}^2 \geq {\bigl\langle Pg, e \bigr\rangle}^2 =
{\bigl\langle g, Pe \bigr\rangle}^2.$$ Finally, $Pe = e$, since $e$ is invariant under the action of the unitary operators $U_i$, and so $${\bigl\langle g, Pe \bigr\rangle}^2 = {\bigl\langle g, e \bigr\rangle}^2 = {\mu(A)}^2.$$ Combining the three displayed (in)equalities gives the desired result.
The consequences of the preceding theorem are twofold. First, when applied to the case of a single measure-preserving transformation, the two inequalities in Theorem \[thm:Consequence\] show precisely that the sets $${\bigl\{f({\vec{z}_{\thinspace}}) \thinspace\big\vert\thinspace {\vec{z}_{\thinspace}}\in {{\mathbb{Z}}^{n}} \bigr\}}$$ and $${\bigl\{f(w_{\alpha}^1, \dotsc, w_\alpha^n) \thinspace\big\vert\thinspace \alpha \in \mathcal{F} \bigr\}}$$ are sets of nice recurrence; here $f$ may be any polynomial in ${{\mathrm{Int} \lbrack {\vec{z}_{\thinspace}}\rbrack}}$ satisfying $f(0) = 0$, and $w_{\bullet}^j$, with $j = 1, \dotsc, n$, can be arbitrary IP-sets.
Secondly, we can exploit the fact that the idempotent ultrafilters in Theorem \[thm:Consequence\] may be chosen arbitrarily. Under the assumptions made above, for any $\epsilon > 0$, the set $$R_{\epsilon} = {\bigl\{{\vec{z}_{\thinspace}}\in {{\mathbb{Z}}^{n}} \thinspace\big\vert\thinspace \mu\bigl(A \cap T^{-1} A\bigr) \geq {\mu(A)}^2 -
\epsilon \bigr\}}$$ has to be contained in every idempotent ultrafilter in $\beta{{\mathbb{Z}}^{n}}$. The reader will remember that this is equivalent to saying that $R_{\epsilon}$ is IP\*, that is, intersects every IP-set of ${{\mathbb{Z}}^{n}}$. In particular, every $R_{\epsilon}$ is a syndetic set, because the IP\* property implies syndeticity. A special case of this result is Khintchine’s recurrence theorem, which states that for a single measure-preserving transformation $T$, the sets of nice returns $${\bigl\{n \in {\mathbb{N}}\thinspace\big\vert\thinspace \mu\bigl(A \cap T^{-1} A\bigr) \geq {\mu(A)}^2 - \epsilon \bigr\}}$$ are syndetic. Several other applications may be found in the original paper [@BFM].
|
---
abstract: 'Observations of ultra-diffuse galaxies NGC1052-DF2 and -DF4 show they contain little dark matter, challenging our understanding of galaxy formation. Using controlled N-body simulations, we explore the possibility to reproduce their properties through tidal stripping from elliptical galaxy NGC1052, in both cold dark matter (CDM) and self-interacting dark matter (SIDM) scenarios. To explain the dark matter deficiency, we find that a CDM halo must have a very low concentration so that it can lose sufficient mass in the tides. In contrast, SIDM favors a higher and more reasonable concentration as core formation enhances tidal mass loss. Stellar distributions in our SIDM benchmarks are more diffuse than CDM one, and hence the former provide a better match to the data. We further show that the presence of stellar particles is critical for preventing the halos from being totally disrupted and discuss its implications. Our results indicate that the dark matter-deficient galaxies may provide important tests for the nature of dark matter.'
author:
- Daneng Yang
- 'Hai-Bo Yu'
- Haipeng An
bibliography:
- 'reference.bib'
title: 'Self-Interacting Dark Matter and the Origin of NGC1052-DF2 and -DF4'
---
[**Introduction.**]{} Dark matter plays a crucial role in galaxy formation and evolution [@White:1977jf]. In the standard scenario, a luminous galaxy is hosted by a dark matter halo, which dominates the overall mass of the galactic system. However, using globular clusters as a tracer, the Dragonfly team shows ultra-diffuse galaxy NGC1052-DF2 (DF2) contains little dark matter [@vanDokkum:2018vup]. Its total stellar mass is $2\times10^8~{\rm M_\odot}$, while the estimated total dynamical mass within $7.6~{\rm kpc}$ is less than $3.4 \times 10^8~{\rm M_{\odot}}$. Intriguingly, the team has discovered a second galaxy lacking dark matter [@2019ApJ...874L...5V], NGC1052-DF4 (DF4). Both galaxies are members of the NGC 1052 group [@2019arXiv191007529D; @2018ApJ...864L..18V; @2018RNAAS...2..146B] and they exhibit many similar properties. In particular, their dark matter contents are at least a factor of $\sim400$ lower than expected from the canonical stellar-to-halo mass relation [@Moster:2009fk; @Behroozi:2012iw].
The dark matter deficiency of the galaxies may be related to their environments. Both DF2 and DF4 are likely to be satellite galaxies of massive elliptical galaxy NGC 1052, which has a distance of $20~{\rm Mpc}$ from the earth. A satellite can experience significant mass loss due to tidal stripping after evolving in its host’s tides [@Hayashi:2002qv; @Penarrubia:2007zx; @Penarrubia:2010jk; @Errani:2015ep; @vandenBosch:2017ynq; @Sameie:2019zfo; @Kahlhoefer:2019oyt]. It has been shown that cosmological hydrodynamical cold dark matter (CDM) simulations can produce analogs of dark matter-deficient galaxies due to tidal stripping [@2019MNRAS.488.3298J; @Yu:2018wxs; @Haslbauer:2019gzq], but it is hard to find simulated galaxies that can match observations of DF2 and DF4 [@Haslbauer:2019cpl]. Ref. [@Ogiya:2018jww] constructs a DF2-like system based on controlled N-body simulations. It argues that a cored dark matter halo is required to match with the observations, as a CDM cuspy halo does not lose sufficient mass in the NGC 1052’s tides. In addition, stars also expand more significantly in a cored halo in response to tidal stripping [@2019MNRAS.485..382C].
In this paper, we study realizations of DF2 and DF4-like galaxies in the self-interacting dark matter (SIDM) scenario [@Spergel:1999mh; @Kaplinghat:2015aga]; see [@Tulin:2017ara] for a recent review. Dark matter self-interactions can thermalize the inner halo and naturally lead to a density core in the inner halo for low-surface brightness galaxies, see, e.g., [@Vogelsberger:2012ku; @Rocha:2012jg; @Zavala:2012us; @Vogelsberger:2015gpr; @Robertson:2016qef; @Banerjee:2019bjp; @Nadler:2020ulu]. Recent studies show that SIDM may provide a unified explanation to dark matter distributions in galactic systems over a wide mass range, including satellite galaxies in the Milky Way [@Vogelsberger:2012ku; @Valli:2017ktb; @Sameie:2019zfo; @Kahlhoefer:2019oyt], spiral galaxies in the field [@Kamada:2016euw; @Creasey:2016jaq; @Ren:2018jpt; @Kaplinghat:2019dhn] and galaxy clusters [@Kaplinghat:2015aga]. It is intriguing to see how the newly-observed dark matter-deficient galaxies can shed further light on the nature of dark matter.
Using controlled N-body simulations, we model evolution of satellite galaxies in the tidal field of NGC 1052 and study their properties in both SIDM and CDM scenarios. After choosing a radial orbit to enhance tidal stripping, we impose observational constraints from the dark matter-deficient galaxies and derive conditions on initial halo parameters for the satellites. We will show that SIDM is more likely to populate DF2 and DF4-like galaxies than CDM, in terms of reproducing their little dark matter contents and diffuse stellar distributions. We further demonstrate that stellar particles can prevent a satellite halo from being totally disrupted in the tidal field. This may help distinguish halo models predicted in SIDM and those in CDM with strong baryonic feedback. We find a cored CDM halo modified by the feedback can be destroyed within much less than $10~{\rm Gyr}$.
  
[**Simulation setup.**]{} We model host galaxy NGC 1052 with a static spherical potential including both halo and stellar components. Assuming a Navarro-Frenk-White (NFW) density profile [@Navarro:1996gj] for the host halo, we fix the characteristic scale density and radius as $\rho_{\rm s}=1.6\times10^6~{\rm M_\odot/kpc^3}$ and $r_{\rm s}=80~{\rm kpc}$, respectively, similar to those used in [@Ogiya:2018jww]. The total halo mass is $M_{200}=1.1\times10^{13}~{\rm M_\odot}$. The luminosity of the galaxy follows a 2D S[é]{}rsic profile [@1963BAAA....6...41S] with the index parameter $n=3.4$ and the effective radius $R_{\rm e}=2.06~{\rm kpc}$; the total stellar mass is $10^{11}~{\rm M_\odot}$ [@2017MNRAS.464.4611F]. In our simulations, we use a Hernquist profile $\rho_{\rm H}=\rho_{\rm h}/[r/r_{\rm h}(1+r/r_{\rm h})^3]$ [@Hernquist:1990be] to model the stellar distribution, where we take $\rho_{\rm h}=1.1\times10^{10}~{\rm M_\odot/kpc^3}$ and $r_{\rm h}=1.2~{\rm kpc}$ such that both Hernquist and 3D deprojected S[é]{}rsic profiles have the same total enclosed mass and half-mass radius. For an NGC 1052-like system, where stars dominate the central regions, an SIDM halo profile can be similar to an NFW one, because SIDM thermalization with the baryonic potential increases the inner halo density [@Kaplinghat:2013xca; @Elbert:2016dbb; @Sameie:2018chj; @Robertson:2017mgj; @Fitts:2018ycl]. Thus our halo model for the host is also valid if dark matter has self-interactions. We also have checked that the effect of dynamical friction is negligible for the purpose of this work and the approach with a static potential is well justified.
For the satellite system, we use live particles. The [initial]{} halo and stellar masses are chosen to be $M_{200}=6.0\times 10^{10}~{\rm M_\odot}$ and $M_\star=3.2\times10^{8}~{\rm M_\odot}$, respectively, and their ratio is consistent with expected from the canonical stellar-to-halo mass relation [@Moster:2009fk]. We assume the halo follows an NFW profile. We perform a coarse scan of the concentration parameter $c_{200}$ from $4$ to $10$, and find a proper value such that simulated satellites after tidal evolution can match with observed dark matter contents of DF2 and DF4. While for the stellar component, we use a Plummer profile $\rho_{\rm P}=\rho_{\rm p}/[1+(r/r_{\rm p})^2]^{5/2}$ [@1911MNRAS..71..460P] to model its initial distribution, where $\rho_{\rm p}=5.8\times10^7~{\rm M_\odot/kpc^3}$ and $r_{\rm p}=1.1~{\rm kpc}$.
We perform both SIDM and CDM simulations. For the former, we consider two values of the self-scattering cross section per mass, $\sigma/m=3~{\rm cm^2/g}$ (SIDM3) and $5~{\rm cm^2/g}$ (SIDM5), consistent with the ones used to explain dark matter distributions in field spiral and Milky Way satellite galaxies [@Ren:2018jpt; @Sameie:2019zfo]. We use public code `GADGET-2` [@Springel:2005mi; @Springel:2000yr] to perform simulations. To model dark matter self-interactions, we have developed and implemented an SIDM module based on the Monte Carlo-based algorithm as in [@Kochanek:2000pi]. We have checked our code for a test halo and found that the simulated density and velocity-dispersion profiles well agree with the results obtained using a semi-analytical model [@Kaplinghat:2015aga], which has been calibrated to other SIDM simulations, see [@Ren:2018jpt]. For the satellite system, we use code `SpherIC` [@GarrisonKimmel:2013aq] to generate initial conditions. The mass of simulated particles is $10^4~{\rm M_\odot}$ for both halo and stellar components, and the softening length is $50~{\rm pc}$.
[**Orbital parameters.**]{} We confine orbits of the satellite in a plane in our simulations. Initially, it is placed at the apocenter, which is $380~{\rm kpc}$ away from the center of the host, and has a tangential velocity of $9~{\rm km/s}$. We find that the orbital period of the satellite is about $2.5~{\rm Gyr}$, the pericenter is $2.8~{\rm kpc}$, and the velocity at the passage is $740~{\rm km/s}$. The orbits we choose are rather radial so that the stripping effects are significant.
We determine a timescale for the final snapshot from the following consideration. DF2 has an $80~{\rm kpc}$ projected distance from NGC 1052 and a relative velocity of $293~{\rm km/s}$ along the line-of-sight direction [@vanDokkum:2018vup]. Suppose the angle between the host–satellite plane and the line-of-sight direction is $\theta$ and the satellite’s orbit is nearly radial, as in our setup, we have the relations, $d\sin\theta\approx80~{\rm kpc}$ and $v\cos\theta\approx293~{\rm km/s}$. For $t\approx9.4~{\rm Gyr}$ and $11~{\rm Gyr}$, corresponding to the moments right before and after passing the apocenter, respectively, our simulated satellites satisfy condition $\left({80~{\rm kpc}}/{d}\right)^2+\left({293~{\rm km/s}}/{v}\right)^2\approx1$. The difference between the two snapshots is small, and we show results with $t=11~{\rm Gyr}$.
 
[**Mass profiles.**]{} Our simulations search for upper limits of $c_{200}$ such that the simulated satellites can match overall with the observations. We find three benchmark cases, $c_{200}=4,~7,$ and $10$ for CDM, SIDM3 and SIDM5, respectively. Fig. \[fig:mass\] shows their enclosed dark matter and stellar masses vs radii at $t=0~{\rm Gyr}$ and $11~{\rm Gyr}$, as well as the final total mass profiles. For all three benchmarks, the simulated halos experience significant mass loss in the tidal field of NGC 1052 and the final halo masses become almost a constant for $r\gtrsim4~{\rm kpc}$, approximately $M_{\rm DM}\approx1.5\times10^8~{\rm M_\odot}$, reduced by a factor of $400$ compared to the initial value, $6.0\times10^{8}~{\rm M_\odot}$. While tidal mass loss for the stars is much more mild, resulting in a total stellar mass of $M_{\rm star}\approx 1.3\times10^8~{\rm M_\odot}$. The mass ratio is $M_{\rm DM}/M_{\rm stars}\sim1$ at $t=11~{\rm Gyr}$ after tidal evolution.
Observationally, DF2 has a total stellar mass of $2\times10^{8}~{\rm M_\odot}$ and the dynamical mass within $7.6~{\rm kpc}$ is $M_{\rm dyn}\lesssim3.4\times10^{8}~{\rm M_\odot}$ [@vanDokkum:2018vup], based on the analysis of globular clusters using the tracer mass estimator method [@Watkins:2010fe]. Other studies show the upper limit of $M_{\rm dyn}$ could be higher due to uncertainties [@Martin:2018ijt; @2019MNRAS.484..245L; @2019MNRAS.486.1192T; @Hayashi:2018emo; @2019MNRAS.484..510N]. Further measurements using stellar spectroscopy show that its stellar and dynamical masses within the half-light radius $R_{1/2}=2.7~{\rm kpc}$ are $M_{\rm star}=(1.0\pm0.2)\times10^8~{\rm M_\odot}$ and $M_{\rm dyn}=(1.3\pm0.8)\times10^8~{\rm M_\odot}$ [@Danieli:2019zyi], respectively; see also . For DF4, $M_{\rm star}=(1.5\pm0.4)\times10^8~{\rm M_\odot}$ and $M_{\rm dyn}=0.4^{+1.2}_{-0.3}\times10^8~{\rm M_\odot}$ within $7~{\rm kpc}$ [@2019ApJ...874L...5V]. We see our three benchmarks well reproduce the observations. For a reference, we display the upper limits of $M_{\rm dyn}$ for DF2 from Ref. [@vanDokkum:2018vup] in Fig. \[fig:mass\], $3.2\times10^8~{\rm M_\odot}$ and $3.4\times10^8~{\rm M_\odot}$, within $3.1~{\rm kpc}$ and $7.6~{\rm kpc}$, respectively.
[**Halo concentration.**]{} Using tailored simulations, we have shown that the tidal interactions can cause low dark matter contents of DF2 and DF4 in both CDM and SIDM scenarios. It is important to note the benchmark halos have different concentrations. The CDM one has the lowest $c_{200}=4$, even though we have chosen a radial orbital to enhance the mass loss. Using the concentration-mass relation at $z=0$ [@Dutton:2014xda] as a reference, we find it corresponds to $4\sigma$ below the median. While SIDM favors a higher concentration. SIDM3 has $c_{200}=7$, $1.8\sigma$ below the median; SIDM5 $c_{200}=10$, $0.4\sigma$ below.
We see that the dark matter-deficient galaxies are more likely to be realized in SIDM than in CDM. Since the inner density cusp in a CDM halo is resilient to tidal stripping, a low concentration is required. In contrast, dark matter self-interactions can thermalize the inner halo and push dark matter from inner to outer regions, lowering the inner gravitation potential. Thus, SIDM satellite halos are more prone to tidal stripping (if there is no core collapse; see [@Sameie:2019zfo; @Kahlhoefer:2019oyt]), and a higher and more reasonable $c_{200}$ value can match with the observations.
For comparison, we also perform CDM runs for the halos with $c_{200}=7$ and $10$. At $t=11~{\rm Gyr}$, their total stellar masses are close to $2\times10^8~{\rm M_\odot}$; while the halo masses are approximately $3.2\times 10^8~{\rm M_\odot}$ and $6.6\times10^{8}~{\rm M_\odot}$ for the CDM runs with $c_{200}=7$ and $10$, respectively; see Fig. \[fig:mass\] for their total enclosed mass profiles (dotted). Apparently, they fit the data much worse than their SIDM counterparts.
[**Stellar distributions.**]{} Fig. \[fig:stellar\] (left) shows stellar density profiles at $t=0~{\rm Gyr}$ and $11~{\rm Gyr}$ for the benchmarks. From our simulations, we find the half-mass radii are $R_{1/2}=1.3~{\rm kpc},~1.7~{\rm kpc}$ and $2.3~{\rm kpc}$ for CDM, SIDM3 and SIDM5 benchmarks, respectively. We further use a 3D deprojected Sésic profile to fit the simulated stellar distributions within $5~{\rm kpc}$ and find a good agreement. The inferred Sésic indices, characterizing the stellar concentration, are $n=1.2,~0.91$ and $0.65$ for CDM, SIDM3 and SIDM5, respectively; the associated effective radii $R_{\rm e}$ are consistent with the half-mass radii from simulations ($R_{1/2}\approx4/3R_{\rm e}$ [@Wolf:2009tu]).
As $\sigma/m$ increases, the stellar distributions become more diffuse and baryon concentration decreases. This is because SIDM core formation leads to a shallow gravitational potential and the stars expands more significantly through tidal stripping; see also [@2019MNRAS.485..382C]. The measured 3D half-light radii and Sésic indices are $R_{1/2}=2.7~{\rm kpc}$ and $n=0.6$ for DF2 [@vanDokkum:2018vup], and $R_{1/2}=2.0~{\rm kpc}$ and $n=0.79$ for DF4 [@2019ApJ...874L...5V]. Thus, our SIDM benchmarks provide a better match with the observed stellar distributions than the CDM one, although all three cases have similar $M_{\rm DM}/M_{\rm star}$ ratios, as shown in Fig. \[fig:mass\].
[**Tidal evolution.**]{} Fig. \[fig:stellar\] (right) shows the bound halo masses vs. evolution time for the benchmarks (solid). All the halos pass through the pericenter, $2.8~{\rm kpc}$ from the center of the host, four times over $11~{\rm Gyr}$. Their masses are similar right after the first pericenter passage, but differ significantly afterwards before the last one. The SIDM5 halo has the lowest rate of tidal mass loss, as it has the highest concentration. At this middle stage, the concentration is the dominant factor controlling the mass loss. But, after the fourth passage, the SIDM5 halo mass drops most significantly, because it has the highest cross section, resulting in the largest density core. Eventually, all three halos have about the same mass at $t=11~{\rm Gyr}$.
We find the stars are crucial for preventing the satellite halos from being completely disrupted in the tidal field. Fig. \[fig:stellar\] (right) also shows the evolution of the bound halo masses [*without*]{} including stellar particles in simulations. All of them are destroyed right after their third pericenter passages at $t=7~{\rm Gyr}$. Even though the SIDM5 halo has a concentration of $c_{200}=10$, close to the median as in [@Dutton:2014xda], dark matter self-interactions make it vulnerable to tidal disruption in the absence of the stars. For comparison, we take the same initial halo, $c_{200}=10$, and perform CDM runs without stellar particles. The halo survives from total disruption and its final bound mass is $2.5\times10^{8}~{\rm M_\odot}$, even higher than predicted in the SIDM5 run with the stars.
Our results have implications for understanding the population of the dark matter-deficient galaxies. If CDM satellite halos host DF2 and DF4 galaxies, they must be on the very low end of the concentration distribution. We expect that a large number of satellite halos with higher concentrations would populate the NGC 1052 group. They would have deep gravitational potentials to collect gas to form stars and prevent them from total disruption. It seems odd that we have observed only two that contain little dark matter. On the other hand, SIDM satellite halos have higher concentrations towards the median, and they are more prone to tidal disruption if the stellar distributions are not concentrated enough. Thus, it is natural to expect that luminous satellite galaxies in the NGC 1052 group are rare in SIDM.
[**Discussion.**]{} Dark matter self-interaction is not the only mechanism that can create density cores. Hydrodynamical simulations show that strong baryonic feedback may create cores in CDM halos [@Navarro:1996bv; @Governato:2009bg; @2013MNRAS.429.3068T; @DiCintio:2013qxa; @Chan:2015tna; @Read:2015sta; @2016MNRAS.456.3542T; @Fitts:2016usl; @Hopkins:2017ycn; @2018MNRAS.473.4392S]. For DF2-like systems we consider, the ratio of initial stellar-to-halo masses is $\sim5\times10^{-3}$, at which the feedback has the strongest impact and the core size is the largest [@DiCintio:2013qxa; @2013MNRAS.429.3068T]. Ref. [@Ogiya:2018jww] considers a cored CDM halo motivated by those simulations and finds it may reproduce DF2 observations after evolving in the NGC 1052’s tides, although the halo has a median concentration. It also assumes a steep initial density profile for stars, which dominate in mass for $r\lesssim1.5~{\rm kpc}$.
However, CDM simulations that create dark matter cores would also produce diffuse stellar distributions, as stellar particles behave like collisionless dark matter particles in the simulations [@Kaplinghat:2019dhn]. We take the cored CDM halo as in [@Ogiya:2018jww] and perform simulations without including stellar particles. The halo is totally destroyed at $t\approx4~{\rm Gyr}$ and $8~{\rm Gyr}$, when our orbital parameters and those in [@Ogiya:2018jww] are used, respectively. Thus, it is not clear whether cored CDM halos modified by the feedback can reproduce the dark matter-deficient galaxies. A related issue is that if the feedback is strong enough to produce dark matter cores, the predicted stellar distributions are too diffuse to accommodate field dwarf galaxies with high stellar densities [@Kaplinghat:2019dhn].
It is useful to recall tensions of CDM in explaining other galactic systems. For low-surface brightness galaxies in the field, where the tidal effects are absent, NFW halos are too concentrated overall to fit their slowly rising rotation curves [@Flores:1994gz; @Moore:1994yx; @Persic:1995ru; @KuziodeNaray:2007qi; @Oh:2011; @Oh:2015xoa; @Oman:2015xda; @Santos-Santos:2019vrw]. The inner halo profiles of well-resolved galaxy clusters are shallower than predicted in CDM [@Newman:2012nw]. The most massive satellite halos in CDM simulations of Milky Way-like systems are too dense to host the observed brightest spheroidal galaxies [@BoylanKolchin:2011de; @BoylanKolchin:2011dk; @Kaplinghat:2019svz]. All these tensions may have a common cause, i.e., inner CDM halos are too dense, and they can be resolved if dark matter has strong self-interactions; see [@Tulin:2017ara] for an extensive discussion. In this work, we demonstrate that the problem persists in explaining the dark matter-deficient galaxies, and SIDM may again offer a solution.
[**Conclusions.**]{} We have studied realizations of the dark matter-deficient galaxies through tidal stripping. Both CDM and SIDM halos can lose the majority of their masses in the NGC 1052’s tides, drastically increasing the ratio of stellar-to-halo masses in accord with the observations. In CDM, the halo must have a very low concentration to explain the dark matter deficiency. In contrast, an SIDM halo can have a higher and more reasonable concentration, as collisional thermalization leads to core formation, boosting tidal mass loss. Our SIDM benchmarks also predict more diffuse stellar distributions, resulting in better agreement with measurements. We have shown the newly-observed DF2 and DF4 are more naturally to be realized in SIDM than in CDM scenarios.
We have also tested a cored CDM satellite halo, motivated by simulations with strong baryonic feedback, and found that it cannot survive in the host’s tides. This further makes SIDM a compelling case for explaining observations of the dark matter-deficient galaxies through tidal stripping, although more thorough investigations along these lines are required. There are other promising directions we can explore in the future. It is interesting to study correlations between orbital and halo parameters of the satellites. Cosmological simulations are necessary to understand formation and growth of NGC 1052-like systems. Observations of more dark matter-deficient galaxies, see, e.g. [@Guo:2019wgb], will further help test the nature of dark matter.
We would like to thank Ran Huo and Go Ogiya for useful discussions. HBY was supported by the U.S. Department of Energy under Grant No. de-sc0008541 and in part by the U.S. National Science Foundation under Grant No. NSF PHY-1748958 through the “From Inflation to the Hot Big Bang" KITP program. HA was supported by NSFC under Grant No. 11975134, the National Key Research and Development Program of China under Grant No.2017YFA0402204 and Tsinghua University Initiative Scientific Research Program.
|
---
address: 'Kazan Federal University, Russia'
author:
- 'K.Khadiev'
---
Preliminaries
=============
The $k$-OBDD and OBDD models are well known models of branching programs. Good source for a different models of branching programs is the book by Ingo Wegener [@we00].
The branching program $P$ over a set $X$ of $n$ Boolean variables is a directed acyclic graph with a source node and sink nodes. Sink nodes are labeled by $1$ (Accept) or $0$ (Reject). Each inner node $v$ is associated with a variable $x\in X$ and has two outgoing edges labeled $x=0$ and $x=1$ respectively. An input $\nu\in
\{0,1\}^n$ determines a computation (consistent) path of from the source node of $P$ to a one of the sink nodes of $P$. We denote $P(\nu)$ the label of sink finally reached by $P$ on the input $\nu$. The input $\nu$ is accepted or rejected if $P(\nu)=1$ or $P(\nu)=0$ respectively.
Program $P$ computes (presents) Boolean function $f(X)$ ($f:\{0,1\}^n \rightarrow \{0,1\}$) if $f(\nu)=P(\nu)$ for all $\nu
\in \{0,1\}^n$.
A branching program is [*leveled*]{} if the nodes can be partitioned into levels $V_1, \ldots,
V_\ell$ and a level $V_{\ell+1}$ such that the nodes in $V_{\ell+1}$ are the sink nodes, nodes in each level $V_j$ with $j \le \ell$ have outgoing edges only to nodes in the next level $V_{j+1}$.
The [*width*]{} $w(P)$ of leveled branching program $P$ is the maximum of number of nodes in levels of $P$: $ w(P)=\max_{1\le
j\le \ell}|V_j|.$
A leveled branching program is called [*oblivious*]{} if all inner nodes of one level are labeled by the same variable. A branching program is called [*read once*]{} if each variable is tested on each path only once.
The oblivious leveled read once branching program is also called Ordinary Binary Decision Diagram (OBDD).
A branching program $P$ is called $k$-OBDD with order $\theta(P)$ if it consists of $k$ layers and each $i$-th layer is OBDD with the same order $\theta(P)$. In nondeterministic case it is denoted $k$-NOBDD.
The [*size*]{} $s(P)$ of branching program $P$ is a number of nodes of program $P$. Note, that for $k$-OBDD and $k$-NOBDD following is right: $ s(P)<w(P) \cdot n \cdot k $.
There are many paper which explore width and size as measure of complexity of classes. Most of them investigate exponential difference between models of Branching Program. Models with less restrictions than $k$-OBDD like non-deterministic, probabilistic and others also were explored, for example in papers [@brs93; @AGKMP05; @A97; @ak98; @bssw96; @hs2000; @hs2003; @sau2000; @tha98]. More precise width hierarchy is presented in the paper.
We denote $\mathsf{k-OBDD_{w}}$ is the sets of Boolean functions that have representation as $k$-OBDD of width $w$. We denote $\mathsf{k-OBDD_{POLY}}$ and $\mathsf{k-OBDD_{EXP}}$ is the sets of Boolean functions that have representation as $k$-OBDD of polynomial and exponential width respectively. In [@bssw96] was shown that $\mathsf{k-OBDD_{POLY}}\subsetneq\mathsf{k-OBDD_{EXP}}$. Result in this paper is following.
\[h-kobdd\] For integer $k=k(n),w=w(n)$ such that $2kw(2w + \lceil \log k
\rceil + \lceil \log 2w \rceil)<n, k\geq 2, w\geq 64$ we have $\mathsf{k-OBDD_{\lfloor w/16
\rfloor-3}}\subsetneq\mathsf{k-OBDD_{w}}$.
Analogosly hierarchies was considered for OBDD in paper [@agky14] and for two way non-uniform automata in cite[ky14]{}. This kind of automata can be considered like special type of branching programs.
Proof of this Theorem is presented in following section. It based on lower bound which presented in [@ak13].
Proof of Theorem \[h-kobdd\]
============================
We start with needed definitions and notations.
Let $\pi=(\{x_{j_1},\dots, x_{j_u}\}, \{x_{i_1},\dots, x_{i_v}\})=(X_A,X_B)$ be a partition of the set $X$ into two parts $X_A$ and $X_B=X\backslash X_A$. Below we will use equivalent notations $f(X)$ and $f(X_A, X_B)$.
Let $f|_\rho$ be subfunction of $f$, where $\rho$ is mapping $\rho:X_A \to \{0,1\}^{|X_A|}$. Function $f|_\rho$ is obtained from $f$ by applying $\rho$. We denote $N^\pi(f)$ to be amount of different subfunctions with respect to partition $\pi$.
Let $\Theta(n)$ be the set of all permutations of $\{1,\dots,n\}$. We say, that partition $\pi$ agrees with permutation $\theta=(j_1,\dots, j_n)\in \Theta(n)$, if for some $u$, $1<u<n$ the following is right: $\pi=(\{x_{j_1},\dots,
x_{j_u}\},\{x_{j_{u+1}},\dots, x_{j_n}\})$. We denote $\Pi(\theta)$ a set of all partitions which agrees with $\theta$.
Let $ N^\theta(f)= \max_{\pi\in \Pi(\theta)} N^\pi(f), \qquad
N(f)=\min_{\theta\in \Theta(n)}N^\theta(f). $ Proof of Theorem \[h-kobdd\] based on following Lemmas and complexity properties of Boolean [*Shuffled Address Function*]{} $SAF_{k,w}(X)$.
Let us define Boolean function $SAF_{k,w}(X):\{0,1\}^n\to \{0,1\}$ for integer $k=k(n)$ and $w=w(n)$ such that $$2kw(2w + \lceil \log k \rceil + \lceil \log 2w \rceil)<n.\label{kw}$$ We divide input variables to $2kw$ blocks. There are $\lceil
n/(2kw)\rceil =a$ variables in each block. After that we divide each block to [*address*]{} and [*value*]{} variables. First $\lceil\log k\rceil + \lceil\log 2w\rceil$ variables of block are [*address*]{} and other $a-\lceil\log k\rceil + \lceil\log 2w\rceil=b$ variables of block are [*value*]{}.
We call $x^{p}_{0},\dots,x^{p}_{b-1}$ [*value*]{} variables of $p$-th block and $y^{p}_{0},\dots,y^{p}_{\lceil\log k\rceil + \lceil\log 2w\rceil}$ are [*address*]{} variables, for $p\in\{0,\dots,2kw-1\}$.
Boolean function $SAF_{k,w}(X)$ is iterative process based on definition of following six functions:
Function $AdrK:\{0,1\}^n\times\{0,\dots,2kw-1\}\to \{0,\dots,k-1\}$ obtains firsts part of block’s address. This block will be used only in step of iteration which number is computed using this function:
$$AdrK(X,p)=\sum_{j=0}^{\lceil\log k\rceil-1}y^{p}_{j}\cdot 2^{j} (mod\textrm{
}k).$$
Function $AdrW:\{0,1\}^n\times\{0,\dots,2kw-1\}\to \{0,\dots,2w-1\}$ obtains second part of block’s address. It is the address of block within one step of iteration:
$$AdrW(X,p)=\sum_{j=0}^{\lceil\log 2w\rceil-1}y^{p}_{j+\lceil\log k\rceil}\cdot 2^{j} (mod\textrm{
}2w).$$
Function $Ind:\{0,1\}^n\times\{0,\dots,2w-1\}\times\{0,\dots,k-1\}\to \{0,\dots,2kw-1\}$ obtains number of block by number of step and address within this step of iteration:
$$Ind(X,i,t) = \left\{ \begin{array}{ll}
p, & \textrm{where $p$ is minimal number of block such that}\\
& \textrm{$AdrK(X,p)=t$ and $AdrW(X,p)=i$}, \\
-1, & \textrm{if there are no such $p$}.
\end{array} \right.$$
Function $Val:\{0,1\}^n\times\{0,\dots,2w-1\}\times\{1,\dots,k\}\to \{-1,\dots,w-1\}$ obtains value of block which have address $i$ within $t$-th step of iteration:
$$Val(X,i,t) = \left\{ \begin{array}{ll}
\sum_{j=0}^{b-1}x^{p}_{j} (mod\textrm{ }w), & \textrm{where }p=Ind(X,i,t)\textrm{, for $p\geq 0$}, \\
-1, & \textrm{if }Ind(X,i,t)<0.
\end{array} \right.$$
Two functions $Step_1$ and $Step_2$ obtain value of $t$-th step of iteration. Function $Step_1:\{0,1\}^n\times\{0,\dots,k-1\}\to \{-1,w\dots,2w-1\}$ obtains base for value of step of iteration:
$$Step_1(X,t) = \left\{ \begin{array}{ll}
-1, & \textrm{if } Step_2(X,t-1)=-1, \\
0, & \textrm{if } t=-1,\\
Val(X,Step_2(X,t-1),t) + w, & \textrm{otherwise}.
\end{array} \right.$$
Function $Step_2:\{0,1\}^n\times\{0,\dots,k-1\}\to \{-1,\dots,w-1\}$ obtain value of $t$-th step of iteration:
$$Step_2(X,t) = \left\{ \begin{array}{ll}
-1, & \textrm{if } Step_1(X,t)=-1, \\
0, & \textrm{if } t=-1\\
Val(X,Step_1(X,t),t), & \textrm{otherwise}.
\end{array} \right.$$
Note that address of current block is computed on previous step.
Result of Boolean function $SAF_{k,w}(X)$ is computed by following way:
$$SAF_{k,w}(X) = \left\{ \begin{array}{ll}
0, & \textrm{if } Step_2(X,k-1)\leq 0, \\
1, & \textrm{otherwise}.
\end{array} \right.$$
Let us discuss complexity properties of this function in Lemma \[fkw1\] and Lemma \[fkw2\]. Proof of Lemma \[fkw1\] uses following technical Lemmas \[good-set\] and \[good-input\].
\[good-set\] Let integer $k=k(n)$ and $w=w(n)$ are such that inequality (\[kw\]) holds. Let partition $\pi=(X_A,X_B)$ is such that $X_A$ contains at least $w$ [*value*]{} variables from exactly $kw$ blocks. Then $X_B$ contains at least $w$ [*value*]{} variables from exactly $kw$ blocks.
[[*Proof.*]{}]{}Let $I_A=\{i:$ $X_A$ contains at least $w$ [*value*]{} variables from $i$-th block$\}$. And let $i'\not\in I_A$ then $X_A$ contains at most $w-1$ [*value*]{} variables from $i'$-th block. Hence $X_B$ contains at least $b-(w-1)$ [*value*]{} variables from $i'$-th block. By (\[kw\]) we have:
$$b-(w-1)=(n/(2kw)-(\lceil \log k \rceil + \lceil \log 2w \rceil)-(w-1)>$$$$>(2w + \lceil \log k \rceil +
\lceil \log 2w \rceil)-(\lceil \log k \rceil + \lceil \log 2w \rceil)-(w-1)=2w-(w-1)=w+1.$$
Let set $I=\{0,\dots,2kw-1\}$ is numbers of all blocks and $i'\in
I\backslash I_A$. Note that $|I\backslash I_A|=2kw- kw =kw$. [$\Box$\
]{}
Let us choose any order $\theta\in \Theta(n)$. And we choose partition $\pi=(X_A,X_B)\in \Pi(\theta)$ such that $X_A$ contains at least $w$ [*value*]{} variables from exactly $kw$ blocks. Let $I_A=\{i:$ $X_A$ contains at least $w$ [*value*]{} variables from $i$-th block$\}$ and $I_B=\{0,\dots,2kw-1\}\backslash I_A$. By Lemma \[good-set\] we have $|I_B|=kw$.
For input $\nu$ we have partition $(\sigma,\gamma)$ with respect to $\pi$. We define sets $\Sigma\subset\{{0,1\}^{|X_A|}}$ and $\Gamma\subset\{{0,1\}^{|X_B|}}$ for input with respect to $\pi$, that satisfies the following conditions: for $\sigma,\sigma'\in\Sigma$, $\gamma\in\Gamma$ and $\nu=(\sigma,\gamma)$, $\nu'=(\sigma',\gamma)$ we have
- for any $r\in\{0,\dots,k-1\}$ and $z\in\{0,\dots, w-1\}$ it is true that $Ind(\nu,z,r)\in I_A$;
- for any $r\in\{0,\dots,k-1\}$ and $z\in\{w,\dots, 2w-1\}$ it is true that $Ind(\nu,z,r)\in I_B$;
- there are $r\in\{1,\dots,k-1\}$, $z\in\{0,\dots, w-3\}$, such that $Val(\nu',z,r)\neq
Val(\nu,z,r)$;
- value of $x^{p}_{j}$ is $0$, for any $p\in I_B$ and $x^{p}_{j}\in X_A$;
- value of $x^{p}_{j}$ is $0$, for any $p\in I_A$ and $x^{p}_{j}\in X_B$;
- following statement is right: $$\label{nu1}
Val(\nu,w-2,t)=2w-2, Val(\nu',w-1,t)=2w-1, \textrm{ for }0\leq t\leq
k-1;$$ $$\label{nu2}
Val(\nu,2w-2,t)=w-2,Val(\nu,2w-1,t)=w-1 \textrm{ for }0\leq t\leq
k-2;$$
- for $p=Ind(\nu,2w-1,k-1)$ and $p'=Ind(\nu,2w-2,k-1)$ following statement is right: $$\label{nu3}
Val(\nu, 2w-1,k-1)=0\quad\quad Val(\nu, 2w-2,k-1)=1.$$
Let us show needed property of this sets.
\[good-input\] Sets $\Sigma$ and $\Gamma$ such that for any sequence $v=(v_0,\dots,v_{ 2(k-1)(w-2)-1})$, for $v_i\in\{0,\dots, w-1\}$, there are $\sigma\in \Sigma$ and $\gamma\in\Gamma$ such that: for each $i\in \{0,\dots, (k-1)(w-2)-1\}$ there are $r_i\in\{1,\dots,k-1\}$ and $z_i\in\{0,\dots, w-3\}$ such that $Val(\nu,z_i,r_i)=a_i$, and for each $i\in \{(k-1)(w-2),\dots,
2(k-1)(w-2)-1\}$ there are $r_i\in\{1,\dots,k-1\}$ and $z_i\in\{w,\dots, 2w-3\}$ such that $Val(\nu,z_i,r_i)=a_i$.
[[*Proof.*]{}]{}Let $p_i\in I_A$, such that $p_i=Ind(\nu,z_i,r_i)$, for $i\in \{0,\dots, (k-1)(w-2)-1\}$. Let us remind that value of $x^{p_i}_{j}$ is $0$ for any $x^{p_i}_{j}\in X_B$. Hence value of $Val(\nu,z_i,r_i)$ depends only on variables from $X_A$. At least $w$ [*value*]{} variables of $p_i$-th block belong to $X_A$. Hence we can choose input $\sigma$ with $a_i$ $1$’s in [*value*]{} variables of $p_i$-th block which belongs to $X_A$.
Input $\gamma\in\Gamma$ and $i\in \{(k-1)(w-2),\dots, 2(k-1)(w-2)-1\}$ we can proof by the same way. [$\Box$\
]{}
\[fkw1\] For integer $k=k(n)$, $w=w(n)$ and Boolean function $SAF_{k,w}$, such that inequality (\[kw\]) holds, the following statement is right: $N(SAF_{k,w})\geq w^{(k-1)(w-2)}$.
[[*Proof.*]{}]{}Let us choose any order $\theta\in \Theta(n)$. And we choose partition $\pi=(X_A,X_B)\in \Pi(\theta)$ such that $X_A$ contains at least $w$ [*value*]{} variables from exactly $kw$ blocks. Let us consider two different inputs $\sigma,\sigma'\in
\Sigma$ and corresponding mappings $\tau$ and $\tau'$. Let us show that subfunctions $SAF_{k,w}|_\tau$ and $SAF_{k,w}|_{\tau'}$ are different. Let $r\in\{1,\dots,k-2\}$ and $z\in\{0,\dots, w-3\}$ are such that $s'=Val(\nu',z,r)\neq Val(\nu,z,r)=s$. Let us choose $\gamma\in \Gamma$ such that $Val(\nu,s+w,r)=w-1,$ $Val(\nu',s'+w,r)=w-2$ and $Val(\nu,i,r-1)=Val(\nu',i,r-1)=z$, where $i\in \{w,\dots,2w-1\}$.
It means $Step_2(\nu,r-1)=Step_2(\nu',r-1)=z$ and $Step_2(\nu,r)=w-1, Step_2(\nu',r)=w-2$. Also conditions (\[nu1\]), (\[nu2\]) mean that $Step_2(\nu,t)=w-1, Step_2(\nu',t)=w-2$, for $r< t\leq k$. Hence $Step_1(\nu,k-1)= 2w-2, Step_1(\nu',k-1)= 2w-1$ and by (\[nu3\]) we have $SAF_{k,w}(\nu)\neq SAF_{k,w}(\nu')$.
Let $r=k-1$, $z\in\{0,\dots, w-3\}$ such that $s'=Val(\nu',z,r)\neq
Val(\nu,z,r)=s$. Let us choose $\gamma\in \Gamma$ such that $Val(\nu,s+w,r)=1$ ,$Val(\nu',s'+w,r)=0$. Therefore $SAF_{k,w}|_\tau(\gamma)\neq SAF_{k,w}|_{\tau'}(\gamma)$ also $SAF_{k,w}|_\tau\neq SAF_{k,w}|_{\tau'}$.
Let us compute $|\Sigma|$. For $\sigma\in\Sigma$ by Lemma \[good-input\] we can get each value of $Val(\nu,i,t)$ for $0\leq
i \leq w-3$ and $1\leq t \leq k-1$. It means $|\Sigma|\geq
w^{(k-1)(w-2)}$. Therefore $N^{\pi}(SAF_{k,w})\geq w^{(k-1)(w-2)}$ and by definition of $N(SAF_{k,w})$ we have $N(SAF_{k,w})\geq
w^{(k-1)(w-2)}$. [$\Box$\
]{}
\[fkw2\] There is $2k$-OBDD $P$ of width $3w+1$ which computes $SAF_{k,w}$
[[*Proof.*]{}]{}Let us construct $P$. Let us use natural order $(1,\dots,n)$ and in each $(2t-1)$-th layer $P$ computes $Step_1(X,t-1)$ and in each $(2t)$-th layer it computes $Step_2(X,t-1)$. Let us consider computation on input $\nu\in\{0,1\}^n$.
Let us consider layer $2t-1$. The first level contains $w$ nodes for store each value of function $Step_2(\nu,t-2)$. For $i$-th node of first level program $P$ checks each block with the following conditions $AdrK(\nu,j)=t-1$ and $AdrW(\nu,j)=i$. If this condition is true then $P$ computes $Val(\nu,i,t-1)$ by this $j$-th block. The result of computation by this $j$-th block is the value of $Step_1(\nu,t-1)$. If this condition is false $P$ goes to next block without branching.
Note that computing of $Val(\nu,i,t-1)$ does not depend on $i$ if we know $j$. And it means the part for computing of $Val(\nu,i,t-1)$ is common for different $i$.
In each level program $P$ has $w+1$ nodes for result of layer. After computing of $Step_1(\nu,t-1)$ by block $j$ program $P$ goes to one of result of layer nodes. From result of layer nodes $P$ goes to end of layer without branching, because result of layer is already obtained. If block $j$ such that $AdrK(\nu,j)=t-1$ and $AdrW(\nu,j)=i$ are not founded then $P$ goes to $-1$ result of layer node and from this node $P$ goes to $0$ result of program node without branching.
Let us consider layer $2t$. The first level has $w$ nodes for store each value of function $Step_1(\nu,t-1)$. For $i$-th node of first level program $P$ checks each block for the following condition $AdrK(\nu,j)=t-1$ and $AdrW(\nu,j)=i+w$. If this condition is true then $P$ computes $Val(\nu,i+w,t-1)$ by this $j$-th block. The result of computation by this $j$-th block is the value of $Step_2(\nu,t-1)$. If this condition is false $P$ goes to next block without branching.
In each level program $P$ has $w+1$ nodes for result of the layer. After computing of $Step_2(\nu,t-1)$ by block $j$ program $P$ goes to one of result of layer nodes.
In last layer program $P$ computes $Val(\nu,i+w,k-1)$ and if $Val(\nu,i+w,k-1)=0$ then $P$ answers $0$ and answers $1$ otherwise.
Let us compute width of program. The block checking procedure needs only $2$ nodes in level. Hence for each value of $i$ we need $2w$ nodes in checking levels. Computing of $Val(\nu,i,t-1)$ and $Val(\nu,i+w,t-1)$ needs $w$ nodes in non checking levels. And $w$ nodes for going to next block in case the block is not needed for non checking levels. And result of layer nodes needs $w+1$ nodes. Therefore we have at most $3w+1$ nodes on each layer. [$\Box$\
]{}
From paper [@ak13] we have the following lower bound.
\[th-main-ak13\] Let function $f(X)$ is computed by $k$-OBDD $P$ of width $w$, then $N(f) \leq w^{(k-1)w+1}. $
Finally we complite the proof of Theorem \[h-kobdd\]. It is obvious that $\mathsf{k-OBDD_{\lfloor
w/16\rfloor-3}}\subseteq\mathsf{k-OBDD_{w}}$. Let us show inequality of this classes. Let us look at function $SAF_{\lceil
k/3\rceil,\lceil w/4 \rceil}$. By Lemma \[fkw2\] we have $SAF_{\lceil k/3\rceil,\lceil w/4 \rceil}\in \mathsf{k-OBDD_{w}}$. By Lemma \[fkw1\] $N(SAF_{\lceil k/3\rceil,\lceil w/4
\rceil})\geq(\lceil w/4 \rceil)^{(\lceil k/3\rceil-1)(\lceil w/4
\rceil-2)}$.
Let us compute $N(SAF_{\lceil k/4\rceil,\lceil w/5 \rceil}) /(\lfloor w/16\rfloor-3)^{(k-1)(\lfloor w/16\rfloor-3)+1}$.
$$\frac{N(SAF_{\lceil k/3\rceil,\lceil w/4 \rceil})}{(\lfloor w/16\rfloor-3)^{(k-1)(\lfloor w/20\rfloor-3)+1}}
\geq\frac{(\lceil w/4 \rceil)^{(\lceil k/3\rceil-1)(\lceil w/4 \rceil-2)}}{(\lfloor w/16\rfloor-3)^{(k-1)(\lfloor w/16\rfloor-3)+1}}=$$ $$=2^{(\lceil k/3\rceil-1)(\lceil w/4 \rceil-2)\log (\lceil w/4 \rceil) - ((k-1)(\lfloor w/16\rfloor-3)+1)\log (\lfloor w/16\rfloor-3)}\geq$$ $$\geq 2^{(\lceil k/3\rceil-1)(\lceil w/4 \rceil-2)\log (\lceil w/4 \rceil) - (k-1)(\lfloor w/16\rfloor-2)\log (\lfloor w/16\rfloor-3)}>$$ $$>2^{\frac{1}{4}(k-1)(\lceil w/4 \rceil-2)\log (\lceil w/4 \rceil) - (k-1)(\lfloor w/16\rfloor-2)\log (\lfloor w/16\rfloor-3)}>$$ $$>2^{(k-1)(\lceil w/16 \rceil-2)\log (\lceil w/4 \rceil) - (k-1)(\lfloor w/16\rfloor-2)\log (\lfloor w/16\rfloor-3)}>1$$
Hence $N(SAF_{\lceil k/3\rceil,\lceil w/4 \rceil})>(\lfloor
w/16\rfloor-3)^{(k-1)(\lfloor w/16\rfloor-3)+1}$ and by Theorem \[th-main-ak13\] we have $SAF_{\lceil k/3\rceil,\lceil w/4
\rceil}\notin \mathsf{k-OBDD_{\lfloor w/16\rfloor-3}}$. [$\Box$\
]{}
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|
---
abstract: 'We review the current observational status of string cosmology when confronted with experimental datasets. We begin by defining common observational parameters and discuss how they are determined for a given model. Then we review the observable footprints of several string theoretic models, discussing the significance of various potential signals. Throughout we comment on present and future prospects of finding evidence for string theory in cosmology, and on significant issues for the future.'
address:
- '$^1$ Astronomy Unit, School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London, E1 4NS, UK'
- '$^2$ Department of Physics and Astronomy, University of Victoria, Victoria, BC, V8P 1A1, Canada'
author:
- 'David J. Mulryne$^1$ and John Ward$^2$'
title: Towards an Observational Appraisal of String Cosmology
---
Introduction
============
Thanks to important advances in experimental astrophysics, the past two decades has seen modern cosmology become a high precision, data rich science (see for example [@Astier:2005qq; @Perlmutter:1998np; @Riess:2004nr; @Riess:1998cb; @Komatsu:2010fb; @Spergel:2006hy; @Spergel:2003cb; @Tegmark:2003ud; @Tegmark:2006az]). Observations suggest that our universe is consistent with a flat, so called ‘$\Lambda$CDM universe’, and have confirmed that our favoured theory for the origin of structure, cosmological inflation [@inflation] is well supported by the data.
Given the extremely high energy scales present in the early universe, and huge distances probed by large scale cosmological evolution, cosmological observables may be sensitive to Planck scale physics. String theory is a leading contender for a theory of the ultraviolet (UV) completion of quantum field theory and gravity, and hence one within which Planck scale effects can be addressed. It is natural, therefore, to ask what the consequences of cosmology are for string theory, and visa versa. Thus the notion of ‘string cosmology’ [@Lidsey:1999mc] was born.
Ideally string cosmology is the *direct* application of string theory to understanding the evolution of the universe. Then, through comparison with data, we might hope to experimentally test the theory. In practice, however, this is too ambitious and generally models are constructed using ideas and intuition from string theory, which are then confronted with observation. Although such a program has been remarkably successful, the inherent stringiness of the models is seldom explicitly present, and it is difficult to argue that constraining models is a direct probe of string theory itself.
On the other hand, particular theories can be sensitive to UV physics, even if not probing it directly, and certain self-interaction potentials for the inflaton can arise in string motivated models which are rather unlikely to arise from pure field theory constructions. Moreover, the self consistency of string inflation models can certainly be probed by observation. Furthermore, while a less developed area of research, string cosmology extends beyond the inflationary epoch. It may, for example, offer convincing alternatives to inflation, which flow more directly from the UV complete nature of the theory. It is also possible to generate cosmic [*super*]{}strings which could be directly detected by observation. String theory may even have a role to play in understanding why our universe is accelerating today.
With all this in mind, the purpose of the present review is to ask how far we have come in our quest to probe string theory using cosmology, and to address questions such as: will we ever see observational signatures of string theory in cosmology? And what are the most promising signals to look for?
We structure the review as follows. In Section \[sec:Obs\], we review how inflationary models are confronted with observation, and the strength of present and future constraints, as well as discussing other observations relevant to probing string theory. In Section \[sec:Test\], we discuss how string theory models are being tested by observation, discussing what would constitute evidence of string theory in light of the issues of naturalness and robustness of models, and review a number of inflationary models and others together with their observational predictions. Finally we conclude in Section \[sec:Discuss\] by highlighting what the key issues are for the future.
Observations and discriminators of early universe models {#sec:Obs}
========================================================
There exists an extremely well developed framework for determining how well a given inflationary model (or alternative) fits experimental data, which we will now review. We then discuss current and future observational constraints, as well as other observations of interest for string cosmology.
Discriminators of the very early universe {#subsec:dis}
-----------------------------------------
Inflation is the accelerated expansion of spacetime; during which quantum fluctuations of the metric and matter are ‘stretched’ to large scales, and subsequently become the origin of cosmic structure (for useful reviews see for example [@Lyth:2009zz; @Lyth:1998xn; @Guth:2005zr; @Lidsey:1995np]). The fundamental scalar quantity is the primordial curvature perturbation $\zeta$ and tensor fluctuations are parametrised by their amplitude, $T$ (see for example [@Bardeen:1983qw; @perts]). Isocurvature perturbations may also be produced and persist after inflation, but are not inevitable and are disfavoured by current data. In the absence of isocurvature perturbations, a given wavenumber, $k$, of $\zeta$ is conserved once stretched to super-horizon scales, $k<aH$, where $a$ is the scalar factor and $H=\dot{a}/a$ [@Bardeen:1983qw; @Lyth:1984gv; @Wands:2000dp; @Rigopoulos:2003ak; @Lyth:2004gb]. The stochastic properties of these perturbations are probed by observation. The full power spectrum, parametrising the two-point function, can be a powerful tool, but is generally parametrised about some pivot scale, $k_*$ in terms of a number of key parameters. The first are the square of the amplitude of the scalar and tensor modes, denoted ${\cal P}_\zeta^*$ and ${\cal P}^*_T$ respectively, which lead to the ratio $r={\cal P}^*_T/{\cal P}^*_\zeta$. This parameter is important because a detection would directly probe the energy scale of inflation. The next key parameters are the spectral tilts; $n_s(k)-1 = d \ln {\cal P}_\zeta / d \ln k|_*$ for scalar perturbations, and $n_T(k) = d \ln {\cal P}_T/ d \ln k|_*$ for tensors. One can then continue to define a running, the derivative of the tilt with respect to $\ln k$, and higher derivative parameters if required.
Further information is available from studying statistics beyond the two-point function. The three-point function, which vanishes for Gaussian perturbations, is parametrised by the bispectrum, $B_\zeta (k,k',k'')$, generally normalised by the square of the power spectrum to give the reduced bispectrum or ${f_{\rm NL}}(k,k',k'')$ parameter. One can then continue to the trispectrum (four-point function), running of ${f_{\rm NL}}$ and so on. Currently meaningful constraints exist only for the subset of parameters $\{r, n_s, {f_{\rm NL}}\}$ and the running of $n_s$.
Single field inflationary models are the most widely studied, because of their simplicity, and are characterised by the generalised action $$\label{action}
S = \int d^4 x \sqrt{-g} \left[\frac{{m_{pl}}^2}{2} R + P(\phi,X) \right ]\,,$$ where $R$ is the Ricci scalar, minimal coupling has been assumed and $X = -\frac{1}{2}g^{\mu \nu} \nabla_\mu \phi \nabla_\nu \phi$. For inflation driven by this action it has been found that [@Garriga:1999vw] $${\cal P}_\zeta = \frac{1}{8 \pi^2 {m_{pl}}^2} \frac{H^2}{c_s \epsilon}\,,\hspace{0.4cm} {\cal P}_T = \frac{2}{\pi^2} \frac{H^2}{{m_{pl}}^2}$$ where $\epsilon = -\dot{H}/{H^2}$, $c_s$ is sound speed of scalar fluctuations, $c_s^2 = P_{,X}/(P_{,X}+2XP_{,XX})$, and all expressions are evaluated at the scale $k^*=aH/c_s$. We note that accelerated expansion requires $\epsilon <1$, and $c_s =1$ corresponds to an canonical scalar field with modes travelling at the speed of light, for which $\epsilon \approx \frac{m_{pl}^2}{2} (V'/V)^2$. Moreover one finds $$r=16 c_s \epsilon \,,~~ (1-n_s)=2\epsilon +\frac{\dot{\epsilon}}{\epsilon H} + \frac{\dot{c_s}}{c_s H}\,,~~ n_t=-2\epsilon$$ and for equilateral triangles $k \sim k' \sim k''$ [@Maldacena:2002vr; @Seery:2005gb; @Chen:2006nt] $$\label{fnl}
{f_{\rm NL}}^{\rm eq}=\frac{5}{81} \left ( \frac{1}{c_s}-1-2\Lambda\right) - \frac{35}{108} \left (\frac{1}{c_s}-1\right)\,,$$ where, $\Lambda = (X^2 P_{,XX}+ \frac{2}{3}X^3 P_{,XXX})/(X P_{,X}+2X^2 P_{,XX})$. All shapes of ${f_{\rm NL}}$ are negligible for canonical single field models with $P=X$.
To predict the observational parameters from a given model of inflation, we must find the values that parameters ($\epsilon$, $c_s$ etc.) took when $k^*=aH/c_s$. This is dependent on the number of e-folds ${\cal N^*} = \ln(a_{\rm end}/a^*)$, which in turn depends on post inflationary physics. A reasonable range is ${\cal N^*} \approx 54 \pm 7$ [@Liddle:2003as], but values *well* outside this range are possible. Since the observational footprint of any given model will be sensitive to $\cal N^*$, to properly compare a given model with observation one must generally determine its *post-inflationary* behaviour. Unfortunately this is not known in most models.
Space will not permit a careful review of how observational predictions are made for every string cosmology model we will discuss, so here, following [@Alabidi:2008ej; @Alabidi:2010sf] (where the reader can turn for a fuller discussion of observational constraints), we include two illustrative examples of canonical models (which cover a large number in the literature): $$\label{examples}
{\rm(a)}~V= V_0 \left [1-\left ( \frac{\phi}{\lambda} \right)^p\right]\,,~~~~ {\rm (b)}~V= V_0 \left (\frac{\phi}{\lambda}\right)^{p}\,,$$ where $V_0$, $\lambda$ and $p$ are constants. Assuming the potential maintains this form over the entire inflationary evolution, and using the approximate expression $d {\phi}/ d {\cal N} \approx -\sqrt{2 \epsilon {m_{pl}}^2}$, which follows when $\epsilon\ll1$, one can find the field value and thus all relevant quantities ${\cal N^*}$ e-folds before the end of inflation (defined as $\epsilon = 1$). Potential (a) represents a small field model for which the range of field values traversed during inflation is $|\Delta \phi| < {m_{pl}}$, and (b) represents a large field model where $|\Delta \phi| > {m_{pl}}$, raising the usual issue of corrections to the potential. Following the procedure outlined, for (a) one finds that $r$ is negligible (typical of small scale models since one can in general express $r=8 (d\phi/d{\cal N})^2/{m_{pl}}^2$ [@Lyth:1996im]), and $n_s = 1-2 \left( \frac{p-1}{p-2}\right)\frac{1}{\cal N^*}$ (the case of $p=-\infty$ corresponds to the potential $V=V_0 (1- e^{-a \phi/{m_{pl}}})$, and $p=0$ to $V=V_0(1+A\ln(\phi/B))$). For (b) one finds $1-n_s=\frac{2+p}{2 {\cal N^*}}$ and $r=\frac{8[{\cal N^*}(1-n_s)-1]}{\cal N^*}$ (where we have considered $p>0$). The relation between $n_s$ and $r$ is important because the corresponding parameter space is well constrained. Note, however, that such simple expressions follow from the simplicity of the potential. Were there additional (potentially unknown) terms, such simple relations would not exist. There are, therefore, two key lessons of this discussion. Firstly observables depend on ${\cal N}^*$, which we don’t know apriori and is not itself an observable. And secondly, simple relations between parameters are possible but will be spoilt if the form of the potential is altered by further terms arising from quantum corrections, and which may introduce new parameters.
Further complications arise if more than one field is light at horizon crossing, since isocurvature modes will be produced. No observational evidence that such modes existed has been found, but were it to be it would rule out single field inflation. When isocurvature modes are present, the curvature perturbation and its statistics may evolve on super-horizon scales during inflation if the field space path curves [@Gordon:2000hv]. Even if isocurvature modes decay before or during reheating, a curved path during inflation will alter the relation between observational parameters and the value of $\epsilon^*$ etc. at horizon crossing (even if the path only curves after modes around the pivot scale exit the horizon). The best developed method to account for this is the $\delta {\cal N}$ formalism [@Starobinsky:1986fxa; @Sasaki:1995aw; @Lyth:2005fi]. Space restricts us from providing the full details, but for canonical inflation the amplitude of the power spectrum is given by ${\cal P}_\zeta = {\cal N}_{,i}{\cal N}_{,i} H^2/(4 \pi^2)$, the tilt by $1- n_s = 2\epsilon^* - 2\dot{\phi_i}^* {\cal N}_{,i j} {\cal N}_{,j}/(H^* {\cal N}_{,k} {\cal N}_{,k})$, and the most important contribution to the reduced bispectrum in the squeezed limit ($k\sim k'<<k''$) is ${f_{\rm NL}}^{\rm loc} = 5/6 {\cal N}_{,i} {\cal N}_{,i j} {\cal N}_{,j}/({\cal N}_{,k} {\cal N}_{,k})^2$, which can be large for certain models. Here ${\cal N}$ is the number of e-folds from horizon crossing to a constant energy density hypersurface once the evolution has become adiabatic, roman numerals label the ${\cal M}$ light fields, and the subscripts denote derivatives with respect to changes in the field values at horizon crossing. In multi-field models, therefore, simple relations for quantities such as $n_s$ are only available for the simplest trajectories, and moreover, a ${\cal M}-1$ dimensional surface now leads to any given ${\cal N}^*$.
Observational constraints and other signatures
----------------------------------------------
Observations of the CMB constitutes the most important tool at our disposal to constrain the defined parameters. Normalisation of the CMB anisotropy requires ${\cal P}_\zeta =2.42 \times 10^{-9}$ [@Komatsu:2010fb]. Precise constraints on the observational parameters depend on how many parameters are included in the statistical analysis, and what other data sets are included. For example, at the $68\%$ confidence level, if the running of the scalar spectral index and $r$ are assumed to be zero, the WMAP-$7$ [@Komatsu:2010fb] data alone implies $n_s \approx 0.967 \pm 0.014$. If $r$ is also included the data gives $n_s \approx 0.982\pm 0.02$ and $r<0.36$ (at the $95 \%$ confidence level), while allowing for running of the scalar spectral index leads to $n_s \approx 1.027 \pm 0.05$ and $d n_s / d \ln k \approx -0.034\pm 0.026$ ($r$ taken to be zero). Moreover, it is important to recognise that the analysis assumes the absence of other contributions to density fluctuations, such as cosmic (super)strings. When these were included, a recent study found a blue spectrum ($n_s>1$) could be accommodated [@Bevis:2007qz] ($n_s=1.00\pm0.03$ with a maximum $11\%$ contribution to power from cosmic strings). An important outcome, therefore, is to recognise that while statements such as the WMAP data favours a red ($n_s<1$) spectrum are common, this is highly analysis dependent.
WMAP also constrains non-Gaussianity, with $-10<{f_{\rm NL}}^{\rm loc}<74$ and $-214 < {f_{\rm NL}}^{\rm eq} <266$ at the $95\%$ confidence level.
The Planck satellite [@:2011ah] currently taking data will hopefully improve these constraints considerably, and in particular, in the absence of a detection one expects limits of roughly $|{f_{\rm NL}}|<5$, and $r<0.05$ and error bars on $n_s$ at least an order of magnitude better than WMAP. The proposed CMBPOL mission [@Baumann:2008aq] (designed specifically to look at the polarisation of the CMB) could probe down to $r<0.01$. There are other exciting future possibilities, such as observation of 21cm radiation [@Khatri:2009aw], which probes the structure of the universe during the cosmic dark ages before re-ionisation, and could give limits of $|{f_{\rm NL}}|<1$. Other important observations which constrain primordial perturbations over different scales are the various surveys of large scale structure, which are often combined with WMAP data, and in particular can give constraints on the running of parameters (see for example [@Dalal:2007cu]).
CMB polarization is also an important discriminator in its own right. Scalar perturbations generate only E-modes, whilst tensor perturbations generate both E and B-modes. Vector perturbations also generate B-modes (the E-mode being negligible with respect to the B-mode), and while highly suppressed during inflation are sourced by cosmic strings. Thus the detection of B-modes would automatically lead to exciting new knowledge about the universe. One caveat is that we must assume there is no axionic coupling to the photon through terms of the form $ \sigma a F \wedge F$, since this can rotate the E-mode into a B-mode with mixing angle given by $\Delta \theta \sim \sigma \Delta a $. Current WMAP bounds on this angle at the $68 \%$ confidence level are $$\Delta \theta = -1.1 \rm{deg} \pm 1.4 \rm{deg} (\rm{stat}) \pm 1.5 \rm{deg} (\rm{syst})$$ which is consistent with it vanishing, but further observations are clearly required [^1].
Observational cosmology is an incredibly rich field, and important data for string cosmology may lurk in numerous other observations. As we will discuss, gravitational lensing - both strongly lensed images and weak lensing - could contain information about cosmic strings, as could micro lensing surveys [@Das:2011ak]. Moreover, data is available on the peculiar velocities of clusters of galaxies, which probe the laws of gravity on the largest scales [@Feldman:2009es]. Not to mention the improving data from supernovae which played the crucial role in determining the need for a dark energy component (acting like a cosmological constant) to accelerate the universe today [@Guy:2010bc]. The potential evolution of dark energy will be a key focus of future observations. Finally, we note there is potential for the direct observation of gravitational waves by ground or space based interferometers (LIGO and LISA) [@:2010yba; @Babak:2010ej]. Though we will likely have to wait many years before an experiment has the sensitivity to be relevant to inflationary gravitational waves (BBO) [@BBO], constraints relevant to cosmic strings already exist [@Wyman:2005tu].
Testing string theory using cosmology {#sec:Test}
=====================================
The overall goal of the string cosmology program is to use cosmology as a testing ground for string theory. The twin aims are to understand whether and how string theory constructions can explain observed properties of the universe, and, more excitingly, to determine whether there might be a *signal* of string theory in observations. The holy grail would be an observational confirmation through the direct detection of something genuinely ‘stringy’. It is possible, perhaps even likely, though that there will be no smoking gun, rather, evidence for string theory might come in a less dramatic form; for example by providing a natural, convincing explanation for something already observed but not properly understood. An example would be a truly compelling model of inflation (or an alternative). It may transpire, however, that all we can achieve is consistency of models with observables, and nothing more. Yet even if this were so, a well motivated string inflationary model which passed successive observational tests of this type, at the expensive of other models failing, could come to be seen as evidence of string theory itself. Likely such evidence would need to be augmented by other observations. For example if such a model additionally predicted cosmic superstrings and some evidence of cosmic strings was found, it would become doubly appealing.
In light of these points, as we discuss aspects of string cosmology – and models of the early universe in particular – we will keep three issues in mind. The first is naturalness of the model. This is a hard concept to make precise but, for example, a model for which a vast range of parameters or initial conditions is allowed, but for which only a vanishingly small range leads to a consistent cosmology, could be viewed as unnatural. The second is testability. Is the model sufficiently well developed that all its parameters are derivable and constrained by observation? Or does it require elements that are plausible, but not yet consistently implemented? Ideally to test models we should have more observable parameters than model parameters, if not we can only probe combinations of model parameters. Finally a model could be predictive, in the limited sense that it leads to a ‘stringy’ prediction, which could not come from, or would be hard to produce, in the absence of stringy physics.
String Inflation
----------------
First we consider string inflationary models. We aim to review a representative selection, discuss how they confront observation and attempt to address how well they measure up against the issues of naturalness, testability and predictivity. We will be particularly interested in those that include concrete calculations of world-sheet or loop corrections, since these are both a sign of the maturity of a model, and will likely lead to reliable signals and consistency relations. We will also focus on observable features which seem to appear more naturally in a string theory setting.
Before embarking on this path, we first note that while many parameters appearing in a model, such as background fluxes, are inherently stringy, we cannot measure them directly cosmologically because we restrict ourselves to an effective inflaton action. Such parameters are absorbed by field redefinitions, giving rise to additional degeneracies, and it is difficult to argue that by probing a given model we are truly probing the original stringy parameters. Moreover, we adopt a critical position that although supergravities are the low energy limits of string theory, an observable signature within supergravity is not (in itself) proof of a string theory signal.
[***Modular inflation.***]{} Most work in this area has focused on flux compactification of type IIB supergravity. Such compactifications preserve at least one, but in general many, massless moduli due to the no-scale structure of the classical theory. These fields can acquire a mass through non-perturbative contributions to the superpotential, such as those arising from gaugino condensation on wrapped $D$-branes. However such terms can only be calculated explicitly in string theory in a handful of cases, since they typically depend on moduli that have been integrated out of the theory. The first attempt to fix all moduli was the KKLT construction [@Kachru:2003aw] which considered one complex modulus. The resulting potential requires an additional uplifting term provided by an anti-$D3$ ($\bar{D3})$ brane at the tip of the warped throat, to obtain a $dS$ vacuum, and doesn’t lead to a consistent inflationary scenario, but the basic procedure underlies many other models, and including more than one moduli can lead to viable inflationary models.
One such scenario, tree level in loops but including world-sheet corrections, is Racetrack inflation on the $\mathbb{CP}^4[1,1,1,6,9]$ Calabi-Yau (CY) three-fold [@BlancoPillado:2006he; @BlancoPillado:2009nw]. The complicated ‘Racetrack’ potential arises from competition between competing terms in the non-perturbative superpotential, with two scalar fields driving inflation [@BlancoPillado:2004ns]. Inflation is possible with particular initial conditions, but not generic, on this potential and satisfying the WMAP normalisation of the power spectrum proves challenging. The authors restricted themselves to variations of the constant term in the superpotential $(W_0)$, and found that maximising the manifold volume led to $n_s \sim 0.95$ for a straight trajectory evolving from the saddle point and emulating a small field model, with negligible ${f_{\rm NL}}$ and $r$. While a useful proof of principal, meaningfully probing the parameter space using observations for this model would be extremely difficult.
Another model of interest is based on the large volume scenario (LVS) [@Conlon:2005ki; @Conlon:2005jm] with one or more of the geometric moduli identified as the inflaton [^2]. The world-sheet corrections ensure that the volume is stabilised at exponentially large values after inflation. There are several different models in this class including Kähler inflation, Roulette inflation [@Bond:2006nc] and Fibre inflation [@Cicoli:2008gp] which all have exponential potentials. Both Kähler and Roulette inflation lead to small tensor models $r \sim \mathcal{O}(10^{-10})$, with the simplest Kähler model having the form $V=V_0 ( 1 - e^{-a\phi})$, and hence leads to $n_s = 1-2/{\mathcal{N}}^*$ (this corresponds to the $p=-\infty$ small field example of Section \[subsec:dis\]). In Roulette models the inflaton is associated with a classical trajectory through field space (perpendicular to the isocurvature trajectory) [@Bond:2006nc]. However inflationary trajectories typically have $\epsilon^* \sim 0$, which suggests conflict with the WMAP data. In the more general multi-field scenario, results indicate a larger red-tilted power spectrum with $|{f_{\rm NL}}| < 0.1$. Fibre inflation consists of a $K3$-manifold fibred over a $\mathbb{CP}1$, allowing for the explicit inclusion of one-loop corrections. These corrections are quantified by $R = 16 AC/B^2 $, where $A, C$ are terms explicitly arising from loop effects. Sufficient e-folds are obtained for small $R$, with ${\mathcal{N}}=60$ occurring for $R = 3 \times 10^{-5}$. Therefore in the limit that $R \to 0$ one finds the model independent footprint[^3] $r \simeq 6 (n_s-1)^2$ which is within current WMAP bounds. In the opposing regime we find $r \simeq (32/3) R^{2/3}, n_s \simeq 1 - 4 R^{2/3}$ which implies $r \leq 0.01, n_s \leq 0.996$ at ${\mathcal{N}}=60$. A two-field model was also constructed in this class with similar results [@Cicoli:2008gp].
These large volume models are well motivated, considered natural, and are testable at least for single field constructions, though they do not yet predict any genuinely stringy signatures. One should note that loop corrections are not universal, although their general form is known [@Berg:2007wt], and must be computed for each CY. The prototypical case is $\mathbb{CP}^4[1,1,1,6,9]$, where the leading order perturbative and non-perturbative world-sheet corrections, and the one and two-loop terms are known [@Misra:2007cq]. The non-perturbative corrections ensure a $dS$ vacuum without the need for $\bar{D3}$ branes, however the loop corrections were argued to be *sub-dominant* with respect to the world-sheet corrections. The inflaton here [@Misra:2007cq] is a linear combination of NS-NS axions, and inflation occurs at a saddle-point where $\cal N$ depends explicitly on the degree of the genus-zero holomorphic curve. Subsequent work computed cosmological observables, which are sensitive to the volume $(\mathcal{V})$ and the $D3$-instanton number $(n^s)$. Favourable scenarios require $\mathcal{V} \gg 1, n^s \sim 10$, which yields $|f_{nl}| \sim 10^{-2}, r \sim 10^{-4}$ and $|n_s-1| \sim 10^{-3}$ [@Misra:2007cq]. Although one can compute various (soft) susy-breaking masses and Yukawa couplings, which are themselves expected to be experimentally constrained, direct cosmological observation will require the inclusion of loop corrections to distinguish these models from field theory.
Brane inflation models
----------------------
$D$-branes play a key role in modern string theory, so it is natural to consider their cosmological consequences. These branes are described by non-linear Dirac-Born-Infeld (DBI) theory. The most popular class of cosmological models exist in IIB, where a mobile $D$ brane travels relativistically down a warped throat towards attractive $\bar{D}3$-brane charge [@Alishahiha:2004eh]. In the simplest case $P(\phi, X)$ is of the form $$P=-T(\phi) \sqrt{1-2X T(\phi)^{-1}} + T(\phi) - V(\phi)$$ with $T$ the warped brane tension and $V(\phi)$ the scalar potential. The non-linear nature of DBI-inflation ensures that $c_s$ can become small, leading to large equilateral ${f_{\rm NL}}$ (Eq.\[fnl\]). These models are potentially testable [@Lidsey:2007gq], predictive and moreover the speed limit imposed by the warped geometry was originally thought to make inflation extremely natural. Combined, these features have lead to significant interest in such models. The brane, however, can only travel a finite distance $\Delta \phi$ due to the finite length of the warped throat [@Chen:2006hs; @Baumann:2006cd], which translates to an upper limit on $r$, as discussed in Section \[sec:Obs\]. When implemented with the above $D3$-brane action, assuming the simplest throat constructions, one finds [@Baumann:2006cd; @Lidsey:2006ia] $$r_{*} < \frac{32}{{\mathcal{N}}{\mathcal{N}}_{eff}^2}(c_s P_{,X})_{*}$$ where $30<{\mathcal{N}}_{eff}<60$ is expected. Moreover, when combined with Eq.(\[fnl\]) and other observational constraints, this implies ${f_{\rm NL}}$ would have to be outside the WMAP bounds [@Lidsey:2006ia]. A result which is independent of the scalar potential. While this is disappointing, it highlights that we are now genuinely able to confront string theoretic models in an increasingly powerful way. Before calculations of ${f_{\rm NL}}$ were performed, and the WMAP constraints available, this model would have appeared viable at the level of the power spectrum. Now despite its appeal, in its simplest form at least, it can be ruled out. Note that small field DBI-inflation may evade this bound while still leading to an ${f_{\rm NL}}$ signal within reach of Planck (see for example [@Bean:2007eh]). Moreover complex models can be constructed which relax the above bounds and include, angular modes, wrapped branes, multiple branes and multi-field theories [@Becker:2007ui] - though the issue of naturalness must be raised in this context. One such proposal, independent of the scalar potential using multiple/wrapped branes, links the tensor-scalar ratio directly to ${f_{\rm NL}}$ via $$r_{*} < - \frac{5}{\mathcal{N}^2_{eff}} \frac{f_{nl}}{(n-1)\sqrt{N}}$$ where $N$ is the $D3$-brane charge of the $AdS_5 \times X_5$ geometry, and $n$ is the number of branes. Such a model is clearly ruled out if $f_{nl}$ is observed to be zero, or has positive sign. In the case of wrapped $D(3+2k)$-branes, it was found that the backreaction becomes more important for higher $k$. The wrapped $D7$-brane bound becomes $$\frac{(1-n_s)}{8} < r < \frac{2 16 \pi}{3^4} \frac{K^4 P_s^2}{\rm{Vol}(X_5) (\Delta {\mathcal{N}})^6}\left(1 + \frac{1}{3 f_{nl}} \right)$$ which can be satisfied for $K \ge 46$, where $K \in \mathbb{Z}$ is the NS-NS flux at the tip of the throat. There are many possible extensions, and it is likely to be a major area of inflationary model building for some time to come.
Another open string model embedded into IIB compactifications is that of $D3/D7$-brane inflation [@Dasgupta:2002ew; @Haack:2008yb], where the compact manifold is $K3 \times T^2/Z_2$. The inflaton is related to the distance between the two types of brane on the orbifold, and its potential is generated by the presence of non self-dual flux on the $D7$-brane. This generates a non-zero D-term potential with Fayet-Illioupolous (FI) parameter $\xi$. The resulting mechanism is a stringy version of hybrid inflation, and inclusion of loop effects leads to the footprint [@Haack:2008yb] $$n_s = 1 -\alpha \left(1 + \frac{1}{(1 - e^{-\alpha {\mathcal{N}}})} \right), \hspace{1cm} \alpha = \frac{4 m^2}{g_s^2 \xi^2}$$ An interesting consequence of many brane models is the creation of cosmic superstrings, which is perhaps the most predictive of all possibly observable stringy physics. This model is a particularly interesting example for which strings with a tension spectrum $G \mu \sim \xi/4$ are produced, and which could be be used to constrain or fix the FI parameter. Under the assumption that cosmic strings contribute at most $\mathcal{O}(11\%)$ to the power spectrum, this implies $r < 10^{-4} g_s^2$, which is vanishingly small for perturbative strings. Increasing the scalar mass tends to suppress the cosmic string contribution, but shifts $n_s$ further towards unity, and towards unfavoured WMAP values. This is another interesting example of how combinations of observables can probe or constrain a model. One interesting recent development in brane inflation models, has been to consider corrections to the inflaton potential, for brane models in which the brane is moving non-relativistically, from compactification effects in the throat [@Kachru:2003sx; @Baumann:2010sx]. Such a calculation has been done for a class of the simplest $D3$/$\bar{D}3$ models discussed above. At leading order the inflaton potential is generated by the Coulombic interaction between these branes, however the corrections due to a single angular mode can be included resulting in the following potential $$V(\phi) = V_0 (\phi) + M_p^2 H_0^2 \left(\left( \frac{\phi}{M_p}\right)^2 - a_{\Delta} \left( \frac{\phi}{M_p}\right)^{\Delta}\right), \hspace{0.6cm}
a_{\Delta} = c_{\Delta} \left( \frac{M_p}{\phi_{UV}}\right)^{\Delta}$$ where $V_0 (\phi)$ generally includes all terms that yield negligible corrections to $\eta$. Note that $\Delta$ corresponds to the eigenvalues of the compact Laplacian, and the smallest eigenvalue ($\Delta =3/2$) is expected to dominate. However if symmetries forbid the existence of such a term, the next possible contribution comes from quadratic modes $\Delta = 2$. Such a model is relatively generic in that it relies on the computation of the Laplacian in the non-compact throat, rather than on the full details of the compact space. The potential in the case of $\Delta = 2$ has been well studied since it is of the form $V(\phi) = V_0(\phi) + \beta H_0^2 \phi^2$. Slow roll inflation requires $\beta \ll1$, because the potential becomes steep as $\beta \to 1$ [^4]. The inflationary footprint is $$n_s -1 \sim \frac{2 \beta}{3} \left(1 - \frac{5}{(e^{2\beta {\mathcal{N}}}-1)} \right)$$ with $r(\beta \sim 0.1) \sim 10^{-4}$, significantly larger than the KKLMMT model where $r\sim 10^{-9}$ [@Kachru:2003sx]. The full phenomenology is discussed in [@Firouzjahi:2005dh] where they used the WMAP data to bound the parameter $\beta$, however for fully UV complete scenarios we expect this to be fixed.
Finally we mention a $D$-term inflationary model in IIA, arising from the intersection of four brane stacks in a phenomenological configuration [@Dutta:2007cr]. The inflaton connects two different brane stacks, and has a one-loop potential of the form $$V(\phi) = g^2 \xi^2 \left(1 + \frac{g^2}{4 \pi^2} \ln \left( \frac{\lambda \phi^2}{\Lambda^2}\right) \right)$$ with scalar index $n_s = 1 - 1/{\mathcal{N}}^*$ (this is the $p=0$ small field example of Section \[subsec:dis\]). The FI-term $\xi$ sets the scale for the power spectrum, and the WMAP normalisation imposes $\xi \sim (10^{15} \rm{Gev})^2$ assuming $g^2 \sim 10^{-2}$. Any cosmic strings formed in this process have a tension $G \mu \sim \xi M_p^{-2}$, which is fortunately well below the current observable threshold.
[***Axion monodromy.***]{} An interesting proposal which has developed from the brane models we have been discussing relies on axion monodromy [@Silverstein:2008sg]. This requires a $D5$-brane to be present in a type IIB compactification, wrapping some two-cycle ($\Sigma_2$) and carrying NS-NS flux on the worldvolume [^5]. One can associate an axion to this flux through a term $b = 2\pi \int_{\Sigma} B$. Computation of the brane action in a particular compactification, results in a scalar (inflaton) potential that is linear in $b$ (provided that it is larger than the size of the compact cycle), and therefore gives rise to a linear inflaton potential of the form of Eq.(\[examples\]) (b) with $p=1$. Such a potential is strongly disfavored from a field theory perspective, and therefore could be considered a signature of stringy physics. Since a relation between $r$ and $n_s$ exists for this model it is testable without knowing ${\mathcal{N}}^*$. For ${\mathcal{N}}^* \sim 60$, one finds $r \sim 0.07, n_S \sim 0.975$ [@Silverstein:2008sg], within current WMAP bounds. Compactification of the model using $D4$-branes on a Nil-manifold results in a fractional power law potential of the form $\phi^{2/3}$, the predictions of which follow from Eq.(\[examples\]) (a) with $p=2/3$. Again, such models are disfavored in field theory[^6]. One other interesting feature is that, dependent on the details of the compactification, the potential may have a superimposed small oscillation from instanton effects, which would lead to an oscillatory feature in the power spectrum [@Flauger:2009ab] and more pronounced oscillatory features in the bi-spectrum [@Chen:2008wn; @Hannestad:2009yx]. If, for example, the level of the effect was too small to affect the power-spectrum relations discussed above, but could be seen in the bi-spectrum, this combined evidence would be powerfully predictive.
[***Tachyon models.***]{} One of the simplest, and most popular, models is that of tachyon inflation, driven by the condensation of an open string mode on a non-BPS $D$-brane [@Sen:2003mv]. Early constructions were unable to satisfy observational bounds because the tachyon mass was too large, however once warped models were developed this constraint could be evaded [@Kofman:2002rh; @Choudhury:2002xu]. Although the action is non-linear, tachyon inflation does not generate large $f_{nl}$ because inflation ends before (ultra)-relativistic effects become dominant. A step towards a concrete UV embedding of this theory was developed in [@Cremades:2005ir; @Panda:2007ie] where they considered a non-BPS $D6$ in a geometry generated by $D3$ flux. The scalar index was found to be $0.94<n_s < 0.97$ for a string coupling in the range $0.1<g_s <0.34$, which suggested that larger $D3$-flux would lead to better agreement with experiment.
[***Non-local Inflation***]{} Thus far the models we have considered have been in the context of low energy supergravity. In the case of $D$-brane and tachyon actions we have considered models which contain terms of higher order in $X$, but none of the models retains higher derivatives. Generally there will be an infinite tower of such higher derivatives which, at energies above the string scale, cannot be ignored. A radical proposal, referred to as non-local inflation, aims to study the effect of such a tower in a limited way. One example uses the action for the tachyon from a toy model of string theory, the P-adic string, where the world-sheet coordinates are restricted to the set of P-adic numbers[^7]. Other settings include the action for the tachyon derived from truncated cubic string field theory (CSFT) (see for example [@Arefeva:2001ps]), which can also only be considered as a toy model. If non-local effects are generic, studying these models [*may*]{} still tell us something interesting about possible stringy observables. A general non-local scalar field action takes the form ${\cal L}_{\phi} = \phi G(\Box)\phi-V(\phi)$, where $\Box$ is the d’Alembertian operator, and $G$ an arbitrary analytic function. Considering this Lagrangian, one discovers that inflation can proceed even if $V$ is naively too steep for slow-roll to be supported, in a sense the additional derivatives act as friction terms. For the p-adic string, $G(x)=-\gamma^4 \exp(-\alpha x)$, and $V(\phi) = \gamma^4 \phi^{p+1}/(1+p)$ where $\alpha = \ln p/(2m_s^2)$ and $\gamma^4 = (m_s^4 /g_s^2 )(p^2 /(p-1))$. The potential is naively too steep for large $p$, however the effects of the infinite number of derivatives leads to a consistent dual hilltop inflationary model [@Barnaby:2006hi; @Lidsey:2007wa]. While it is too early to say that inflation is more natural once higher derivative effects are taken into account, it is an intriguing possibility. A final remark is that initial calculations suggest this model can give rise to large non-Gaussianity, with [@Barnaby:2006hi] $$f^{\rm eq}_{nl} = \frac{5({\mathcal{N}}-2)}{24 \ln p} \sqrt{p r}|n_s-1|^{3/2} \frac{(p-1)}{(p+1)}, \hspace{0.4cm} r = \frac{(p+1)}{2p}|n_s-1|e^{-{\mathcal{N}}|n_s-1|}$$ where $r$ decreases for larger values of $p$, $r$ is unobservably small of order $\mathcal{O}(10^{-3})$, $f_{nl}$ clearly scales as $\sqrt{p r}$, and for large $p,r$ becomes independent of $p$, allowing ${f_{\rm NL}}$ to become large. One issue is that determining the precise end of inflation (and hence precise observational parameters) requires knowledge of the dynamics of the fully non-linear regime, which is extremely difficult [@Mulryne:2008iq]. Interestingly the shape of the non-Gaussianity is different from DBI models, peaking on squeezed triangles, similar to the shape produced by multi-field models. This signature is interesting, but whether it can be distinguished from the multi-field models is unclear, and it may require knowledge of higher order statistics such as the trispectrum. The scalar index is red with current calculations giving $$|n_s-1| \sim \frac{4}{3} \left( \frac{m_s}{H}\right)^2$$ and $H> m_s$. Such a condition is acceptable in this model because of the possible UV completion and, as discussed, is the source of the novel features present.
[***Assisted inflation models.***]{} A number of the models above may contain multiple fields when we move beyond the minimal scenarios. Typically a small number of fields are considered, both in order to keep the calculation tractable and because a larger number of fields implies more freedom and hence less opportunity for models to be probed by observations, or make robust predictions. In the limit where a very large number of fields are present however, interesting effects can occur. In certain models; many fields with potentials which are naively too steep to give rise to inflation can act in a collective, assisted manner enabling inflation to proceed [@Liddle:1998jc; @Kanti:1999vt]. This also allows each individual field to travel sub-Planckian distances. Furthermore the collective behaviour can appear very similar to inflation sourced by one field moving over a much larger distance, and hence $r$ can be large enough to be observable. From one point of view such scenarios appear natural, since the assisted behaviour relaxes the conditions each individual potential must satisfy, and they also have the potential to be testable and predictive, because the very large number of fields can enable a statistical approach to making predictions.
One such model of interest is N-flation [@Dimopoulos:2005ac], which considers a large number of axion fields, each paired with a modulus of the compactification, to act collectively to source inflation. With coupling neglected, each axion has a sinusoidal self-interaction potential of the form $V_n(\phi_n) =\Lambda_n^4 \left(1- \cos(2 \pi \phi_n / f_n)\right)$, which appears like a quadratic potential when expanded around a minimum, with $m_n=2\pi\Lambda_n^2/f_n$. Then, if the masses are identical, the theory is effectively that of a single field sourced by a quadratic potential of the form of Eq.(\[examples\]) (b) with $p=2$ and hence $n_s-1 = -2/{\mathcal{N}}$, $r= 8/{\mathcal{N}}$. For ${\mathcal{N}}\sim 60$, and with $f<{m_{pl}}$, the number of axions required is typically thousands. The observational signature changes if the axion masses are not all identical, and a more realistic approach is to have masses distributed according to a Marcenko-Pastur probability distribution $p(m^2) = \sqrt{(b-m^2)(m^2-a)}/(2\pi m^2 \beta \sigma^2)$, where $a<m^2<b$, $a=\sigma^2 (1-\sqrt{\beta})^2$,$b= \sigma^2(1+\sqrt{\beta})^2$, $\sigma^2 = \langle m^2 \rangle$ and $\beta\sim 1/2$ is typical, and depends on the dimension of the Kähler and complex moduli spaces [@Easther:2005zr; @Piao:2006nm]. Remarkably this statistical approach is surprisingly testable. When comparing to observations we must also fix initial conditions for the various fields. One approach is to also do this randomly, and one finds average values for the spectral index are typically lower than with equal masses, $n_s \approx 0.95$ for $50$ e-folds, this being insensitive to the distribution from which the initial conditions are drawn. $r$ is independent of the model parameters and given by the same expression as above and the non-Gaussianity negligible [@Kim:2007bc]. An intriguing recent development has been the observation that when the full axion potential is considered, a large local ${f_{\rm NL}}$ can be produced, with ${f_{\rm NL}}^{\rm loc} \sim 10$ when all axions are taken to be identical, and $f= {m_{pl}}$ [@Kim:2010ud], though in this case $r$ is negligible and $n_s$ slightly lower again, putting the model in tension with WMAP. This result has been calculated using the $\delta N$ formalism, and may be altered in the light of numerical simulations [@mulryne], and should be tested in more realistic settings and mass distributions. We note that statistical approaches may well have a role to play when confronting other complicated string theory models with observation.
[***M-theory models***]{} A robust scenario in Heterotic M-theory reproduces the results of assisted power law inflation where the potential is exponential and $a(t) \sim a_0 t^p$. Inflation here occurs before moduli stabilisation and is driven by the non-perturbative dynamics of $N$ five-branes along the orbifold direction [@Becker:2005sg]. Under a set of reasonable assumptions, the instanton generated scalar potential is always the steepest direction in field space, which would be unable to support single field inflation. The scalar index takes the expected form $n_s = 1-2/p$ where $p =N^3 + \ldots$ which is used to fix the number of branes using the WMAP data. Whilst the $R^4$ corrections are known, their implementation is difficult since they compete with higher order instanton effects - spoiling the simplicity of the model. However their inclusion could break the field theory degeneracy and point to a unique signature of M-theory. Moreover moduli stabilization and subsequent reheating in this model will no doubt further constrain the parameter space, and test the viability of such a scenario.
A related model arises with only a single five-brane wrapped on the orbifold [@Buchbinder:2004nt], where the inflaton is identified with the real part of the five-brane modulus $(x)$. For $x \ll1$ the five-brane is localised near the visible sector, and inclusion of a FI-term in the *hidden* sector uplifts the stabilised vacuum to $dS$. Slow roll inflation (with no back-reaction) occurs in this regime with ${\mathcal{N}}\sim \eta^{-1} \ln (x_i/x_f)$, where $x_i, x_f$ are the initial and final positions of the brane, and $\eta$ is the slow roll parameter which can be expressed as a ratio of the fluxes arising from the superpotential. With $\eta =0.1, x_i = 10^4 x_f$ we find ${\mathcal{N}}\sim 80$ and $\mathcal{P}_s^2 \sim 10^{-10}$ which agrees with WMAP. Inflation ends once the five-brane dissolves into the visible sector via instanton transition, this in turn excites vector bundle moduli resulting in a shift of the cosmological constant. For larger values of $x$ other moduli will be destabilised from their vacua, and may subsequently lead to a novel inflationary footprint.
Alternative models
------------------
Inflation is by far the most developed, and promising, theory for the origin of structure in the universe. But in the context of string cosmology, other scenarios exist which could be more natural. We briefly mention below some attempts to develop such alternatives.
[***Ekpyrotic model.***]{} The Ekpyrotic model is an alternative to inflation [@Khoury:2001wf; @Lehners:2008vx; @Lehners:2011kr]. Instead of generating perturbations during an exponential expansion, successive $k$ modes exit the horizon during a slow contraction. Predictions are made predominantly within an effective field theory, but the model can be embedded in Heterotic M-theory. In the original single field models the inflaton is associated with the distance between the two $5D$ ‘end of the world’ branes located at the orbifold fixed points [^8], and has a steep negative potential, not directly derived from the theory. As the field evolves, the universe collapses and these branes approach one another. During the collapse the spectrum of $\zeta$ perturbations is extremely blue and not phenomenologically viable [@Lyth:2001pf], however an almost scale invariant spectrum can be produced in the Newtonian potential [@Gratton:2003pe]. Standard hot big bang cosmology is recovered when these branes collide, and it is possible that scale invariant perturbations get imprinted on $\zeta$, but this requires going beyond the $4$-dimensional description [@McFadden:2005mq] and is a potential weakness of the model. An alternative suggestion is to consider a two-field model, the second field arising from the volume modulus of the internal dimensions [@Lehners:2007ac; @Koyama:2007ag; @Buchbinder:2007ad]. If both fields have steep negative potentials parametrised by $V_i= e^{-c_i(\phi) \phi}$, then (in field space) the shape of the potential looks like a ridge. If the trajectory is finely tuned such that the inflaton rolls down this ridge, then the isocurvature perturbation produced during collapse is close to scale invariant. If the trajectory subsequently curves, either by the trajectory naturally falling off the ridge or by ‘bouncing’ off a boundary in field space, then this isocurvature perturbation can be converted into $\zeta$. The model predicts, $n_s -1 = \epsilon^{-1}(2- \partial \ln \epsilon / \partial {\mathcal{N}})$ where the first term is blue-tilted and the second term red-tilted. Tensors are unobservable, which means that detection of almost scale invariant gravitational waves will rule out Ekpyrosis, and strongly favour the simplest inflationary models.
The conversion of isocurvature to curvature perturbations has a secondary effect, which is to produce a large value of ${f_{\rm NL}}$ in the squeezed shape typical of multi-field models. For the simplest case where the conversion occurs by naturally falling of the ridge, $f_{nl} \sim -5 c_1^2/12$, where $1$ labels the field which dominates at late times, and can be calculated using the $\delta N$ formalism [@Koyama:2007if]. Clearly large non-Gaussianities can be generated if $c_1 \gg 1$, and moreover $c_1\gg10$ is required for consistency of the spectral index with WMAP data, and therefore the level of non-Gaussianity is in severe tension with observation. In the case where the trajectories ‘bounce’, positive and negative values of ${f_{\rm NL}}$ are possible and the amplitude depends on how suddenly the bounce occurs [@Buchbinder:2007at; @Lehners:2007wc; @Lehners:2010fy]. Interestingly in search of a robust predictive signal, the authors have considered higher order statistics [@Lehners:2009ja], and though no meaningful constraints currently exist, future observations may probe the scenario in this way. A final comment on this scenario is that the initial conditions are extremely finely tuned. While mechanisms have been suggested to alleviate this tuning in a pre-ekpyrotic phase [@Buchbinder:2007tw], it is hard to not to consider the evolution rather unnatural. On the other hand, because of the special initial conditions required to make the model work, in contrast to multi-field inflationary models, it is extremely testable and potentially predictive.
[***String gas cosmology.***]{} A novel program which both aims to understand the effect of the extended nature of strings on the early universe, and has attempted to replace inflation with an alternative mechanism of generating scale invariant perturbations, is that of string gas cosmology [@Battefeld:2005av; @Brandenberger:2008nx]. This involves coupling the graviton and dilaton to a string gas (which may also include other degrees of freedom such as branes) and using T-duality to interchange winding and momentum modes. The model predicts a slightly red scalar index, but a blue tilt for gravitational waves which allows the theory to be ruled out. Interestingly the theory favours the Heterotic string due to existence of enhanced symmetries necessary for moduli stabilization.
[***Pre-big bang.***]{} Older models of the early (stringy) universe restricted themselves purely to the dilaton sector, at leading order in world-sheet and string loops (with $V(\phi)=0$). Application of generalised T-duality led to the existence of scale-factor duality which aimed to resolve the Big Bang singularity by replacing it by an epoch of high, but finite, curvature [@Gasperini:2007ar]. At early times, before this ’Big Bang’, we find $g_s \ll 1$ allowing us to probe the perturbative string vacuum without worrying about loop corrections. The coupling increases as we pass through this singularity until it becomes constant at late times. However this simple picture does not account for the observed perturbations, the dilaton perturbations leading to a strongly blue spectrum, instead one must consider a curvaton mechanism driven by an axionic field dual to $B_{\mu nu}$. In turn this drives the production of both a graviton and dilaton background, where the dilaton mass can be $m \ge 10^{-23}eV$, which is detectable (in principle). The curvaton potential is assumed to be quadratic, in which case the predictions are the same as canonical $m^2 \phi^2$ inflation and satisfy the WMAP data. Interestingly the type I string is favoured over the Heterotic string in such models due to the difference in primordial magnetic seed production.
Reheating
---------
Reheating is a significantly less developed topic in comparison to inflationary model building, but hugely important. Indeed in many instances there are only vague ideas as to the existence/location of the standard model sector. As we have already discussed this lack of post-inflationary knowledge means ${\mathcal{N}}$ is not fully determined, and hence observational predictions ambiguous. Reheating is also interesting in its own right, and although the energy scale involved is significantly smaller than that associated with inflation, one may still hope that there is sensitivity to the extended nature of the superstring.
A landmark paper [@Frey:2005jk] considered the case of (warped) closed and open string reheating. The closed string sector was purely in the supergravity limit, and despite leading to interesting results, they argued that it was hard to distinguish between string theory and Kaluza-Klein physics unless one could examine the $H/M$ expansion order by order. In the open string case, the strings were argued to redshift like matter. Both approaches further suggested the formation of long string networks during inflation, which could be detectable in the CMB.
More recent papers discuss the case of Kähler inflation in two different classes of closed string model [@Cicoli:2010ha], depending on whether the inflaton is the size of a blow-up mode of the Calabi-Yau (BI model), or whether it is the size of the $K3$ fibre in Fibre inflation (FI model). Both cases involve the leading order $\alpha'$ corrections, although $g_s$ corrections are argued to be decoupled from the theory. The results indicate that there will always be hidden sectors present, that the BI model requires a much higher level of fine-tuning than FI since the hidden sector must wrap the same four-cycle as the inflaton in the former scenario - however despite the higher degree of tuning, the FI scenario leads to a small reheat temperature which is incompatible with TeV scale SUSY. As such, it should be disfavored. The final conclusion was that the hidden and visible sectors are not directly coupled, and the degrees of freedom in the visible sector cannot be more strongly coupled than its hidden sector counterparts. They further identified two generic problems with such a reheating mechanism; i) Inflationary energy will be transferred to the hidden sector. This is not a problem if the hidden sector degrees of freedom are relativistic, but does require a curvaton mechanism to generate the perturbations in the visible sector. ii) There may be overproduction of hidden sector dark matter, which would spoil BBN.
Reheating of the axion monodromy scenario has also been explored, at least in the IIA framework where the $D4$-brane unwinds [@Brandenberger:2008kn]. The $D4$-brane passes through a $D6$-brane, however the open string modes present during the collision act as a braking force. This interaction was described by a toy field theory model, and suggested that no energy was transferred during these collisions. All the reheating energy is dumped into the final collision event, resulting in a high reheating temperature. However backreactive effects were not considered, and may spoil the simple field theory picture.
Cosmic Strings
--------------
A striking prediction of several string scenarios is the formation of cosmic super-strings during or even *after* inflation [@Copeland:2003bj]. Such objects confront observation in a number of ways. First they contribute to primordial density perturbations, and CMB analysis indicates that they cannot account for more than $11\%$ of the power [@Bevis:2007qz], limiting their tension to $G\mu<2.1\times10^{-7}$ [@Fraisse:2006xc]. Moreover, the vector-mode perturbations they source will lead to CMB polarisation potentially observable by the Planck satellite or future missions [@Pogosian:2007gi]. They can strongly gravitationally lens background objects in a distinctive way, but as yet no candidate lensing event has been seen, and through vector perturbations they will rotate images observed in weak lensing surveys [@Thomas:2009bm]. The most promising way in which they can be detected, however, is through a gravitational wave signal produced by cusps on the strings, potentially detectable for tensions as low as $G\mu \sim 10^{-10}$, though whether this signal will be observable by LIGO, LISA or BBO is model dependent [@Polchinski:2007qc]. Current limits come from pulsar timing bounds, which lead to $G\mu<1.5\times 10^{-8} c^{-3/2}$, where $c$ is the number of cusps per string loop.
The discovery of evidence for cosmic super-strings would be extremely powerful evidence for string theory. This requires, however, that they be distinguished from standard cosmic strings, and this is extremely challenging (see for example [@Copeland:2009ga]). One important difference is that intersecting field theory strings recombine with probability $P$, passing through one another with probability $1-P$. Numerical simulation, along with theoretical calculation, suggests that $P \sim 1$ to a remarkably high degree for strings of the same type. For F-F strings it turns out that $P \sim g_s^2$, suggesting they will pass through one another rather than reconnecting. If $P$ and $\mu$ can be determined through gravitational wave observation, these possibilities can be distinguished. For F-D or D-D interactions, the value of $P$ is less constrained, valued in the range $P \in 10^{-3} \ldots 1$. Therefore the perturbative F-F interaction could provide the best direct evidence for string theory [@Copeland:2003bj; @Sakellariadou:2008ie]. Cosmic super-strings of different type may also combine to form $(p,q)$-strings or even networks [@Schwarz:1995dk; @Copeland:2006if]. Such objects have a distinctive tension spectrum which is remarkably difficult to recreate using perturbative field theory.
In Heterotic M-theory one can consider three different types of cosmic strings [@Gwyn:2008fe], a membrane wrapped on the $x^{11}$ direction, a five-brane wrapped on a four-cycle $(\Sigma_4)$ of the internal space or a five-brane wrapped on the product space $\Sigma_3 \times x^{11}$ - where $\Sigma_3$ is a three-cycle. Such strings are formed after the inflationary phases discussed in [@Becker:2005sg; @Buchbinder:2004nt]. The membrane tension is too large and is strongly dis-favoured. The only stable five-brane string is the one wrapped on $\Sigma_4$ because one must turn on a gauge field to cancel the anomaly term, which must live on the boundary and only the brane wrapped on $\Sigma_4$ can be stabilised. Such a string can be superconducting and generates seed magnetic fields that are coherent on *all* cosmological scales. Those fields at large scales cannot be amplified by a dynamo mechanism and therefore will have a weak, coherent amplitude, which may be detectable in future experiments.
Discussion {#sec:Discuss}
==========
In this paper we have provided a critical, non-exhaustive, review of the current observational status of string theory using cosmological data. We have emphasised that inflationary model building has been a success, in the sense that string theory can accommodate inflation, and that the footprints of many different models are testable and conform with WMAP data. Planck will offer a considerably more stringent test, and will undoubtedly rule out many models. Most models, however, exist as field theories and do not make direct predictions for how stringy corrections lead to shifts in observables. Never-the-less we have reviewed a number of signals, such as non-Gaussianity, and relations between parameters, which might in combination with other considerations be evidence for stringy models. There are even some indications that inflation might be more natural in certain stringy settings. Alternatives to inflation are less well developed, but may also offer predictions which allow them to be probed observationally.
Reheating in inflationary models which includes stringy corrections will be different to those in field theory, and have been examined in several specific instances with important phenomenological consequences. However much more work needs to be done, particularly in M-theory, if these models are to be falsifiable. The cleanest signal for string theory still remains the detection of a cosmic superstring through the colliding F-F channel, although more work needs to be done on understanding how such strings scale during the reheating phase, and whether the dynamics of network formation is likely to be important.
We hope our review has highlighted the need for future work in a number of key areas. First, we note that parts of a model are often studied and compared with observation in isolation. For example inflationary predictions, and subsequent evolution including the reheating scale are generally treated separately, while in reality the later impacts on the former through ${\mathcal{N}}^*$. Likewise the production of cosmic strings is often treated separately and compared with observation independently of inflationary constraints, while the presence of both will alter the constraints considerably. This points to the need for a more holistic treatment, and for work on complicated questions. Another key issue is correctly accounting for the presence of more than one inflationary field, typical of complex models. Perhaps most importantly is the issue of the inclusion of higher order corrections, leading to robust cosmological tests.
Aside from these issues there exist other ways in which string theory could be cosmologically tested. One proposal is to embed inflation in non-geometric flux compactifications. Since non-geometric fluxes arise from multiple T-dualities, they are inherently stringy, therefore observables should directly probe the underlying theory. Work along these lines [@Flauger:2008ad] determined a no-go theorem for massive IIA with metric flux. Only the $\mathbb{Z}_2 \times \mathbb{Z}_2$ case evaded this stringent theorem, but did not lead to slow-roll inflation. Future models are likely to evade this theorem, and it will be interesting to determine their footprints.
Finally, we note that there are many other areas of string cosmology we have not been able to discuss, such as corrections to the power spectra due to new physics at high energy scales [@Danielsson:2002kx], or effects from large extra dimensions. It is likely that work on several fronts will be necessary to determine which footprints are inherently stringy. Although thus far there are no truly convincing models or models with signals uniquely predictive of string theory, there remains much work to be done and the future for this field remains bright.
Acknowledgments {#acknowledgments .unnumbered}
===============
DJM is supported by the Science and Technology Facilities Council grant ST/H002855/1. We would like to thank Reza Tavakol and Xingang Chen for useful comments.
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[^1]: Such a coupling can arise in string models through the inclusion of Wess-Zumino (WZ) terms.
[^2]: This is a Kähler modulus in IIB and a complex structure modulus in IIA.
[^3]: Meaning that the observables only depend on the slow roll parameters.
[^4]: The KKLMMT model [@Kachru:2003sx] corresponds to $\beta =0$.
[^5]: An S-dual system involving NS$5$-branes can also be constructed, however the axion now arises from integration of the RR flux.
[^6]: One caveat here is that linear (and fractional) models can be found in the SUGRA literature [@Takahashi:2010ky] and therefore the sub-leading corrections present in the axion monodromy framework will be important in breaking the degeneracy with such models.
[^7]: This assumption was argued to be relaxed in to consider any positive integer $(p)$ [@Freund:1987kt].
[^8]: The additional $6$ dimensions being compactified on a small scale.
|
---
abstract: |
In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. While Hadwiger’s conjecture does not hold for list-coloring, the linear weakening is conjectured to be true. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and thus is $O(t\sqrt{\log t})$-list-colorable.
Recently, the authors and Song proved that every graph with no $K_t$ minor is $O(t(\log t)^{\beta})$-colorable for every $\beta > \frac 1 4$. Here, we build on that result to show that every graph with no $K_t$ minor is $O(t(\log t)^{\beta})$-list-colorable for every $\beta > \frac 1 4$.
Our main new tool is an upper bound on the number of vertices in highly connected $K_t$-minor-free graphs: We prove that for every $\beta > \frac 1 4$, every $\Omega(t(\log t)^{\beta})$-connected graph with no $K_t$ minor has $O(t (\log t)^{7/4})$ vertices.
author:
- 'Sergey Norin[^1]'
- 'Luke Postle[^2]'
bibliography:
- '../snorin.bib'
title: 'Connectivity and choosability of graphs with no $K_t$ minor'
---
*Dedicated to the memory of Robin Thomas*
Introduction
============
All graphs in this paper are finite and simple. Given graphs $H$ and $G$, we say that $G$ has *an $H$ minor* if a graph isomorphic to $H$ can be obtained from a subgraph of $G$ by contracting edges. We denote the complete graph on $t$ vertices by $K_t$.
In 1943 Hadwiger made the following famous conjecture.
\[Hadwiger\] For every integer $t \geq 0$, every graph with no $K_{t+1}$ minor is $t$-colorable.
Hadwiger’s conjecture is widely considered among the most important problems in graph theory and has motivated numerous developments in graph coloring and graph minor theory. For an overview of major progress we refer the reader to [@NPS19], and to the recent survey by Seymour [@Sey16Survey] for further background.
The following natural weakening of Hadwiger’s conjecture has been considered by several researchers.
\[c:LinHadwiger\] There exists $C>0$ such that for every integer $t \geq 1$, every graph with no $K_{t}$ minor is $Ct$-colorable.
For many decades, the best general bound on the number of colors needed to properly color every graph with no $K_t$ minor has been $O(t\sqrt{\log{t}})$, a result obtained independently by Kostochka [@Kostochka82; @Kostochka84] and Thomason [@Thomason84] in the 1980s. The results of [@Kostochka82; @Kostochka84; @Thomason84] bound the “degeneracy" of graphs with no $K_t$ minor. Recall that a graph $G$ is *$d$-degenerate* if every non-null subgraph of $G$ contains a vertex of degree at most $d$. A standard inductive argument shows that every $d$-degenerate graph is $(d+1)$-colorable. Thus the following bound on the degeneracy of graphs with no $K_t$ minor gives a corresponding bound on their chromatic number and even their list chromatic number.
\[t:KT\] Every graph with no $K_t$ minor is $O(t\sqrt{\log{t}})$-degenerate.
Very recently, authors and Song [@NPS19] improved the bound implied by \[t:KT\] with the following theorem.
\[t:ordinaryHadwiger\] For every $\beta > \frac 1 4$, every graph with no $K_t$ minor is $O(t (\log t)^{\beta})$-colorable.
In [@NorSong19Odd] Song and the first author extended \[t:ordinaryHadwiger\] to odd minors.
In this paper we extend \[t:ordinaryHadwiger\] in a different direction – to list coloring. Let $\{L(v)\}_{v \in V(G)}$ be an assignment of lists of colors to vertices of a graph $G$. We say that $G$ is *$L$-list colorable* if there is a choice of colors $\{c(v)\}_{v \in V(G)}$ such that $c(v) \in L(v)$, and $c(v) \neq c(u)$ for every $uv \in E(G)$. A graph $G$ is said to be *$k$-list colorable* if $G$ is $L$-list colorable for every list assignment $\{L(v)\}_{v \in V(G)}$ such that $|L(v)| \geq k$ for every $v \in V(G)$. Clearly every $k$-list colorable graph is $k$-colorable, but the converse does not hold.
Voigt [@Voigt93] has shown that there exist planar graphs which are not $4$-list colorable. Generalizing the result of [@Voigt93], Barát, Joret and Wood [@BJW11] constructed graphs with no $K_{3t+2}$ minor which are not $4t$-list colorable for every $t \geq 1$. These results leave open the possibility that Linear Hadwiger’s Conjecture holds for list coloring, as conjectured by Kawarabayashi and Mohar [@KawMoh07].
\[c:ListHadwiger\] There exists $C>0$ such that for every integer $t \geq 1$, every graph with no $K_{t}$ minor is $Ct$-list colorable.
\[t:KT\] implies that every graph with no $K_t$ minor is $O(t\sqrt{\log{t}})$-list colorable, which until now was the best known upper bound for general $t$.
Our main result extends \[t:ordinaryHadwiger\] to list colorings.
[thm]{}[Main]{}\[t:main\] For every $\beta > \frac 1 4$, every graph with no $K_t$ minor is $O(t (\log t)^{\beta})$-list-colorable.
In the course of proving Theorem \[t:main\], we also prove a remarkably small upper bound on the number of vertices in $K_t$-minor-free graphs with connectivity $O(t(\log t)^\beta)$ for every $\beta > 1/4$ as follows.
[thm]{}[Connect]{}\[t:connect\] For every $\delta > 0$ and $1/2 \geq \beta > 1/4$, there exists $C=C_{\ref{t:connect}}(\beta,\delta)>0$ such that if $G$ is $Ct (\log t)^{\beta}$-connected and has no $K_t$ minor then ${\textup{\textsf{v}}}(G) \leq t(\log t)^{3-5\beta + \delta}$.
Note that Böhme et al.[@BKMM09] proved a variant of \[t:connect\] for graphs with connectivity linear in $t$. Namely, they show that for every $t$ there exists $N(t)$ such that every $\lceil \frac{31}{2}(t+1) \rceil$-connected graph $G$ with no $K_t$ minor satisfies ${\textup{\textsf{v}}}(G) \leq N(t)$. Their proof, however, relies on the Robertson-Seymour graph minor structure theorem and does not provide a reasonable bound for $N(t)$.
### Outline of Paper {#outline-of-paper .unnumbered}
The proof of \[t:connect\] reuses the main tools used to establish \[t:ordinaryHadwiger\] in [@NPS19]. \[t:newforced\] below shows that any dense enough graph with no $K_t$ minor contains a reasonably small subgraph with essentially the same density. Meanwhile, \[t:minorfrompieces\] guarantees that any graph with appropriately high connectivity containing many such dense subgraphs has a $K_t$ minor. These and other necessary tools are introduced in \[s:prelim\].
To play these two results against each other, we need a new ingredient: an extension of \[t:KT\] to upper bound the density of asymmetric bipartite graphs with no $K_t$ minor. We prove such a bound in \[s:density\]. In \[s:connect\] we use this bound to derive \[t:connect\].
In \[s:lower\] we use random constructions to show that the bounds in \[s:density\] are tight up to the constant factor and establish lower bounds on the maximum size of a graph with no $K_t$ minor and given connectivity.
In \[s:alon\] we generalize a bound of Alon [@Alon92] on choosability of complete multipartite graphs to prove a bound on choosability of a graph in terms of its number of vertices and Hall ratio. In \[s:final\] we use this bound and \[t:connect\] to establish \[t:main\]. \[s:remarks\] contains concluding remarks.
### Notation {#notation .unnumbered}
We use largely standard graph-theoretical notation. We denote by ${\textup{\textsf{v}}}(G)$ and ${\textup{\textsf{e}}}(G)$ the number of vertices and edges of a graph $G$, respectively, and denote by ${\textup{\textsf{d}}}(G)={\textup{\textsf{e}}}(G)/{\textup{\textsf{v}}}(G)$ the *density* of a non-null graph $G$. We use $\chi_{\ell}(G)$ to denote the list chromatic number of $G$, and $\kappa(G)$ to denote the (vertex) connectivity of $G$. We write $H \prec G$ if $G$ has an $H$ minor. We denote by $G[X]$ the subgraph of $G$ induced by a set $X \subseteq V(G)$. For disjoint subsetes $A,B\subseteq V(G)$, we let $G(A,B)$ denote the bipartite subgraph induced by $G$ on the parts $(A,B)$. For $F \subseteq E(G)$ we denote by $G/F$ the minor of $G$ obtained by contracting the edges of $F$.
For a positive integer $n$, let $[n]$ denote the set $\{1,2,\ldots,n\}$. The logarithms in the paper are natural unless specified otherwise.
We say that vertex-disjoint subgraphs $H$ and $H'$ of a graph $G$ are *adjacent* if there exists an edge of $G$ with one end in $V(H)$ and the other in $V(H')$, and $H$ and $H'$ are *non-adjacent*, otherwise.
A collection ${\mathcal{X}} = \{X_1,X_2,\ldots,X_h\}$ of pairwise disjoint subsets of $V(G)$ is a *model of a graph $H$ in a graph $G$* if $G[X_i]$ is connected for every $i \in [h]$, and there exists a bijection $\phi: V(H) \to [h]$, such that $G[X_{\phi(u)}]$ and $G[X_{\phi(v)}]$ are adjacent for every $uv \in E(H)$. It is well-known and not hard to see that $G$ has an $H$ minor if and only if there exists a model of $H$ in $G$.
Preliminaries and Previous Results {#s:prelim}
==================================
For this paper, we will need two classical results on $K_t$ minor-free graphs: the first, a lower bound on their independence number; the second, an upper bound on their density.
\[t:DucMey\] Every graph $G$ with no $K_t$ minor has an independent set of size at least $\frac{{\textup{\textsf{v}}}(G)}{2(t-1)}$.
\[t:density\] Let $t \geq 2$ be an integer. Then every graph $G$ with ${\textup{\textsf{d}}}(G) \geq 3.2 t \sqrt{\log t}$ has a $K_t$ minor.
We also need the following results from Norin, Postle and Song [@NPS19].
\[t:newforced\] For every $\delta > 0$ there exists $C=C_{\ref{t:newforced}}(\delta) > 0$ such that for every $D > 0$ the following holds. Let $G$ be a graph with ${\textup{\textsf{d}}}(G) \ge C$, and let $s=D/{\textup{\textsf{d}}}(G)$. Then $G$ contains at least one of the following:
(i)
: a minor $J$ with ${\textup{\textsf{d}}}(J) \geq D$, or
(ii)
: a subgraph $H$ with ${\textup{\textsf{v}}}(H) \leq s^{1+\delta} CD$ and ${\textup{\textsf{d}}}(H) \geq s^{-\delta}{\textup{\textsf{d}}}(G)/C$.
\[t:minorfrompieces\] For every $\beta\in [\frac 1 4, \frac 1 2]$, there exists $C=C_{\ref{t:minorfrompieces}} >1$ satisfying the following. Let $G$ be a graph with $\kappa(G) \geq Ct(\log t)^{\beta}$, and let $r \geq (\log t)^{1-2\beta}/2$ be an integer. If there exist pairwise vertex disjoint subgraphs $H_1,H_2,\ldots,H_{r}$ of $G$ such that ${\textup{\textsf{d}}}(H_i) \geq Ct(\log t)^{\beta}$ for every $i \in [r]$ then $G$ has a $K_t$ minor.
Note that Theorem \[t:minorfrompieces\] was stated only for $\beta = \frac 1 4$ in [@NPS19], however the same proof works for every $\beta \in [\frac 1 4, \frac 1 2]$.
Asymmetric density {#s:density}
==================
In this section we use variants of arguments of Thomason [@Thomason84; @Thomason01] to establish an upper bound on the density of assymetric bipartite graphs with no $K_t$ minor.
\[l:dense\] Let $t$ be a positive integer, let $G$ be a graph with $n ={\textup{\textsf{v}}}(G) \geq 9t$, let $q= 1- \frac{{\textup{\textsf{e}}}(G)}{\binom{n}{2}}$ and $l = \left\lfloor \frac{n}{9t} \right\rfloor$. If $$\label{e:dense} 6t(70q)^{l^2} \leq 1,$$ then $G$ has a $K_t$ minor.
By definiton of $q$, $G$ contains $\binom{n}{2}q$ non-edges. Thus there exists a set $Z$ of $\lfloor n/3 \rfloor $ vertices of $G$ such that each vertex in $Z$ has at most $2qn$ non-neighbors in $G$.
Given $v \in Z$, consider $X \subseteq Z - \{v\}$ with $|X|=l$ chosen uniformly at random. Then the probability that $v$ has no neighbor in $X$ is at most $(2qn/(|Z|-1))^l \leq (7q)^l$.
It follows that if $X \subseteq Z$ with $|X|=l$ is chosen uniformly at random, then the expected number of vertices in $Z-X$ with no neighbor in $X$ is at most $n(7q)^l$. We say that a set $X$ is *good* if at most $3n(7q)^l$ vertices in $Z-X$ have no neighbor in $X$. By Markov’s inequality the probability that the set $X$ as above is good is at least $2/3$.
Given a good set $X \subseteq Z$, suppose that a set $Y$ of size $l$ is selected from $Z-X$ uniformly at random. Then the probability that no vertex of $Y$ is adjacent to a vertex of $X$ is at most $$\left( \frac{3n(7q)^{l}}{|Z|-l} \right)^l \leq (70q)^{l^2}.$$
We now select disjoint subsets $X_1,X_2,\ldots,X_{2t},Y_1,Y_2,\ldots,Y_{t}$ of $Z$ such that $|X_i|,|Y_j|=l$ uniformly at random. We say that a pair $(i,j) \in [2t] \times [t]$ is *unfulfilled* if there does not exist $\{u,v\} \in E(G)$ with $u \in X_i$, $v \in Y_j$. We say that $X_i$ is *perfect* if $(i,j)$ is not unfulfilled for every $j \in [t]$.
By the calculations above, if $X_i$ is good then the expected number of unfulfilled pairs $(i,j)$ is at most $10^l(7q)^{l^2}t \leq 1/6$ by (\[e:dense\]). Therefore the probability that $X_i$ is perfect is at least $1/2$. Thus there exists a choice of sets $\{X_i\}_{i \in [2t]}, \{Y_j\}_{j \in [t]}$ as above, such that at least $t$ of the sets $X_1,X_2,\ldots,X_{2t}$ are perfect. Thus we may assume that $X_1,\ldots,X_t$ are perfect.
Note that every two non-adjacent vertices in $Z$ have at least $(1-4q)n \geq 2/3n$ common neighbors. In particular, every two such vertices have more than $|Z|$ common neighbors in $V(G)-Z$. Thus we can greedily construct pairwise disjoint $B_1,B_2,\ldots,B_t \subseteq V(G)$ such that $X_i \cup Y_i \subseteq B_i$, and $G[B_i]$ is connected for every $i \in [t]$ . These sets form a model of $K_t$ as desired.
\[t:logbip\] There exists $C=C_{\ref{t:logbip}}>0$ such that for every $t \geq 3$ and every bipartite graph $G$ with bipartition $(A,B)$ and no $K_t$ minor we have $$\label{e:logbip}
{\textup{\textsf{e}}}(G) \le C t\sqrt{\log t} \sqrt{|A||B|} + (t-2){\textup{\textsf{v}}}(G).$$
We show that $C = 6400 > 4(20)^2(1+\log(20))$ satisfies the lemma. Suppose for a contradiction that there exists a bipartite graph $G$ with bipartition $(A,B)$ with no $K_t$ minor such that (\[e:logbip\]) does not hold. Choose such $G$ with ${\textup{\textsf{v}}}(G)$ minimum. Let $\alpha=\sqrt{|A|/|B|}$ and consider $v \in A$. By the choice of $G$, we have $${\textup{\textsf{e}}}(G \setminus v) \le C t\sqrt{\log t} \sqrt{(|A|- 1)|B|} + (t-2)({\textup{\textsf{v}}}(G) - 1),$$ and so $$\begin{aligned}
\deg(v) &= {\textup{\textsf{e}}}(G) - {\textup{\textsf{e}}}(G \setminus v) \\ &\geq C t\sqrt{\log t} (\sqrt{|A||B|} - \sqrt{(|A|-1)|B|}) + t-2 \\ &\geq \frac{C}{2} \alpha^{-1}t\sqrt{\log t} + t-2. \end{aligned}$$ Similarly, $$\deg(v)\geq \frac{C}{2} \alpha t\sqrt{\log t} + t-2$$ for every $v \in B$. Assume $|A| \geq |B|$, without loss of generality. Then there exists $v_0 \in A$ such that $\deg(v_0) \leq 7 t \sqrt{\log{t}} \leq \frac{C}{4} \alpha t\sqrt{\log t},$ as otherwise $G$ has a $K_t$ minor by Theorem \[t:density\].
Fix an arbitrary pair of neighbors $u_1,u_2 \in B$ of $v_0$ and consider the graph $G'$ obtained from $G$ by deleting $v_0$ and identifying $u_1$ and $u_2$. As $G'$ is a minor of $G$, we have that $G'$ has no $K_t$ minor, and so $${\textup{\textsf{e}}}(G') \leq C t\sqrt{\log t} \sqrt{(|A|- 1)(|B|-1)} + (t-2)({\textup{\textsf{v}}}(G) - 2),$$ by the choice of $G$. Let $d(u_1,u_2)$ denote the number of common neighbors of $u_1$ and $u_2$ in $A - \{v_0\}$. As ${\textup{\textsf{e}}}(G)-{\textup{\textsf{e}}}(G') = \deg(v_0) + d(u_1,u_2)$ the bounds on ${\textup{\textsf{e}}}(G),{\textup{\textsf{e}}}(G')$ and $\deg(v_0)$ above imply that $$d(u_1,u_2) \geq \frac{C}{4} \alpha t\sqrt{\log t} =: s.$$
Let $n = \lceil \alpha^{-1}t\sqrt{\log t} + t-2 \rceil \geq t-1$, and let $X$ be a set of $n$ arbitrary neighbors of $v_0$. For every $v \in A - v_0$ such that $v$ has a neighbor in $X$, we choose such a neighbor $u$ uniformly independently at random, and contract $v$ onto $u$. Let $H$ be the random graph induced on $X$ obtained via this procedure. The probability that any two given vertices in $X$ are non-adjacent in $H$ is at most $$q:=\left(1-\frac{2}{n}\right)^{s} \le e^{-2s/n}.$$ If $\binom{n}{2}q < 1$, then with positive probability $H$ is complete, and so $G$ contains a complete minor on $n+1$ vertices, a contradiction. Thus we assume that $\binom{n}{2}q \geq 1$, implying $2\log n \geq 2s/n$. Moreover, $sn \geq Ct^2\log{t}/4$ by definition of $s$ and $n$. It follows that $n^2 \log{n} \geq Ct^2\log{t}/4$ implying $n \geq 20t$.[^3] As $\alpha \leq 1$ from definition of $n$ we have $n \leq 2t\sqrt{\log t}$. Combining these inequalities we have and so $2s/n \geq C/8$ and $q < 1/(70)^2$.
Let $l = \lfloor \frac{n}{9t} \rfloor$. Then $$\label{e:l}
l \geq \frac{n}{18t} \geq \frac{1}{18}\alpha^{-1}\sqrt{\log t}$$ and $$\begin{aligned}
(70q)^{l^2} &\leq q^{l^2/2} \leq \exp\left( -\frac{sl^2}{n}\right)
\leq \exp\left(-\frac{sl}{18t}\right) \\ &\leq \exp\left(-\frac{C}{72} \log t\right)\leq \frac{1}{t^3} \leq \frac{1}{6t}.
\end{aligned}$$ Thus (\[e:dense\]) holds for $H$, and thus $H$ contains a $K_t$ minor by \[l:dense\], a contradiction.
Note that the graph $K_{a,t-2}$ has no $K_t$ minor for any integer $a$ showing that the term $(t-2){\textup{\textsf{v}}}(G)$ in (\[e:logbip\]) is necessary. In \[s:lower\] we show that the bound in \[t:logbip\] is tight for all values of $|A|,|B|$ up to the constant factor.
Proof of \[t:connect\] {#s:connect}
======================
In this section, we prove Theorem \[t:connect\], which we restate for convenience.
We assume without loss of generality that $\delta < 1/4$. It suffices to show that there exist $C, t_0=t_0(\delta)$, such that for all positive integers $t \geq t_0$, every graph $G$ with $\kappa(G) \geq Ct(\log t)^{\beta}$ and no $K_t$ minor satisfies $${\textup{\textsf{v}}}(G) \le t(\log t)^{3-5\beta+\delta}.$$
Let $\delta' = \delta/3$. Let $C_1 = C_{\ref{t:newforced}}(\delta')$, and let $C=\max\{ 4 C_{\ref{t:logbip}}, C_{\ref{t:minorfrompieces}}\}$. We choose $t_0 \gg C,C_1,1/\delta$ implicitly to satisfy the inequalities appearing throughout the proof.
Let $k = Ct(\log t)^{\beta}$ and let $G$ be a graph with $\kappa(G) \geq k$ and no $K_t$ minor. Choose a maximal collection $H_1,H_2,\ldots,H_{r}$ of pairwise vertex disjoint subgraphs of $G$ such that ${\textup{\textsf{d}}}(H_i) \geq Ct(\log t)^{\beta-\delta'}$ and ${\textup{\textsf{v}}}(H_i) \leq t(\log t)^{1-\beta+\delta'}$. Since $G$ has no $K_t$ minor, it follows from \[t:minorfrompieces\] that $r < (\log t)^{1-2\beta+2\delta'}/2$. Let $X = \cup_{i \in [r]}V(H_i)$. Then $|X| < t (\log t)^{2-3\beta+3\delta'} = t(\log t)^{2-3\beta + \delta}$.
Let $G' = G\setminus X$. First suppose that ${\textup{\textsf{d}}}(G')\ge k/4$. Let $D= 3.2 t \sqrt{\log t}$. We apply \[t:newforced\] to $\delta'$, $D$ and $G'$. If $G'$ has a minor $J$ with ${\textup{\textsf{d}}}(J) \geq D$ then $G'$ has a $K_{t}$ minor by \[t:density\], contradicting the choice of $G$. Thus there exists a subgraph $H$ of $G'$ such that ${\textup{\textsf{v}}}(H) \leq s^{1+\delta'} C_1D$ and ${\textup{\textsf{d}}}(H) \geq s^{-\delta'}{\textup{\textsf{d}}}(G')/C_1$, where $s=D/{\textup{\textsf{d}}}(G') \leq 13(\log{t})^{1/2-\beta}$. It is easy to check that for large enough $t$ the above conditions imply ${\textup{\textsf{d}}}(H) \geq Ct(\log t)^{\beta-\delta'}$ and ${\textup{\textsf{v}}}(H) \leq t(\log t)^{1-\beta+\delta'}$. Thus the collection $\{H_1,H_2,\ldots,H_{r},H\}$ contradicts the maximality of $\{H_1,H_2,\ldots,H_{r}\}$.
So we may assume that ${\textup{\textsf{d}}}(G') < k/4$. That is, ${\textup{\textsf{e}}}(G') < (k/4) {\textup{\textsf{v}}}(G')$. Since $\kappa(G) \geq k$, every vertex in $V(G')$ has degree at least $k$ in $G$. It follows that $$\label{e:connect1}
e(G(X, V(G'))) \ge \frac{k}{2} {\textup{\textsf{v}}}(G').$$ Yet since $G$ has no $K_t$ minor, we have by \[t:logbip\] applied to $G(X,V(G'))$ that $$\label{e:connect2}
e(G(X, V(G'))) \le C_{\ref{t:logbip}} t\sqrt{\log t}\sqrt{|X|{\textup{\textsf{v}}}(G')} + t(|X|+{\textup{\textsf{v}}}(G')).$$ If ${\textup{\textsf{v}}}(G') \leq |X|$ then $${\textup{\textsf{v}}}(G) \leq 2t(\log t)^{2-3\beta + \delta} \leq t(\log t)^{3-5\beta+\delta}$$ for sufficiently large $t$, as desired. Thus we assume ${\textup{\textsf{v}}}(G') \geq |X|$. Combining (\[e:connect1\]) and (\[e:connect2\]) we have $$\label{e:connect3} (k/2-2t){\textup{\textsf{v}}}(G') \leq C_{\ref{t:logbip}}t\sqrt{\log t}\sqrt{|X|{\textup{\textsf{v}}}(G')}.$$ Assuming that $t$ is large enough, we have that $k \geq 8t$, and so $k/2-2t \geq k/4$. Thus the above implies $${\textup{\textsf{v}}}(G') \leq (4 C_{\ref{t:logbip}})^2t^2\log t \cdot\frac{|X|}{k^2} \leq \frac{|X|\log{t}}{(\log{t})^{2\beta}} \leq t(\log t)^{3-5\beta + \delta},$$ as desired.
Lower bounds {#s:lower}
============
In this section we prove lower bounds on the density of asymmetric bipartite graphs with no $K_t$ minor and on the size of such graphs with given connectivity.
For $0 \leq p \leq 1$ and pair of integers $a,b >0$ we denote by ${\bf G}(a,b,p)$ a random bipartite graph with bipartition $(A,B)$ where $A$ and $B$ are disjoint sets with $|A|=a$, $|B|=b$ and the edges between $A$ and $B$ are chosen independently at random with probability $p$. The next lemma mirrors a computation first used by Bollobas, Caitlin and Erdős [@BCE80] to compute the size of the largest minor in a random graph.
\[l:random\] For every ${\varepsilon}>0$ there exists $t_0$, such that for all $0 < p < 1$ and integers $t \geq t_0, a,b \geq 0$ such that $ab \leq (1-{\varepsilon})\frac{1}{-2\log (1-p)} t^2\log t$, we have $$\Pr [K_{t} \mathrm{\;is \;a \:minor \:of\;} {\bf G}(a,b,p) ] \leq e^{-t^{{\varepsilon}}/3}.$$
Let $(A,B)$ be the bipartition of $G={\bf G}(a,b)$ as in the definition, and let $(A_1,\ldots,A_t)$ and $(B_1, \ldots,B_t)$ be partitions of $A$ and $B$, respectively. Let $a_i = |A_i|, b_i = |B_i|$ for $i \in [t]$. Let $q = 1-p$. Then the probability that $G$ does not contain an edge from $A_i \cup B_i$ to $A_j \cup B_j$ is $q^{a_ib_j+a_jb_i}$. Thus we can upper bound the probability that $\{A_i \cup B_i\}_{i \in [t]}$ is a model of $K_t$ in $G$ by $$\begin{aligned}
\prod_{\{i,j\} \subseteq [t]}&\left(1 - q^{a_ib_j+a_jb_i} \right) \leq \exp \left(-\sum_{ \{i,j\} \subseteq [t]}q^{a_ib_j+a_jb_i} \right) \\
& \leq \exp\left( -\binom{t}{2} q^{(\sum_{{\{i,j\} \subseteq [t]}}(a_ib_j+a_jb_i))/\binom{t}{2}}\right) \\ &\leq \exp\left( -\binom{t}{2} q^{ab/\binom{t}{2}}\right)\leq \exp\left( -\binom{t}{2} q^{-(1-{\varepsilon})\log t/\log q} \right) \\ &= \exp\left(-\frac{(t-1)t^{{\varepsilon}}}{2}\right).
\end{aligned}$$ Suppose $a \geq b$ without loss of generality. If $b \leq t-2$ then $G$ has no $K_t$ minor. Thus we may assume $b \geq t-1 $, implying $a+b \leq t\log t$ for large enough $t$. The number of partitions $(A_1,\ldots,A_t)$ and $(B_1, \ldots,B_t)$ as above can then be loosely upper bounded by $t^{a+b} \leq \exp(t\log^2{t})$. By the union bound we deduce that the probability that $G$ has a $K_t$ minor is at most $$\exp\left(t\log^2{t}-\frac{(t-1)t^{{\varepsilon}}}{2}\right) \leq \exp\left(-t^{{\varepsilon}}/3\right),$$ as desired, where the last inequality holds for $t$ large enough.
Let $$\lambda := \max_{x >0} \frac{1-e^{-x}}{\sqrt{x}}=0.63817\ldots.$$ be the constant which appears, in particular, in the optimal bound on the asymptotic density of graphs with no $K_t$ minor established by Thomason [@Thomason01].
\[c:random\] For every ${\varepsilon}>0$ there exists $C$, such that for all integers $a,b \geq t \geq C$ such that $$\label{e:ab}
ab \geq C t^2\log t$$ there exists a bipartite graph $G$ with bipartition $(A,B)$ such that $|A|=a,|B|=b$, $G$ has no $K_t$ minor and $${\textup{\textsf{e}}}(G) \geq (1-{\varepsilon}) \frac{\lambda}{\sqrt{2}} t\sqrt{\log t}\sqrt{|A||B| }.$$
Let $C$ be chosen implicitly to satisfy the inequalities throughout the proof, and let ${\varepsilon}' ={\varepsilon}/4$.
Assume $a \geq b$, without loss of generality. Note that $K_{a,t-2}$ has no $K_t$ minor, and hence the corollary holds if $$(1-{\varepsilon}) \frac{\lambda}{\sqrt{2}} t\sqrt{\log t}\sqrt{ab} \leq (t-2)a .$$ Thus we may assume $$\label{e:b}
b \log t \geq a$$ given $C$ is large enough.
Let $x$ be such that $\lambda = \frac{1-e^{-x}}{\sqrt{x}}$, and let $p = 1-e^{-x}$. Let $$k = \left\lceil \sqrt{(1-{\varepsilon}') \frac{-2\log (1-p)ab}{t^2\log t} }\right\rceil.$$ Let $a'= \lfloor a/k \rfloor,b'= \lfloor b/k \rfloor$. By (\[e:ab\]) and (\[e:b\]), we have $b' \geq 1/{\varepsilon}'$ given that $C$ is large enough. In particular, this implies that $b' \geq (1-{\varepsilon}')b/k$ and $a' \geq (1-{\varepsilon}')a/k$.
By the Chernoff bound $$\Pr [ {\textup{\textsf{e}}}({\bf G}(a',b',p)) \leq (1-{\varepsilon}')pa'b ' ] \leq e^{- ({\varepsilon}')^2 pa'b'/2} \leq e^{-p/2}.$$ Combining this observation \[l:random\] we deduce that for large enough $C$, there exists a bipartite graph $G'$ with no $K_t$ minor and a bipartition $(A',B')$ such that $|A'|=a',|B'|=b'$ and ${\textup{\textsf{e}}}(G') \geq (1-{\varepsilon}')pa'b '$.
We obtain $G$ by taking $k$ vertex disjoint copies of $G'$ (and adding isolated vertices if necessary). Then $$\begin{aligned}
{\textup{\textsf{e}}}(G) &\geq (1-{\varepsilon}')kpa'b ' \geq (1-{\varepsilon}')^3p \frac{ab}{k} \\ &\geq (1-{\varepsilon}')^4 p \frac{ab}{\sqrt{\frac{-2\log (1-p)ab}{t^2\log t}}} \\ & = (1 - {\varepsilon}')^4\frac{1}{\sqrt 2}\frac{1-e^{-x}}{\sqrt{x}}t\sqrt{\log t}\sqrt{ab} \\ &\geq
(1-{\varepsilon}) \frac{\lambda}{\sqrt{2}} t\sqrt{\log t}\sqrt{|A||B| },
\end{aligned}$$ as desired.
\[c:random\] shows that the bound in \[t:logbip\] is tight up to the constant factor. We believe that the constant in \[c:random\] is likely asymptotically optimal.
Next we establish a lower bound on the size of graphs with given connectivity and no $K_t$ minor. A standard easy argument shows that with high probability ${\bf G}(a,b,1/2)$ is $(1 - o(1))(b/2)$-connected for $a \geq b$, as long as $a$ is not too large compared to $b$, as formalised in the next lemma.
\[l:random2\] For every $0 < {\varepsilon}< 1$ and all integers $a \geq b \geq 1$ such that $a(a+1) \leq \exp({\varepsilon}^2b/32)$ we have $$\Pr \left[\kappa({\bf G}(a,b,1/2)) < (1 - {\varepsilon})\frac{b}{2}\right] \leq \exp(-{\varepsilon}^2b/64)$$
Again let $(A,B)$ be the bipartition of $G={\bf G}(a,b)$ as in the definition, and let $k = (1-{\varepsilon})b/2$. By the Chernoff bound $$\Pr \left [\deg(v) < k \right] \leq \exp(-{\varepsilon}^2b/8),$$ for every $v \in A$, and the probability that a pair of vertices $v_1,v_2 \in A$ share at most $k/2$ neighbors is at most $\exp(-{\varepsilon}^2b/32).$ Analogous bounds with $b$ replaced by $a$ hold for vertices in $B$. Thus with probability at least $$1 - a(a+1)\exp(-{\varepsilon}^2b/32) \geq 1 - \exp(-{\varepsilon}^2b/64)$$ every vertex of $G$ has degree at least $k$ and every pair of vertices of $G$ on the same side of the bipartion share more than $k/2$ neighbors.
These properties are sufficient to guarantee that $\kappa(G) \geq k$ implying the lemma. Indeed, consider $X \subseteq V(G)$ with $|X| < k$ and assume first $|A \cap X|< k/2$. Then every pair of vertices of $B - X$ share a neighbor in $A - X$, and so $B-X$ lies in a single component of $G \setminus X$. As every vertex in $A-X$ has a neighbor in $B-X$ it follows that $G \setminus X$ is connected. The case $|B \cap X|< k/2$ is completely analogous.
\[c:random2\] There exist ${\varepsilon}, t_0 >0$ such that for all integers $t \geq t_0$ and every integer $k \le {\varepsilon}t \sqrt{\log t}$ there exists a graph $G$ with $\kappa(G) \geq k$ and $${\textup{\textsf{v}}}(G) \geq {\varepsilon}\frac{t^2\log t}{k}.$$
If $k \leq t-2$ then $K_{a,t-2}$ satisfies the corollary for $a$ large enough. Otherwise, let $b = 3k$ and let $a = \lceil \frac{1}{6} \frac{t^2\log t}{k} \rceil$. Then by Lemmas \[l:random\] and \[l:random2\] $G = {\bf G}(a,b,1/2)$ has no $K_t$ minor and satisfies $\kappa(G) \geq k$ for large enough $t$ and small enough ${\varepsilon}$. As ${\textup{\textsf{v}}}(G) \geq a \geq \frac{1}{6} \frac{t^2\log t}{k}$ the corollary follows for ${\varepsilon}\leq 1/6$.
It follows from \[c:random2\] that the bound $t(\log t)^{3-5\beta +o(1)}$ in \[t:connect\] can not be improved beyond $O(t(\log t)^{1-\beta})$.
List coloring vs. Hall ratio {#s:alon}
============================
Let $K_{m*r}$ denote the complete $r$-partite graph with $m$ vertices in every part. Alon [@Alon92] has proved the following.
\[t:alon\] There exists $C_{\ref{t:alon}} > 0$ such that for every $m \geq 2$ $$\chi_l(K_{m*r}) \leq C_{\ref{t:alon}}r \log(m).$$
The *Hall ratio* of a graph $G$ is defined to be $\max_{H\subseteq G} \left\lceil \frac{{\textup{\textsf{v}}}(H)}{\alpha(H)} \right\rceil$. We use Theorem \[t:alon\] to prove the following theorem relating the list chromatic number of a graph with its Hall ratio and number of vertices.
\[t:listHall\] There exists $C =C_{\ref{t:listHall}} > 0$ satisfying the following. Let $\rho \geq 3$, and let $G$ be a graph with the Hall ratio at most $\rho$, and let $n = {\textup{\textsf{v}}}(G)$. If $n \geq 2\rho$, then $$\chi_l(G) \leq C\rho\log^2\left(\frac{n}{\rho}\right).$$
We show by induction on $n$ that $C=\max\{16 C_{\ref{t:alon}}, \frac{3e}{\log 2}\}$ satisfies the theorem. The theorem clearly holds for $n \leq 3e\rho$ for this choice of $C$, so we assume that $n \geq 3e\rho$ for the induction step.
Consider an assignment of lists $\{L(v)\}_{v \in V(G)}$ of colors of size $l \geq C\rho\log^2(\frac{n}{\rho})$ to the vertices of $G$. Select a subset $L_1$ of colors by choosing every color independently at random with probability $1/\log\left(\frac{n}{\rho}\right)$. Let $L_1(v)$ denote the set of colors in $L_1$ assigned to $v$. By the Chernoff bound, we have $$\Pr\left [ |L_1(v)| \leq \frac{C}{2}\rho \log\left(\frac{n}{\rho}\right)\right] \leq \exp\left(-\frac{1}{8}C\rho\log\left(\frac{n}{\rho}\right)\right) < \frac{1}{2n},\footnote{The last inequality holds as $C \geq 8$ and $$\rho \log\left(\frac{n}{\rho}\right) \geq \rho + \log\left(\frac{n}{\rho}\right) + 1 \geq \log{2}+\log{\rho} + \log\left(\frac{n}{\rho}\right) = \log(2n)$$}$$ and, similarly, $$\Pr \left[ |L_1(v)| \geq \frac{3}{2} C\rho\log\left(\frac{n}{\rho}\right) \right] < \frac{1}{2n}.$$ Thus by the union bound, with positive probability none of these events happen for any vertex $v$ of $G$. So we may assume that for every vertex $v$ of $G$, we have $$\frac{C}{2}\rho\log\left(\frac{n}{\rho}\right) \leq |L_1(v)| \leq \frac{3}{2}C\rho\log\left(\frac{n}{\rho}\right).$$
Let $s = \left\lfloor \frac{n}{e\rho}\right\rfloor$. We repeatedly select disjoint independent sets $X_1,X_2,\ldots,X_k$ in $G$ of size $s$, where $k = \left\lceil \left(1 -\frac{1}{e}\right)\frac{n}{s} \right\rceil$. This is possible, as $G - \cup_{j=1}^{i}X_j$ is a subgraph of $G$ on at least $n-(k-1)s \geq n/e$ vertices for every $i \in [k-1]$, and so $G - \cup_{j=1}^{i}X_j$ contains an independent set of size at least $s$ by definition of $\rho$.
Note that $s \geq 3$ by the choice of $n$, and so $s \geq \frac{3n}{4e \rho}$. Thus $$k \leq \left\lceil \frac{4(e-1)}{3} \rho \right\rceil \leq 4 \rho.$$ Let $X = \cup_{i=1}^{k} X_i$. By Theorem \[t:alon\] and the above bounds on $k$ and $s$, we have $$\chi_l(G[X]) \leq C_{\ref{t:alon}}k\log{s} \leq 4C_{\ref{t:alon}}\rho \left(\log\left(\frac{n}{\rho}\right) + 2 \right) \leq 8C_{\ref{t:alon}} \rho \log\left(\frac{n}{\rho}\right)$$ and so there exists an $L_1$-coloring $\phi_1$ of $G[X]$,as $C \geq 16 C_{\ref{t:alon}}$.
By the induction hypothesis, we have $$\begin{aligned}
\chi_l(G \setminus X) &\leq C\rho\log^2\left(\frac{n}{e\rho}\right) = C \rho \left(\log\left(\frac{n}{\rho}\right) - 1 \right)^2\\
&= C\rho\log^2\left(\frac{n}{\rho}\right) - 2 C \rho \log\left(\frac{n}{\rho}\right) + C \rho\\
&\leq C\rho\log^2\left(\frac{n}{\rho}\right) - \frac{3}2{}C\rho\log\left(\frac{n}{\rho}\right),
\end{aligned}$$ where the last inequality follows since $n\ge e^2\rho$. Thus $G\setminus X$ has an $L_2$-coloring $\phi_2$, where $L_2(v) = L(v)\setminus L_1(v)$ for every $v\in V(G\setminus X)$. But then $\phi_1\cup \phi_2$ is an $L$-coloring of $G$ as desired.
\[c:listsmall\] There exists $C= C_{\ref{c:listsmall}} > 0$ satisfying the following. If $G$ is a graph with no $K_t$ minor for some $t \ge 2$ and ${\textup{\textsf{v}}}(G) \geq 4t$, then $$\chi_l(G) \leq Ct\log^2\left(\frac{{\textup{\textsf{v}}}(G)}{2t}\right).$$
It follows from Theorem \[t:DucMey\] that the Hall ratio of $G$ is at most $2t$. The corollary now follows from Theorem \[t:listHall\] with $\rho = 2t$.
Proof of \[t:main\] {#s:final}
===================
Before we prove Theorem \[t:main\], we first need the following definition and lemma. If $G$ is a graph and $X\subseteq V(G)$, then the *coboundary* of $X$ in $G$ is $(\bigcup_{v\in X} N(v))\setminus X$.
\[l:contract\] Let $k\ge 1$. If $G$ is a non-empty graph with minimum degree $d\ge 6k$, then there exists a non-empty $X\subseteq V(G)$ and a matching $M$ from the coboundary $Y$ of $X$ to $X$ that saturates $Y$ such that $|Y|\le 3k$ and $G[X\cup Y]/M$ is $k$-connected.
Suppose not. Let $X$ be a non-empty subset of $V(G)$ such that the coboundary $Y$ of $X$ has size at most $3k$ and subject to that $|X|$ is minimized. Such an $X$ exists as $V(G)$ has empty coboundary.
We claim that there exists a matching $M$ from $Y$ to $X$ that saturates $Y$. Suppose not. By Hall’s theorem, there exists $S\subseteq Y$, such that $|N(S)\cap X| < |S|$. If $N(S)\cap X = X$, then $|X| < 3k$ and hence every vertex in $X$ has degree at most $|X|+|Y|-1 < 6k \leq d$, a contradiction. So we may assume that $X' = X \setminus (N(S)\cap X)\ne \emptyset$. But then $X'$ has a coboundary of size at most $|N(S)\cap X| + |Y\setminus S| < |Y| \le 3k$ and $|X'| < |X|$, contradicting the minimality of $|X|$. This proves the claim.
Let $G' = G[X\cup Y]/M$. If $G'$ is $k$-connected, then the desired outcome of the lemma holds, contradicting that $G$ is a counterexample. So we may assume that $G'$ is not $k$-connected. More formally, that is, there exist $A',B'\subseteq V(G')$ such that $A'\setminus B', B'\setminus A' \ne \emptyset$, $A'\cup B' = V(G')$, $|A'\cap B'| \leq k-1$, and ${\textup{\textsf{e}}}(G'(A'\setminus B', B'\setminus A'))=0$. Let $A$ be the subset of $V(G)$ corresponding to $A'$ in $G'$, and similarly let $B$ be the subset of $V(G)$ corresponding to $B'$ in $G'$.
Since $A'\setminus B'\ne \emptyset$ and $M$ saturates $Y$, it follows that $(A\cap X)\setminus B \ne\emptyset$. Similarly, $(B\cap X)\setminus A \ne \emptyset$.
Now $|A\cap B \cap Y| \leq |A'\cap B'| \leq k-1$. Hence $|A\cap Y| + |B\cap Y| \le |Y| + |A\cap B\cap Y| \le 4k-1$. Moreover, $|A\cap B\cap X|\leq k-1$.
Yet if $|A\cap (Y\cup B)| \le 3k$, then $A\setminus (Y\cup B) = (A\cap X)\setminus B$ contradicts the minimality of $X$. So we may assume that $|A\cap (Y\cup B)| > 3k$. But then $|A\cap Y| > 2k$. Similarly, $|B\cap Y| > 2k$ as otherwise $B\setminus (Y\cup A) = (B\cap X)\setminus A$ contradicts the minimality of $X$. But now $|A\cap Y| + |B\cap Y| > 4k$, a contradiction.
This yields the following structural corollary for $K_t$-minor-free graphs.
\[c:connect\] For every $\delta > 0$ and $1/2 \geq \beta > 1/4$, there exists $C=C_{\ref{c:connect}}(\beta,\delta)>0$ such that for every $t \geq 3$ if a non-empty graph $G$ has no $K_t$ minor, then there exists a non-empty subset $X$ of $V(G)$ such that coboundary of $X$ has size at most $Ct(\log t)^{\beta}$ and $|X| \le t(\log t)^{3-5\beta + \delta}$.
We show that $C=6(C_{\ref{t:connect}}(\beta,\delta)+1)$ satisfies the corollary. Let $d=Ct(\log t)^{\beta}$. If there exists a vertex $v\in V(G)$ with degree at most $d$ in $G$, then $X=\{v\}$ is as desired. So we may assume that $G$ has minimum degree at least $d$. Let $k = \lfloor d/6 \rfloor \geq C_{\ref{t:connect}}(\beta,\delta) t(\log t)^{\beta}$. By Lemma \[l:contract\], there exists a non-empty $X\subseteq V(G)$ and a matching $M$ from the coboundary $Y$ of $X$ to $X$ that saturates $Y$ such that $|Y|\le 3k < d$ and $G'=G[X\cup Y]/M$ is $k$-connected. By Theorem \[t:connect\], ${\textup{\textsf{v}}}(G') \le t (\log t)^{3-5\beta+\delta}$. Hence $|X| \le t (\log t)^{3-5\beta+\delta}$ as desired.
We are now ready to prove Theorem \[t:main\], which we restate for convenience.
Let $C=C_{\ref{c:connect}}(\beta,1)$. We show that for $t \gg C$, every graph $G$ with no $K_t$ minor is $2\lceil Ct(\log t)^{\beta} \rceil$-list-colorable, which implies the theorem. Let $d = \lceil Ct(\log t)^{\beta} \rceil$. Suppose for a contradiction that there exists $G$ with no $K_t$ minor and a $2d$-list assignment $L$ such that $G$ is not $L$-colorable, and choose such a graph $G$ with ${\textup{\textsf{v}}}(G)$ minimum. By the choice of $C$, there exists a non-empty $X\subseteq V(G)$ such that coboundary $Y$ of $X$ has size at most $d$ and $|X| \le Ct(\log t)^{4-5\beta}$. By minimality, there exists an $L$-coloring $\phi$ of $G-X$. For each $v\in X$, let $L'(v) = L(v)\setminus \{\phi(w): w\in N(v)\setminus X\}$. Since $N(v)\setminus X\subseteq Y$ for each $v\in X$ by definition of coboundary, we have that $|N(v)\setminus X| \le |Y|\le d$. Hence for each $v\in X$, $|L'(v)|\ge |L(v)|- |Y| \ge d$. By Corollary \[c:listsmall\], we have that $\chi_{\ell}(G[X]) \le C_{\ref{c:listsmall}}t \log^2(C(\log t)^{4-5\beta}) \le d$ for large enough $t$. Hence $G[X]$ has an $L'$-coloring $\phi'$. But now $\phi\cup \phi'$ is an $L$-coloring of $G$, a contradiction.
Further Improvements {#s:remarks}
====================
The central obstacle in improving the bound on the chromatic number (and the list chromatic number) of graphs with no $K_t$ minors using our methods is the absence of the analogue of \[t:connect\] for graphs of connectivity $o(t (\log t)^{1/4})$. To determine the limits of this strategy it would be interesting to answer the following question.
For which $\beta >0 $, does there exist $C>0$ such that for every integer $t \geq 3$, every graph $G$ with $\kappa(G) = \Omega(t(\log t)^\beta)$ and no $K_t$ minor satisfies ${\textup{\textsf{v}}}(G) \leq t (\log t)^C$?[^4]
### Acknowledgement. {#acknowledgement. .unnumbered}
We thank Zi-Xia Song for valuable comments.
[^1]: Department of Mathematics and Statistics, McGill University. Email: [sergey.norin@mcgill.ca]{}. Supported by an NSERC Discovery grant.
[^2]: Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada. Email: [lpostle@uwaterloo.ca]{}. Canada Research Chair in Graph Theory. Partially supported by NSERC under Discovery Grant No. 2019-04304, the Ontario Early Researcher Awards program and the Canada Research Chairs program.
[^3]: Otherwise, $n^2 \log{n} \leq (20)^2 t^2(\log t + \log 20) \leq (20)^2(\log(20)+1) t^2\log{t} < Ct^2\log{t}/4$.
[^4]: The last condition can be replaced by weaker, but less transparent inequality $${\textup{\textsf{v}}}(G) \leq t e^{o(\log t)}.$$
|
---
abstract: 'We make explicit computations in the formal symplectic geometry of Kontsevich and determine the Euler characteristics of the three cases, namely commutative, Lie and associative ones, up to certain weights. From these, we obtain some non-triviality results in each case. In particular, we determine the [*integral*]{} Euler characteristics of the outer automorphism groups $\mathrm{Out}\, F_n$ of free groups for all $n\leq 10$ and prove the existence of plenty of rational cohomology classes of [*odd*]{} degrees. We also clarify the relationship of the commutative graph homology with finite type invariants of homology $3$-spheres as well as the leaf cohomology classes for transversely symplectic foliations. Furthermore we prove the existence of several new [*non-trivalent*]{} graph homology classes of [*odd*]{} degrees. Based on these computations, we propose a few conjectures and problems on the graph homology and the characteristic classes of the moduli spaces of graphs as well as curves.'
address:
- 'Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan'
- 'Department of Mathematics, Akita University, 1-1 Tegata-Gakuenmachi, Akita, 010-8502, Japan'
author:
- Shigeyuki Morita
- Takuya Sakasai
- Masaaki Suzuki
title: Computations in formal symplectic geometry and characteristic classes of moduli spaces
---
Introduction and statements of the main results {#sec:intro}
===============================================
In celebrated papers [@kontsevich1][@kontsevich2], Kontsevich considered three infinite dimensional Lie algebras, namely commutative, Lie and associative ones. He proved that the stable homology group of each of these Lie algebras is isomorphic to a free graded commutative algebra generated by the stable homology group of $\mathfrak{sp}(2g,{\mathbb{Q}})$ as $g$ tends to infinity together with certain set of generators which he described explicitly. It is the totality of the [*graph homologies*]{} for the commutative case, the totality of the cohomology groups of the outer automorphism groups of free groups, denoted by $\mathrm{Out}\,F_n$, for the Lie case, and the totality of the cohomology groups of the moduli spaces of curves with [*unlabeled*]{} marked points, denoted by $\mathbf{M}_g^m/\mathfrak{S}_m$, for the associative case.
As for the commutative (reps. associative) case, Kontsevich described a general method of constructing cycles of the corresponding graph complex by making use of finite dimensional Lie (reps. $A_\infty$) algebras with non-degenerate invariant scalar products. In the Lie case, however, he mentioned that no non-trivial class was obtained by similar construction. In the associative case, he also introduced a dual construction of producing cocycles starting from a differential associative algebra with non-degenerate odd scalar product and trivial cohomology. Certain detailed description and generalizations of these methods have been given by several authors including Hamilton, Lazarev (see e.g. [@hl][@h]) and others.
However there have been known only a few results which deduce new information about the graph homology and cohomology groups of $\mathrm{Out}\,F_n$ or $\mathbf{M}_g^m/\mathfrak{S}_m$ by making a direct use of the above theorem of Kontsevich. First, as for the Lie case, in [@morita99] the first named author defined a series of certain unstable homology classes of $\mathrm{Out}\,F_n$ by using his trace maps introduced in [@morita93]. Only the first three classes are known to be non-trivial (see Conant and Vogtmann [@cov] and Gray [@gr]). Second, recently Conant, Kassabov and Vogtmann [@ckv] made a remarkable new development in this direction and defined many more classes. Thirdly, in the commutative case, the existence of two graph homology classes of [*odd*]{} degrees was proved, one in Gerlits [@ge] and the other in Conant, Gerlits and Vogtmann[@cgv]. Fourthly, as for the associative case, in [@morita08] a series of certain unstable homology classes for genus $1$ moduli spaces was introduced, all of which have been proved to be non-trivial by Conant [@con]. Finally, in our recent paper [@mss1] we determined the stable abelianization of the Lie algebra in the associative case. As an application of this result, we obtained a new proof of an unpublished result of Harer. Church, Farb and Putman [@cfp1] gave a different proof.
The purpose of this paper is to continue these lines of investigations. We obtain new results in each of the three cases.
To be more precise, let $\Sigma_{g,1}$ be a compact oriented surface of genus $g\geq 1$ with one boundary component and we denote its first rational homology group $H_1(\Sigma_{g,1};{\mathbb{Q}})$ simply by $H_{\mathbb{Q}}$. It can be regarded as the standard symplectic vector space of dimension $2g$ induced from the intersection pairing on it. Let $\mathfrak{c}_g$ denote the graded Lie algebra consisting of Hamiltonian polynomial vector fields, without constant terms, on $H_{\mathbb{Q}}\otimes{\mathbb{R}}\cong{\mathbb{R}}^{2g}$ with rational coefficients. The homogeneous degree $k$ part, denoted by $\mathfrak{c}_g(k)$, can be naturally identified with $S^{k+2}H_{\mathbb{Q}}$ where $S^{k}H_{\mathbb{Q}}$ denotes the $k$-th symmetric power of $H_{\mathbb{Q}}$. Let $\mathfrak{c}^+_g$ be the ideal of $\mathfrak{c}_g$ consisting of Hamiltonian polynomial vector fields without linear terms. Next, we denote by $\mathcal{L}_{H_{\mathbb{Q}}}$ the free Lie algebra generated by $H_{\mathbb{Q}}$. Let $\mathfrak{h}_{g,1}$ be the graded Lie algebra consisting of [*symplectic*]{} derivations of $\mathcal{L}_{H_{\mathbb{Q}}}$ and let $\mathfrak{h}^+_{g,1}$ be the ideal consisting of derivations with [*positive*]{} degrees. This Lie algebra was introduced in the theory of Johnson homomorphisms before the work of Kontsevich (see [@morita89]) and has been investigated extensively. We use our notation for this Lie algebra. The notation $\mathfrak{h}_g$ is reserved for the case of a closed surface (see Remark \[rem:t\]) while $\mathfrak{h}_{g,1}$ corresponds to genus $g$ compact surface with one boundary component.
Finally, let $T_0H_{\mathbb{Q}}$ denote the free associative algebra without unit generated by $H_{\mathbb{Q}}$. Let $\mathfrak{a}_{g}$ be the graded Lie algebra consisting of [*symplectic*]{} derivations of $T_0H_{\mathbb{Q}}$ and let $\mathfrak{a}^+_{g}$ be the ideal consisting of derivations with [*positive*]{} degrees.
We denote by $\mathrm{Sp}(2g,{\mathbb{Q}})$ the symplectic group which we sometimes denote simply by $\mathrm{Sp}$. If we fix a symplectic basis of $H_{\mathbb{Q}}$, then the space $H_{\mathbb{Q}}$ can be regarded as the standard representation of $\mathrm{Sp}(2g,{\mathbb{Q}})$. Each piece $\mathfrak{c}_g(k), \mathfrak{h}_{g,1}(k), \mathfrak{a}_{g}(k)$, of the three graded Lie algebras, is naturally an $\mathrm{Sp}$-module so that it has an irreducible decomposition. It is known that this decomposition stabilizes when $g$ is sufficiently large.
Now we describe our main results. We determine the dimensions of the chain complexes which compute the $\mathrm{Sp}$-invariant stable homology of the three Lie algebras $\mathfrak{c}^+_g, \mathfrak{h}^+_{g,1}, \mathfrak{a}^+_g$ up to certain weights (see Tables \[tab:c\], \[tab:h\] and \[tab:a\]). From this, we determine the Euler characteristic of each weight summand and the result is given as follows. For the definition of [*weight*]{}, see Section $2$.
The Euler characteristics $\chi$ of the $\mathrm{Sp}$-invariant stable homologies of the three Lie algebras $\mathfrak{c}^+_g, \mathfrak{h}^+_{g,1}, \mathfrak{a}^+_g$ up to weight $20$, $18$ or $16$ are given as follows. $$\begin{aligned}
\mathrm{(i)}\ & \chi(H_*(\mathfrak{c}^+_{\infty})^{\mathrm{Sp}}_{w})=1,2,3,6,8,14,20,32,44, 68\quad (w=2,4,\ldots,20)\\
\mathrm{(ii)}\ & \chi(H_*(\mathfrak{h}^+_{\infty,1})^{\mathrm{Sp}}_{w})=1,2,4,6,10,16,23,13,-96\quad (w=2,4,\ldots,18)\\
\mathrm{(iii)}\ & \chi(H_*(\mathfrak{a}^+_{\infty})^{\mathrm{Sp}}_{w})=2,5,12,24,50,100, 188, 347\quad (w=2,4,\ldots,16).\end{aligned}$$ \[thm:chi\]
By combining Theorem \[thm:chi\] above with the description of the generators of the stable homologies due to Kontsevich, we obtain the following result. Part $\mathrm{(ii)}$ proves, in particular, the existence of [*odd*]{} dimensional rational homology classes of the outer automorphism groups of free groups for the [*first*]{} time.
The integral Euler characteristics $e$ of the primitive part of $\mathrm{Sp}$-invariant stable homologies of the three Lie algebras $\mathfrak{c}^+_g, \mathfrak{h}^+_{g,1}, \mathfrak{a}^+_g$, up to weight $20$, $18$ or $16$, are given as follows. $$\begin{aligned}
\mathrm{(i)}\ & e(G_*^{(n)})=1,1,1,2,1,2,2,2,1,3\quad (n=2,3,\ldots,11; w=2n-2)\\
\mathrm{(ii)}\ & e(\mathrm{Out}\, F_{n})=1,1,2,1,2,1,1,-21,-124\quad (n=2,3,\ldots,10; w=2n-2)\\
\mathrm{(iii)}\ & \sum_{\begin{subarray}{c}
2g-2+m=n\\ m>0
\end{subarray}}
e(\mathbf{M}_g^m/\mathfrak{S}_m)=2,2,4,2,6,6, 6, 1\quad (n=1,2,\ldots,8; w=2n)\end{aligned}$$ \[thm:chip\]
Here $G_*^{(n)}$ denotes the graph complex due to Kontsevich which is defined in terms of graphs with the Euler characteristic $(1-n)$, $\mathrm{Out}\,F_n$ denotes the outer automorphism group of the free group $F_n$ of rank $n$, $\mathbf{M}_g^m$ denotes the moduli space of curves of genus $g$ with labeled $m$ marked points and $\mathfrak{S}_m$ denotes the $m$-th symmetric group.
The commutative case of the graph homology has deep connections with two important subjects in topology. One is the theory of [*finite type*]{} invariants for homology $3$-spheres initiated by Ohtsuki [@oh]. The other is the theory of characteristic classes of [*transversely symplectic*]{} foliations going back to Gelfand, Kalinin and Fuks [@gkf] and more recently developed by Kontsevich [@kontsevich3] and further in [@metoki][@km]. On the other hand, a beautiful connection between these two theories was found by Garoufalidis and Nakamura [@gn].
Let $\mathcal{A}(\phi)$ denote the commutative algebra generated by vertex oriented connected trivalent graphs modulo the two relations, one is the $\mathrm{(AS)}$ relation and the other is the $\mathrm{(IHX)}$ relation. This algebra plays a fundamental role in the former theory above. In fact, Le [@l] and Garoufalidis and Ohtsuki [@go] proved that the graded algebra associated to the Ohtsuki filtration on the space of all the homology $3$-spheres is isomorphic to $\mathcal{A}(\phi)$ which is a polynomial algebra generated by the subspaces $\mathcal{A}^{(2n-2)}_{\mathrm{conn}} \ (n=2,3,\ldots)$ corresponding to connected graphs with $(2n-2)$ vertices. Furthermore the completion $\widehat{\mathcal{A}}(\phi)$ of $\mathcal{A}(\phi)$ with respect to its gradings serves as the target of the $\mathrm{LMO}$ invariant introduced in [@lmo].
Based on a result of Garoufalidis and Nakamura cited above, we show that the top homology group $H_{2n-2}(G_*^{(n)})$ of $G_*^{(n)}$ is canonically isomorphic to $\mathcal{A}^{(2n-2)}_{\mathrm{conn}}$ so that $\mathcal{A}(\phi)$ can be embedded into $H_*(\mathfrak{c}^+_\infty)^{\mathrm{Sp}}$ as a bigraded subalgebra. We define $\mathcal{E}$ to be the [*complementary*]{} bigraded algebra (see Definition \[def:e\] for details) so that we have an isomorphism $H_*(\mathfrak{c}^+_\infty)^{\mathrm{Sp}}\cong\mathcal{A}(\phi)\otimes
\mathcal{E}$ of bigraded algebras. This bigraded algebra $\mathcal{E}$ can be interpreted as the space of all the graph homology classes represented by [*non-trivalent*]{} graphs. In the context of the theory of stable leaf cohomology classes for transversely symplectic foliations, it can also be interpreted as the dual space of all the [*exotic*]{} characteristic classes. Here by [*exotic*]{} we mean that the class depends on higher jets than the connection as well as the curvature forms by which the usual secondary characteristic classes are defined. See Section $5$ for details.
Now we can deduce the following result from Theorem \[thm:chip\] (i).
There exists an isomorphism $$H_*(\mathfrak{c}^+_\infty)^{\mathrm{Sp}}\cong\mathcal{A}(\phi)\otimes
\mathcal{E}$$ of bigraded algebras. If we denote by $P\mathcal{E}$ the primitive part of $\mathcal{E}$, then the Euler characteristic of its weight $w$-part $P\mathcal{E}_w$ is given by $$e(P\mathcal{E}_w)=0,0,0,0,-1,-1,-2,-3,-5,-5\quad (w=2,4,\ldots,20).$$ It follows that there exist several odd dimensional non-trivalent graph homology classes, as well as exotic stable leaf cohomology classes for transversely symplectic foliations, in each of the weights $w=10,12,\ldots,20$. \[thm:ce\]
The above theorem in the range $w\leq 10$ is essentially due to Gerlits [@ge], Theorem 4.1, and the case $w=12$ is due to Conant, Gerlits and Vogtmann [@cgv], Theorem 5.1. In fact, in the former paper the author computed, among other things, $H_*(G_*^{(n)})$ for all $n\leq 6$ and the case $n=7$ was treated in the latter paper. If we combine the former result with the above connection with the theory of foliations, we can conclude the existence of a certain [*exotic*]{} stable leaf cohomology class of transversely symplectic foliations of degree $7$ and weight $10$. This is the [*first*]{} appearance of such classes.
Next we consider Theorem \[thm:chip\] $\mathrm{(ii)}$. As is well-known, there is a beautiful formula for the [*rational*]{} Euler characteristics of the mapping class groups due to Harer-Zagier [@hz] and Penner [@p]. In the case of $\mathrm{Out}\,F_{n}$, Smillie and Vogtmann [@sv] obtained a generating function for the rational Euler characteristics $\chi(\mathrm{Out}\,F_{n})$ and computed them for all $n\leq 100$. They are all negative and they conjecture that they are always negative. However, compared to the case of the mapping class groups, there still remain many open problems. For example, the relation between the rational and the [*integral*]{} Euler characteristics seems to be not very well understood. We will compare our computation above with the result of Smillie and Vogtmann cited above in Table \[tab:chie\] and we observe a very interesting behavior of the two numbers for the first time. See Section $6$ for details of this as well as other discussions of our results.
Finally we consider the case of $\mathfrak{a}_g$. In this case, Kontsevich proved that the primitive part of $H_*(\mathfrak{a}^+_\infty)^{\mathrm{Sp}}$ corresponds to the totality of the $\mathfrak{S}_m$-invariant rational cohomology groups $H^*(\mathbf{M}_{g}^m;{\mathbb{Q}})^{\mathfrak{S}_m}$ of the moduli spaces $\mathbf{M}_g^m$ of genus $g$ curves with $m$ marked points for all $g, m$ with $2g-2+m>0, m\geq 1$. There have been known many results concerning the cohomology of these moduli spaces for the cases of low genera $g=0,1,2,3,4$ due to Getzler-Kapranov [@gk], Getzler [@getz98][@getz99], Looijenga [@loo], Tommasi [@tom05][@tom06], Gorsky [@go1][@go2], Bergström [@b] and others. In Section $7$, we will check that our computation of the Euler characteristics is consistent with these known results or deduced from them by explicit computations, in the range $2g-2+m\leq 8$.
In our forthcoming paper [@mss2a], which is a sequel to this paper, we will extend both of Theorem \[thm:chi\] $\mathrm{(iii)}$ and Theorem \[thm:chip\] $\mathrm{(iii)}$ from $w=16$ to $500$ by adopting a completely different method. More precisely, we use a formula of Gorsky [@go2] for the equivariant Euler characteristics of the moduli space of curves to obtain certain closed formulas which enable us to determine the above values. However, in this paper we only describe the values which we deduced from Table \[tab:a\] in order to compare with the other two cases. See Section $7$ for more details about this point.
To compute the graph homologies directly, we have to enumerate certain types of graphs. However, according to the number of vertices increases, the difficulty of the problem of deciding whether two given graphs are isomorphic to each other grows very rapidly. In view of this, we adopted a method in a pure framework of symplectic representation theory. This has a disadvantage that the dimensions which we have to compute are considerably larger than the graph theoretical method, because there is no effective way to distinguish between connected and disconnected graphs in the framework of representation theory. In order to overcome this difficulty, we made various devise to lighten the burden imposed on computers. More precisely, our task is to determine the dimensions of the $\mathrm{Sp}$-invariant subspaces of various $\mathrm{Sp}$-modules. Theoretically there is no problem here because we know the characters of these modules completely. However, the problem lies in the huge size of the data as well as the time which computers need. We have developed several our own programs on the computer software Mathematica which realize theoretical considerations in the representation theory. See Section $4$ for details. We have also made an extensive use of the computer program LiE to obtain irreducible decompositions of various $\mathrm{Sp}$-modules. [*Acknowledgement*]{} The authors would like to thank Christophe Soulé for helpful information about the (non)vanishing of the Borel regulator classes. Thanks are also due to James Conant and Alastair Hamilton for informing the authors their works [@cgv] and [@h]. The first named author would like to thank Richard Hain and Hiroaki Nakamura for enlightening discussions about the symplectic representation theory related to the mapping class group in the $1990$’s.
The authors were partially supported by KAKENHI (No. 24740040 and No. 24740035), Japan Society for the Promotion of Science, Japan.
Preliminaries {#sec:p}
=============
In this section, we prepare a few facts about the (co)homology of graded Lie algebras which will be needed in our later considerations.
Let $\mathfrak{g}=\oplus_{k=0}^{\infty}\mathfrak{g}(k)$ be a graded Lie algebra over ${\mathbb{Q}}$ and let $\mathfrak{g}^+=\oplus_{k=1}^{\infty}\mathfrak{g}(k)$ be its ideal consisting of all the elements of $\mathfrak{g}$ with [*positive*]{} gradings. We assume that each piece $\mathfrak{g}(k)$ is finite dimensional for all $k$. Then the chain complex $C_*(\mathfrak{g})$ of $\mathfrak{g}$ splits into the direct sum $$C_*(\mathfrak{g})=\bigoplus_{w=0}^\infty C_*^{(w)}(\mathfrak{g})$$ of [*finite dimensional*]{} subcomplexes $C_*^{(w)}(\mathfrak{g})=\oplus_{i=0}^w C_i^{(w)}(\mathfrak{g})$ where $$C_i^{(w)}(\mathfrak{g})=
\bigoplus_{\begin{subarray}{c}
i_0+ i_1+\cdots + i_w=i\\
i_1+2 i_2+\cdots +w i_w=w
\end{subarray}}
\wedge^{i_0} (\mathfrak{g}(0))\otimes\wedge^{i_1} (\mathfrak{g}(1))\otimes\cdots\otimes
\wedge^{i_w} (\mathfrak{g}(w))$$ so that $C_i^{(w)}(\mathfrak{g})=0$ for $i> w+\frac{1}{2} d(d-1)\ (d=\mathrm{dim}\, \mathfrak{g}(0))$. This induces a bigraded structure on the homology group $H_*(\mathfrak{g})$ described as $$H_i(\mathfrak{g})=\bigoplus_{w=0}^\infty H_i(\mathfrak{g})_{w}$$ where $H_i(\mathfrak{g})_{w}=H_i(C_*^{(w)}(\mathfrak{g}))$. We call $H_i(\mathfrak{g})_{w}$ the [*weight*]{} $w$-part of $H_i(\mathfrak{g})$. Let $\widehat{\mathfrak{g}}$ be the completion of $\mathfrak{g}$ with respect to the grading and let $H^*_c(\widehat{\mathfrak{g}})$ be the [*continuous*]{} cohomology. Then we have $$H^k_c(\widehat{\mathfrak{g}})\cong \bigoplus_{w=0}^\infty\ \left(H_k(\mathfrak{g})_w\right)^*.$$
Now suppose that $\mathfrak{g}$ is an $\mathrm{Sp}$-graded Lie algebra by which we mean that each piece $\mathfrak{g}(k)$ is a finite dimensional representation of $\mathrm{Sp}(2g,{\mathbb{Q}})$ for some fixed $g$ such that the bracket operation $\mathfrak{g}(i)\otimes\mathfrak{g}(j)\rightarrow\mathfrak{g}(i+j)$ is a morphism of $\mathrm{Sp}$-modules for any $i,j$. We further assume that $\mathfrak{g}(0)=\mathfrak{sp}(2g,{\mathbb{Q}})\cong S^2H_{\mathbb{Q}}$. Then we have a split extension $$0\rightarrow \mathfrak{g}^+\rightarrow \mathfrak{g}\rightarrow\mathfrak{sp}(2g,{\mathbb{Q}})\rightarrow 0$$ of Lie algebras. The $E^2$-term of the Hochschild-Serre spectral sequence for the homology of $\mathfrak{g}$ is given by $$E^2_{p,q}=H_p(\mathfrak{sp}(2g,{\mathbb{Q}});H_q(\mathfrak{g}^+)).$$ By the assumption, the chain complex $C_*(\mathfrak{g}^+)$ decomposes into the direct sum of subcomplexes corresponding to $\mathrm{Sp}$-irreducible components. It follows that the homology group $H_q(\mathfrak{g}^+)$ also decomposes into the $\mathrm{Sp}$-irreducible components. In particular, we have the $\mathrm{Sp}$-invariant part which we denote by $H_q(\mathfrak{g}^+)^{\mathrm{Sp}}$. Now we consider the situation where the Borel vanishing theorem [@borel1][@borel2] applies so that $H_p(\mathfrak{sp}(2g,{\mathbb{Q}});H_q(\mathfrak{g}^+)_{\lambda})=0$ for any $p>0$ and for any $\mathrm{Sp}$-irreducible component $\lambda$ different from the trivial representation. Under this situation, we can conclude that the spectral sequence collapses at the $E^2$-term and we have an isomorphism $$H_*(\mathfrak{g})\cong H_*(\mathfrak{sp}(2g;{\mathbb{Q}}))\otimes H_*(\mathfrak{g}^+)^{\mathrm{Sp}}.$$ Such a situation occurs in the case of three Lie algebras treated in this paper. More precisely, we set $\mathfrak{g}_g$ to be one of $\mathfrak{c}_g, \mathfrak{h}_{g,1}$ or $\mathfrak{a}_g$. Then we have natural embeddings $\mathfrak{g}_g\subset \mathfrak{g}_{g+1}$ so that we can consider the union (or equivalently the direct limit) $\mathfrak{g}_\infty=\lim_{g\to\infty}\mathfrak{g}_g$. Also we have its ideal $\mathfrak{g}^+_\infty$. The homology groups of them are given by $$H_*(\mathfrak{g}_\infty)=\lim_{g\to\infty} H_*(\mathfrak{g}_g),\quad
H_*(\mathfrak{g}^+_\infty)=\lim_{g\to\infty} H_*(\mathfrak{g}^+_g).$$ Since it is well known that the $\mathrm{Sp}$-irreducible decompositions of $\mathfrak{g}_g(k)$ stabilizes as $g$ goes to $\infty$, we can apply the preceding argument to conclude that $$H_*(\mathfrak{g}_\infty)\cong H_*(\mathfrak{sp}(\infty,{\mathbb{Q}}))\otimes H_*(\mathfrak{g}^+_\infty)^{\mathrm{Sp}},\quad H_*(\mathfrak{g}^+_\infty)^{\mathrm{Sp}}=
\bigoplus_{w=1}^\infty H_*(\mathfrak{g}^+_\infty)^{\mathrm{Sp}}_w.$$ Similar formulas are also valid for the continuous cohomology, although we have to be careful here because projective limit arises rather than the direct limit.
Since $H_*(\mathfrak{g}^+_\infty)^{\mathrm{Sp}}_w$ is finite dimensional by the assumption, we have its Euler characteristic $\chi(H_*(\mathfrak{g}^+_\infty)^{\mathrm{Sp}}_w)$. We call $$g(t)=1+\sum_{w=1}^\infty \chi(H_*(\mathfrak{g}^+_\infty)^{\mathrm{Sp}}_w) \ t^w
\in {\mathbb{Z}}[[t]]$$ the [*weight*]{} generating function for the $\mathrm{Sp}$-invariant stable homology group of the $\mathrm{Sp}$-Lie algebra $\mathfrak{g}^+$. Observe here that if we replace $\mathfrak{g}^+$ with $\mathfrak{g}$ here, then we obtain the trivial function $1$ because $\chi(H_*(\mathfrak{sp}(2g,{\mathbb{Q}})))=0$.
We mention that such a kind of generating function was first considered by Perchik [@p] in the context of the [*unstable*]{} Gelfand-Fuks cohomology of the Lie algebra of formal Hamiltonian vector fields ($\mathfrak{ham}_{2g}$ in the notation of [@kontsevich3] and Section $5$ below) and later in [@km] in the context of the [*stable*]{} Gelfand-Fuks cohomology of $\lim_{g\to\infty}\mathfrak{ham}_{2g}^0=\widehat{\mathfrak{c}}_{\infty}\otimes{\mathbb{R}}$.
In the case where $\mathfrak{g}_g$ is one of the three Lie algebras considered in this paper, we can consider the [*weight*]{} generating function for the $\mathrm{Sp}$-invariant homology group of the limit Lie algebra $\mathfrak{g}^+_\infty$.
For later use, we generalize the definition of the [*weight*]{} generating function in a broader context as follows.
Let $$\mathcal{K}=\bigoplus_{d,w=0}^\infty \mathcal{K}_{d,w}$$ be a bigraded algebra over ${\mathbb{Q}}$ such that the multiplication $$\mathcal{K}_{d,w}\otimes \mathcal{K}_{d',w'}\longrightarrow \mathcal{K}_{d+d',w+w'}$$ is graded commutative with respect to $d$ (called the degree) and the weight $w$ part $$\mathcal{K}_w=\bigoplus_d K_{d,w}$$ is finite dimensional for any $w$. We define the Euler characteristic $\chi(\mathcal{K}_w)$ by $$\chi(\mathcal{K}_w)=\sum_{d} (-1)^d \mathrm{dim}\ K_{d,w}.$$ We also assume that $K_{0,0}={\mathbb{Q}}$. Hereafter we always assume the above conditions whenever we mention weight generating functions of bigraded algebras. It is easy to see that if there are given two bigraded algebras $\mathcal{K},\mathcal{K}'$ which satisfy the above conditions, then the tensor product $\mathcal{K}\otimes\mathcal{K}'$ also satisfy them with respect to the induced bigradings on it.
We define the [*weight*]{} generating function $k(t)$ for a bigraded algebra $\mathcal{K}$ as above to be $$k(t)=1+\sum_{w=1}^\infty \chi(\mathcal{K}_w)\ t^w.$$
Let $P={\mathbb{Q}}[x_1,x_2,\ldots]$ be the polynomial algebra on given variables $x_i$ with degree $2d_i$. If we set the weight to be equal to the degree, then the weight generating function for $P$ is given by $$p(t)=\prod_{i} (1+t^{2d_i}+t^{4d_i}+\cdots)
=\prod_{i}(1-t^{2d_i})^{-1}.$$ Let $E=\wedge^*[y_1,y_2,\ldots]$ be the exterior algebra on given variables $y_i$ with degree $2s_i-1$. If we set the weight to be equal to the degree, then the weight generating function for $E$ is given by $$e(t)=\prod_i (1-t^{2s_i-1}).$$ \[ex:w\]
Let $\mathcal{K}$ be a bigraded algebra and assume that it is a free graded commutative algebra with respect to the grading by degrees. Let $P\mathcal{K}$ be the subspace consisting of primitive elements. For each $d, w$, set $PK_{d,w}=P\mathcal{K}\cap K_{d,w}$ and define $$\chi^{\mathrm{pr}}_w(\mathcal{K})=\sum_{d} (-1)^d \mathrm{dim}\ PK_{d,w}.$$ Then we have $$k(t)=\prod_{w=1}^\infty (1-t^w)^{-\chi^{\mathrm{pr}}_w(\mathcal{K})}.$$ \[prop:wp\]
It is easy to see that the weight generating function of the tensor product of two graded commutative bialgebras is the product of those of each bigraded algebra. Since $\mathcal{K}$ is free by the assumption, it is the tensor product of subalgebras generated by $P\mathcal{K}$. Consider the product of weight generating functions of two graded commutative algebras each of which is generated by an element whose weight is the same whereas the degree is complementary, namely one is even and the other is odd. Then by Example \[ex:w\], we see that this product is the constant function $1$. Hence the weight generating function of $\mathcal{K}$ depends only on $\chi^{\mathrm{pr}}_w(\mathcal{K})$ and the claim follows.
The number $\chi^{\mathrm{pr}}_w(\mathcal{K})$ can be interpreted as the Euler characteristic of the space of primitive elements of weight $w$, or equivalently as the number of new generators of weight $w$ with [*even*]{} degrees minus that of new generators of weight $w$ with [*odd*]{} degrees.
Let $\mathcal{K}$ be a bigraded algebra and assume that it is a free graded commutative algebra with respect to the grading by degrees. Assume that the weight generating function $k(t)$ is determined up to weight $w_0$. Then we can determine the numbers $\chi^{\mathrm{pr}}_w(\mathcal{K})\ (w=1,2,\ldots,w_0)$ inductively by the following recursive formula $$\chi^{\mathrm{pr}}_w(\mathcal{K})=[k(t)]_{t^w}-
\left[\prod_{i=1}^{w-1} (1-t^i)^{-\chi^{\mathrm{pr}}_{i}(\mathcal{K})}\right]_{t^w}
\quad (w=1,2,\ldots,w_0)$$ where $[f(t)]_{t^w}$ denotes the coefficient of $t^w$ in a given formal power series $f(t)\in {\mathbb{Z}}[[t]]$. \[prop:wc\]
This follows from Proposition \[prop:wp\].
Computation of the irreducible decompositions {#sec:irrep}
=============================================
In this section we describe our explicit determination of the stable irreducible decompositions of $\mathfrak{h}_{g,1}(k), \mathfrak{a}_g(k)$ up to certain degrees. At present, we have determined them for all $k\leq 20$. As already mentioned in the introduction, in the commutative case, we have an isomorphism $\mathfrak{c}_g(k)\cong S^{k+2} H_{\mathbb{Q}}$ which is known to be an irreducible representation for any $k$.
To describe our result, we fix our notations. Any Young diagram ${\lambda}=[{\lambda}_1\cdots{\lambda}_h]$ with $k$ boxes defines an irreducible representation of the symmetric group $\mathfrak{S}_k$ which we denote by ${\lambda}_{\mathfrak{S}_k}$ or sometimes simply by the same symbol ${\lambda}$. Thus $[k]$ corresponds to the trivial representation and $[1^k]$ the alternating representation. Here we use a simplified notation to express Young diagrams. For example $[422]$ will be denoted by $[42^2]$. For any Young diagram ${\lambda}=[{\lambda}_1\cdots{\lambda}_h]$ as above and for any $n\geq h$, let $
{\lambda}_{\mathrm{GL}}
$ be the corresponding irreducible representation of $\mathrm{GL}(n,{\mathbb{Q}})$. Similarly for any $g\geq h$, let $
{\lambda}_{\mathrm{Sp}}
$ be the corresponding irreducible representation of $\mathrm{Sp}(2g,{\mathbb{Q}})$.
Our method of computing the stable irreducible decompositions for the Lie case $\mathfrak{h}_{g,1}$ as well as the associative case $\mathfrak{a}_g$ can be described as follows. For the former case, we use the following result of Kontsevich.
\[thm:k\] Let $W_k$ be the $\mathfrak{S}_{k+2}$-module with character $\chi_k(1^{k+2})=k!, \chi_k(1^1a^b)=(b-1)! a^{b-1} \mu(a),
\chi_k(a^b)=-(b-1)! a^{b-1} \mu(a)$, and $\chi_k$ vanishes on all the other conjugacy classes, where $\mu$ denotes the Möbius function. Then there exists an isomorphism $$\mathfrak{h}_{g,1}(k)\cong H_{\mathbb{Q}}^{\otimes (k+2)}\otimes_{A_{k+2}} W_k$$ of $\mathrm{Sp}(2g,{\mathbb{Q}})$-modules, where $A_{k+2}={\mathbb{Q}}\mathfrak{S}_{k+2}$ denotes the group algebra of $\mathfrak{S}_{k+2}$.
We have given in [@mss2] a simple proof of this result using only the standard representation theory (see also [@es]). As an immediate corollary to this theorem, we obtain the following.
\[cor:h\] Let ${\lambda}=[{\lambda}_1\cdots{\lambda}_{h}]$ be a Young diagram with $(k+2)$ boxes and let ${\lambda}_{\mathfrak{S}_{k+2}}$ be the corresponding irreducible representation of the symmetric group $\mathfrak{S}_{k+2}$. Then the multiplicity $m_{\lambda}$ of $V_{\lambda}=H_{\mathbb{Q}}^{\otimes (k+2)}\otimes_{A_{k+2}}{\lambda}_{\mathfrak{S}_{k+2}}$ in $\mathfrak{h}_{g,1}(k)$ is expressed as $$m_{\lambda}=\frac{1}{(k+2)!} \sum_{\gamma\in \mathfrak{S}_{k+2}}
\chi_k (\gamma)\chi_{\lambda}(\gamma)$$ where $\chi_{\lambda}$ denotes the character of ${\lambda}_{\mathfrak{S}_{k+2}}$.
For the case of $\mathfrak{a}_g(k)$, we have the following result.
\[prop:a\] Let ${\lambda}=[{\lambda}_1\cdots{\lambda}_{h}]$ be a Young diagram with $(k+2)$ boxes and let ${\lambda}_{\mathfrak{S}_{k+2}}$ be the corresponding irreducible representation of the symmetric group $\mathfrak{S}_{k+2}$. Then the multiplicity $n_{\lambda}$ of $V_{\lambda}=H_{\mathbb{Q}}^{\otimes (k+2)}\otimes_{A_{k+2}}{\lambda}_{\mathfrak{S}_{k+2}}$ in $\mathfrak{a}_{g}(k)$ is expressed as $$n_{\lambda}=\frac{1}{k+2} \sum_{i=1}^{k+2}
\chi_{\lambda}(\sigma_{k+2}^i)$$ where $\sigma_{k+2}\in\mathfrak{S}_{k+2}$ denotes the cyclic permutation $(12\cdots k+2)$ of order $k+2$.
As is well known, for any $k$ we have an isomorphism $$H_{\mathbb{Q}}^{\otimes (k+2)}\cong \bigoplus_{|{\lambda}|=k+2}\ (\mathrm{dim}\, {\lambda}_{\mathfrak{S}_{k+2}}) V_{\lambda}$$ as $\mathrm{GL}(2g,{\mathbb{Q}})$-modules, where $|{\lambda}|$ denotes the number of the boxes of the Young diagram ${\lambda}$. On the other hand, we have an isomorphism $$\mathfrak{a}_g(k)=\left(H_{\mathbb{Q}}^{\otimes (k+2)}\right)^{{\mathbb{Z}}/(k+2)}$$ where the cyclic group ${\mathbb{Z}}/(k+2)$ of order $k+2$ acts on $H_{\mathbb{Q}}^{\otimes (k+2)}$ by cyclic permutations. Then the claim follows by considering the restriction to the subgroup ${\mathbb{Z}}/(k+2)\subset \mathfrak{S}_{k+2}$ and applying the standard argument.
Our explicit irreducible decompositions of $\mathfrak{h}_{g,1}(k)$ and $\mathfrak{a}_g(k)$ as $\mathrm{Sp}(2g,{\mathbb{Q}})$-modules are done as follows. First we determine the irreducible decompositions of these modules as $\mathrm{GL}(2g,{\mathbb{Q}})$-modules. For this, we use Corollary \[cor:h\] and Proposition \[prop:a\], respectively, to compute the multiplicities $m_{\lambda}, n_{\lambda}$ by applying the formula of Frobenius which expresses the value $\chi_{\lambda}(\gamma)$ for any given element $\gamma\in\mathfrak{S}_{k+2}$ as the coefficient of a certain polynomial $f_{\lambda}$ with respect to a certain monomial $x_{\lambda}$ (see e.g. [@fh], Frobenius Formula 4.10). We made a systematic computer computation by using this formula. Recall here that we must take the number of variables for the polynomial $f_{\lambda}$ at least as large as the number $h({\lambda})$ of rows of the Young diagram ${\lambda}$. Hence, the necessary data will get larger and larger as the number $h({\lambda})$ increases. To overcome this difficulty, we adopted the following simple argument. Let ${\lambda}'$ denote the [*conjugate*]{} Young diagram of any given one ${\lambda}$. Then, as is well known, we have an isomorphism $${\lambda}'_{\mathfrak{S}_{k+2}}\cong {\lambda}_{\mathfrak{S}_{k+2}}\otimes [1^{k+2}]$$ where $[1^{k+2}]$ denotes the $1$-dimensional alternating representation. Since, for any given $\mathfrak{S}_{k+2}$-module $W$, its multiplicity of an irreducible representation ${\lambda}$ is equal to that of ${\lambda}'$ in the conjugate representation $W'=W\otimes [1^{k+2}]$, we can easily deduce the following.
In the same situation as in Corollary \[cor:h\] and Proposition \[prop:a\], we have $$m_{\lambda}=\frac{1}{(k+2)!} \sum_{\gamma\in \mathfrak{S}_{k+2}}
(\mathrm{sgn}\;\gamma)\;\chi_k (\gamma)\chi_{{\lambda}'}(\gamma)$$ $$n_{\lambda}=\frac{1}{k+2} \sum_{i=1}^{k+2}\, (\mathrm{sgn}\;\sigma_{k+2}^i)\,
\chi_{{\lambda}'}(\sigma_{k+2}^i)$$ where $\mathrm{sgn}\;\gamma$ denotes the sign of $\gamma$.
Since the two numbers $h({\lambda}')$ and $h({\lambda})$ are so to speak [*complementary*]{} to each other (e.g. $h([1^k]')=1$ while $h([1^k])=k$), we can make computer computations roughly twice as much compared to the situation where we do not use this method. Also certain symmetries in the structure of $\mathfrak{h}_{g,1}$ as well as $\mathfrak{a}_g$ which we found in [@mss2] decrease necessary computations considerably.
Next we use the known formula of decomposing a given irreducible $\mathrm{GL}(2g,{\mathbb{Q}})$-module ${\lambda}_{GL}$ into $\mathrm{Sp}(2g,{\mathbb{Q}})$-irreducible components (see e.g. formula (25.39) in [@fh]). We made a computer program of this formula and by using it we made a database which contains the $\mathrm{Sp}(2g,{\mathbb{Q}})$-irreducible decomposition of all the $\mathrm{GL}(2g,{\mathbb{Q}})$-modules ${\lambda}_{GL}$ with the number $|{\lambda}|$ of boxes of the Young diagram ${\lambda}$ less than or equal to $30$ (there are $28628$ such Young diagrams).
As mentioned already, we have so far determined the $\mathrm{Sp}(2g,{\mathbb{Q}})$-irreducible decompositions of $\mathfrak{h}_{g,1}(k)$ and $\mathfrak{a}_{g}(k)$ for all $k\leq 20$ by making use of the above method. Although here we omit the description of the results, see Tables \[tab:i1\] and \[tab:i2\] in which we express the dimensions of the $\mathrm{Sp}$-invariant subspaces $\mathfrak{h}_{g,1}(k)^{\mathrm{Sp}}$ for all $k\leq 20$. These tables contain more precise information on these subspaces. Namely they contain a complete description how these subspaces degenerate according as the genus $g$ decreases from the stable range one by one to the final case $g=1$.
Computation of the dimensions of the $Sp$-invariant subspaces of various $Sp$-modules {#sec:di}
=====================================================================================
In this section, we describe several methods which we developed in our computer computations. We have to determine the dimensions of the subspaces consisting of the $\mathrm{Sp}$-invariant elements of various $\mathrm{Sp}$-modules such as $$\left(\wedge^{d_1} \mathfrak{c}_g(i_1)\right)\otimes\cdots\otimes
\left(\wedge^{d_s} \mathfrak{c}_g(i_s)\right)$$ or the corresponding modules where we replace $\mathfrak{c}_g$ by $\mathfrak{h}_{g,1}$ or $\mathfrak{a}_g$. We mention that the character of any of these modules is known so that [*theoretically*]{} there is no problem. More precisely, we can adopt the method given in [@km], which treated the case of $\mathfrak{c}_g$ by extending the original one due to Perchik [@p], to the other two case as well to obtain closed formulas for the above dimensions of $\mathrm{Sp}$-invariants. Unfortunately however, these formulas are too complicated so that when we use computers to obtain explicit values, the memory problem arises in a very early stage. We have developed other methods described as follows.
Method (I) ($\mathrm{GL}$-decomposition of tensor products)
There is a formula, called the Littlewood-Richardson rule, which gives the irreducible decomposition of the tensor product of any two $\mathrm{GL}$-modules and a similar formula is known for the case of $\mathrm{Sp}$-modules (see e.g. [@fh]). However, the latter formula is considerably more complicated than the former one. In view of this, we postpone the $\mathrm{Sp}$-irreducible decomposition as late as possible and we make the $\mathrm{GL}$-irreducible decomposition as far as possible. We made a computer program for the Littlewood-Richardson rule and apply it in various stages in our computation.
Method (II) ($\mathrm{Sp}$-decomposition of $\mathrm{GL}$-modules)
There is a combinatorial formula which gives the $\mathrm{Sp}$-irreducible decomposition of any irreducible $\mathrm{GL}$-module ${\lambda}_{\mathrm{GL}}$, namely the restriction law corresponding to the pair $\mathrm{Sp}(2g,{\mathbb{Q}})\subset \mathrm{GL}(2g,{\mathbb{Q}})$ (see [@fh]). We made a computer program for this procedure and apply it in various stages in our work.
Method (III) (Counting the number of Young diagrams with [*multiple double floors*]{})
Let us call a Young diagram ${\lambda}$ [*with multiple double floors*]{} if it has the form ${\lambda}=[{\lambda}_1{\lambda}_1\cdots{\lambda}_s{\lambda}_s]$. It is easy to see that ${\lambda}$ is such a Young diagram if and only if its conjugate Young diagram ${\lambda}'$ is of [*even type*]{} in the sense that all the numbers appearing in it are even integers. Now at the final stage of counting the dimension of the $\mathrm{Sp}$-invariant subspace of a $\mathrm{GL}$-module $V$, we can determine the required number without performing the $\mathrm{Sp}$-decomposition of $V$ by adopting the following method.
Let $V$ be a $\mathrm{GL}(2g,{\mathbb{Q}})$-module and let $V^{\mathrm{Sp}}$ denote the subspace consisting of $\mathrm{Sp}(2g,Q)$-invariant elements of $V$ considered as an $\mathrm{Sp}(2g,{\mathbb{Q}})$-module. Also let $$V=\bigoplus_{{\lambda}} m_{\lambda}(V)\ {\lambda}_{\mathrm{Sp}}$$ be the $\mathrm{Sp}$-irreducible decomposition of $V$. Then we have the equality $$\mathrm{dim}\, V^{\mathrm{Sp}}
=\sum_{{\lambda}: \text{multiple double floors}} m_{\lambda}(V).$$
This follows from the fact that $$\mathrm{dim}\, ({\lambda}_{\mathrm{GL}})^{\mathrm{Sp}}=
\begin{cases}
&1\quad (\text{${\lambda}$: multiple double floors})\\
&0\quad (\text{otherwise})
\end{cases}$$ which follows from the restriction law corresponding to the pair $\mathrm{Sp}(2g,{\mathbb{Q}})\subset \mathrm{GL}(2g,{\mathbb{Q}})$.
We made a computer program which counts the number of Young diagrams with multiple double floors in any linear combination of Young diagrams.
Method (IV) (Counting pairs of Young diagrams with the same shape)
The difficulty in applying our program of performing the Littlewood-Richardson rule for the tensor product $V_1\otimes V_2$ increases according to the numbers of boxes of the Young diagrams appearing in the irreducible decompositions of $V_i$ get larger and larger. In case we cannot obtain the result within an appropriate time, we adopt this method which depends on the following fact.
Let $V_1,V_2$ be two $\mathrm{GL}(2g,{\mathbb{Q}})$-modules and let $$V_i=\bigoplus_{{\lambda}} m_{\lambda}(V_i)\ {\lambda}_{\mathrm{Sp}} \quad (i=1,2)$$ be the $\mathrm{Sp}$-irreducible decompositions of $V_i$. Then we have the equality $$\mathrm{dim}\, (V_1\otimes V_2)^{\mathrm{Sp}}
=\sum_{\lambda}m_{\lambda}(V_1) m_{\lambda}(V_2).$$
This follows from the well known fact that $$\mathrm{dim}\, ({\lambda}_{\mathrm{Sp}}\otimes \mu_{\mathrm{Sp}})^{\mathrm{Sp}}=
\begin{cases}
&1\quad ({\lambda}=\mu)\\
&0\quad (\text{otherwise})
\end{cases}
.$$
Here is another similar formula.
Let $V$ be an $\mathrm{Sp}(2g,{\mathbb{Q}})$-module and let $$V=\bigoplus_{{\lambda}} m_{\lambda}(V)\ {\lambda}_{\mathrm{Sp}}$$ be its $\mathrm{Sp}$-irreducible decomposition. Then we have the equality $$\mathrm{dim}\, (\wedge^2 V)^{\mathrm{Sp}}
=\frac{1}{2}\left(\sum_{\text{$|{\lambda}|$: odd}} m_{\lambda}(V) (m_{\lambda}(V)+1)+
\sum_{\text{$|{\lambda}|$: even}} (m_{\lambda}(V)-1) m_{\lambda}(V) \right).$$
This follows from the well known fact that $$\mathrm{dim}\, (\wedge^2\,{\lambda}_{\mathrm{Sp}})^{\mathrm{Sp}}=
\begin{cases}
&1\quad (\text{$|{\lambda}|$: odd})\\
&0\quad (\text{$|{\lambda}|$: even})
\end{cases}
.$$
We made a computer program which counts the number of pairs with the same Young diagrams in any two linear combination of Young diagrams. We can use this method to check the accuracy of our computations by applying it to plural expressions $$V=V_1\otimes V_2=V'_1\otimes V'_2$$ as tensor products of the same $\mathrm{GL}(2g,{\mathbb{Q}})$-module $V$.
Method (V) (Adams operations)
The most difficult part in our computation is the determination of the $\mathrm{GL}$ as well as $\mathrm{Sp}$ irreducible decomposition of the exterior powers $\wedge^k [1^3]_{\mathrm{GL}}$ of $\mathfrak{h}_{g,1}(1)=[1^3]_{\mathrm{GL}}$.
As is well known, the character of $\wedge^k [1]_{\mathrm{GL}}=[1^k]_{\mathrm{GL}}$ is given by $$\mathrm{ch}([1^k]_{\mathrm{GL}})=E_k(x_1,...,x_N)$$ where $E_k$ denotes the $k$-th elementary symmetric polynomial and $N$ denotes some fixed large number. In particular $\mathrm{ch}([1^3]_{\mathrm{GL}})=E_3$. Then the character of $\wedge^k [1^3]_{\mathrm{GL}}$ can be written as $$\mathrm{ch}(\wedge^k [1^3]_{\mathrm{GL}})
=E_k(E_3)$$ where $E_k(E_3)$ denotes the $k$-th elementary symmetric polynomial with respect to the new variables $\{x_ix_jx_k\}_{i<j<k}$. Here we apply the well-known classical combinatorial algorithm to express any symmetric polynomial as a polynomial on the elementary symmetric polynomial to obtain a formula for the character of $\wedge^k [1^3]_{\mathrm{GL}}$. For example $\mathrm{ch}(\wedge^2 [1^3]_{\mathrm{GL}})
=E_6+E_2E_4-E_1E_5$. Then we apply Method (I) to obtain the $\mathrm{GL}$-irreducible decomposition and further apply Method (II) to obtain the $\mathrm{Sp}$-irreducible decomposition. For example we have $$\begin{aligned}
\wedge^2 [1^3]_{\mathrm{GL}}&=[1^6]_{\mathrm{GL}}+[2^21^2]_{\mathrm{GL}}\\
&=[1^6]_{\mathrm{Sp}}+[2^21^2]_{\mathrm{Sp}}+2 [1^4]_{\mathrm{Sp}}+[21^2]_{\mathrm{Sp}}+[2^2]_{\mathrm{Sp}}+3 [1^2]_{\mathrm{Sp}}+2 [0]_{\mathrm{Sp}}\end{aligned}$$ where $[0]_{\mathrm{Sp}}$ denotes the trivial representation. For large $k$, we used the computer software LiE to obtain the irreducible decompositions. However, because of the memory problem we could obtain the $\mathrm{GL}$-irreducible decomposition of $\wedge^k [1^3]_{\mathrm{GL}}$ only up to $k=6$ or so. To overcome this difficulty, we used the Adams operations $\psi^k\ (k=1,2,\ldots)$ which satisfy the identity $$\wedge^k V=\frac{1}{k}\left(\wedge^{k-1}V\otimes V-\wedge^{k-2}V\otimes \psi^2(V)+\cdots +(-1)^{k-1}\psi^k(V)\right)$$ on any representation $V$. It turns out that the computer computation of the Adams operation is much easier than that of the exterior powers. By utilizing this merit of the Adams operations, we have determined so far the $\mathrm{GL}$-irreducible decomposition of $\wedge^k [1^3]_{\mathrm{GL}}$ for $k\leq 10$ and the $\mathrm{Sp}$-irreducible decomposition of $\wedge^k [1^3]_{\mathrm{GL}}$ for $k\leq 9$. As for the computation of the dimensions of the $\mathrm{Sp}$-invariant subspaces of $\mathrm{GL}$-modules with the form $\wedge^k [1^3]_{\mathrm{GL}}\otimes V$ for certain $V$, we can go further up to $k=16$ or so. Here we express $\wedge^k [1^3]_{\mathrm{GL}} (k=10, 11,\ldots)$ in terms of $\wedge^k [1^3]_{\mathrm{GL}} (k=1, 2, \ldots ,9)$ and $\psi^k [1^3]_{\mathrm{GL}} (k=1,2,\ldots)$ and apply the preceding methods. The coefficients of this expression are complicated rational numbers rather than the integers. From this fact, we obtain an [*extra merit*]{} of this method. Namely, we can check the accuracy of the computation just by confirming the answer to be an [*integer*]{} because it is most likely that any small mistake in the computation would force that the output is [*not*]{} an integer.
Method (VI) (Counting the number of graphs with a prescribed type)
In [@morita96], a certain linear mapping $${\mathbb{Q}}\langle\text{isomorphism class of trivalent graph with $2k$-verticies}\rangle
\rightarrow
\left(\wedge^{2k}[1^3]_{\mathrm{GL}}\right)^{\mathrm{Sp}}$$ was introduced by making use of a classical result of Weyl, which is an isomorphism in the stable range. Here the left hand side denotes the vector space generated by the isomorphism classes of trivalent graphs with $2k$ vertices where we allow a trivalent graph to have multi-edges and/or loops. In the theory of enumeration of graphs, the numbers of such trivalent graphs are known for $k\leq 16$ by making use of the result of Read [@read].
Method (VII) (Checking the accuracy of computations)
We have adopted a few checking procedure to confirm the accuracy of our computations. As for the irreducible decompositions, we have checked that the dimension of the resulting decomposition coincides with that of the original module by applying the Weyl character formula. As for the dimension counting of various $\mathrm{Sp}$-invariant subspaces, we carried out plural different ways of computations and checked that the answers coincide to each other.
In short, our strategy is a mixture of theoretical considerations and computer computations. By combining the above Methods (I)-(VII) in various ways, we made explicit computer computations the results of which will be given in the following three sections.
The case of $\mathfrak{c}_g$ and the graph homology as well as transversely symplectic foliations {#sec:c}
=================================================================================================
First we consider the commutative case. From the point of view of explicit computations, this case of $\mathfrak{c}_g$ is the simplest among the three Lie algebras because each piece $\mathfrak{c}_g(k)\cong S^{k+2} H_{\mathbb{Q}}$ is a single irreducible $\mathrm{Sp}(2g,{\mathbb{Q}})$-module. However, its stable (co)homology is far from being well understood and there are big mysteries here. Before describing them, the result of our computation for this case is depicted in Table \[tab:c\].
[|c|r|r|r|r|r|r|r|r|r|r|]{} $w$ & $2$ & $4$ & $6$ & $8$ & $10$ & $12$ & $14$ & $16$ & $18$ & $20$\
$C_1$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$\
$C_2$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$\
$C_3$ & & $1$ & $1$ & $4$ & $3$ & $8$ & $6$ & $12$ & $10$ & $17$\
$C_4$ & & $3$ & $0$ & $16$ & $20$ & $63$ & $78$ & $164$ & $205$ & $355$\
$C_5$ & & & $4$ & $20$ & $112$ & $271$ & $748$ & $1484$ & $3103$ & $5447$\
$C_6$ & & & $7$ & $15$ & $269$ & $1013$ & $3964$ & $11047$ & $29423$ & $67611$\
$C_7$ & & & & $25$ & $310$ & $2784$ & $14034$ & $59153$ & $200982$ & $613281$\
$C_8$ & & & & $24$ & $223$ & $4690$ & $36530$ & $220693$ & $1023318$ & $4068707$\
$C_9$ & & & & & $166$ & $4683$ & $68504$ & $592111$ & $3862954$ & $20226716$\
$C_{10}$ & & & & & $86$ & $2963$ & $87552$ & $1167459$ & $10828229$ & $76399055$\
$C_{11}$ & & & & & & $1395$ & $73358$ & $1682134$ & $22709573$ & $220634704$\
$C_{12}$ & & & & & & $426$ & $39797$ & $1727415$ & $35748802$ & $488935936$\
$C_{13}$ & & & & & & & $13984$ & $1221607$ & $41935536$ & $832479480$\
$C_{14}$ & & & & & & & $2732$ & $570419$ & $35952084$ & $1085617203$\
$C_{15}$ & & & & & & & & $164365$ & $21796235$ & $1073488879$\
$C_{16}$ & & & & & & & & $23701$ & $8867266$ & $789223120$\
$C_{17}$ & & & & & & & & & $2199842$ & $417233525$\
$C_{18}$ & & & & & & & & & $258951$ & $149905889$\
$C_{19}$ & & & & & & & & & & $32900910$\
$C_{20}$ & & & & & & & & & & $3365151$\
$\text{total}$ & $1$ & $4$ & $13$ & $104$ & $1190$ & $18296$ & $341288$ & $7441764$ & $185416514$ & $5195165986$\
$\chi$ & $1$ & $2$ & $3$ & $6$ & $8$ & $14$ & $20$ & $32$ & $44$ & $68$\
\[tab:c\]
Here $C_k$ of the weight $w$ part denotes $$\lim_{g\to\infty}\
\mathrm{dim}
\left(
\bigoplus_{\begin{subarray}{c}
i_1+ i_2+\cdots + i_w=k\\
i_1+2 i_2+\cdots +w i_w=w
\end{subarray}}
\wedge^{i_1} (S^{3} H_{\mathbb{Q}})\otimes\wedge^{i_2} (S^{4} H_{\mathbb{Q}})\otimes\cdots\otimes
\wedge^{i_w} (S^{w+2} H_{\mathbb{Q}})\right)^{\mathrm{Sp}}$$ so that we have a [*finite*]{} dimensional chain complex $0\to C_{w}\to\cdots\to C_{1}\to 0$. Also $\chi$ denotes the Euler characteristic of this chain complex, namely the weight $w$ part of the $\mathrm{Sp}$-invariant stable homology $
H_*(\mathfrak{c}^+_\infty)^{\mathrm{Sp}}_w.
$
This follows from Table \[tab:c\].
Let $G_*^{(n)}\ (n\geq 2)$ be the graph complex defined by Kontsevich, which is a chain complex of dimension $2n-2$.
\[thm:kontsevich-c\] There exists an isomorphism $$PH_k (\mathfrak{c}^+_{\infty})^{\mathrm{Sp}}_{2n}
\cong
H_{k} (G_*^{(n+1)}).$$ \[thm:kc\]
The weight generating function, denoted by $c(t)$, for the $\mathrm{Sp}$-invariant stable homology group $H_*(\mathfrak{c}^+_{\infty})^{\mathrm{Sp}}$ is given by $$c(t)=\prod_{n=2}^\infty (1-t^{2n-2})^{- e(G_*^{(n)})}$$ where $e(G_*^{(n)})$ denotes the Euler characteristic of $H_*(G_*^{(n)})$. \[prop:ct\]
This follows from Theorem \[thm:kc\] and Proposition \[prop:wp\].
By Theorem \[thm:chi\] (i), we see that the weight generating function $$c(t)=\sum_{w=0}^\infty \chi(H_*(\mathfrak{c}^+_\infty)^{\mathrm{Sp}}_w)t^w$$ for $H_*(\mathfrak{c}^+_\infty)^{\mathrm{Sp}}$, up to weight $20$, is given by $$c(t)=1+t^2+2t^4+3t^6+6t^8+8t^{10}+14t^{12}+20t^{14}+32t^{16}+44t^{18}+68t^{20}+\cdots.$$ By applying Proposition \[prop:wc\], we can inductively determine the integral Euler characteristics of the primitive parts, namely $e(G_*^{(n)})$. If we put $$\begin{aligned}
\bar{c}(t)=(1-t^2)^{-1}&(1-t^4)^{-1}(1-t^6)^{-1}(1-t^8)^{-2}\\
&(1-t^{10})^{-1}(1-t^{12})^{-2}(1-t^{14})^{-2}(1-t^{16})^{-2}(1-t^{18})^{-1}(1-t^{20})^{-3},\end{aligned}$$ then we see that $$c(t)-\bar{c}(t)\equiv 0\,
\bmod t^{21}.$$ By Proposition \[prop:ct\], we can now conclude that $e(G_*^{(n)})=1,1,1,2,1,2,2,2,1,3$ for $n=2,3,\ldots,11$, respectively. The result is depicted in the fourth row of Table \[tab:cn\]
[|c|r|r|r|r|r|r|r|r|r|r|]{} $w$ & $2$ & $4$ & $6$ & $8$ & $10$ & $12$ & $14$ & $16$ & $18$ & $20$\
$\chi$ & $1$ & $2$ & $3$ & $6$ & $8$ & $14$ & $20$ & $32$ & $44$ & $68$\
$\text{$\chi$ of lower terms}$ & $0$ & $1$ & $2$ & $4$ & $7$ & $12$ & $18$ & $30$ & $43$ & $65$\
$\text{$\chi$ of primitive part}$ & $1$ & $1$ & $1$ & $2$ & $1$ & $2$ & $2$ & $2$ & $1$ & $3$\
$\mathrm{dim}\,\mathcal{A}(\phi)^{(w)}$ & $1$ & $2$ & $3$ & $6$ & $9$ & $16$ & $25$ & $42$ & $65$ & $105$\
$\text{generators for $\mathcal{A}(\phi)$}$ & $1$ & $1$ & $1$ & $2$ & $2$ & $3$ & $4$ & $5$ & $6$ & $8$\
$\text{$\chi$ of primitive part of $\mathcal{E}$}$ & $0$ & $0$ & $0$ & $0$ & $-1$ & $-1$ & $-2$ & $-3$ & $-5$ & $-5$\
\[tab:cn\]
As was already mentioned in the introduction, there are deep connections of this case with two important subjects in topology. Namely the theory of [*finite type*]{} invariants of homology $3$-spheres as well as $3$-manifolds and the theory of characteristic classes of [*transversely symplectic*]{} foliations.
In [@gn] (Theorem 2), Garoufalidis and Nakamura proved the following beautiful result. Stably there exists an isomorphism $$\mathcal{A}(\phi)
\cong \left(\wedge^* (S^3H_{\mathbb{Q}})/([4]_{\mathrm{Sp}})\right)^{\mathrm{Sp}}$$ of graded algebras. Here $[4]_{\mathrm{Sp}}=S^4H_{\mathbb{Q}}\subset \wedge^2 (S^3H_{\mathbb{Q}})$ denotes a certain summand and $([4]_{\mathrm{Sp}})$ denotes the ideal of $\wedge^* (S^3H_{\mathbb{Q}})$ generated by it.
Now in the chain complex computing $H_*(\mathfrak{c}^+_g)_2$ (the weight $2$ part), the boundary operator is the Poisson bracket $$\wedge^2(S^3H_{\mathbb{Q}}) \overset{\partial}{\longrightarrow} S^4H_{\mathbb{Q}}$$ which is easily seen to be [*surjective*]{}. Passing to the dual, the cochain complex computing $H_c^*(\widehat{\mathfrak{c}}^+_g)_2$ is $$S^4H_{\mathbb{Q}}\overset{\delta}{\longrightarrow} \wedge^2(S^3H_{\mathbb{Q}})$$ which is easily seen to be [*injective*]{}. Here recall that any finite dimensional $\mathrm{Sp}$-module is canonically isomorphic to its dual module. Since the multiplicity of $S^4 H_{\mathbb{Q}}$ in $\wedge^2(S^3H_{\mathbb{Q}})$ is one as already pointed out in [@gn], this is the same summand as above. By the definition of the Lie algebra cohomology, we can now conclude that the final part of the cochain complex computing $H_c^*(\widehat{\mathfrak{c}}^+_g)^{\mathrm{Sp}}_{2k}$ is $$\cdots \rightarrow
\left(S^4H_{\mathbb{Q}}\otimes \wedge^{2k-2}(S^3H_{\mathbb{Q}})\right)^{\mathrm{Sp}}
\xrightarrow{\wedge (\delta\otimes\mathrm{id})}
\left(\wedge^{2k}(S^3H_{\mathbb{Q}})\right)^{\mathrm{Sp}}\rightarrow 0$$ where the last non-trivial homomorphism can be identified with $\delta\otimes\mathrm{id}$ followed by the wedge product $\wedge$. This is because $\delta(\alpha\wedge\beta)=(\delta \alpha)\wedge\beta
+(-1)^{\mathrm{deg}\,\alpha}\alpha\wedge(\delta\beta)$ in general and $\delta \beta=0$ for any $\beta\in S^3H_{\mathbb{Q}}$ in the present case. Now the top cohomology group $H_c^{2k}(\widehat{\mathfrak{c}}^+_g)^{\mathrm{Sp}}_{2k}$ is the cokernel of the above homomorphism and clearly the image of $\wedge(\delta\otimes\mathrm{id})$ coincides with the ideal $\left(S^4H_{\mathbb{Q}}\right)$. We can now apply the result of Garoufalidis and Nakamura above to conclude that $$H_c^{2k}(\widehat{\mathfrak{c}}^+_g)^{\mathrm{Sp}}_{2k}\cong \mathcal{A}(\phi)^{(2k)}.$$ Passing to the dual, we obtain an isomorphism $$H_{2k}(\mathfrak{c}^+_g)^{\mathrm{Sp}}_{2k}\cong \mathcal{A}(\phi)^{(2k)}$$ of the top homology group. By restricting to the primitive part, we obtain the following result.
There exists an isomorphism $$H_{2n-2}(G_*^{(n)})\cong \mathcal{A}(\phi)^{(2n-2)}_{\mathrm{conn}}.$$ \[prop:ce\]
Thus we obtain an injective homomorphism $$\mathcal{A}(\phi)\rightarrow H_*(\mathfrak{c}^+_\infty)^{\mathrm{Sp}}$$ and let us consider $\mathcal{A}(\phi)$ as a subalgebra of $H_*(\mathfrak{c}^+_\infty)^{\mathrm{Sp}}$.
Let $\mathcal{A}^+(\phi)$ denote the subalgebra of $\mathcal{A}(\phi)$ consisting of all the elements with [*positive*]{} degrees and let $\mathcal{I}(\mathcal{A}^+(\phi))$ denote the ideal of $H_*(\mathfrak{c}^+_\infty)^{\mathrm{Sp}}$ generated by $\mathcal{A}^+(\phi)$. Now set $$\mathcal{E}=H_*(\mathfrak{c}^+_\infty)^{\mathrm{Sp}}/\mathcal{I}(\mathcal{A}^+(\phi))$$ which is a free graded commutative algebra with respect to the degree. It is also equipped with the second grading induced by the weights. \[def:e\]
By the definition, clearly we have an isomorphism $$H_*(\mathfrak{c}^+_\infty)^{\mathrm{Sp}}\cong \mathcal{A}^+(\phi)\otimes \mathcal{E}$$ of bigraded algebras.
Although the structure of the polynomial algebra $
\mathcal{A}(\phi)
$ is far from being understood, it is known that the numbers of generators for this algebra are $1,1,1,2,2,3,4,5,6,8,9,\ldots$ for degrees $w=2,4,\ldots, 22$ (see [@ohe]) and the generating function of this algebra is $$\phi(t)=1+t^2+2t^4+3t^6+6t^8+9t^{10}+16t^{12}+25t^{14}+42t^{16}+65t^{18}+105t^{20}+161t^{22}+\cdots.$$ We write these values in the fifth and the sixth rows of Table \[tab:cn\]. Then by subtracting the sixth row from the fourth row of Table \[tab:cn\], we can determine the first several terms of the weight generating function for the bigraded algebra $\mathcal{E}$ to be $$e(t)=1-t^{10}-t^{12}-2 t^{14}-3t^{16}-5t^{18}-5t^{20}+\cdots.$$ Of course we should have the identity $c(t)=\phi(t) e(t)$ which is easy to check. This completes the proof.
In the framework of our bigraded algebra $\mathcal{E}$, the results of Gerlits [@ge] (Theorem 4.1.) as well as Conant, Gerlits and Vogtmann [@cgv] (Theorem 5.1.) can be described as follows. Namely, $\mathcal{E}_w=0$ for all $w=2,\ldots,8$ and $\mathcal{E}_{10}\cong {\mathbb{Q}}, \mathcal{E}_{12}\cong {\mathbb{Q}}$ are spanned by certain elements in $PH_7(\mathfrak{c}^+_\infty)^{\mathrm{Sp}}_{10}$ and $PH_9(\mathfrak{c}^+_\infty)^{\mathrm{Sp}}_{12}$, respectively.
The free graded algebra $\mathcal{E}$ is [*infinitely*]{} generated. Furthermore there exist infinitely many generators with [*odd*]{} degrees. \[conj:e\]
Construct explicit cycles lying in $P\mathcal{E}$.
Next we describe the connection of the commutative case with the theory of characteristic classes of transversely symplectic foliations. Let $\mathfrak{ham}_{2g}$ denote the Lie algebra consisting of all the formal Hamiltonian vector fields on ${\mathbb{R}}^{2g}$ with respect to the standard symplectic form. In [@kontsevich3], Kontsevich considered two Lie subalgebras $$\mathfrak{ham}^1_{2g}\subset \mathfrak{ham}^0_{2g}\subset \mathfrak{ham}_{2g}$$ where $\mathfrak{ham}^0_{2g}$ and $\mathfrak{ham}^1_{2g}$ denote the Lie subalgebra consisting of formal Hamiltonian vector fields without constant terms and without constant as well as linear terms, respectively. He gave a geometric meaning to the Gelfand-Fuks cohomology $$H_{GF}^*(\mathfrak{ham}^0_{2g},\mathrm{Sp}(2g,{\mathbb{R}}))\cong
H^*_{GF}(\mathfrak{ham}^1_{2g})^{\mathrm{Sp}}$$ as follows. Let $\mathcal{F}$ be a transversely symplectic foliation on a smooth manifold $M$ of codimension $2g$ and let $H^*_{\mathcal{F}}(M)$ be the associated foliated cohomology group. Then he constructed a homomorphism $$H^*_{GF}(\mathfrak{ham}^1_{2g})^{\mathrm{Sp}}\rightarrow H^*_{\mathcal{F}}(M).$$ Now it is easy to see that the Lie algebras $\mathfrak{ham}^0_{2g}, \mathfrak{ham}^1_{2g}$ are nothing other than the completions of $\mathfrak{c}_g\otimes{\mathbb{R}}, \mathfrak{c}^+_g\otimes{\mathbb{R}}$ with respect to the natural gradings so that we can write $$\mathfrak{ham}^0_{2g}=\widehat{\mathfrak{c}}_g\otimes{\mathbb{R}},\quad
\mathfrak{ham}^1_{2g}=\widehat{\mathfrak{c}}^+_g\otimes{\mathbb{R}}.$$ It follows that we have a homomorphism $$H^*_c(\widehat{\mathfrak{c}}^+_g)^{\mathrm{Sp}}\otimes{\mathbb{R}}\cong
H^*_{GF}(\mathfrak{ham}^1_{2g})^{\mathrm{Sp}}\rightarrow H^*_{\mathcal{F}}(M)$$ for any transversely symplectic foliation $(M,\mathcal{F})$. Let $\mathfrak{c}^+_g\rightarrow \mathfrak{c}_g(1)=S^3H_{\mathbb{Q}}$ be the projection. Then the composition $$H^*(S^3H_{\mathbb{Q}})^{\mathrm{Sp}}\rightarrow H^*_c(\widehat{\mathfrak{c}}^+_g)^{\mathrm{Sp}}\otimes{\mathbb{R}}\rightarrow H^*_{\mathcal{F}}(M)$$ produces the [*usual*]{} leaf cohomology classes in the sense that they are expressed by differential forms involving only the connection form and the curvature form including the Pontrjagin forms. It follows that our bigraded algebra $\mathcal{E}$ can be interpreted as the dual of the space of all the [*exotic*]{} stable leaf cohomology classes, as already mentioned in the introduction.
Study the geometric meaning of the classes in $\mathcal{E}$ in the context of universal characteristic classes for [*odd*]{} dimensional manifold bundles as well as characteristic classes for transversely symplectic foliations.
The case of $\mathfrak{h}_{g,1}$ and the outer automorphism groups of free groups {#sec:l}
=================================================================================
Next we consider the Lie case. The result of our computation for this case is depicted in Table \[tab:h\].
[|c|r|r|r|r|r|r|r|r|r|]{} $w$ & $2$ & $4$ & $6$ & $8$ & $10$ & $12$ & $14$ & $16$ & $18$\
$C_1$ & $1$ & $0$ & $5$ & $3$ & $108$ & $650$ & $8817$ & $111148$ & $1729657$\
$C_2$ & $2$ & $0$ & $10$ & $66$ & $580$ & $6621$ & $84756$ & $1281253$ & $21671535$\
$C_3$ & & $6$ & $7$ & $239$ & $1928$ & $29219$ & $424358$ & $7286710$ & $137344661$\
$C_4$ & & $8$ & $16$ & $342$ & $4946$ & $78443$ & $1400274$ & $27097563$ & $575398310$\
$C_5$ & & & $41$ & $293$ & $8375$ & $152310$ & $3289532$ &$73457788$ & $1766236662$\
$C_6$ & & & $31$ & $287$ & $8887$ & $227058$ & $5780112$ & $152604335$ & $4190265424$\
$C_7$ & & & & $294$ & $6536$ & $254063$ & $7885801$ & $249166200$ & $7923956179$\
$C_8$ & & & & $140$ & $4175$ & $206753$ & $8491679$ & $324662115$ & $12158481555$\
$C_9$ & & & & & $2353$ & $123990$ & $7160718$ & $340745360$ & $15284159637$\
$C_{10}$ & & & & & $722$ & $58302$ & $4634679$ & $288478215$ & $15809478819$\
$C_{11}$ & & & & & & $21368$ & $2269538$ & $195270880$ & $13456339409$\
$C_{12}$ & & & & & & $4439$ & $836620$ & $103755671$ & $7824793027$\
$C_{13}$ & & & & & & & $221987$ & $42207231$ & $5290518430$\
$C_{14}$ & & & & & & & $32654$ & $12701040$ & $2368530727$\
$C_{15}$ & & & & & & & & $2624381$ & $816469677$\
$C_{16}$ & & & & & & & & $289519$ & $206593733$\
$C_{17}$ & & & & & & & & & $34966981$\
$C_{18}$ & & & & & & & & & $3054067$\
$\text{total}$ & $3$ & $14$ & $110$ & $1664$ & $38610$ & $1163216$ & $42521525$ & $1821739409$ & $89423442490$\
$\chi$ & $1$ & $2$ & $4$ & $6$ & $10$ & $16$ & $23$ & $13$ & $-96$\
\[tab:h\]
This follows from Table \[tab:h\].
\[thm:kontsevich-h\] There exists an isomorphism $$PH_k (\mathfrak{h}_{\infty,1})_{2n}
\cong
H^{2n-k} (\mathrm{Out}\,F_{n+1};{\mathbb{Q}}).$$ \[thm:kh\]
The weight generating function, denoted by $h(t)$, for the $\mathrm{Sp}$-invariant stable homology group $H_*(\mathfrak{h}^+_{\infty,1})^{\mathrm{Sp}}$ is given by $$h(t)=\prod_{n=2}^\infty (1-t^{2n-2})^{- e(\mathrm{Out}\, F_n)}$$ where $e(\mathrm{Out}\, F_n)$ denotes the [*integral*]{} Euler characteristic of $\mathrm{Out}\, F_n$. \[prop:ht\]
This follows from Theorem \[thm:kh\] and Proposition \[prop:wp\].
By Theorem \[thm:chi\] (ii), we see that the weight generating function $$h(t)=\sum_{w=0}^\infty \chi(H_*(\mathfrak{h}^+_{\infty,1})^{\mathrm{Sp}}_w)t^w$$ for $H_*(\mathfrak{h}^+_{\infty,1})^{\mathrm{Sp}}$, up to weight $18$, is given by $$h(t)=1+t^2+2t^4+4t^6+6t^8+10t^{10}+16t^{12}+23t^{14}+13t^{16}-96t^{18}+\cdots.$$ By applying Proposition \[prop:wc\], we can inductively determine the Euler characteristics of the primitive parts, namely $e(\mathrm{Out}\, F_n)$. If we put $$\begin{aligned}
\bar{h}(t)=(1-t^2)^{-1}(1-t^4)^{-1}&(1-t^6)^{-2}(1-t^8)^{-1}\\
&(1-t^{10})^{-2}(1-t^{12})^{-1}(1-t^{14})^{-1}(1-t^{16})^{21}(1-t^{18})^{124},\end{aligned}$$ then we see that $$h(t)-\bar{h}(t)\equiv 0
\bmod t^{19}.$$ By Proposition \[prop:ht\], we can now conclude that $e(\mathrm{Out}\, F_n)=1,1,2,1,2,1,1,-21,-124$ for $n=2,3,\ldots,10$, respectively. The result is depicted in the fourth row of Table \[tab:hn\].
[|c|r|r|r|r|r|r|r|r|r|]{} $w$ & $2$ & $4$ & $6$ & $8$ & $10$ & $12$ & $14$ & $16$ & $18$\
$\chi$ & $1$ & $2$ & $4$ & $6$ & $10$ & $16$ & $23$ & $13$ & $-96$\
$\text{$\chi$ of lower terms}$ & $0$ & $1$ & $2$ & $5$ & $8$ & $15$ & $22$ & $34$ & $28$\
$\text{$\chi$ of primitive part}$ & $1$ & $1$ & $2$ & $1$ & $2$ & $1$ & $1$ & $-21$ & $-124$\
\[tab:hn\]
Thus we see that there are many [*odd*]{} dimensional non-trivial rational cohomology classes of $\mathrm{Out}\,F_9$ as well as $\mathrm{Out}\,F_{10}$. Before this result, very few results have been known about the rational cohomology group of $\mathrm{Out}\,F_{n}$. As for the cases $n\leq 6$, by the works of Hatcher and Vogtmann [@hv] as well as Ohashi [@o], the only non-trivial cohomology groups are $H^4(\mathrm{Out}\,F_{4};{\mathbb{Q}})\cong{\mathbb{Q}}$ and $H^8(\mathrm{Out}\,F_{6};{\mathbb{Q}})\cong{\mathbb{Q}}$. On the other hand, by making use of the trace maps introduced in [@morita93] which give a large abelian quotient of $\mathfrak{h}^+_{g,1}$, the first named author defined many rational homology classes of $\mathrm{Out}\, F_n$ in [@morita99][@morita06], the most important classes being a series of homology classes $$\mu_k\in H_{4k}(\mathrm{Out}\, F_{2k+2};{\mathbb{Q}}) \ (k=1,2,\ldots).$$ It was conjectured in [@morita99] that these will be all non-trivial. However, at present only the first three classes are known to be non-trivial, $\mu_1$ in [@morita99], $\mu_2$ by Conant and Vogtmann [@cov] and $\mu_3$ by Gray [@gr]. Conant and Vogtmann also gave a geometric construction of many homology classes in the framework of the [*Outer Space*]{} of Culler and Vogtmann [@cuv].
As already mentioned in the introduction, recently, Conant, Kassabov and Vogtmann [@ckv] proved a remarkable result about the structure of $\mathfrak{h}_{g,1}$. They found a deep connection with the theory of elliptic modular forms by which they show the existence of a large new abelianization beyond the trace maps. In particular, they defined many new cohomology classes in $H^2_c(\widehat{\mathfrak{h}}^+_{\infty,1})^{\mathrm{Sp}}_{2w}$ whenever the dimension of the cusp forms of some weight $w$ is larger than $1$, the first one being $w=24$. These classes then produce, by Theorem \[thm:kh\], rational homology classes of $\mathrm{Out}\, F_n$ the first of which lies in $H_{46}(\mathrm{Out}\, F_{25};{\mathbb{Q}})$.
Now we go back to the case of $\mathrm{Out}\, F_7$ which is the unknown case with the smallest rank. By our result Theorem \[thm:chip\] $\mathrm{(ii)}$, the Euler characteristic of this group is $1$ and it is an interesting problem to determine whether the rational cohomology group of this group is trivial or not. See Problem \[prob:st\] for this. Next we consider $\mathrm{Out}\, F_8$. Again by Theorem \[thm:chip\] $\mathrm{(ii)}$, $e(\mathrm{Out}\, F_8)=1$. On the other hand, Gray [@gr] proved that $\mu_3\not=0\in H_{12}(\mathrm{Out}\, F_8;{\mathbb{Q}})$. It follows that there exists at least one [*odd*]{} dimensional rational homology class. Here we propose a candidate of such a class in the following proposition ($\gamma_1\in H_{11}(\mathrm{Out}\, F_8;{\mathbb{Q}})$ is our candidate). For this, we use the summands $H_1(\mathfrak{h}^+_{\infty,1})_{2k+4}\supset [2k+1,1]_{\mathrm{Sp}}\ (k=1,2,\ldots)$ which are part of the new abelianizations found by Conant, Kassabov and Vogtmann cited above. By an explicit computation motivated by their result, we have proved that $H_1(\mathfrak{h}^+_{g,1})_{6}\cong [31]_{\mathrm{Sp}}$.
For any $k\geq 1$, we have an isomorphism $$\left([31]_{\mathrm{Sp}}\otimes [2k+1]_{\mathrm{Sp}}\otimes[2k+3]_{\mathrm{Sp}}\right)^{\mathrm{Sp}}
\cong{\mathbb{Q}}$$ so that we obtain a series of (co)homology classes $$\gamma_k\in PH_c^{3}(\widehat{\mathfrak{h}}_{\infty,1})^{\mathrm{Sp}}_{4k+10}
\overset{\mathrm{Kontsevich}}
{\cong} H_{4k+7}(\mathrm{Out}\,F_{2k+6};{\mathbb{Q}}) \quad (k=1,2,\ldots).$$
By the Littlewood-Richardson rule, it is easy to see that $$[2k+1]_{\mathrm{Sp}}\otimes [2k+3]_{\mathrm{Sp}}
\cong
[4k+4]_{\mathrm{GL}}\oplus [4k+3,1]_{\mathrm{GL}}\oplus \cdots\oplus
[2k+3,2k+1]_{\mathrm{GL}}.$$ On the other hand, among the ${\mathrm{Sp}}$-irreducible decompositions of the ${\mathrm{GL}}$-irreducible summands on the right hand side, only the last one $[2k+3,2k+1]_{\mathrm{GL}}$ contains $[31]_{\mathrm{Sp}}$ and the multiplicity is $1$. The claim follows.
These classes are all non-trivial. In particular, $H_3(\mathfrak{h}_{\infty,1})$ is [*infinite*]{} dimensional.
Also if we combine the trace components $[2k+1]_{\mathrm{Sp}}$ with the new components $[2\ell+1,1]_{\mathrm{Sp}},...$, we obtain huge amount of (co)homology classes of $\mathfrak{h}_{\infty,1}$.
Next we consider the problem of comparison between the [*rational*]{} and the [*integral*]{} Euler characteristics of $\mathrm{Out}\,F_{n}$. The second row of Table \[tab:chie\] is taken from Smillie and Vogtmann [@sv] where we write the values to the second decimal places and the third row is our Theorem \[thm:chip\] $\mathrm{(ii)}$.
[|c|c|c|c|c|c|c|c|c|c|]{} $n$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$\
$\chi$ & $-0.04$ & $-0.02$ & $-0.02$ & $-0.06$ & $-0.20$ & $-0.87$ & $-4.58$ & $-28.52$ & $-205.83$\
$e$ & $1$ & $1$ & $2$ & $1$ & $2$ & $1$ & $1$ & $-21$ & $-124$\
\[tab:chie\]
Study the relation between $\chi(\mathrm{Out}\,F_{n})$ and $e(\mathrm{Out}\,F_{n})$.
We refer to the book [@f] edited by Farb, in particular Bridson and Vogtmann [@bv], as well as Farb [@f1] for various problems concerning $\mathrm{Out}\, F_n$, mapping class groups, $\mathrm{GL}(n,{\mathbb{Z}})$ and other related groups.
The case of $\mathfrak{a}_g$ and the moduli spaces of curves {#sec:a}
============================================================
Finally we consider the case of $\mathfrak{a}_g$. From the point of view of computations, this case is the most heavy one among the three Lie algebras as can be seen by comparing the size of the numbers in the former two tables Tables \[tab:c\] and \[tab:h\] with the present one depicted in Table \[tab:a\].
As was already mentioned in Section $1$, in [@mss2a] we determine the values $\chi(H_*(\mathfrak{a}^+_{\infty})^{\mathrm{Sp}}_{w})$ for all $w\leq 500$ by a completely different argument which makes use of a formula of Gorsky [@go2] for the equivariant Euler characteristics of the moduli spaces of curves $\mathbf{M}_g^m$. We confirm that the two values for $w\leq 16$ are the same. We think that this coincidence serves as a strong evidence for the accuracy of our computations in the other two cases $\mathfrak{c}_\infty, \mathfrak{h}_{\infty,1}$. We mention that the existence of the formula of Gorsky depends heavily on the fact that the totality of $\mathbf{M}_g^m$ for various $g$ and $m$ makes a beautiful unified world. It seems unlikely that similar formulas will be found in the other two cases, at least in a near future.
[|c|r|r|r|r|r|r|r|r|]{} $w$ & $2$ & $4$ & $6$ & $8$ & $10$ & $12$ & $14$ & $16$\
$C_1$ & $1$ & $2$ & $17$ & $88$ & $897$ & $9562$ & $127071$ & $1912970$\
$C_2$ & $3$ & $8$ & $111$ & $1146$ & $14735$ & $212965$ & $3483545$ & $63522967$\
$C_3$ & & $18$ & $289$ & $5561$ & $99285$ & $1918401$ & $39558275$ & $880137499$\
$C_4$ & & $17$ & $403$ & $13653$ & $366878$ & $9590016$ & $253890290$ & $6966037951$\
$C_5$ & & & $320$ & $19138$ & $827528$ & $30225682$ & $1047033554$ & $35904134757$\
$C_6$ & & & $124$ & $15860$ & $1193367$ & $63894814$ & $2967604968$ & $129283963277$\
$C_7$ & & & & $7466$ & $1111456$ & $93211250$ & $6001387476$ & $339000966002$\
$C_8$ & & & & $1618$ & $651577$ & $94398768$ & $8825700683$ & $663520078156$\
$C_9$ & & & & & $220905$ & $65356859$ & $9484791225$ & $982320832329$\
$C_{10}$ & & & & & $33564$ & $29594121$ & $7384704777$ & $1104356533575$\
$C_{11}$ & & & & & & $7925093$ & $4061192184$ & $938047301852$\
$C_{12}$ & & & & & & $956263$ & $1497800877$ & $592700462357$\
$C_{13}$ & & & & & & & $332831365$ & $270228006160$\
$C_{14}$ & & & & & & & $33736198$ & $84077896041$\
$C_{15}$ & & & & & & & & $15987868100$\
$C_{16}$ & & & & & & & & $1402665692$\
$\text{total}$ & $4$ & $45$ & $1264$ & $64530$ & $4520192$ & $397293794$ & $41933842488$ & $5164742319685$\
$\chi$ & $2$ & $5$ & $12$ & $24$ & $50$ & $100$ & $188$ & $347$\
\[tab:a\]
This follows from Table \[tab:a\].
\[thm:kontsevich-a\] There exists an isomorphism $$PH_k (\mathfrak{a}^+_{\infty})_{2n}
\cong \bigoplus_{\begin{subarray}{c}
2g-2+m=n\\ m>0
\end{subarray}}
H^{2n-k} (\mathbf{M}_g^m;{\mathbb{Q}})^{\mathfrak{S}_m}.$$ \[thm:ka\]
The weight generating function, denoted by $a(t)$, for the $\mathrm{Sp}$-invariant stable homology group $H_*(\mathfrak{a}^+_{\infty})^{\mathrm{Sp}}$ is given by $$a(t)=\prod_{n=1}^\infty (1-t^{2n})^{- a^{\mathrm{pr}}_{2n}}$$ where $$a^{\mathrm{pr}}_{2n}=
\sum_{\begin{subarray}{c}
2g-2+m=n\\ m>0
\end{subarray}}
e(\mathbf{M}_g^m/\mathfrak{S}_m).$$ \[prop:at\]
This follows from Theorem \[thm:ka\] and Proposition \[prop:wp\].
By Theorem \[thm:chi\] (iii), we see that the weight generating function $$a(t)=\sum_{w=0}^\infty \chi(H_*(\mathfrak{a}^+_{\infty})^{\mathrm{Sp}}_w)t^w$$ for $H_*(\mathfrak{a}^+_{\infty})^{\mathrm{Sp}}$, up to weight $16$, is given by $$a(t)=1+2t^2+5t^4+12t^6+24t^8+50t^{10}+100t^{12}+188t^{14}+347t^{16}+\cdots.$$ By applying Proposition \[prop:wc\], we can inductively determine the Euler characteristics of the primitive parts, namely $e(\mathbf{M}_g^m/\mathfrak{S}_m)$. If we put $$\begin{aligned}
\bar{a}(t)=(1-t^2)^{-2}(1-t^4)^{-2}&(1-t^6)^{-4}(1-t^8)^{-2}\\
&(1-t^{10})^{-6}(1-t^{12})^{-6}(1-t^{14})^{-6}(1-t^{16})^{-1},\end{aligned}$$ then we see that $$a(t)-\bar{a}(t)\equiv 0
\bmod t^{17}.$$ By Proposition \[prop:at\], we can now conclude that $a^{\mathrm{pr}}_{2n}=2,2,4,2,6,6,6,1$ for $n=1,2,\ldots,8$, respectively. The result is depicted in the fourth row of Table \[tab:an\] as well as the following proposition \[prop:ass\].
[|c|r|r|r|r|r|r|r|r|]{} $w$ & $2$ & $4$ & $6$ & $8$ & $10$ & $12$ & $14$ & $16$\
$\chi$ & $2$ & $5$ & $12$ & $24$ & $50$ & $100$ & $188$ & $347$\
$\text{$\chi$ of lower terms}$ & $0$ & $3$ & $8$ & $22$ & $44$ & $94$ & $182$ & $346$\
$\text{$\chi$ of primitive part}$ & $2$ & $2$ & $4$ & $2$ & $6$ & $6$ & $6$ & $1$\
\[tab:an\]
We have the following equalities. $$\begin{aligned}
\mathrm{(1)}\quad &e(\mathbf{M}_0^3/\mathfrak{S}_3)+e(\mathbf{M}_1^1) =2\\
\mathrm{(2)}\quad &e(\mathbf{M}_0^4/\mathfrak{S}_4)+e(\mathbf{M}_1^2/\mathfrak{S}_2) =2\\
\mathrm{(3)}\quad &e(\mathbf{M}_0^5/\mathfrak{S}_5)+e(\mathbf{M}_1^3/\mathfrak{S}_3)+e(\mathbf{M}_2^1) =4\\
\mathrm{(4)}\quad &e(\mathbf{M}_0^6/\mathfrak{S}_6)+e(\mathbf{M}_1^4/\mathfrak{S}_4)+e(\mathbf{M}_2^2/\mathfrak{S}_2) =2\\
\mathrm{(5)}\quad &e(\mathbf{M}_0^7/\mathfrak{S}_7)+e(\mathbf{M}_1^5/\mathfrak{S}_5)+e(\mathbf{M}_2^3/\mathfrak{S}_3)+e(\mathbf{M}_3^1) =6\\
\mathrm{(6)}\quad &e(\mathbf{M}_0^8/\mathfrak{S}_8)+e(\mathbf{M}_1^6/\mathfrak{S}_6)+e(\mathbf{M}_2^4/\mathfrak{S}_4)+e(\mathbf{M}_3^2/\mathfrak{S}_2) =6\\
\mathrm{(7)}\quad &e(\mathbf{M}_0^9/\mathfrak{S}_9)+e(\mathbf{M}_1^7/\mathfrak{S}_7)+e(\mathbf{M}_2^5/\mathfrak{S}_5)+e(\mathbf{M}_3^3/\mathfrak{S}_3)+e(\mathbf{M}_4^1) =6\\
\mathrm{(8)}\quad &e(\mathbf{M}_0^{10}/\mathfrak{S}_{10})+e(\mathbf{M}_1^8/\mathfrak{S}_8)+e(\mathbf{M}_2^6/\mathfrak{S}_6)+e(\mathbf{M}_3^4/\mathfrak{S}_4)+e(\mathbf{M}_4^2/\mathfrak{S}_2) =1.\end{aligned}$$ \[prop:ass\]
Now we check that our result above is consistent with the known results. By Getzler [@getz99], $e(\mathbf{M}_0^m/\mathfrak{S}_m)=1$ for all $m\geq 3$. Also he determined $H^*(\mathbf{M}_1^m)$ as an $\mathfrak{S}_m$-module. In particular, he obtained the following formula $$\begin{aligned}
\sum_{m=1}^\infty e(\mathbf{M}_1^m/\mathfrak{S}_m) x^m&=
(x+x^2+x^3)\frac{(1-x^4-2x^8-x^{12}+x^{16})}{(1-x^8)(1-x^{12})}\\
&=x+x^2+x^3-x^5-x^6-x^7-x^9-x^{10}-x^{11}-x^{13}-x^{14}-\cdots\end{aligned}$$ so that $e(\mathbf{M}_1^m/\mathfrak{S}_m)=1,1,1,0,-1,-1,-1,0,-1$ for $m=1,\ldots,9$. It is well known that $e(\mathbf{M}_2^1)=2$ and Getzler proved $e(\mathbf{M}_2^2)=1$. The fifth equality $\mathrm{(5)}$ (case of $w=10$) was first proved by Getzler and Kapranov [@gk], and then in [@getz98] it was shown that $e(\mathbf{M}_2^3/\mathfrak{S}_3)=0$ by using the result of Looijenga [@loo] determining $H^*(\mathbf{M}_3;{\mathbb{Q}}), H^*(\mathbf{M}^1_3;{\mathbb{Q}})$, especially $e(\mathbf{M}^1_3)=6$. Tommasi [@tom05th] (see also [@tom05][@tom06]) determined $H^*(\mathbf{M}_4,{\mathbb{Q}})$ as well as the equivariant Hodge Euler characteristics of $\mathbf{M}_2^4$ and $\mathbf{M}_3^2$ and in particular $e(\mathbf{M}_2^4/\mathfrak{S}_4)=1$ and $e(\mathbf{M}_3^2/\mathfrak{S}_2)=5$. The sixth equality $\mathrm{(6)}$ is consistent with these results. Next we consider the seventh equality. By Harer and Zagier [@hz], $e(\mathbf{M}_4^1)=2$. Gorsky [@go1] (Theorem $2$) extended the work of Getzler and obtained a formula for the $\mathfrak{S}_m$-equivariant Euler characteristic for $\mathbf{M}_2^m$. More precisely he obtained a formula for the generating function $\sum_{m=0}^\infty e^{\mathfrak{S}_m}(\mathbf{M}_2^m) t^m$ in terms of Newton’s power sum polynomials. He then made a computer computation and determined $e^{\mathfrak{S}_m}(\mathbf{M}_2^m)$ explicitly for all $m\leq 4$ which coincide with the former results of Getzler and Tommasi cited above. By making use of our Method (I) and Method (V) described in Section $4$, we extended Gorsky’s computation to obtain closed formulas for the cases $5\leq m\leq 35$. Here we describe the results for $m=5,6,7,8$. $$\begin{aligned}
e^{\mathfrak{S}_5}(\mathbf{M}_2^5)&=2[5]-2[32]+2[41]\\
e^{\mathfrak{S}_6}(\mathbf{M}_2^6)&=-2[3^2]+2[51]-3[2^3]-2[321]+2[41^2]+[2^21^2]-[21^4]-[1^6]\\
e^{\mathfrak{S}_7}(\mathbf{M}_2^7)&=-2[7]-2[43]+2[52]-2[61]-4[32^2]-2[3^21]+4[421]+2[51^2]-2[2^31]+\\
&4[321^2]+2[41^3]-2[21^5]-2[1^7]\\
e^{\mathfrak{S}_8}(\mathbf{M}_2^8)&=-[8]-3[4^2]-4[53]+2[62]-3[71]-7[3^22]+4[42^2]-7[431]+2[521]-\\
&7[61^2]-4[2^4]-3[32^21]-2[3^21^2]-[421^2]-3[51^3]-7[2^31^2]+2[41^4]-3[2^21^4]-\\
&5[21^6]-3[1^8]\end{aligned}$$ It is amusing to calculate the dimensions of the above expression which give the integral Euler characteristics of $\mathbf{M}_2^m$. The results are $0,-24$ for $m=5, 6$ and $(-1)^{n+1} (n+1)!/240$ for $m\geq 7$ which coincide with the known values obtained by Harer and Zagier [@hz]. On the other hand, as for the coefficients of the trivial representation $[m]$, it can be shown that Gorsky’s formula implies $$\begin{aligned}
\sum_{m=0}^\infty &e(\mathbf{M}_2^m/\mathfrak{S}_m) x^m=
-\frac{1}{240}(1+x)^{-2}-\frac{1}{240}(1+x)^{6}(1+x^2)^{-4}+\frac{1}{12}(1+x)^{2}(1+x^2)^{-2}\\
&-\frac{1}{12}(1+x)^{4}(1+x^3)^{-2}-\frac{1}{8}(1+x)^{2}(1+x^2)^{2}(1+x^4)^{-2}+\frac{2}{5}(1+x)^{3}(1+x^5)^{-1}\\
&-\frac{1}{12}(1+x^2)^{2}(1+x^3)^{2}(1+x^6)^{-2}
+\frac{1}{6}(1+x)^{2}(1+x^2)(1+x^6)^{-1}\\
&+\frac{1}{4}(1+x)^{2}(1+x^4)(1+x^8)^{-1}
+\frac{2}{5}(1+x)(1+x^2)(1+x^5)(1+x^{10})^{-1}\\
&=1+2x+x^2+x^4+2x^5-2x^7-x^8-x^{10}+2x^{12}+2x^{13}-3x^{14}+\cdots\end{aligned}$$ so that we find $e(\mathbf{M}_2^m/\mathfrak{S}_m)=2,0,-2,-1,0,-1,0,2,2,-3,\ldots$ for $m=5,6,\ldots,14,\ldots$. By substituting these known values in the seventh equality $\mathrm{(7)}$, we obtain $e(\mathbf{M}_3^3/\mathfrak{S}_3)=2$. This should be consistent with the work of Bergström [@b] determining $H^*(\overline{\mathbf{M}}_3^m;{\mathbb{Q}})$ as well as $H^*(\mathbf{M}_3^m;{\mathbb{Q}})$ as an $\mathfrak{S}_m$-module for all $m\leq 5$, although the latter is not described explicitly. From the eighth equality $\mathrm{(8)}$ together with the above results, we can conclude $
e(\mathbf{M}_3^4/\mathfrak{S}_4)+e(\mathbf{M}_4^2/\mathfrak{S}_2)=0.
$ Now Gorsky [@go2] (Theorem $3$) extended his own result cited above to obtain a formula for the equivariant Euler characteristics of [*all*]{} the moduli spaces $\mathbf{M}_g^m$ again in terms of Newton power sum polynomials. It is an extensive generalization of the results of Getzler cited above as well as the formula of Harer and Zagier [@hz] for the [*integral*]{} Euler characteristics of the moduli space of curves. By using Gorsky’s formula, we obtain explicit closed formulas for the equivariant Euler characteristic $e^{\mathfrak{S}_m} (\mathbf{M}_g^m)$ as well as the generating functions $$\sum_{m=0}^\infty e(\mathbf{M}_g^m/\mathfrak{S}_m) x^m$$ for $g\leq 125$. Here we describe the values for $g=3,4; m\leq 10$. $$\begin{aligned}
&e(\mathbf{M}_3^m/\mathfrak{S}_m)=3,6,5,2,0,0,1,0,-1,2,2 \quad (m=0,1,\ldots,10)\\
&e(\mathbf{M}_4^m/\mathfrak{S}_m)=2,2,0,2,0,2,10,6,-19,-12,34 \quad (m=0,1,\ldots,10).\end{aligned}$$ In particular $e(\mathbf{M}_3^4/\mathfrak{S}_4)=e(\mathbf{M}_4^2/\mathfrak{S}_2)=0$. Thus we find that our results are completely consistent with known results in algebraic geometry, or can be deduced from them by explicit computations. In our forthcoming paper [@mss2a], we will further discuss these formulas.
We mention that the integral Euler characteristic of the moduli space $\mathbf{M}_g^m$, rather than its quotient $\mathbf{M}_g^m/\mathfrak{S}_m$, is known by Harer and Zagier [@hz] as well as Bini and Harer [@bh] up to certain values of $g,m$. In particular, $e(\mathbf{M}_3^4)=4, e(\mathbf{M}_4^2)=-2$.
In our paper [@mss1], we proved that the stable abelianization of $\mathfrak{a}_g$ is trivial, namely $H_1(\mathfrak{a}^+_\infty)^{\mathrm{Sp}}=0$ and deduced from it the vanishing result, $H^{4g-5}({\mathcal{M}_g};{\mathbb{Q}})=0$, of the top rational cohomology group of the mapping class group ${\mathcal{M}_g}$ of a closed oriented surface of genus $g$ for all $g\geq 2$. See also Church, Farb and Putman [@cfp1].
For each $k\geq 2$, determine whether the $\mathrm{Sp}$-invariant stable homology group $H_k(\mathfrak{a}^+_{\infty})^{\mathrm{Sp}}$ is [*finite*]{} dimensional or not, especially for the low values $k=2,3,\ldots$.
Kontsevich made a conjecture in [@kontsevich1] that the stable homology of each of the infinite dimensional Lie algebras he considered is [*finite*]{} dimensional in each degree. We would like to mention the following implication of this conjecture in the case of $\mathfrak{a}_g$. Namely, if $H_k(\mathfrak{a}^+_{\infty})^{\mathrm{Sp}}$ is finite dimensional, then $$H_{4g-4-k}(\mathbf{M}_{g};{\mathbb{Q}}) =0\;
\text{for all $g$ but finitely many possible exceptions}.$$ This is because, if the assumption is valid, then $PH_c^k$ is also finite dimensional so that $H_{4g-4+2m-k}(\mathbf{M}^m_{g};{\mathbb{Q}})^{\mathfrak{S}_m}=0$ for all $g$ and $m>0$ except for finitely many values. If we put $m=1$ here, we see that $H_{4g-2-k}(\mathbf{M}^1_{g};{\mathbb{Q}})=0$ for all $g$ but finitely many exceptions. Now if $H_{4g-4-k}(\mathbf{M}_g;{\mathbb{Q}})\not=0$ for some $g\geq 2$, then as was proved in [@morita87] that the homomorphism $$H^{*}(\mathbf{M}_g;{\mathbb{Q}}) \xrightarrow{p^*}
H^{*}(\mathbf{M}^1_{g};{\mathbb{Q}}) \xrightarrow{\cup\ e}
H^{*+2}(\mathbf{M}^1_{g};{\mathbb{Q}})$$ is injective for any $g\geq 2$ where $e\in H^2(\mathbf{M}^1_{g};{\mathbb{Q}})$ denotes the Euler class. It follows that $H_{4g-2-k}(\mathbf{M}^1_{g};{\mathbb{Q}})\not=0$ for such $g$. The claim follows.
Looijenga [@loo] determined the rational homology groups of both of $\mathbf{M}_3, \mathbf{M}_3^1$ and in particular he found an unstable cohomology class in $H^6(\mathbf{M}_3;{\mathbb{Q}})$ which is the first known unstable class of the moduli spaces [*without*]{} punctures. He also showed that $H^8(\mathbf{M}_3^1;{\mathbb{Q}})\cong{\mathbb{Q}}, H^6(\mathbf{M}_3^1;{\mathbb{Q}})\cong{\mathbb{Q}}$. It follows that $PH_2(\mathfrak{a}^+_\infty)^{\mathrm{Sp}}_{10}\cong{\mathbb{Q}}$ and $PH_4(\mathfrak{a}^+_\infty)^{\mathrm{Sp}}_{10}\supset {\mathbb{Q}}$. It is a very important problem to determine whether $H_2(\mathfrak{a}^+_{\infty})^{\mathrm{Sp}}$ is finite dimensional or not.
Recently Church, Farb and Putman [@cfp2] proposed a new stability conjecture about the unstable cohomology of $\mathrm{SL}(n,{\mathbb{Z}}), \mathrm{Aut}\, F_n$ and the mapping class groups. In the case of the mapping class groups, our argument above implies the following. Namely, if the above conjecture of Kontsevich is true, then their conjecture also holds in the form that all the groups are trivial.
Dimensions of the $Sp$-invariant subspaces $\mathfrak{h}_{g,1}(2k)^{\mathrm{Sp}}$ {#sec:i}
=================================================================================
As an application of our consideration, we obtain a complete description of how the $\mathrm{Sp}$-invariant subspaces $\mathfrak{h}_{g,1}(k)^{\mathrm{Sp}}$ degenerate with respect to $g$. It turns out that this degeneration is perfectly compatible with the orthogonal direct sum decomposition of $\mathfrak{h}_{g,1}(k)^{\mathrm{Sp}}$ with respect to the canonical metric on it introduced in [@morita11]. The results for all $k\leq 20$ are depicted in Tables \[tab:i1\] and \[tab:i2\]. In the tables, the symbol $*$ denotes that the dimension of the $\mathrm{Sp}$-invariant subspaces [*stabilizes*]{} there with respect to the genus $g$. In general, we can show that the stable range is given by $$\begin{aligned}
&\dim\, \mathfrak{h}_{k,1}(2k)^{\mathrm{Sp}}=\dim\, \mathfrak{h}_{k+1,1}(2k)^{\mathrm{Sp}}=\cdots\quad (\text{$k$ odd})\\
&\dim\, \mathfrak{h}_{k-1,1}(2k)^{\mathrm{Sp}}=\dim\, \mathfrak{h}_{k,1}(2k)^{\mathrm{Sp}}=\cdots\quad (\text{$k$ even}).\end{aligned}$$
[|c|r|r|r|r|r|r|r|r|r|]{} & $g=1$ & $g=2$ & $g=3$ & $g=4$ & $g=5$\
$\mathfrak{h}_{g,1}(2)^{\mathrm{Sp}}$ & $1*$ & $1$ & $1$ & $1$ & $1$\
$\mathfrak{h}_{g,1}(4)^{\mathrm{Sp}}$ & $0*$ & $0$ & $0$ & $0$ & $0$\
$\mathfrak{h}_{g,1}(6)^{\mathrm{Sp}}$ & $1$ & $4$ & $5*$ & $5$ & $5$\
$\mathfrak{h}_{g,1}(8)^{\mathrm{Sp}}$ & $0$ & $2$ & $3*$ & $3$ & $3$\
$\mathfrak{h}_{g,1}(10)^{\mathrm{Sp}}$ & $3$ & $51$ & $97$ & $107$ & $108*$\
$\mathfrak{h}_{g,1}(12)^{\mathrm{Sp}}$ & $0$ & $190$ & $544$ & $643$ & $650*$\
$\mathfrak{h}_{g,1}(14)^{\mathrm{Sp}}$ & $11$ & $1691$ & $6471$ & $8505$ & $8795$\
$\mathfrak{h}_{g,1}(16)^{\mathrm{Sp}}$ & $10$ & $11842$ & $69544$ & $104190$ & $110610$\
$\mathfrak{h}_{g,1}(18)^{\mathrm{Sp}}$ & $57$ & $100908$ & $888099$ & $1548984$ & $1710798$\
$\mathfrak{h}_{g,1}(20)^{\mathrm{Sp}}$ & $108$ & $869798$ & $12057806$ & $25062360$ & $29129790$\
\[tab:i1\]
[|c|r|r|r|r|r|r|r|r|r|]{} & $g=6$ & $g=7$ & $g=8$ & $g=9$\
$\mathfrak{h}_{g,1}(14)^{\mathrm{Sp}}$ & $8816$ & $ 8817*$ & $8817$ & $8817$\
$\mathfrak{h}_{g,1}(16)^{\mathrm{Sp}}$ & $111131$ & $111148*$ & $111148$ & $111148$\
$\mathfrak{h}_{g,1}(18)^{\mathrm{Sp}}$ & $1728591$ & $1729620$ & $1729656$ & $1729657*$\
$\mathfrak{h}_{g,1}(20)^{\mathrm{Sp}}$ & $29688027$ & $29728348$ & $29729957$ & $29729988*$\
\[tab:i2\]
Details will be given in our forthcoming paper [@mss3] where we will discuss how the Lie bracket operation on $\mathfrak{h}_{g,1}^{\mathrm{Sp}}$ is related to the above description of the orthogonal direct sum decomposition as well as the degeneration. This should be important in the investigation of the [*arithmetic*]{} mapping class group.
Concluding remarks and problems {#sec:r}
===============================
In this section, we discuss differences between the three cases $\mathfrak{g}_\infty=\mathfrak{c}_\infty, \mathfrak{h}_{\infty,1}, \mathfrak{a}_\infty$.
First we consider the [*lowest weight*]{} part of the continuous cohomology by which we mean the image of the homomorphism $$H_c^*(\widehat{\mathfrak{g}}_\infty(1))^{\mathrm{Sp}}\rightarrow H^*_c(\mathfrak{g}^+_\infty)^{\mathrm{Sp}}$$ which is induced by the Lie algebra homomorphism $\mathfrak{g}_\infty \rightarrow \mathfrak{g}_\infty(1)$. In the commutative case, this [*lowest weight*]{} cohomology is precisely the dual of $\mathcal{A}(\phi)\subset H_*(\mathfrak{c}^+_\infty)^{\mathrm{Sp}}$ so that it is still mysterious. On the other hand, in the other two cases, it is completely understood because the [*lowest weight*]{} cohomology is precisely the totality of $H_0(\mathrm{Out}\,F_n;{\mathbb{Q}})\ (n\geq 2)$ for the Lie case and the totality of $H_0(\mathbf{M}_g^m;{\mathbb{Q}})\ (2g-2+m> 0, m\geq 1)$ for the associative case. We mention here that the [*lowest weight*]{} cohomology of the Lie algebras $\mathfrak{h}^+_{g,*}$ and $\mathfrak{h}^+_{g}$ surject onto the tautological algebras $\mathcal{R}^*(\mathbf{M}_{g}^1)$ and $\mathcal{R}^*(\mathbf{M}_{g})$, respectively, by the results of [@morita96],[@kam]. Further, it was conjectured in [@morita06] that these homomorphisms are isomorphisms. \[rem:t\]
In the commutative case, it is easy to see that the homomorphism $\mathfrak{c}^+_g\rightarrow \mathfrak{c}_g(1)=S^3H_{\mathbb{Q}}$ is nothing other than the [*abelianization*]{} of the Lie algebra $ \mathfrak{c}^+_g$ because it is fairy easy to see that the Poisson bracket $\mathfrak{c}_g(k)\otimes \mathfrak{c}_g(1)\rightarrow\mathfrak{c}_g(k+1)$ is surjective for any $k\geq 1$. Hence the lowest weight cohomology in this case is the same as those classes which are induced from the abelianization, namely the image of the homomorphism $$H^*_c\left(\widehat{H}_1(\mathfrak{c}^+_{\infty})^{\mathrm{Sp}}\right)\rightarrow
H^*_c(\widehat{\mathfrak{c}}^+_{\infty})^{\mathrm{Sp}}.$$ Theorem \[thm:ce\] shows that this homomorphism is [*not*]{} surjective and furthermore we conjecture that the cokernel is [*infinitely*]{} generated (see Conjecture \[conj:e\]). In the associative case, in our former paper [@mss1] we have determined the [*stable*]{} abelianization of $\mathfrak{a}_\infty$ which turned out to be very [*small*]{}. Also the associative version of the above homomorphism is far from being surjective. In the Lie case, the known abelianization of $\mathfrak{h}^+_{\infty,1}$ turns out to be already very large by [@morita93][@ckv] (although the final answer is not yet known) and many cohomology classes have been defined by making use of it. On the other hand, we proved that $e(\mathrm{Out}\, F_{10})=-124$ (Theorem \[thm:chip\] $\mathrm{(ii)}$) and it seems that this number is too large to be covered by the above construction. Because of this, we propose the following problem.
Prove that the natural homomorphism $$H^*_c\left(\widehat{H}_1(\mathfrak{h}^+_{\infty,1})^{\mathrm{Sp}}\right)^{\mathrm{Sp}}\rightarrow
H^*_c(\widehat{\mathfrak{h}}^+_{\infty,1})^{\mathrm{Sp}}$$ is [*not*]{} surjective.
In view of the fact that a single irreducible piece $\mathfrak{h}_g(1)=[1^3]_{\mathrm{Sp}}$ gives rise to the whole tautological algebra $\mathcal{R}^*(\mathbf{M}_{g})$ as already mentioned in the preceding remark, it seems worthwhile to consider the following problem.
Let $\mathrm{Tr}(2k+1): \mathfrak{h}^+_{g,1}\rightarrow S^{2k+1} H_{\mathbb{Q}}$ be the $(2k+1)$-st trace map defined in [@morita93] and let $$\varprojlim_{g\to\infty}\ PH^{2n}(S^{2k+1} H_{\mathbb{Q}})^{\mathrm{Sp}}_{2n(2k+1)}\rightarrow
PH^{2n}_c(\widehat{\mathfrak{h}}^+_{\infty,1})^{\mathrm{Sp}}_{2n(2k+1)}\cong H_{4nk}(\mathrm{Out}\,F_{2nk+n+1};{\mathbb{Q}})$$ be the homomorphism induced from the above [*single*]{} trace homomorphism. It is easy to see that the left hand side is non-trivial for any $n\geq 1$ and $k\geq 1$. Determine whether the classes in the image of this homomorphism are non-trivial or not. \[prob:st\]
The case $n=1$ corresponds to the original conjecture proposed in [@morita99] expecting the non-triviality of the classes $\mu_k$. In view of our result that $e(\mathrm{Out}\, F_7)=1$, the case $n=2, k=1$ which asks whether the homomorphism $
\varprojlim_{g\to\infty}\ PH^{4}(S^{3} H_{\mathbb{Q}})^{\mathrm{Sp}}_{12}\cong{\mathbb{Q}}^2\rightarrow
PH^{4}_c(\widehat{\mathfrak{h}}^+_{\infty,1})^{\mathrm{Sp}}_{12}\cong H_{8}(\mathrm{Out}\,F_{7};{\mathbb{Q}})
$ is non-trivial or not, should be an important test case.
In this paper, we consider only the Euler characteristics of various chain complexes. However we are planning to study the boundary operators as well. In fact, we already have a proof of the non-triviality $\mu_2\not= 0$, which was first proved by Conant and Vogtmann [@cov], in our context. Also it will be nice if one could construct cycles corresponding to the unstable cohomology classes found by Looijenga and/or Tommasi.
We expect that there should be a close relation between the cohomology of $\mathrm{Out}\,F_n$ and that of $\mathrm{GL}(n,{\mathbb{Z}})$. For example, Elbaz-Vincent, Gangl and Soulé [@egs] recently calculated the rational cohomology of $\mathrm{GL}(n,{\mathbb{Z}})$ for $n=5,6,7$ and it will be a very interesting problem to compare these results with the known results about $H^*(\mathrm{Out}\, F_6;{\mathbb{Q}})$. Also we have a conjectural [*geometric meaning*]{} of the classes $\mu_k\in H_{4k}(\mathrm{Out}\,F_{2k+2};{\mathbb{Q}})$. More precisely, we expect that these classes can be interpreted as certain [*secondary*]{} classes associated with the difference between two reasons for the Borel regulator classes $\beta_k\in H^{4k+1}(\mathrm{GL}(N,{\mathbb{Z}});{\mathbb{R}})$ to vanish first in $H^{4k+1}(\mathrm{Out}\, F_N;{\mathbb{R}})$ by the vanishing theorem of Igusa [@i] (see also Galatius [@ga]) and secondly vanish in $H^{4k+1}(\mathrm{GL}(N^*_k,{\mathbb{Z}});{\mathbb{R}})$ for certain [*unknown*]{} critical rank $N^*_k$ depending on $k$ (we conjecture that $N^*_k=2k+2$). We mention that Bismut and Lott [@bl] proved that $\beta_k$ vanishes in $H^{4k+1}(\mathrm{GL}(2k+1,{\mathbb{Z}});{\mathbb{R}})$ (we thank Soulé for this information) and the above value for $N_k^*$ is the next one after $2k+1$. This expectation is consistent with the result of Conant and Vogtmann [@covs] where they proved that $\mu_k$ vanishes after one stabilization.
Our [*ultimate*]{} goal is to enhance our study of the cohomology of the Lie algebra $\mathfrak{h}_{g,1}$ to those of the groups $\mathcal{H}_{g,1}^{\mathrm{smooth}}, \mathcal{H}_{g,1}^{\mathrm{top}}$ of homology cobordism classes of homology cylinders over the surfaces, both in the [*smooth*]{} as well as the [*topological*]{} categories and also the group $\mathrm{Aut}_0 F^{\mathrm{acy}}_{2g}$ which was introduced by the second named author in [@s]. The group $\mathcal{H}_{g,1}^{\mathrm{smooth}}$ was introduced by Garoufalidis and Levine [@gl] as an [*enlargement*]{} of the mapping class group $\mathcal{M}_{g,1}$. For basic facts as well as various problems about this group, we refer to their paper cited above. We would like to clarify the difference between the above three groups by investigating suitable characteristic classes.
[30]{}
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